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\def \O {\mathcal{O}}
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\def \Q {\mathbb{Q}}
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\def \Supp {\mathrm{Supp}\,}
\def \mult {\mathrm{mult}}
\def \Bs {\mathrm{Bs}\,}
\def \Bir {\mathrm{Bir}}
\def \Aut {\mathrm{Aut}}
\def \Pic {\mathrm{Pic}\,}
\def \Sing {\mathrm{Sing}\,}
\def \lct {\mathrm{lct}}
\def \diam {\mathrm{diam}}
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\author{Ivan Cheltsov and Constantin Shramov\\ \\ With an appendix by Jean-Pierre Demailly}


\title{Log canonical thresholds of smooth Fano threefolds}



\address{University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK}%
\email{cheltsov@yahoo.com, shramov@mccme.ru}

\thanks{The first author was supported by
the grants NSF DMS-0701465 and EPSRC EP/E048412/1,
the~second~author was supported by the grants RFFI
No.~08-01-00395-a, N.Sh.-1987.2008.1 and EPSRC EP/E048412/1.}




\begin{document}

\begin{abstract}
We compute global log canonical thresholds of some smooth Fano threefolds.
\end{abstract}

\maketitle

\tableofcontents



\section{Introduction}
\label{section:intro}

The multiplicity of a non-zero polynomial
$\phi\in\mathbb{C}[z_1,\ldots, z_n]$ in the origin $O\in
\mathbb{C}^n$ is the number
$$
\mathrm{min}\left\{m\in\mathbb{Z}_{\ge 0}\ \Big|\ \frac{\partial^m
 \phi\big(z_{1},\ldots,z_{n}\big)}{\partial^{m_1} z_1\partial^{m_2} z_2\ldots\partial^{m_n}
 z_n}\big(O\big)\ne 0
 \right\}\in\mathbb{Z}_{\ge 0}.
$$
There is a similar but more subtle invariant
$$
c_{0}\big(\phi\big)=\mathrm{sup}\left\{\eps\in\mathbb{Q}\ \Big|\
\text{the function}\  \frac{1}{\ \ \big|\phi\big|^{2\eps}}~\text{is
locally integrable near $O\in\mathbb{C}^n$}\right\}\in\mathbb{Q}_{\ge 0}
\cup\{+\infty\},%
$$
which is called the complex singularity exponent of the polynomial
$\phi$ at the point $O$.

\begin{example}
\label{remark:sum-of-powers} Let
$\phi=\sum_{i=1}^{n}z_{i}^{m_{i}}$, where $m_{1},\ldots,m_{n}$ are
positive integers. Then
$$
c_{0}(\phi)=\mathrm{min}\left(1,\ \sum_{i=1}^{n} \frac{1}{m_{i}}\right).%
$$
\end{example}

\begin{example}
\label{example:normal-crossing} Let
$\phi=\prod_{i=1}^{n}z_{i}^{m_{i}}$, where $m_{1},\ldots,m_{n}$
are positive integers. Then
$$
c_{0}(\phi)=\mathrm{min}\left(\frac{1}{m_{1}},\frac{1}{m_{2}},\ldots,\frac{1}{m_{n}}\right).%
$$
\end{example}

Let $X$ be a variety\footnote{All varieties are assumed to be
complex, algebraic, projective and normal if the opposite is not
stated explicitly.} with at most log terminal singularities (see
\cite{KMM}), let $Z\subseteq X$ be a closed subvariety, and let
$D$ be an effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on
the variety $X$. Then the number
$$
\mathrm{lct}_{Z}\big(X,D\big)=\mathrm{sup}\left\{\lambda\in\mathbb{Q}\
\Big|\ \text{the log pair}\
 \big(X, \lambda D\big)\ \text{is log canonical along}~Z\right\}\in\mathbb{Q}%
$$
is called a log canonical threshold of the divisor $D$ along $Z$.
It follows from \cite{Ko97} that
$$
\mathrm{lct}_{O}\Big(\mathbb{C}^n,\big(\phi=0\big)\Big)=c_0\big(\phi\big),
$$
so that $\mathrm{lct}_{Z}(X,D)$ is an algebraic
counterpart of the number $c_{0}(\phi)$. One has
$$
\mathrm{lct}_{X}\big(X,D\big)=\mathrm{inf}\left\{\mathrm{lct}_P\big(X,D\big)\
\Big\vert\ P\in
 X\right\}=\mathrm{sup}\left\{\lambda\in\mathbb{Q}\ \Big|\ \text{the log pair}\ \big(X, \lambda
 D\big)\
 \text{is log canonical}\right\},%
$$
and, for simplicity, we put
$\mathrm{lct}(X,D)=\mathrm{lct}_X(X,D)$.

\begin{example}
\label{example:cubics} Suppose that $X=\mathbb{P}^{2}$ and
$D\in|\mathcal{O}_{\mathbb{P}^{2}}(3)|$. Then
$$
\mathrm{lct}\big(X,D\big)=\left\{%
\aligned
&1\ \text{if}\ D\ \text{is a smooth curve},\\%
&1\ \text{if}\ D\ \text{is a curve with ordinary double points},\\%
&5/6\ \text{if}\ D\ \text{is a curve with one cuspidal point},\\%
&3/4\ \text{if}\ D\ \text{consists of a conic and a line that are tangent},\\%
&2/3\ \text{if}\ D\ \text{consists of three lines intersecting at one point},\\%
&1/2\ \text{if}\ \mathrm{Supp}\big(D\big)\ \text{consists of two lines},\\%
&1/3\ \text{if}\ \mathrm{Supp}\big(D\big)\ \text{consists of one line}.\\%
\endaligned\right.%
$$
\end{example}

Suppose, in addition, that the variety $X$ is a Fano variety (see
\cite{IsPr99}).

\begin{definition}
\label{definition:threshold} Global log canonical threshold of the
Fano variety $X$ is the number
$$
\mathrm{lct}\big(X\big)=\mathrm{inf}\left\{\mathrm{lct}\big(X,D\big)\ \Big\vert\ D\ \text{is an effective $\mathbb{Q}$-divisor on $X$ such that}\ D\ \sim_{\mathbb{Q}} -K_{X}\right\}\geqslant 0.%
$$
\end{definition}

\begin{remark}
\label{remark:extension} To define the number
$\mathrm{lct}(X)\in\mathbb{R}$, we only need to assume that
$$
\Big|-nK_X\Big|\neq\varnothing
$$
for some $n\gg 0$. This property is shared by many varieties
(toric varieties, weak Fano varieties), but all the currently
known applications are related to the case when $-K_{X}$ is ample.
\end{remark}


The number $\mathrm{lct}(X)$ is an algebraic counterpart of the
$\alpha$-invariant introduced in \cite{Ti87}. One~has
$$
\mathrm{lct}\big(X\big)=\mathrm{sup}\left\{\eps\in\mathbb{Q}\ \left|\ %
\aligned
&\text{the log pair}\ \left(X, \frac{\eps}{n}D\right)\ \text{is log canonical for}\\
&\text{every divisor}\ D\in\big|-nK_{X}\big|\ \text{and all}\ n\in\mathbb{Z}_{>0}\\
\endaligned\right.\right\}.%
$$
Recall that every Fano variety $X$ is rationally connected (see
\cite{Sho00}, \cite{Zh06}). Thus, the group $\mathrm{Pic}(X)$ is
torsion free. Then
$$
\mathrm{lct}\big(X\big)=\mathrm{sup}\left\{\lambda\in\mathbb{Q}\ \left|\ %
\aligned
&\text{the log pair}\ \Big(X, \lambda D\Big)\ \text{is log canonical}\\
&\text{for every effective $\mathbb{Q}$-divisor}\ D\qlineq -K_{X}\\
\endaligned\right.\right\}.%%
$$


\begin{example}
\label{example:hypersurface-index-big} Let $X$ be a smooth
hypersurface in $\mathbb{P}^{n}$ of degree $m<n$. Then
$$
\mathrm{lct}\big(X\big)=\frac{1}{n+1-m}%
$$
as shown in \cite{Ch01b} (see Corollary~\ref{corollary:lct-ci-big-index}). In
particular, the equality $\mathrm{lct}(\mathbb{P}^{n})=1/(n+1)$ holds.
\end{example}

\begin{example}
\label{example:Hwang} Let $X$ be a rational homogeneous space such
that $-K_{X}\sim rD$ and
$$
\mathrm{Pic}\big(X\big)=\mathbb{Z}\big[D\big],
$$
where $D$ is an ample divisor and $r\in\mathbb{Z}_{>0}$. Then
$\mathrm{lct}(X)=1/r$ (see \cite{Hw06a}, \cite{Hw06b}).
\end{example}

\begin{example}
\label{example:Pukhlikov} Let $X$ be a general intersection of
hypersurfaces $F_{1},\ldots, F_{k}\subset\mathbb{P}^{n}$ such that
$$
\sum_{i=1}^{k}\mathrm{deg}\big(F_{i}\big)=n\geqslant 5k+1\geqslant 11,%
$$
where
$\mathrm{deg}(F_{k})\geqslant\ldots\geqslant\mathrm{deg}(F_{1})\geqslant
2$ and $\mathrm{deg}(F_{k})\geqslant 8$. Then $\mathrm{lct}(X)=1$
(see \cite{Pu06}).
\end{example}

In general, the number $\mathrm{lct}(X)$ depends on small
deformations of the variety $X$.

\begin{example}
\label{example:sextic-double-solid-explicit} Let $X$ be a smooth
hypersurface in $\mathbb{P}(1,1,1,1,3)$ of degree $6$. Then
$$
\mathrm{lct}\big(X\big)\in\left\{\frac{5}{6},\frac{43}{50},\frac{13}{15},\frac{33}{38},\frac{7}{8},\frac{33}{38},\frac{8}{9}, \frac{9}{10},\frac{11}{12},\frac{13}{14},\frac{15}{16},\frac{17}{18},\frac{19}{20},\frac{21}{22},\frac{29}{30},1\right\}%
$$
by \cite{Pu04d} and \cite{ChPaWo}, and all these values are attained.
\end{example}

\begin{example}
\label{example:double-cover} Let $X$ be a smooth hypersurface in
$\mathbb{P}(1^{n+1},n)$ of degree $2n$. The inequalities
$$
1\geqslant\mathrm{lct}\big(X\big)\geqslant \frac{2n-1}{2n}%
$$
hold (see \cite{ChPaWo}). But the equality $\mathrm{lct}(X)=1$
holds if $X$ is general and $n\geqslant 3$.
\end{example}

\begin{example}
\label{example:Cheltsov-Park} Let $X$ be a smooth hypersurface in
$\mathbb{P}^{n}$ of degree $n\geqslant 2$. Then the inequalities
$$
1\geqslant\mathrm{lct}\big(X\big)\geqslant \frac{n-1}{n}%
$$
hold (see \cite{Ch01b}). Then it follows from \cite{Pu04d} and \cite{ChPaWo}
that
$$
\mathrm{lct}\big(X\big)\geqslant\left\{%
\aligned
&1\ \text{if}\ n\geqslant 6,\\%
&22/25\ \text{if}\ n=5,\\%
&16/21\ \text{if}\ n=4,\\%
&3/4\ \text{if}\ n=3,\\%
\endaligned\right.%
$$
whenver $X$ is general. But $\mathrm{lct}(X)=1-1/n$ if $X$ contains a cone of
dimension $n-2$.
\end{example}

\begin{example}
\label{example:IHES} Let $X$ be a quasismooth hypersurface in
$\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of
degree~$\sum_{i=1}^{4}a_{i}$ such that $X$ has at most terminal
singularities (see \cite{KMM}), where $a_{1}\leqslant
a_{2}\leqslant a_{3}\leqslant a_{4}$. Then
$$
-K_{X}\qlineq\mathcal{O}_{\mathbb{P}(1,\,a_{1},\,a_{2},\,a_{3},\,a_{4})}\big(1\big)\Big\vert_{X}
$$
and there are $95$ possibilities for the quadruple
$(a_{1},a_{2},a_{3},a_{4})$ (see \cite{IF00}, \cite{JoKo01}). Then
$$
1\geqslant\mathrm{lct}\big(X\big)\geqslant\left\{%
\aligned
&16/21\ \text{if}\ a_{1}=a_{2}=a_{3}=a_{4}=1,\\%
&7/9\ \text{if}\ (a_{1},a_{2},a_{3},a_{4})=(1,1,1,2),\\%
&4/5\ \text{if}\ (a_{1},a_{2},a_{3},a_{4})=(1,1,2,2),\\%
&6/7\ \text{if}\ (a_{1},a_{2},a_{3},a_{4})=(1,1,2,3),\\%
&1\ \text{in the remaining cases},\\%
\endaligned\right.%
$$
if $X$ is general (see  \cite{Ch07a}, \cite{ChPaWo}, \cite{Ch08},
\cite{Ch08d}). The global log canonical threshold of the
hypersurface
$$
w^{2}=t^{3}+z^{9}+y^{18}+x^{18}\subset\mathbb{P}\big(1,1,2,6,9\big)\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t,w\big]\Big)
$$
is equal to $17/18$ (see  \cite{Ch07a}), where
$\mathrm{wt}(x)=\mathrm{wt}(y)=1$, $\mathrm{wt}(z)=2$,
$\mathrm{wt}(t)=6$, $\mathrm{wt}(w)=9$.%
\end{example}


\begin{example}
\label{example:WPS} It follows from Lemma~\ref{lemma:toric} that
$$
\mathrm{lct}\Big(\mathbb{P}\big(a_{0},a_{1},\ldots,a_{n}\big)\Big)=\frac{a_{0}}{\sum_{i=0}^{n}a_{i}},%
$$
where  $\mathbb{P}(a_{0},a_{1},\ldots,a_{n})$ is well-formed (see
\cite{IF00}), and $a_{0}\leqslant a_{1}\leqslant\ldots\leqslant
a_{n}$.
\end{example}

\begin{example}
\label{example:double-cover-big-index} Let $X$ be a smooth
hypersurface in $\mathbb{P}(1^{n+1},d)$ of degree $2d$.
Then
$$
\mathrm{lct}\big(X\big)=\frac{1}{n+1-d}%
$$
in the case when the inequalities $2\leqslant d\leqslant n-1$ hold
(see Proposition~20 in \cite{Ch08a}).
\end{example}

\begin{example}
\label{example:del-Pezzos} Let $X$ be smooth surface del Pezzo. It
follows from \cite{Ch07b} that
$$
\mathrm{lct}\big(X\big)=\left\{%
\aligned
&1\ \text{if}\ K_{X}^{2}=1\ \text{and}\ |-K_{X}|\ \text{contains no cuspidal curves},\\%
&5/6\ \text{if}\ K_{X}^{2}=1\ \text{and}\ |-K_{X}|\ \text{contains a cuspidal curve},\\%
&5/6\ \text{if}\ K_{X}^{2}=2\ \text{and}\ |-K_{X}|\ \text{contains no tacnodal curves},\\%
&3/4\ \text{if}\ K_{X}^{2}=2\ \text{and}\ |-K_{X}|\ \text{contains a tacnodal curve},\\%
&3/4\ \text{if}\ X\ \text{is a cubic in}\ \mathbb{P}^{3}\ \text{with no Eckardt points},\\%
&2/3\ \text{if either}\ X\ \text{is a cubic in}\ \mathbb{P}^{3}\
\text{with an Eckardt point, or
 $K_{X}^{2}=4$},\\%
&1/2\ \text{if}\ X\cong\mathbb{P}^{1}\times\mathbb{P}^{1}\ \text{or}\ K_{X}^{2}\in\{5,6\},\\%
&1/3\ \text{in the remaining cases}.\\%
\endaligned\right.%
$$
\end{example}

It would be interesting to compute global log canonical thresholds
of del Pezzo surfaces with at most canonical singularities that
are of Picard rank one, which has been classified in \cite{Fur86}.

\begin{example}
\label{example:singular-cubics} Let $X$ be a singular cubic
surface in $\mathbb{P}^{3}$ such that $X$ has at most canonical
singularities. The singularities of the surface $X$ are classified
in \cite{BW79}. It follows from \cite{Ch07c} that
$$
\mathrm{lct}\big(X\big)=\left\{%
\aligned
&2/3\ \text{if}\ \mathrm{Sing}\big(X\big)=\big\{\mathbb{A}_{1}\big\},\\%
&1/3\ \text{if}\ \mathrm{Sing}\big(X\big)\supseteq\big\{\mathbb{A}_{4}\big\},\\
&1/3\ \text{if}\ \mathrm{Sing}\big(X\big)=\big\{\mathbb{D}_{4}\big\},\\
&1/3\ \text{if}\ \mathrm{Sing}\big(X\big)\supseteq\big\{\mathbb{A}_{2},\mathbb{A}_{2}\big\},\\%
&1/4\ \text{if}\ \mathrm{Sing}\big(X\big)\supseteq\big\{\mathbb{A}_{5}\big\},\\
&1/4\ \text{if}\ \mathrm{Sing}\big(X\big)=\big\{\mathbb{D}_{5}\big\},\\%
&1/6\ \text{if}\ \mathrm{Sing}\big(X\big)=\big\{\mathbb{E}_{6}\big\},\\%
&1/2\ \text{in the remaining cases}.\\%
\endaligned\right.%
$$
\end{example}

It is unknown whether $\mathrm{lct}(X)\in\mathbb{Q}$ or not\footnote{It is even
unknown whether $\mathrm{lct}(X)\in\mathbb{Q}$ or not if $X$ is a del Pezzo
surfaces with log terminal singularities.} (cf. Question~1 in \cite{Ti90b}).

\begin{conjecture}
\label{conjecture:stabilization} There is an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ on the variety $X$ such that
$$
\mathrm{lct}\big(X\big)=\mathrm{lct}\big(X,D\big)\in\mathbb{Q}.%
$$
\end{conjecture}

Let $G\subset\mathrm{Aut}(X)$ be an arbitrary subgroup.

\begin{definition}
\label{definition:G-threshold} Global $G$-invariant log canonical threshold of
the Fano variety $X$ is
$$
\mathrm{lct}\big(X, G\big)=\mathrm{sup}\left\{\eps\in\mathbb{Q}\ \left|\ %
\aligned
&\text{the log pair}\ \left(X, \frac{\eps}{n}\mathcal{D}\right)\ \text{has log canonical singularities for every}\\
&\text{$G$-invariant linear system}\
\mathcal{D}\subset\big|-nK_{X}\big|\ \text{and every}\ n\in\mathbb{Z}_{>0}\\
\endaligned\right.\right\}.%
$$
\end{definition}

\begin{remark}
\label{remark:G-extension} To define the threshold
$\mathrm{lct}(X,G)\in\mathbb{R}\cup\{+\infty\}$, we only need to
assume that
$$
\Big|-nK_X\Big|\neq\varnothing
$$
for some $n\in\mathbb{N}$. But all known applications require $-K_{X}$ to be
ample, and $G$ to be compact.
\end{remark}


In the case when the Fano variety $X$ is smooth and $G$ is compact,
the equality
$$
\mathrm{lct}\big(X,G\big)=\alpha_{G}\big(X\big),
$$
holds (see Appendix~\ref{section:appendix}), where $\alpha_{G}(X)$
is the $\alpha$-invariant introduced in \cite{Ti87}. It is clear
that
$$
\mathrm{lct}\big(X, G\big)=\mathrm{sup}\left\{\lambda\in\mathbb{Q}\ \left|\ %
\aligned
&\text{the log pair}\ \Big(X, \lambda D\Big)\ \text{has log canonical singularities}\\
&\text{for every $G$-invariant effective $\mathbb{Q}$-divisor}\ D\qlineq -K_{X}\\
\endaligned\right.\right\}%
$$
in the case when $|G|<+\infty$. Note that
$0\leqslant\mathrm{lct}(X)\leqslant\mathrm{lct}(X,G)\in\mathbb{R}\cup\{+\infty\}$.


\begin{example}
\label{example:Klein} The simple group
$\mathrm{PGL}(2,\mathrm{F}_{7})$ is a group of automorphisms of
the quartic
$$
x^{3}y+y^{3}z+z^{3}x=0\subset\mathbb{P}^{2}\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z]\Big),
$$
which induces an embedding
$\mathrm{PGL}(2,\mathrm{F}_{7})\subset\mathrm{Aut}(\mathbb{P}^{2})$.
Then $\mathrm{lct}(\mathbb{P}^{2},
\mathrm{PGL}(2,\mathrm{F}_{7}))=4/3$ (see \cite{Ch07b}).
\end{example}

\begin{example}
\label{example:quintic} Let $X$ be a smooth del Pezzo surface such
that $K_{X}^{2}=5$. Then
\begin{itemize}
\item the isomorphism $\mathrm{Aut}(X)\cong\mathrm{S}_{5}$ holds (see \cite{RaSl02}),%
\item the equalities $\mathrm{lct}(X,\mathrm{S}_{5})=\mathrm{lct}(X,\mathrm{A}_{5})=2$ hold (see~\cite{Ch07b}).%
\end{itemize}
\end{example}


\begin{example}
\label{example:Fermat-cubic} Let $X$ be the cubic surface in
$\mathbb{P}^{3}$ given by the equation
$$
x^{3}+y^{3}+z^{3}+t^{3}=0\subset\mathbb{P}^{3}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t\big]\Big),
$$
and let $G=\mathrm{Aut}(X)\cong
\mathbb{Z}_{3}^{3}\rtimes\mathrm{S}_{4}$. Then $\mathrm{lct}(X,
G)=4$ by~\cite{Ch07b}.
\end{example}

The following result was proved in \cite{Ti87}, \cite{Na90},
\cite{DeKo01} (see Appendix).

\begin{theorem}
\label{theorem:KE} Suppose that $X$ has at most quotient
singularities, and the inequality
$$
\mathrm{lct}\big(X,G\big)>\frac{\mathrm{dim}\big(X\big)}{\mathrm{dim}\big(X\big)+1}%
$$
holds. Then $X$ admits an orbifold K\"ahler--Einstein metric.
\end{theorem}


Let us show how to apply Theorem~\ref{theorem:KE} (cf.
Examples~\ref{example:double-cover},~\ref{example:Cheltsov-Park},~\ref{example:IHES}).

\begin{example}
\label{example:Kollar-Johnson-2} Let $X$ be a quasismooth
hypersurface in $\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$ of degree
$\sum_{i=0}^{3}a_{i}-1$, where $a_{0}\leqslant a_{1}\leqslant
a_{2}\leqslant a_{3}$. Then it follows from \cite{DeKo01},
\cite{JoKo01b}, \cite{BoGaNa02}, \cite{Ara02} (cf.
\cite{BoGaNa02b}, \cite{BoGaNa03}) that
\begin{itemize}
\item either the surface $X$ is smooth, which implies that
$$
\big(a_{0},a_{1},a_{2},a_{3}\big)\in\Big\{\big(1,1,1,1\big),\ \big(1,1,1,2\big),\ \big(1,1,2,3\big)\Big\},%
$$
and all possible values of $\mathrm{lct}(X)$ are contained in Example~\ref{example:del-Pezzos},%
\item or  $(a_{0},a_{1},a_{2},a_{3})=(2,2n+1,2n+1,4n+1)$ and $\mathrm{lct}(X)=1$, where $n\in\mathbb{Z}_{>0}$,%
\item or we have the following sporadic possibilities:
\begin{itemize}
\item $(a_{0},a_{1},a_{2},a_{3})=(1,2,3,5)$ and $\mathrm{lct}(X)>2/3$;%
\item $(a_{0},a_{1},a_{2},a_{3})=(1,3,5,7)$ and $\mathrm{lct}(X)>2/3$ if $X\subset\mathbb{P}(1,3,5,7)$ is general;%
\item $(a_{0},a_{1},a_{2},a_{3})=(1,3,5,8)$ and $\mathrm{lct}(X)\geqslant 11/16$ if $X\subset\mathbb{P}(1,3,5,8)$ is general;%
\item the inequality $\mathrm{lct}(X)>1$ holds and
$$
\big(a_{0},a_{1},a_{2},a_{3}\big)\in\left\{\aligned
&\big(2,3,5,9\big),\big(3,3,5,5\big),\big(3,5,7,11\big),\big(3,5,7,14\big),\big(3,5,11,18\big)\\
&\big(5,14,17,21\big), \big(5,19,27,31\big),\big(5,19,27,50\big),\big(7,11,27,37\big)\\
& \big(7,11,27,44\big),\big(9,15,17,20\big), \big(9,15,23,23\big),\big(11,29,39,49\big)\\
& \big(11,49,69,128\big),\big(13,23,35,57\big),\big(13,35,81,128\big)%
\endaligned\right\}.
$$
\end{itemize}
\end{itemize}
\end{example}

\begin{example}
\label{example:Kollar-Johnson-3} Let $X$ be a quasismooth
hypersurface in $\mathbb{P}(a_{0},\ldots,a_{4})$ of degree
$\sum_{i=0}^{4}a_{i}-1$, where $a_{0}\leqslant a_{1}\leqslant
a_{2}\leqslant a_{3}\leqslant a_{4}$. Then it follows from
\cite{JoKo01} that
\begin{itemize}
\item the inequality $\mathrm{lct}(X)>3/4$ holds for at least $1936$ quintuples  $(a_{0},a_{1},a_{2},a_{3},a_{4})$,%
\item the inequality $\mathrm{lct}(X)\geqslant 1$ holds for at least $1605$ quintuples $(a_{0},a_{1},a_{2},a_{3},a_{4})$.%
\end{itemize}
\end{example}


\begin{example}
\label{example:Fermat-hypersurface} Let $X$ be one of the
following smooth Fano varieties:
\begin{itemize}
\item a Fermat hypersurface in $\mathbb{P}^{n}$ of degree $n/2\leqslant d\leqslant n$ (cf. Example~\ref{example:Fermat-cubic});%
\item a general complete intersection of three quadrics in $\mathbb{P}^{6}$ that is given by%
$$
\sum_{i=0}^{6}x_{i}^{2}=\sum_{i=0}^{6}\lambda_{i}x_{i}^{2}=\sum_{i=0}^{6}\mu_{i}x_{i}^{2}=0\subseteq\mathbb{P}^{6}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x_{0},\ldots,x_{6}\big]\Big),
$$
where $\lambda_{i}$ and $\mu_{i}$ are complex numbers;%
\item a smooth complete intersection of two quadrics in $\mathbb{P}^{5}$ that is given by%
$$
\sum_{i=0}^{5}x_{i}^{2}=\sum_{i=0}^{5}\zeta^{i}x_{i}^{2}=0\subseteq\mathbb{P}^{5}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x_{0},\ldots,x_{5}\big]\Big),
$$
where $\zeta$ is a primitive sixth root of unity;%
\item a smooth complete intersection of a quadric and a cubic
in $\mathbb{P}^{5}$ that is given by%
$$
\sum_{i=0}^{5}x_{i}^{3}=\sum_{i=0}^{5}\zeta^{i}x_{i}^{2}=0\subseteq\mathbb{P}^{5}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x_{0},\ldots,x_{5}\big]\Big),
$$
where $\zeta$ is a nontrivial cube root of unity;%
\item a hypersurface in $\mathbb{P}(1^{n+1},q)$ of degree $pq$ that is given by the equation%
$$
w^{p}=\sum_{i=0}^{5}x_{i}^{pq}\subseteq\mathbb{P}\big(1^{n+1},q\big)\cong\mathrm{Proj}\Big(\mathbb{C}\big[x_{0},\ldots,x_{n},w\big]\Big),
$$
such that $pq-q\leqslant n$, where
$\mathrm{wt}(x_{0})=\ldots=\mathrm{wt}(x_{n})=1$,
$\mathrm{wt}(w)=q\in\mathbb{Z}_{>0}$ and $p\in \mathbb{Z}_{>0}$;
\end{itemize}
and let $G=\mathrm{Aut}(X)$. Then $G$ is finite, and the
inequality $\mathrm{lct}(X,G)\geqslant 1$ holds (see \cite{Ti87},
\cite{Na90}).
\end{example}

\begin{example}
\label{example:skew-blow-up} Let $X$ be a blow up of
$\mathbb{P}^{3}$ along a disjoint union of two lines,
let~$G$~be~a~maximal compact subgroup in $\mathrm{Aut}(X)$. Then
the inequality $\mathrm{lct}(X,G)\geqslant 1$ holds by \cite{Na90}
(cf.~Lemma~\ref{lemma:3-25}).
\end{example}


If a variety with quotient singularities admits an orbifold
K\"ahler--Einstein metric, then
\begin{itemize}
\item either its canonical divisor is numerically trivial;%
\item or its canonical divisor is ample (variety of general type);%
\item or its canonical divisor is antiample (Fano variety).%
\end{itemize}

\begin{remark}
\label{remark:Yau-Aubin} Every variety with at most quotient
singularities that has numerically trivial or ample canonical
divisor admits an orbifold K\"ahler--Einstein metric (see
\cite{Aub78}, \cite{Yau78}, \cite{Yau96}).
\end{remark}

There are several known obstructions for the Fano variety $X$ to admit a
K\"ahler--Einstein~met\-ric. For example, if the variety $X$ is smooth, then it
does not admit a K\"ahler--Einstein metric~if
\begin{itemize}
\item either the group $\mathrm{Aut}(X)$ is not reductive (see~\cite{Mat57}),%
\item or the tangent bundle of $X$ is not polystable with respect to $-K_{X}$ (see  \cite{Lub83}),%
\item or the Futaki character of holomorphic vector fields on $X$ does not vanish~(see~\cite{Fu83}),%
\item or the pair $(X, -K_{X})$ is not $K$-semistable (see \cite{Ti97}, \cite{Do02}, \cite{Do05}, \cite{RoTh06}, \cite{RoTh07}).%
\end{itemize}

\begin{example}
\label{example:V7} The following varieties admit no
K\"ahler--Einstein metrics:
\begin{itemize}
\item a blow up of $\mathbb{P}^{2}$ in one or two points (see~\cite{Mat57}),%
\item a smooth Fano threefold
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}\oplus
 \mathcal{O}_{\mathbb{P}^{2}}(1))$ (see  \cite{Stef96}),%
\item a smooth Fano fourfold
$$
\mathbb{P}\Big(\alpha^{*}\big(\mathcal{O}_{\mathbb{P}^{1}}(1)\big)\oplus
 \beta^{*}\big(\mathcal{O}_{\mathbb{P}^{2}}(1)\big)\Big),%
$$
where $\alpha\colon
\mathbb{P}^{1}\times\mathbb{P}^{2}\to\mathbb{P}^{1}$ and
$\beta\colon \mathbb{P}^{1}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$
are natural projections (see \cite{Fu83}).
\end{itemize}
\end{example}

\begin{example}
\label{example:V22} Let $X$ be a smooth Fano threefold such that
$$
\mathrm{Pic}\big(X\big)=\mathbb{Z}\big[-K_{X}\big]
$$
and $-K_{X}^{3}=22$. Then $\mathrm{lct}(X)\leqslant 2/3$ by
Lemma~\ref{lemma:1-10} (see Section~\ref{section:MU}). But
\begin{itemize}
\item the tangent bundle of the threefold $X$ is stable~(see~\cite{Stef96}),%
\item the group $\mathrm{Aut}(X)$ is trivial if the threefold $X$ is general (see \cite{MuuM83}),%
\item there is $X$ such that $\mathrm{Aut}(X)=\{1\}$ and $X$ admits no K\"ahler--Einstein metrics (see \cite{Ti97});%
\item if $\mathrm{Aut}(X)\cong\mathrm{PSL}(2,\mathbb{C})$, then $X$ admits a K\"ahler--Einstein  metric (see  \cite{Do07} and Remark~\ref{remark:MU-SO3}).%
\end{itemize}
\end{example}

The problem of existence of K\"ahler--Einstein metrics on smooth
toric Fano varieties is completely solved. Namely, the following
result holds (see \cite{Mab87}, \cite{BaSe99}, \cite{WaZhu04},
\cite{Ni90}).

\begin{theorem}
\label{theorem:toric} If $X$ is smooth and toric, then the
following conditions are equivalent:
\begin{itemize}
\item the variety $X$ admits a K\"ahler--Einstein metric;%
\item the Futaki character of holomorphic vector field of $X$ vanishes;%
\item the baricenter of the reflexive polytope of $X$ is zero.%
\end{itemize}
\end{theorem}

It should be pointed out that the assertion of
Theorem~\ref{theorem:KE} gives only a sufficient condition~for the
existence of a K\"ahler--Einstein metric on $X$
(cf. \cite{Ma84}, \cite{Ti97}).

\begin{example}
\label{example:cubic-Eckardt-point} Let $X$ be a general cubic
surface in $\mathbb{P}^{3}$ that has an Eckardt point. Then
$$
\mathrm{lct}\Big(X,\ \mathrm{Aut}\big(X\big)\Big)=\mathrm{lct}\big(X\big)=\frac{2}{3}%
$$
by~\cite{Ch07b}, because $\mathrm{Aut}(X)\cong\mathbb{Z}_{2}$ (see
\cite{DoIs06}). But every smooth del Pezzo surface~that has a reductive
automorphism groups admits a K\"ahler--Einstein metric (see \cite{TiYa87},
\cite{Ti90}).
\end{example}

\begin{example}
\label{example:Nadel-Tian-Arezzo-Ghigi-Pirola} Let $X$ be a
general hypersurface in $\mathbb{P}(1^{5},3)$ of degree $6$. Then
$$
\mathrm{lct}\Big(X,\ \mathrm{Aut}\big(X\big)\Big)=\mathrm{lct}\big(X\big)=\frac{1}{2}%
$$
by~\cite{Ch08a}, because $\mathrm{Aut}(X)\cong\mathbb{Z}_{2}$ (see
\cite{MaMon64}). But $X$ admit a K\"ahler--Einstein metric (see
\cite{ArGha06}).
\end{example}


The problem of existence of K\"ahler--Einstein metrics on singular
Fano varities that have quotient singularities is not well studied
even for del Pezzo surfaces with canonical singularities.

\begin{example}
\label{example:singular-cubic} Let $X$ be a cubic surface in
$\mathbb{P}^{3}$. Then
\begin{itemize}
\item the surface $X$ admits a K\"ahler--Einstein metric if $\mathrm{Sing}(X)=\varnothing$ (see   \cite{Ti90}),%
\item the surface $X$ does not admit an orbifold  K\"ahler--Einstein metric in
the case when it has at least one singular point that is not a singular point of type $\mathbb{A}_{1}$ or~$\mathbb{A}_{2}$~(see~\cite{DiTi92}),%
\item the surface that is given by the equation
$$
xyz+xyt+xzt+yzt=0\subset\mathbb{P}^{3}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t\big]\Big)
$$
admits an orbifold K\"ahler--Einstein metric and has $4$ singular points of type $\mathbb{A}_{1}$  (see  \cite{Ch07c}),%
\item the surface that is given by the equation
$$
xyz=t^{3}\subset\mathbb{P}^{3}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t\big]\Big)
$$
admits an orbifold K\"ahler--Einstein metric and has $3$ singular points of type $\mathbb{A}_{2}$ (see  \cite{Ch07c}).%
\end{itemize}
\end{example}

\begin{example}
\label{example:singular-quartic} Let $X$ be a complete
intersection of two quadrics in $\mathbb{P}^{4}$. Then
\begin{itemize}
\item the surface $X$ admits a K\"ahler--Einstein metric if $\mathrm{Sing}(X)=\varnothing$ (see \cite{Ti90}),%
\item the surface $X$ does not admit an orbifold K\"ahler--Einstein metric in the case when it has at least one singular point that is not an ordinary double point~(see~\cite{Jef97}),%
\item if the surface $X\subset\mathbb{P}^{4}$ can be given by the equations
$$
\sum_{i=0}^{4}x_{i}^{2}=\sum_{i=0}^{4}\lambda_{i}x_{i}^{2}=0\subset\mathbb{P}^{4}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x_{0},\ldots,x_{4}\big]\Big)
$$
and $X$ has at most ordinary double points, then
$\mathrm{lct}(X,\mathbb{Z}_{2}^{4})=1$~(see~\cite{MaMu93}).%
\end{itemize}
\end{example}

\begin{remark}
\label{remark:del-Pezzo-degree-1} Let $X$ be a del Pezzo surface
with canonical singularities such that $K_{X}^{2}\leqslant 2$.
Then
\begin{itemize}
\item the surface $X$ admits a K\"ahler--Einstein metric in the following cases:%
\begin{itemize}
\item if $K_{X}^{2}=2$ and $X$ has at most singular points of type $\mathbb{A}_{1}$~or~$\mathbb{A}_{2}$ (see \cite{GiKo05});%
\item if $K_{X}^{2}=1$ and $X$ has at most ordinary double points (see \cite{Ch07b});%
\end{itemize}
\item we expect $X$ to admit no K\"ahler--Einstein metrics if~it~has~bad~singularities~(cf.~\cite{DiTi92},~\cite{Lu99}).%
\end{itemize}
\end{remark}

The numbers $\mathrm{lct}(X)$ and  $\mathrm{lct}(X, G)$ play an
important role in birational geometry.

\begin{example}
\label{example:fiberwise} Let $V$ and $\bar{V}$ be varieties with
at most terminal and $\mathbb{Q}$-factorial singularities, and let
$Z$ be a smooth curve. Suppose that there is a commutative diagram
$$
\xymatrix{
&V\ar@{->}[d]_{\pi}\ar@{-->}[rr]^{\rho}&&\ \bar{V}\ar@{->}[d]^{\bar{\pi}}&\\%
&Z\ar@{=}[rr]&&Z&}
$$
such that $\pi$ and $\bar{\pi}$ are flat morphisms, and $\rho$ is
a birational map that induces an isomorphism
$$
V\setminus X\cong\bar{V}\setminus\bar{X},%
$$
where $X$ and $\bar{X}$ are scheme fibers of $\pi$ and $\bar{\pi}$
over a point $O\in Z$, respectively. Suppose that
\begin{itemize}
\item the fibers $X$ and $\bar{X}$ are irreducible and reduced,
\item the divisors $-K_{V}$ and $-K_{\bar{V}}$ are $\pi$-ample and
$\bar{\pi}$-ample,
 respectively,%
\item the varieties $X$ and $\bar{X}$ have at most log terminal singularities,%
\end{itemize}
and $\rho$ is not an isomorphism. Then it follows from \cite{Pa01}
and \cite{Ch07c} that
\begin{equation}
\label{equation:lct}
\mathrm{lct}\big(X\big)+\mathrm{lct}\big(\bar{X}\big)\leqslant 1,
\end{equation}
where $X$ and $\bar{X}$ are Fano varieties by the adjunction
formula.
\end{example}

In general, the inequality~\ref{equation:lct} is sharp (see
\cite{Pu00cc}, \cite{Gr01MN}, \cite{Gr01UMN}, \cite{Pa03}).

\begin{example}
\label{example:dP3-D4} Let $\pi\colon V\to Z$ be a surjective flat
morphism such that
\begin{itemize}
\item the variety $V$ is a smooth threefold,%
\item the variety $Z$ is a smooth curve,%
\item the divisor $-K_{V}$ is $\pi$-ample,%
\end{itemize}
let $X$ be a scheme fiber of the morphism $\pi$ over a point $O\in
Z$ such that $X$ is a smooth cubic surface in $\mathbb{P}^{3}$
that has an Eckardt poin $P\in X$ (see
Definition~\ref{definition:Eckardt-point}), let
$L_{1},L_{2},L_{3}\subset X$ be the lines that pass through the
point $P$. Then it follows from \cite{Co96} that there is a
commutative diagram
$$
\xymatrix{
&&U\ar@{->}[dl]_{\alpha}\ar@{-->}[rr]^{\psi}&&\bar{U}\ar@{->}[dr]^{\beta}\\%
&V\ar@{->}[dr]_{\pi}\ar@{-->}[rrrr]^{\rho}&&&&\ \bar{V}\ar@{->}[dl]^{\bar{\pi}}&\\%
&&Z\ar@{=}[rr]&&Z&}
$$
such that $\alpha$ is a blow up of the point $P$, the map $\psi$
is an antiflip in the proper transforms of the~curves
$L_{1},L_{2},L_{3}$, and $\beta$~is~a~contraction of the~proper
transform of the fiber $X$. Then
\begin{itemize}
\item the birational map $\rho$ is not an isomorphism,%
\item the threefold $\bar{V}$ has terminal and $\mathbb{Q}$-factorial singularities,%
\item the divisor $-K_{\bar{V}}$ is a Cartier $\bar{\pi}$-ample divisor,%
\item the map $\rho$ induces an isomorphism
$$
V\setminus X\cong\bar{V}\setminus\bar{X},
$$
where $\bar{X}$ is a scheme fiber of $\bar{\pi}$ over the point $O$,%
\item the surface $\bar{X}$ is a cubic surface with a singular point of
type $\mathbb{D}_{4}$.
\end{itemize}
The latter assertion implies that $\mathrm{lct}(X)+\mathrm{lct}(\bar{X})=1$ (see
Examples~\ref{example:del-Pezzos}~and~\ref{example:singular-cubics}).
\end{example}

Global log canonical thresholds can be used to prove that some
Fano varieties are non-rational.

\begin{definition}
\label{definition:birational-superrigidity} We say that the Fano
variety $X$ is  birationally superrigid if
\begin{itemize}
\item the equality $\mathrm{rk}\,\mathrm{Pic}(X)=1$ holds,%
\item the variety $X$ has terminal $\mathbb{Q}$-factorial singularities,%
\item for every linear system $\mathcal{M}$ on the Fano variety
$X$ that does not have fixed components, the log pair $(X,
\lambda \mathcal{M})$ has canonical singularities for
$\lambda\in\mathbb{Q}$ such that $K_{X}+\lambda\mathcal{M}\qlineq
0$.
\end{itemize}
\end{definition}

Birationally superrigid Fano varieties are non-rational~(see
\cite{Co95}, \cite{Is01}, \cite{Ch05umn}, \cite{Pu07umn}).

\begin{remark}
\label{remark:rigidity} If the variety $X$ is birationally
superrigid, then
\begin{itemize}
\item there is no rational dominant map $\rho\colon X\dasharrow Y$
such that
\begin{itemize}
\item a general fiber of the map $\rho$ is rationally connected,%
\item the inequality $\mathrm{dim}(Y)\geqslant 1$ holds,%
\end{itemize}
\item there is no non-biregular birational  map $\rho\colon
X\dasharrow Y$ such that
\begin{itemize}
\item the variety $Y$ has terminal $\mathbb{Q}$-factorial singularities,%
\item the equality $\mathrm{rk}\,\mathrm{Pic}(Y)=1$ holds.%
\end{itemize}
\end{itemize}
\end{remark}

\begin{remark}
If $X$ is
birationally superrigid, then $\dim(X)\neq 2$ (cf. \cite{Ma66}, \cite{Ma67},
\cite{Is96b}).
\end{remark}

\begin{example}
\label{example:unirational} A smooth quartic hypersurface that is
given by the equation
$$
x^{4}+xw^{3}+y^{4}-6y^{2}z^{2}+z^{4}+t^{4}+t^{3}w=0\subset\mathbb{P}^{4}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t,w\big]\Big)
$$
is unirational (see \cite{Is80b}, \cite{Mar00}) and birationally superrigid
(see  \cite{IsMa71}).
\end{example}

\begin{example}
\label{example:Pukhlikov-rigid} The following smooth Fano
varieties are birationally superrigid:
\begin{itemize}
\item a smooth hypersurface in $\mathbb{P}^{n}$ of degree $n$ for
$4\le n\leqslant 12$ (see  \cite{IsMa71}, \cite{Pu87}, \cite{Ch01a}, \cite{Pu02a}, \cite{deFeEiMu03});%
\item a general hypersurface in $\mathbb{P}^{n}$ of degree $n\geqslant 4$ (see \cite{Pu98a});%
\item a smooth hypersurface in $\mathbb{P}(1^{n+1}, n)$ of degree $2n\geqslant 6$ (see \cite{Is80b},~\cite{Pu88a}).%
\item a general complete intersection of hypersurfaces
$F_{1},\ldots, F_{k}\subset\mathbb{P}^{n}$ such that
$$
\sum_{i=1}^{k}\mathrm{deg}\big(F_{i}\big)=n\geqslant 3k+1\geqslant 7,%
$$
where $\mathrm{deg}(F_{k})\geqslant\ldots\geqslant \mathrm{deg}(F_{2})\geqslant\mathrm{deg}(F_{1})\geqslant 2$ (see \cite{Pu01});%
\item a smooth fourfold complete intersection in $\mathbb{P}^{6}$ of degree~$8$ containing~no~planes~(see~\cite{Ch03b}),%
\item a smooth Fano variety $X$ such that there is a double cover
$$
\tau\colon X\longrightarrow V\subset\mathbb{P}^{n}
$$
where $V$ is a hypersurface, $\tau$ is ramified in a divisor
$R\in|\mathcal{O}_{\mathbb{P}^{n}}(2n-2\mathrm{deg}(V))\vert_{V}|$,
and
\begin{itemize}
\item either $\mathrm{deg}(V)=2$ and $n\geqslant 5$ (see \cite{Pu88a}),%
\item or $V$ and $R\in|\mathcal{O}_{\mathbb{P}^{n}}(2n-2\mathrm{deg}(V))\vert_{V}|$ are general and $n\geqslant 5$ (see \cite{Pu00SM}),%
\item or $3\leqslant \mathrm{deg}(V)\leqslant 4$ and $n\geqslant 8$ (see \cite{Ch06MZ}).%
\end{itemize}
\item a sextic hypersurface in $\mathbb{P}^{6}$ with at most ordinary double points (see \cite{Ch07NM}).%
\end{itemize}
\end{example}


\begin{example}
\label{example:sextic-double-solid} Let $\pi\colon
X\to\mathbb{P}^{3}$ be a double cover branched along a surface
$S\subset\mathbb{P}^{3}$ of degree $6$ such that the sextic
surface $S$ has at most ordinary double points.~Then
\begin{itemize}
\item the inequality $|\mathrm{Sing}(S)|\leqslant 65$ holds (see \cite{JaRu97}, \cite{Wa98}),%
\item for any $65\geqslant k\in\mathbb{Z}_{>0}$, there exists
$S\subset\mathbb{P}^{3}$ such that $|\mathrm{Sing}(S)|=k$
(see \cite{CaCe82}, \cite{Ba96}),%
\item the variety $X$ is birationally superrigid in the case when $\mathrm{rk}\,\mathrm{Cl}(X)=1$ (see~\cite{IsMa71},~\cite{ChPa04}),%
\item the equality $\mathrm{rk}\,\mathrm{Cl}(X)=1$ holds if $|\mathrm{Sing}(S)|\leqslant 14$ (see \cite{ChPa04}),%
\item suppose that the surface $S$ is a Barth sextic (see
\cite{Ba96}) that is given by
$$
4\Big(\tau^2x^2-y^2\Big)\Big(\tau^2y^2-z^2\Big)\Big(\tau^2z^2-x^2\Big)=t^2\Big(1+2\tau\Big)\Big(x^2+y^2+z^2-t^{2}\Big)^2\subset\mathbb{P}^{3}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t\big]\Big),%
$$
where $\tau=(1+\sqrt{5})/2$; then $\mathrm{rk}\,\mathrm{Cl}(X)=14$
(see \cite{En99}) and the diagram
$$
\xymatrix{
&Y\ar@{-->}[d]_{\rho}\ar@{^{(}->}[rr]&&\mathbb{P}^4\ar@{-->}[d]^{\psi}&\\%
&X\ar@{->}[rr]_{\pi}&&\mathbb{P}^{3}&}
$$ %
commutes (see \cite{En99}, \cite{Pet98}), where
\begin{itemize}
\item the variety $Y$ is a determinantal quartic threefold in $\mathbb{P}^{4}$ such that $|\mathrm{Sing}(Y)|=42$,%
\item the map $\psi$ is the projection from a singular point of the quartic $Y$,%
\item the map $\rho$ is a birational map,%
\end{itemize}
which implies that the threefold $X$ is rational.
\end{itemize}
\end{example}

The following result is proved in \cite{Pu04d}.

\begin{theorem}
\label{theorem:Pukhlikov} Let $X_{1},X_{2},\ldots,X_{r}$ be
birationally superrigid Fano varieties. Then
\begin{itemize}
\item the variety $X_{1}\times\ldots\times X_{r}$ is non-rational
and
$$
\mathrm{Bir}\Big(X_{1}\times\ldots\times
X_{r}\Big)=\mathrm{Aut}\Big(X_{1}\times\ldots\times
 X_{r}\Big),
$$
\item for every rational dominant map
$$
\rho\colon X_{1}\times\ldots\times X_{r}\dasharrow Y,
$$
whose general fiber is rationally connected, there is a
commutative diagram
$$
\xymatrix{
X_{1}\times\ldots\times X_{r}\ar@{->}[d]_{\pi}\ar@{-->}[rrrrd]^{\rho}\\
X_{i_{1}}\times\ldots\times X_{i_{k}}\ar@{-->}[rrrr]_{\xi}&&&&Y}%
$$
for some $\{i_{1},\ldots,i_{k}\}\subseteq\{1,\ldots,r\}$, where
$\xi$ is a birational map, and $\pi$ is a projection,
\end{itemize}
in the case when the~inequalities $\mathrm{lct}(X_{1})\geqslant 1,
\mathrm{lct}(X_{2})\geqslant 1,\ldots,
\mathrm{lct}(X_{r})\geqslant 1$ hold.
\end{theorem}

Varieties satisfying all hypotheses of
Theorem~\ref{theorem:Pukhlikov} do
exist~(see~Examples~\ref{example:Pukhlikov},~\ref{example:double-cover},~\ref{example:Cheltsov-Park},~\ref{example:Pukhlikov-rigid}).

\begin{example}
\label{example:Cheltsov-Park-Won} Let $X$ be a hypersurface
that is given by
$$
w^{2}=x^6+y^6+z^6+t^6+x^2y^2zt\subset\mathbb{P}\big(1,1,1,1,3\big)\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t,w\big]\Big),
$$
where
$\mathrm{wt}(x)=\mathrm{wt}(y)=\mathrm{wt}(z)=\mathrm{wt}(t)=1$
and $\mathrm{wt}(w)=3$. Then
\begin{itemize}
\item the threefold $X$ is smooth~and~bira\-ti\-onally superrigid~(see~\cite{Is80b}),%
\item it follows from \cite{ChPaWo} that the equality
$\mathrm{lct}(X)=1$ holds.%
\end{itemize}
\end{example}


\begin{example}
\label{example:Pukhlikov-explicit} Let $X$ be a hypersurface
that is given by
$$
w^{2}=\sum_{i=0}^{n}x_{i}^{2n}+\eps\sum_{i=1}^{n-4}\left(\sum_{j=0}^{n}a_{ij}x_{i}\right)^{2n}\subset\mathbb{P}\big(1^{n+1},n\big)\cong\mathrm{Proj}\Big(\mathbb{C}\big[x_{0},x_{1},\ldots,x_{n},w\big]\Big)%
$$
for some $\eps\in\mathbb{C}\ni a_{ij}$, where $\mathrm{wt}(x_{i})=1$ and
$\mathrm{wt}(w)=n$. Suppose that any $n+1$ forms among
$$
x_{0}, x_{1}, \ldots, x_{n},\ \sum_{i=0}^{n}a_{1i}x_{i},\
\sum_{i=0}^{n}a_{2i}x_{i},\ \ldots,\ \sum_{i=0}^{n}a_{n-4\,i}x_{i}
$$
are linearly independent. Put $\Delta=\mathrm{max}\{|a_{ij}|\}$.
Suppose that the inequalities $n\geqslant 8$ and
$$
1>|\eps|\Big(2n\Delta+2\Delta\Big)^{2m}
$$
hold. Then $X$ is bira\-ti\-onally superrigid and
$\mathrm{lct}(X)=1$~(see~\cite{Pu88a}, \cite{Pu07mmj}). The
hypersurface
$$
w^{2}=n^{2n^{2}}\sum_{i=0}^{n}x_{i}^{2n}+\sum_{i=1}^{n-4}\left(\sum_{j=0}^{n}\big(j+1\big)^{i}x_{i}\right)^{2n}
$$
is bira\-ti\-onally superrigid, and its global log canonical
threshold is $1$ (see \cite{Pu07mmj}).
\end{example}

Suppose, in addition, that the subgroup $G\subset\mathrm{Aut}(X)$ is finite.

\begin{definition}
\label{definition:G-birational-superrigidity} The Fano variety $X$ is
$G$-birationally superrigid if
\begin{itemize}
\item the $G$-invariant subgroup of the group $\mathrm{Cl}(X)$ is isomorphic to $\mathbb{Z}$,%
\item the variety $X$ has terminal singularities,%
\item for every $G$-invariant linear system $\mathcal{M}$ on the variety $X$
that has no fixed components, the log pair $(X, \lambda\mathcal{M})$ has
canonical singularities, where $\lambda\in\mathbb{Q}$ is such that
$K_{X}+\lambda\mathcal{M}\qlineq 0$.
\end{itemize}
\end{definition}

If the variety $X$ is $G$-birationally superrigid, then
\begin{itemize}
\item there is no $G$-equivariant rational dominant map $\rho\colon
X\dasharrow Y$ such that
\begin{itemize}
\item general fiber of the map $\rho$ is rationally connected,%
\item the inequality $\mathrm{dim}(Y)\geqslant 1$ holds,%
\end{itemize}
\item there is no $G$-equivariant non-biregular birational  map $\rho\colon
X\dasharrow Y$ such that
\begin{itemize}
\item the $G$-invariant subgroup of the group $\mathrm{Cl}(Y)$ is isomorphic to $\mathbb{Z}$,%
\item the variety $Y$ has terminal singularities,%
\end{itemize}
\end{itemize}

\begin{example}
\label{example:cubic-surface} Let $X$ be a smooth surface in
$\mathbb{P}(1,1,2,3)$ of degree $6$ such that the $G$-invariant subgroup of
the group $\mathrm{Pic}(X)$ is $\mathbb{Z}$. Then $X$ is $G$-birationally
superrigid (see \cite{Ma66}, \cite{Ma67},\cite{Is96b}).
\end{example}

The proof of Theorem~\ref{theorem:Pukhlikov} implies the
following result (see \cite{Ch07b}).

\begin{theorem}
\label{theorem:G-Pukhlikov} Let $X_{i}$ be a Fano variety, and let
$G_{i}\subset\mathrm{Aut}(X_{i})$ be a finite subgroup such that
\begin{itemize}
\item the variety $X_{i}$ is $G_{i}$-birationally superrigid,%
\item the inequality $\mathrm{lct}(X_{i},G_{i})\geqslant 1$ holds,%
\end{itemize}
where $i=1,\ldots,r$. Then the following assertions hold:
\begin{itemize}
\item there is no $G_{1}\times\ldots\times G_{r}$-equivariant birational map $\rho\colon X_{1}\times\ldots\times X_{r}\dasharrow\mathbb{P}^{n}$;%
\item every $G_{1}\times\ldots\times G_{r}$-equivariant birational automorphism of $X_{1}\times\ldots\times X_{r}$ is biregular;%
\item for every $G_{1}\times\ldots\times G_{r}$-equivariant rational dominant
map
$$
\rho\colon X_{1}\times\ldots\times X_{r}\dasharrow Y,
$$
whose general fiber is
rationally connected, there a commutative diagram
$$
\xymatrix{
X_{1}\times\ldots\times X_{r}\ar@{->}[d]_{\pi}\ar@{-->}[rrrrd]^{\rho}\\
X_{i_{1}}\times\ldots\times X_{i_{k}}\ar@{-->}[rrrr]_{\xi}&&&&Y}%
$$
where $\xi$ is a birational~map, $\pi$ is a natural projection,
and $\{i_{1},\ldots,i_{k}\}\subseteq\{1,\ldots,r\}$.
\end{itemize}
\end{theorem}

Varieties satisfying all hypotheses of
Theorem~\ref{theorem:G-Pukhlikov} do exist (see
Example~\ref{example:Fermat-cubic}).

\begin{example}
\label{example:Valentiner} The simple group $\mathrm{A}_{6}$ is a
group of automorphisms of the sextic
$$
10x^{3}y^{3}+9zx^{5}+9zy^{5}+27z^{6}=45x^{2}y^{2}z^{2}+135xyz^{4}\subset\mathbb{P}^{2}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z\big]\Big),
$$
which induces an embedding $\mathrm{A}_{6}\subset\mathrm{Aut}(\mathbb{P}^{2})$.
It follows from \cite{Cra99} that $\mathbb{P}^{2}$ is
$\mathrm{A}_{6}$-birationally~super\-ri\-gid. But the equality
$\mathrm{lct}(\mathbb{P}^{2},\mathrm{A}_{6})=2$ holds (see \cite{Ch07b}). Thus,
there is an induced embedding
$$
\mathrm{A}_{6}\times
\mathrm{A}_{6}\cong\Omega\subset\mathrm{Bir}\big(\mathbb{P}^{4}\big)
$$
such that $\Omega$ is not conjugate to any subgroup in
$\mathrm{Aut}(\mathbb{P}^{4})$ by Theorem~\ref{theorem:G-Pukhlikov}.
\end{example}


\begin{example}
\label{example:icosahedron} Suppose that $X$ be a smooth cubic surface in
$\mathbb{P}^{3}$ that is given by
$$
x^{2}y+xz^{2}+zt^{2}+tx^{2}=0\subset\mathbb{P}^{3}\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t\big]\Big),
$$
which implies that $\mathrm{Aut}(X)\cong\mathrm{S}_{5}$ (see \cite{DoIs06}).
Then it follows from \cite{Ch07b} that
$$
\mathrm{lct}\big(X,\mathrm{S}_{5}\big)=\mathrm{lct}\big(X,\mathrm{A}_{5}\big)=2,
$$
and the surface $X$ is $\mathrm{A}_{5}$-birationally superrigid (see
Example~\ref{example:cubic-surface}).
\end{example}

Let us consider Fano varieties that are very close to being birationally
superrigid.

\begin{definition}
\label{definition:birational-rigidity} The Fano variety $X$ is
birationally rigid\footnote{There are several definitions of
birational rigidity and birational
superrigidity~(see~\cite{Co95},~\cite{Co00},~\cite{Is01},~\cite{Ch05umn},~\cite{Pu07umn}).}
if
\begin{itemize}
\item the equality $\mathrm{rk}\,\mathrm{Pic}(X)=1$ holds,
\item the variety $X$ has $\mathbb{Q}$-factorial and terminal singularities,%
\item for every linear system $\mathcal{M}$ on the Fano variety
$X$ that does not have fixed components, there is birational
automorphism $\xi\in\mathrm{Bir}(X)$ such that the log pair
$$
\Big(X,\ \lambda\,\xi\big(\mathcal{M}\big)\Big)
$$
has canonical singularities, where $\lambda$ is a rational number
such that $K_{X}+\lambda\,\xi(\mathcal{M})\qlineq 0$.
\end{itemize}
\end{definition}


Birationally rigid Fano varieties are non-rational~(see
\cite{Co95}, \cite{Is01}, \cite{Ch05umn}, \cite{Pu07umn}).

\begin{remark}
\label{remark:Castravet} There exists a smooth Fano variety $X$ of dimension
$n$ such that
$$
\mathrm{Pic}\big(X\big)=\mathbb{Z}\big[-K_{X}\big]
$$
and the variety $X$ is not birationally rigid for every $n\geqslant 5$ (see
\cite{Cas07}).
\end{remark}

It is well--known (see \cite{Co95}, \cite{Is01}, \cite{Ch05umn},
\cite{Pu07umn}) that if $X$ is birationally rigid, then
\begin{itemize}
\item there is no rational dominant map $\rho\colon X\dasharrow Y$
such that
\begin{itemize}
\item a general fiber of the map $\rho$ is rationally connected,%
\item the inequality $\mathrm{dim}(Y)\geqslant 1$ holds,%
\end{itemize}
\item there is no birational  map $\rho\colon X\dasharrow Y$ such
that
\begin{itemize}
\item the varieties $Y$ and $X$ are not biregular,%
\item the variety $Y$ has terminal $\mathbb{Q}$-factorial singularities,%
\item the equality $\mathrm{rk}\,\mathrm{Pic}(Y)=1$ holds.%
\end{itemize}
\end{itemize}


\begin{example}
\label{example:rigid-non-superrigid} The following varieties are
birationally rigid but not birationally superrigid:
\begin{itemize}
\item a general complete intersection of a quadric and a cubic in $\mathbb{P}^{5}$ (see  \cite{IsPu96});%
\item a smooth double cover of a quadric in $\mathbb{P}^{4}$ branched over a surface of degree $8$ (see  \cite{Is80b}).%
\end{itemize}
\end{example}

The existence of a birational automorphism in
Definition~\ref{definition:birational-rigidity} is usually
checked for a given set of birational automorphisms. This leads to
the following definition.

\begin{definition}
\label{definition:untwisting} Suppose that $X$ is birationally
rigid. A subset
$\Gamma\subset\mathrm{Bir}(X)$~untwists~all~maximal singularities
if for every linear system $\mathcal{M}$~on the variety $X$ that
has no fixed components, there is a birational automorphism
$\xi\in\Gamma$ such that the log pair
$$
\Big(X,\ \lambda\,\xi\big(\mathcal{M}\big)\Big)
$$
has canonical singularities, where $\lambda$ is a rational number
such that $K_{X}+\lambda\,\xi(\mathcal{M})\qlineq 0$.
\end{definition}

If $X$ is birationally rigid and there is
$\Gamma\subset\mathrm{Bir}(X)$ that untwists all maximal
singularities, then the group $\mathrm{Bir}(X)$ is generated by
$\Gamma$ and $\mathrm{Aut}(X)$.%

\begin{example}
\label{example:singular-hypersurface} Let $X$ be a sufficiently
general hypersurface in $\mathbb{P}^{n}$ of degree $n\geqslant 5$
that has one ordinary singular point $O=\mathrm{Sing}(X)$ of
multiplicity $n-2$. Then the projection
$$
\psi\colon X\dasharrow\mathbb{P}^{n-1}
$$
from the point $O$ induces an involution that untwists all maximal
singularities (see~\cite{Pu02c}).
\end{example}


If $X$ is defined over a perfect field, then
Definition~\ref{definition:birational-rigidity} still makes sense
(see \cite{Ma66}, \cite{Ma67}, \cite{Is01}).

\begin{definition}
\label{definition:birational-rigidity-universal} The variety $X$
is universally birationally rigid if for any variety $U$, the
variety
$$
X\otimes\mathrm{Spec}\Big(\mathbb{C}\big(U\big)\Big)
$$
is birationally rigid over a field of rational functions
$\mathbb{C}(U)$ of the variety $U$.
\end{definition}

\begin{example}
\label{example:double-quadric} Let $X$ be a smooth Fano threefold
such that there is a double cover
$$
\pi\colon X\longrightarrow Q\subset\mathbb{P}^{3},
$$
where $Q$ is a quadric threefold, and $\pi$ is branched in a
surface $S\subset Q$ of degree $8$. Put
$$
\mathcal{C}=\Big\{C\subset X\ \Big\vert\ C\ \text{is a smooth
curve such that}\ -K_{X}\cdot
 C=1\Big\},%
$$
then $\mathcal{C}$ is a one-dimensional family. For every curve
$C\in\mathcal{C}$ there is a commutative diagram
$$
\xymatrix{
X\ar@{-->}[d]_{\psi_{C}}\ar@{->}[rr]^{\pi}&&Q\ar@{-->}[d]^{\phi_{C}}\\
\mathbb{P}^{2}\ar@{=}[rr]&&\mathbb{P}^{2},}%
$$
where $\phi_{C}$ is a projection from the line $\pi(C)$. General
fiber of the map $\psi_{C}$ is an elliptic curve, the map
$\psi_{C}$ induces an elliptic fibration with a section and an
involution $\tau_{C}\in\mathrm{Bir}(X)$. Then
$$
\psi_{C}\in\mathrm{Aut}\big(X\big)\iff C\subset S,
$$
and $S$ contains no curves in $\mathcal{C}$ if $X$ is general. It
follows from \cite{Is80b} that there is an exact sequence
$$
1\longrightarrow\Gamma\longrightarrow\mathrm{Bir}\big(X\big)\longrightarrow\mathrm{Aut}\big(X\big)\longrightarrow
 1,%
$$
where $\Gamma$ is a free product of subgroups that are generated
by  non-biregular birational involutions constructed above. Hence
the Fano variety $X$ is universally birationally rigid (see
\cite{Is80b}).
\end{example}

\begin{example}
\label{example:nodal-quartic} Let $X$ be a quartic threefold in
$\mathbb{P}^{4}$ that has at most ordinary double points.~Then
\begin{itemize}
\item the inequality $|\mathrm{Sing}(X)|\leqslant 45$ holds (see \cite{Va83}),%
\item the variety $X$ is not birationally superrigid if $\mathrm{Sing}(X)\ne\varnothing$,%
\item the variety $X$ is universally birationally rigid if $\mathrm{rk}\,\mathrm{Cl}(X)=1$ (see~\cite{IsMa71},~\cite{Pu88b},~\cite{Me03},~\cite{Shr08}),%
\item the equality $\mathrm{rk}\,\mathrm{Cl}(X)=1$ holds if $|\mathrm{Sing}(X)|\leqslant 8$ (see \cite{Ch06PJM}),%
\item the equality $\mathrm{rk}\,\mathrm{Cl}(X)=1$ holds if $|\mathrm{Sing}(X)|\leqslant 12$ and $X$ contains neither planes of quadric surfaces (see \cite{Sh07}),%
\item in the case when $|\mathrm{Sing}(X)|=45$, it follows from
\cite{JSV90} that $X$ can be given by the equation
$$
w^4-w\Big(x^3+y^3+z^3+t^3\Big)+3xyzt=0\subset\mathbb{P}^4\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t,w\big]\Big),
$$
the quartic $X$ is determinantal and rational, and
$\mathrm{rk}\,\mathrm{Cl}(X)=10$ (see \cite{Pet98}).
\end{itemize}
\end{example}

Birationally superrigid Fano varieties are universally
birationally rigid.

\begin{definition}
\label{definition:untwisting-universal}  Suppose that $X$ is
universally birationally rigid. A subset
$\Gamma\subset\mathrm{Bir}(X)$ universally untwists all maximal
singularities if for every variety $U$ the induced subset
$$
\Gamma\subset\mathrm{Bir}\big(X\big)\subseteq\mathrm{Bir}\left(X\otimes\mathrm{Spec}\Big(\mathbb{C}\big(U\big)\Big)\right)
$$
untwists all maximal singularities on
$X\otimes\mathrm{Spec}(\mathbb{C}(U))$, which is defined over
$\mathbb{C}(U)$.
\end{definition}

An identity map universally untwists all maximal singularities if
$X$ is birationally~superrigid.

\begin{remark}
\label{remark:Kollar} Suppose that $X$~is~bira\-ti\-onally rigid, and
$\mathrm{dim}(X)\ne 1$. Let
$$
\Gamma\subseteq\mathrm{Bir}\big(X\big)
$$
be a subset. It was pointed out by J.\,Koll\'ar that the following conditions
are equivalent:
\begin{itemize}
\item the subset $\Gamma$ universally untwists all maximal singularities;%
\item the subset $\Gamma$ untwists all maximal singularities, and $\mathrm{Bir}(X)$ is countable.%
\end{itemize}
\end{remark}


\begin{example}
\label{example:CPR} In the assumptions of
Example~\ref{example:IHES},~suppose~that~$X$~is~general.~Then
\begin{itemize}
\item the hypersurface $X$ is universally birationally rigid (see \cite{CPR}),%
\item there are involutions
$\tau_{1}, \ldots, \tau_{k}\in\mathrm{Bir}(X)$ such that that the
sequence of groups
$$
1\longrightarrow\big\langle\tau_{1}, \ldots, \tau_{k}\big\rangle\longrightarrow\mathrm{Bir}\big(X\big)\longrightarrow\mathrm{Aut}\big(X\big)\longrightarrow 1%
$$
is exact (see \cite{CPR}, \cite{ChPa05}), where $\langle\tau_{1}, \ldots,
\tau_{k}\rangle$ is a subgroup generated by $\tau_{1},\ldots,\tau_{k}$,%
\item the subgroup $\langle\tau_{1},\ldots,\tau_{k}\rangle$ universally untwists all maximal  singularities (see \cite{CPR}),%
\item all relations between the involutions $\tau_{1},\ldots,\tau_{k}$ are found in \cite{ChPa05},%
\item the papers \cite{Ry06}, \cite{Ry06b}, \cite{ChPa05}, \cite{Ch06e} classify all  maps $\psi\colon X\to\mathbb{P}^{2}$ such that the diagram%
$$
\xymatrix{
&V\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\omega}\\%
X\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{2}}
$$
commutes, where $\alpha$ is a birational morphism, and $\omega$ is an elliptic fibration,%
\item the papers \cite{Ry06}, \cite{Ry06b}, \cite{ChPa05h} classify all  maps $\phi\colon X\to\mathbb{P}^{1}$ such that the diagram%
$$
\xymatrix{
&V\ar@{->}[dl]_{\beta}\ar@{->}[dr]^{\eta}\\%
X\ar@{-->}[rr]_{\phi}&&\mathbb{P}^{1}}
$$
commutes, where $\beta$ is a birational morphism, and $\eta$ is a fibration into $K3$ surfaces.%
\end{itemize}
\end{example}

Let  $X_{1},\ldots,X_{r}$ be Fano varieties that have at most
$\mathbb{Q}$-factorial and terminal singularities, let
$$
\pi_{i}\colon X_{1}\times\ldots\times X_{i-1}\times X_{i}\times
X_{i+1}\times\ldots\times X_{r}\longrightarrow
X_{1}\times\ldots\times X_{i-1}\times \widehat{X_{i}}\times
X_{i+1}\times\ldots\times X_{r}
$$
be a natural projection, and let $\X_{i}$ be a scheme general
fiber of the projection $\pi_{i}$.


\begin{remark}
\label{remark:generic-fiber} There are natural embedding of groups
$$
\prod_{i=1}^{r}\mathrm{Bir}\big(X_{i}\big)\subseteq
\Big\langle\mathrm{Bir}\big(\X_{1}\big), \ldots, \mathrm{Bir}\big(\X_{r}\big)
\Big\rangle\subseteq\mathrm{Bir}\Big(X_{1}\times\ldots\times X_{r}\Big),%
$$
where $\X_{i}$ is defined over the field
$\mathbb{C}(X_{1}\times\ldots\times X_{i-1}\times
\widehat{X_{i}}\times X_{i+1}\times\ldots\times X_{r})$.
\end{remark}

The following generalization of Theorem~\ref{theorem:Pukhlikov}
holds (see \cite{Ch07a}).

\begin{theorem}
\label{theorem:Cheltsov} Suppose that $X_{1},X_{2},\ldots,X_{r}$ are
universally birationally rigid. Then
\begin{itemize}
\item the variety $X_{1}\times\ldots\times X_{r}$ is non-rational
and
$$
\mathrm{Bir}\Big(X_{1}\times\ldots\times X_{r}\Big)=
\Big\langle\mathrm{Bir}\big(\X_{1}\big), \ldots, \mathrm{Bir}\big(\X_{r}\big),
\mathrm{Aut}\Big(X_{1}\times\ldots\times X_{r}\Big)\Big\rangle,%
$$%
\item for every rational dominant map $\rho\colon
X_{1}\times\ldots\times X_{r}\dasharrow Y$, whose general fiber is
rationally connected, there is a subset
$\{i_{1},\ldots,i_{k}\}\subseteq\{1,\ldots,r\}$ and a commutative
diagram
$$
\xymatrix{ X_{1}\times\ldots\times
X_{r}\ar@{->}[d]_{\pi}\ar@{-->}[rr]^{\sigma}&&X_{1}\times\ldots\times
 X_{r}\ar@{-->}[rrd]^{\rho}\\
X_{i_{1}}\times\ldots\times X_{i_{k}}\ar@{-->}[rrrr]_{\xi}&&&&Y,}%
$$
where
$\xi$ and $\sigma$  are birational maps, and $\pi$ is a projection,%
\end{itemize}
in the case when the inequalities $\mathrm{lct}(X_{1})\geqslant 1,
\mathrm{lct}(X_{2})\geqslant 1, \ldots,\mathrm{lct}(X_{r})\geqslant 1$ hold.
\end{theorem}

\begin{corollary}
\label{corollary:Cheltsov} Suppose that there are subgroups
$\Gamma_{1}\subseteq\mathrm{Bir}(X_{1}),\ldots,\Gamma_{r}\subseteq\mathrm{Bir}(X_{r})$
that universally untwists all maximal singularities, and
$\mathrm{lct}(X_{1})\geqslant 1, \mathrm{lct}(X_{2})\geqslant 1,
\ldots,\mathrm{lct}(X_{r})\geqslant 1$. Then
$$
\mathrm{Bir}\Big(X_{1}\times\ldots\times X_{r}\Big)=\Big<\prod_{i=1}^{r}\Gamma_{i},\ \mathrm{Aut}\Big(X_{1}\times\ldots\times X_{r}\Big)\Big>.%
$$
\end{corollary}

The following four examples are implied by
Examples~\ref{example:Cheltsov-Park}, \ref{example:IHES},
\ref{example:Pukhlikov-rigid}, \ref{example:CPR} and
\cite{MaMon64}.

\begin{example}
\label{example:41-41} Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,4,5,10)$ of degree~$20$. The sequence
$$
1\longrightarrow\prod_{i=1}^{m}\Big(\mathbb{Z}_{2}\ast\mathbb{Z}_{2}\Big)\longrightarrow\mathrm{Bir}\Big(\underbrace{X\times\ldots\times X}_{m\ \mathrm{times}}\Big)\longrightarrow\mathrm{S}_{m}\longrightarrow 1%
$$
is exact, where $\mathrm{S}_{m}$ is a permutation group, and
$\mathbb{Z}_{2}\ast\mathbb{Z}_{2}$ is the infinite dihedral group.
\end{example}


\begin{example}
\label{example:20}  Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,3,4,5)$ of degree~$13$. Then
$$
\mathrm{Bir}\Big(X\times V\Big)\cong\mathbb{Z}_{2}\ast\mathbb{Z}_{2}\ast\mathbb{Z}_{2},%
$$
where $V$ is a general hypersurface in $\mathbb{P}^{n}$ of degree
$n\geqslant 6$.
\end{example}

\begin{example}
\label{example:9} Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,2,3,3)$ of degree~$9$. Then
$$
\mathrm{Bir}\Big(X\times V\Big)\cong\Big<a,b,c\ \big\vert\ a^2=b^2=c^2=\big(abc\big)^{2}=1\Big>,%
$$
where $V$ is a general hypersurface in $\mathbb{P}^{n}$ of degree
$n\geqslant 6$.
\end{example}


\begin{example}
\label{example:7}  Let $X$ be a general hypersurface in
$\mathbb{P}(1,1,2,2,3)$ of degree~$8$. Then
$$
\mathrm{Bir}\Big(X\times V\Big)\cong\mathbb{Z}_{2}\ast\mathbb{Z}_{2}\ast\mathbb{Z}_{2}\ast\mathbb{Z}_{2}\ast\mathbb{Z}_{2},%
$$
where $V$ is a general hypersurface in $\mathbb{P}^{n}$ of degree
$n\geqslant 6$.
\end{example}

Suppose now that $X$ is a smooth Fano threefold (see
\cite{IsPr99}). Let
$$
\gimel\big(X\big)\in\Big\{1.1,1.2,\ldots,1.17,2.1,\ldots,2.36,3.1,\ldots,3.31,4.1,\ldots,4.13,5.1,\ldots,5.7,5.8\Big\}
$$
be the ordinal number of the deformation type of the threefold $X$
in the notation of Table~\ref{table:Fanos}.

\begin{remark}
\label{remark:Fanos} The threefold $X$ lies in $105$ deformation families
(see~\cite{Is77},~\cite{Is78},~\cite{MoMu81},~\cite{MoMu83},~\cite{MoMu85},~\cite{MoMu03}).
\end{remark}

The main purpose of this paper is to prove the following result.

\begin{theorem}
\label{theorem:main} The following assertions hold:
\begin{itemize}
\item the equality $\mathrm{lct}(X)=1/5$ holds for $\gimel(X)\in\{2.36, 3.29\}$;%
\item the equality $\mathrm{lct}(X)=1/4$ holds for
$$
\gimel\big(X\big)\in\Big\{1.17, 2.28, 2.30, 2.33, 2.35, 3.23, 3.26, 3.30, 4.12\Big\};
$$%
\item the equality $\mathrm{lct}(X)=1/3$ holds for
$$
\gimel\big(X\big)\in\Big\{1.16, 2.29, 2.31, 2.34, 3.9, 3.18,\ldots, 3.22,3.24, 3.25, 3.28, 3.31, 4.4, 4.8,\ldots 4.11, 5.1, 5.2\Big\};%
$$
\item the equality $\mathrm{lct}(X)=3/7$ holds for $\gimel(X)=4.5$;%
\item the equality $\mathrm{lct}(X)=1/2$ holds for
$$
\gimel\big(X\big)\in\left\{\aligned
&1.11, 1.12, 1.13, 1.14, 1.15, 2.1, 2.3, 2.18, 2.25, 2.27, 2.32, 3.4, 3.10, 3.11, 3.12,\\
&3.14, 3.15, 3,16, 3.17, 3.24, 3.27, 4.1, 4.2, 4.3, 4.6, 4.7, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8%
\endaligned\right\};
$$
\item if the threefold $X$ is general in moduli, then
$$
\mathrm{lct}\big(X\big)=\left\{%
\aligned
&1/3\ \mathrm{if}\ \gimel(X)=2.23,\\%
&1/2\ \mathrm{if}\ \gimel(X)\in\{2.5, 2.8, 2.10, 2.11, 2.14, 2.15, 2.19, 2.24, 2.26, 3.2, 3.5,\ldots, 3.8, 4.13\},\\%
&2/3\ \mathrm{if}\ \gimel(X)=3.3,\\%
&3/4\ \mathrm{if}\ \gimel(X)\in\{2.4, 3.1\},\\%
&1\ \mathrm{if}\ \gimel(X)=1.1.\\%
\endaligned\right.%
$$
\end{itemize}
\end{theorem}


Hence, if the threefold $X$ is general in moduli, then we do not know
$\mathrm{lct}(X)$ only when
$$
\gimel\big(X\big)\in\Big\{1.2,1.3,1.4,1.5,\ldots,1.10,2.2,2.6,2.7,2.9,2.12,2.13,2.16,2.17,2.20,2.21,2.22,3.13\Big\},%
$$
and the generality condition in Theorem~\ref{theorem:main} can not be omitted
in many cases.

\begin{example}
\label{example:4-13} Suppose that $\gimel(X)=4.13$. Note that this deformation
type was omitted in~\cite{MoMu81}, and it has been discovered only twenty years
later (see \cite{MoMu03}). There is a birational morphism
$$
\alpha\colon X\longrightarrow \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1
$$
that contracts a surface $E\subset X$ to a curve $C$ such that
$C\cdot F_{1}=C\cdot F_{2}=1$ and $C\cdot F_{3}=3$, where
$$
F_{1}\cong F_{2}\cong F_{3}\cong\P^1\times\P^1
$$
are fibers of three different projections
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\to\mathbb{P}^{1}$,
respectively. Then
$$
\mathrm{lct}\big(X\big)=1/2
$$
by Theorem~\ref{theorem:main} if $X$ is general. There is a
surface $G\in|F_{1}+F_{2}|$ such that $C\subset G$. Then
$$
-K_{X}\sim 2\bar{G}+E+\bar{F}_{3},
$$
where $\bar{F}_{3}\subset X\supset\bar{G}$ are proper transforms of $F_{3}$ and
$G$, respectively. Then $\mathrm{lct}(X)\leqslant 1/2$.~But
$$
\mathrm{lct}\big(X\big)\leqslant\mathrm{lct}\Big(X,\ 2\bar{G}+E+\bar{F}_{3}\Big)\leqslant 4/9<1/2%
$$
in the case when the intersection $F_{3}\cap C$ consists of a
single point.
\end{example}

We hope that the proof of Theorem~\ref{theorem:main} can be used to
\begin{itemize}
\item  study  the slope stability of the threefold $X$ in the sense of \cite{RoTh06} and \cite{RoTh07},%
\item  study the problem of existence of a K\"ahler--Einstein metric on the threefold $X$,%
\item compute $\mathrm{lct}(X,G)$ for various subgroups
$G\subset\mathrm{Aut}(X)$.%
\end{itemize}

\begin{remark}
\label{remark:stable-Fanos} The stability of the tangent bundle of
$X$ was studied in \cite{Stef96}. It is known that
\begin{itemize}
\item the tangent bundle of $X$ is unstable with respect to
$-K_{X}$ when
$$
\gimel\big(X\big)\in\Big\{2.35, 2.36, 3.29, 3.30, 3.31, 4.11,
4.12\Big\},
$$%
\item the tangent bundle of $X$ is semistable with respect to
$-K_{X}$ when
$$
\gimel\big(X\big)\in\Big\{2.33, 2.34, 3.27, 3.28, 4.10, 5.2,
\ldots, 5.8\Big\},
$$%
\item the tangent bundle of $X$ is stable with respect to $-K_{X}$
when
$$
\gimel\big(X\big)\not\in\Big\{2.33, 2.34, 2.35, 2.36, 3.27, \ldots, 3.31, 4.10, 4.11, 4.12, 5.2, \ldots, 5.8\Big\}.%
$$%
\end{itemize}
\end{remark}

We organize the paper in the following way:
\begin{itemize}
\item in Section~\ref{section:preliminaries}, we consider auxiliary results that are used in the proof of Theorem~\ref{theorem:main};%
\item in Section~\ref{section:MU}, we find the global log canonical threshold of the Mukai--Umemura threefold;%
\item in Section~\ref{section:cubic-surfaces}, we prove Theorem~\ref{theorem:cubic-surfaces} that is required for Example~\ref{example:del-Pezzo-quartic} and Lemma~\ref{lemma:2-4};%
\item in Section~\ref{section:del-Pezzo}, we consider facts on surfaces that are used in the proof of Theorem~\ref{theorem:main};%
\item in Section~\ref{section:toric}, we compute global log canonical thresholds of toric Fano varieties;%
\item in Section~\ref{section:del-Pezzos}, we prove
Theorem~\ref{theorem:main} for smooth Fano threefolds of index
$2$, i.e., for
$$
\gimel\big(X\big)\in\Big\{1.11, 1.12, 1.13, 1.14, 1.15, 2.32, 2.35, 3.27\Big\};%
$$
\item in Section~\ref{section:rho-2}, we prove Theorem~\ref{theorem:main} in the case when $\mathrm{rk}\,\mathrm{Pic}(X)=2$;%
\item in Section~\ref{section:rho-3}, we prove Theorem~\ref{theorem:main} in the case when $\mathrm{rk}\,\mathrm{Pic}(X)=3$;%
\item in Section~\ref{section:rho-4}, we prove Theorem~\ref{theorem:main} in the case when $\mathrm{rk}\,\mathrm{Pic}(X)\geqslant 4$;%
\item in Section~\ref{section:bounds}, we find upper bounds for
$\mathrm{lct}(X)$ in the case when
$$
\gimel\big(X\big)\in\Big\{1.8,1.9,1.10,2.9,2.12,2.13,2.16,2.17,2.20,2.21,2.22,3.13\Big\};%
$$
\item in Section~\ref{section:appendix}, we put Table~\ref{table:Fanos} that
contains the list of all smooth Fano threefolds together with the known values
and bounds for their global log canonical thresholds.
\end{itemize}


We'll use the following notation and conventions throughout the paper.

We use a standard notation $D_1\sim D_2$ (resp., $D_1\qlineq D_2$)
for the linearly equivalent (resp., $\Q$-linearly equivalent)
divisors (resp., $\Q$-divisors). If a divisor (resp., a
$\Q$-divisor) $D$ is linearly equivalent to a line bundle
$\mathcal{L}$ (resp., is $\Q$-linearly equivalent to a divisor
that is linearly equivalent to $\mathcal{L}$), we write
$D\sim\mathcal{L}$ (resp., $D\qlineq\mathcal{L}$). Recall that
$\Q$-linear  equivalence coincides with numerical equivalence in
the case of Fano varieties.

A divisor $D$ on
$\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\ldots\times\mathbb{P}^{n_m}$
is said to be of multidegree
$(a_{1},a_{2},\ldots,a_{m})\in\mathbb{Z}^{m}$ if
$D\sim\sum\limits_{i=1}^m a_iH_i$, where
$$
H_i\sim
\pi_{i}^{*}\Big(\mathcal{O}_{\mathbb{P}^{n_i}}\big(a_{i}\big)\Big),
$$
is a pull-back of a hyperplane section of $\P^{n_i}$ under the
projection
$\pi_{i}\colon\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\ldots\times\mathbb{P}^{n_m}\to\mathbb{P}^{n_i}$
is a natural projection. A curve $C\subset
\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}\times\ldots\times\mathbb{P}^{n_m}$
is said to be of multidegree
$(a_{1},a_{2},\ldots,a_{m})\in\mathbb{Z}^{m}$ if
$$
\pi_{i}^{*}\Big(\mathcal{O}_{\mathbb{P}^{n_i}}\big(1\big)\Big)\cdot C=a_{i}
$$
for $i=1, \ldots, m$. The symbol $\mathbb{F}_{n}$ denotes the
Hirzebruch surface
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$.

We always refer to smooth Fano threefolds using the ordinal number
$\gimel$ introduced in Table~\ref{table:Fanos}.

We are very grateful to J.-P.\,Demailly for writing
Appendix~\ref{section:alpha}. The first author would like to
express his gratitude to IHES (Bures-sur-Yvette, France) and MPIM
(Bonn, Germany) for hospitality.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Preliminaries}
\label{section:preliminaries}

Let $X$ be a variety with log terminal singularities. Let us
consider a $\mathbb{Q}$-divisor
$$
B_{X}=\sum_{i=1}^{r}a_{i}B_{i},
$$
where $B_{i}$ is a prime Weil divisor on the variety $X$, and
$a_{i}$ is an arbitrary non-negative rational number. Suppose that
$B_{X}$ is a $\mathbb{Q}$-Cartier divisor such that
$B_{i}\neq B_{j}$ for $i\neq j$.

Let $\pi\colon\bar{X}\to X$ be a birational morphism such that
$\bar{X}$ is smooth. Put
$$
B_{\bar{X}}=\sum_{i=1}^{r}a_{i}\bar{B}_{i},
$$
where $\bar{B}_{i}$ is a proper transform of the divisor $B_{i}$
on the variety $\bar{X}$. Then
$$
K_{\bar{X}}+B_{\bar{X}}\qlineq\pi^{*}\Big(K_{X}+B_{X}\Big)+\sum_{i=1}^{n}c_{i}E_{i},
$$
where $c_{i}\in\mathbb{Q}$, and $E_{i}$ is an exceptional divisor
of the morphism $\pi$. Suppose that
$$
\left(\bigcup_{i=1}^{r}\bar{B}_{i}\right)\bigcup\left(\bigcup_{i=1}^{n}E_{i}\right)
$$
is a divisor with simple normal crossing. Put
$$
B^{\bar{X}}=B_{\bar{X}}-\sum_{i=1}^{n}c_{i}E_{i}.
$$

\begin{definition}
\label{definition:log-canonical-singularities} The singularities
of $(X, B_{X})$ are log canonical (resp., log terminal) if
\begin{itemize}
\item the inequality $a_{i}\leqslant 1$ holds (resp., the inequality $a_{i}<1$
holds),%
\item the inequality $c_{j}\geqslant -1$ holds (resp., the inequality
$c_{j}>-1$ holds),%
\end{itemize}
for every $i=1,\ldots,r$ and $j=1,\ldots,n$.
\end{definition}

One can show that
Definition~\ref{definition:log-canonical-singularities} does not
depend on the choice of the morphism $\pi$. Put
$$
\mathrm{LCS}\Big(X, B_{X}\Big)=\left(\bigcup_{a_{i}\geqslant
 1}B_{i}\right)\bigcup\left(\bigcup_{c_{i}\leqslant -1}\pi\big(E_{i}\big)\right)\subsetneq X,%
$$
then $\mathrm{LCS}(X,B_{X})$ is called the locus of log canonical
singularities of the log pair $(X, B_{X})$.

\begin{definition}
\label{definition:log-canonical-center} A proper irreducible
subvariety $Y\subsetneq X$ is said to be a center of log canonical
singularities of the log pair $(X, B_{X})$ if one of the following
conditions is satisfied:
\begin{itemize}
\item either the inequality $a_{i}\geqslant 1$ holds and $Y=B_{i}$,%
\item or the inequality $c_{i}\leqslant -1$ holds and $Y=\pi(E_{i})$,%
\end{itemize}
for some choice of the birational morphism $\pi\colon\bar{X}\to
X$.
\end{definition}

Let $\mathbb{LCS}(X, B_{X})$ be the set of all centers of log
canonical singularities of $(X, B_{X})$. Then
$$
Y\in\mathbb{LCS}\Big(X, B_{X}\Big)\Longrightarrow Y\subseteq\mathrm{LCS}\Big(X, B_{X}\Big)%
$$
and $\mathbb{LCS}(X, B_{X})=\varnothing\iff\mathrm{LCS}(X,
B_{X})=\varnothing$ $\iff$ the log pair $(X,B_{X})$ is log
terminal.

\begin{remark}
\label{remark:hyperplane-reduction} Let $\mathcal{H}$ be a linear
system on $X$ that has  no base points, let $H$ be a sufficiently
general divisor in the linear system $\mathcal{H}$, and let
$Y\subsetneq X$ be an irreducible subvariety. Put
$$
Y\Big\vert_{H}=\sum_{i=1}^{m}Z_{i},
$$
where $Z_{i}\subset H$ is an irreducible subvariety. It follows
from Definition~\ref{definition:log-canonical-center} (cf.
Theorem~\ref{theorem:adjunction})~that
$$
Y\in\mathbb{LCS}\Big(X,\ B_{X}\Big)\iff \Big\{Z_{1},\ldots,Z_{m}\Big\}\subseteq \mathbb{LCS}\Big(H,\ B_{X}\Big\vert_{H}\Big).%
$$
\end{remark}

\begin{example}
\label{example:smooth-point-and-log-pull-back} Let $\alpha\colon
V\to X$ be a blow up of a smooth point $O\in X$. Then
$$
B_{V}\qlineq\alpha^{*}\big(B_{X}\big)-\mathrm{mult}_{O}\big(B_{X}\big)E
$$
where $\mathrm{mult}_{O}(B_{X})\in\mathbb{Q}$, and $E$ is the
exceptional divisor of the blow up $\alpha$. Then
$$
\mathrm{mult}_{O}\big(B_{X}\big)>1
$$
if the log pair $(X,B_{X})$ is not log canonical at the point $O$.
Put
$$
B^{V}=B_{V}+\Big(\mathrm{mult}_{O}\big(B_{X}\big)-\mathrm{dim}\big(X\big)+1\Big)E,
$$
and suppose that
$\mathrm{mult}_{O}(B_{X})\geqslant\mathrm{dim}(X)-1$. Then $O\in
\mathbb{LCS}(X, B_{X})$ if and only if
\begin{itemize}
\item either $E\in\mathbb{LCS}(V, B^{V})\iff\mathrm{mult}_{O}(B_{X})\geqslant\mathrm{dim}(X)$,%
\item or there is a subvariety $Z\subsetneq E$ such that $Z\in\mathbb{LCS}(V, B^{V})$.%
\end{itemize}
\end{example}

The locus $\mathrm{LCS}(X,B_{X})\subset X$ can be equipped with a
scheme structure (see \cite{Na90}, \cite{Sho93}). Put
$$
\mathcal{I}\Big(X, B_{X}\Big)=\pi_{*}\Big(\sum_{i=1}^{n}\lceil
c_{i}\rceil
 E_{i}-\sum_{i=1}^{r}\lfloor a_{i}\rfloor \bar{B}_{i}\Big),%
$$
and let $\mathcal{L}(X, B_{X})$ be a subscheme that corresponds to
the ideal sheaf $\mathcal{I}(X, B_{X})$.

\begin{definition}
\label{definition:log-canonical-subscheme} For the log pair
$(X,B_{X})$, we say that
\begin{itemize}
\item the subscheme $\mathcal{L}(X, B_{X})$ is the subscheme of
log canonical singularities of
 $(X,B_{X})$,%
\item the ideal sheaf $\mathcal{I}(X, B_{X})$ is the multiplier ideal sheaf of $(X,B_{X})$.%
\end{itemize}
\end{definition}

It follows from the construction of the subscheme $\mathcal{L}(X,
B_{X})$ that
$$
\mathrm{Supp}\Big(\mathcal{L}\big(X,
B_{X}\big)\Big)=\mathrm{LCS}\Big(X, B_{X}\Big)\subset X.
$$

The following result is the Nadel--Shokurov vanishing theorem (see
\cite{Sho93},~\cite[Theorem~9.4.8]{La04}).

\begin{theorem}
\label{theorem:Shokurov-vanishing} Let $H$ be a nef and big
$\mathbb{Q}$-divisor on $X$ such that
$$
K_{X}+B_{X}+H\qlineq D
$$
for some Cartier divisor $D$ on the variety $X$. Then for every
$i\geqslant 1$
$$
H^{i}\Big(X,\ \mathcal{I}\big(X, B_{X}\big)\otimes D\Big)=0.
$$
\end{theorem}

\begin{proof}
It follows from the Kawamata--Viehweg vanishing theorem (see
\cite{Ko97}) that
$$
R^{i}\pi_{*}\Big(\pi^{*}\big(K_{X}+B_{X}+H\big)+\sum_{i=1}^{n}\lceil
c_{i}\rceil
 E_{i}-\sum_{i=1}^{r}\lfloor a_{i}\rfloor \bar{B}_{i}\Big)=0%
$$
for every $i>0$. It follows from the equality of sheaves
$$
\pi_{*}\Big(\pi^{*}\big(K_{X}+B_{X}+H\big)+\sum_{i=1}^{n}\lceil
c_{i}\rceil
 E_{i}-\sum_{i=1}^{r}\lfloor a_{i}\rfloor \bar{B}_{i}\Big)=\mathcal{I}\big(X, B_{X}\big)\otimes D%
$$
and from the degeneration of a local-to-global spectral sequence
that
$$
H^{i}\Big(X, \mathcal{I}\big(X, B_{X}\big)\otimes
D\Big)=H^{i}\Big(\bar{X},
 \pi^{*}\big(K_{X}+B_{X}+H\big)+\sum_{i=1}^{n}\lceil c_{i}\rceil E_{i}-\sum_{i=1}^{r}\lfloor
 a_{i}\rfloor
 \bar{B}_{i}\Big),%
$$
for every $i\geqslant 0$. But for $i>0$, the cohomology group
$$
H^{i}\Big(\bar{X},
\pi^{*}\big(K_{X}+B_{X}+H\big)+\sum_{i=1}^{n}\lceil c_{i}\rceil
 E_{i}-\sum_{i=1}^{r}\lfloor a_{i}\rfloor \bar{B}_{i}\Big)%
$$
is trivial by the Kawamata--Viehweg vanishing theorem (see
\cite{Ko97}).
\end{proof}

For every Cartier divisor $D$ on the variety $X$, let us consider
the exact sequence of sheaves
$$
0\longrightarrow\mathcal{I}\big(X, B_{X}\big)\otimes
 D\longrightarrow\mathcal{O}_{X}\big(D\big)\longrightarrow\mathcal{O}_{\mathcal{L}(X,\,
 B_{X})}\big(D\big)\longrightarrow 0,%
$$
and let us consider the corresponding exact sequence of cohomology
groups
$$
H^{0}\Big(\mathcal{O}_{X}\big(D\big)\Big)\longrightarrow
H^{0}\Big(\mathcal{O}_{\mathcal{L}(X,\,
 B_{X})}\big(D\big)\Big)\longrightarrow H^{1}\Big(\mathcal{I}\big(X, B_{X}\big)\otimes D\Big).%
$$

\begin{theorem}
\label{theorem:connectedness} Suppose that $-(K_{X}+B_{X})$ is nef
and big. Then $\mathrm{LCS}(X, B_{X})$ is connected.
\end{theorem}

\begin{proof}
Put $D=0$. Then it follows from
Theorem~\ref{theorem:Shokurov-vanishing} that the sequence
$$
\mathbb{C}=H^{0}\Big(\mathcal{O}_{X}\Big)\longrightarrow
H^{0}\Big(\mathcal{O}_{\mathcal{L}(X,\,
 B_{X})}\Big)\longrightarrow H^{1}\Big(\mathcal{I}\big(X, B_{X}\big)\Big)=0%
$$
is exact. Thus, the
locus
$$
\mathrm{LCS}\Big(X,\ B_{X}\Big)=\mathrm{Supp}\Big(\mathcal{L}\big(X,\ B_{X}\big)\Big)%
$$
is connected.
\end{proof}

Let us consider few elementary applications of
Theorem~\ref{theorem:connectedness} (cf.
Example~\ref{example:del-Pezzos}).


\begin{lemma}
\label{lemma:plane} Suppose that $\mathrm{LCS}(X,
B_{X})\ne\varnothing$, where $X\cong\mathbb{P}^{n}$, and
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some rational number $0<\lambda<n/(n+1)$. Then
\begin{itemize}
\item the inequality $\mathrm{dim}(\mathrm{LCS}(X, B_{X}))\ge 1$
holds,
\item the subscheme $\mathcal{L}(X,B_{X})$ does not contain isolated zero-dimensional components.%
\end{itemize}
\end{lemma}

\begin{proof}
Suppose that there is a point $O\in X$ such that
$$
\mathrm{LCS}\Big(X, \lambda B_{X}\Big)=O\cup\Sigma,
$$
where $\Sigma\subset X$ is a possibly empty subset such that
$O\not\in X$.

Let $H$ be a general hyperplane in $X\cong\mathbb{P}^{2}$. Then
the locus
$$
\mathrm{LCS}\Big(X, \lambda B_{X}+H\Big)=O\cup H\cup\Sigma
$$
is disconnected. But the divisor $-(K_{X}+\lambda B_{X}+H)$ is
ample, which contradicts Theorem~\ref{theorem:connectedness}.
\end{proof}

\begin{lemma}
\label{lemma:P3} Suppose that $\mathrm{LCS}(X,
B_{X})\ne\varnothing$, where $X\cong\mathbb{P}^{3}$, and
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some rational number $0<\lambda<1/2$. Then
$\mathbb{LCS}(X, B_{X})$ contains a surface.
\end{lemma}

\begin{proof}
Suppose that $\mathbb{LCS}(X, B_{X})$ contains no surfaces. Let
$S\subset\mathbb{P}^{3}$ be a general plane. The~locus
$$
\LCS\Big(\P^3,\ B_{X}+S\Big)
$$
is connected by Theorem~\ref{theorem:connectedness}. Then $(S,B_{X}\vert_{S})$
is not log terminal by Remark~\ref{remark:hyperplane-reduction}. But the locus
$$
\LCS\Big(S,\ B_{X}\Big\vert_{S}\Big)
$$
consists of finitely many points, which is impossible by
Lemma~\ref{lemma:plane}.
\end{proof}

\begin{lemma}
\label{lemma:quadric-curves} Suppose that $\mathrm{LCS}(X,
B_{X})\ne\varnothing$, where $X$ is a smooth quadric threefold in~$\P^4$,~and
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some rational number $0<\lambda<1/2$. Then
$\mathbb{LCS}(X, B_{X})$ contains a surface.
\end{lemma}

\begin{proof}
Let $L\subset X$ be a general line, let $P_1\in L\ni P_2$ be two general
points, let $H_1$ and $H_2$ be the hyperplane sections of $X$ that are tangent
to $X$ at the points $P_1$ and $P_2$, respectively. Then
$$
\LCS\left(X,\ \lambda B_X+\frac{3}{4}\Big(H_1+H_2\Big)\right)=\LCS\Big(X,\ \lambda B_X\Big)\cup L%
$$
is disconnected, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}

\begin{remark}
One can prove Lemmas~\ref{lemma:P3},~\ref{lemma:quadric-curves} and
\ref{lemma:P1xP2} using the following trick. Suppose that
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some $\lambda\in\mathbb{Q}$ such that $0<\lambda<1/2$, where
$X$ is either $\mathbb{P}^{3}$, or $\P^1\times\P^2$, or a smooth
quadric threefold, and the set $\mathbb{LCS}(X, B_{X})$ contains
no surfaces. Then
$$
\LCS\Big(X,\ B_{X}\Big)\subseteq\Sigma,
$$
where $\Sigma\subset X$ is a (possibly reducible) curve. For a
general $\phi\in\mathrm{Aut}(X)$ we have
$$
\phi\big(\Sigma\big)\cap \Sigma=\varnothing,
$$
which implies that $\LCS(X, \phi(B_{X}))\cap\LCS(X,
B_{X})=\varnothing$. But
$$
\LCS\Big(X,\ \phi\big(B_{X}\big)+B_{X}\Big)=\LCS\Big(X,\ \phi\big(B_{X}\big)\Big)\bigcup\LCS\Big(X,\ B_{X}\Big)%
$$
whenever $\phi$ is sufficiently general. The latter contradicts
Theorem~\ref{theorem:connectedness} since $\lambda<1/2$.
\end{remark}

\begin{lemma}
\label{lemma:V7} Suppose that $\mathrm{LCS}(X,
B_{X})\ne\varnothing$, where $X$ is a blow up of $\P^3$ in one
point,~and
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some rational number $0<\lambda<1/2$. Then
$\mathbb{LCS}(X, B_{X})$ contains a surface.
\end{lemma}

\begin{proof}
Suppose that the set $\mathbb{LCS}(X, B_{X})$ contains no
surfaces. Let
$$
\alpha\colon X\longrightarrow\P^3
$$
be the blow up of a point, and let $E$ be the exceptional divisor
of $\alpha$. In the case when
$$
\LCS\big(X,\ \lambda B_{X}\Big)\not\subseteq E,
$$
we can apply Lemma~\ref{lemma:P3} to the pair
$(\mathbb{P}^{3},\alpha(B_{X}))$ to get a contradiction. Hence
$\LCS(X, B_{X})\subseteq E$.

Let $H\subset\P^3$ a general hyperplane, and let $H_1\subset\P^3\supset H_2$ be
general hyperplanes that pass through $\alpha(E)$. Denote by $\bar{H}$,
$\bar{H}_1$ and $\bar{H}_2$ the proper transforms of these planes on $X$. Then
$$
\LCS\left(X,\
B_{X}+\frac{1}{2}\Big(\bar{H}_1+\bar{H}_2+2\bar{H}\Big)\right)
$$
is disconnected, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}


\begin{lemma}
\label{lemma:singular-quadric-threefold} Suppose that $X$ is a
cone in $\mathbb{P}^{4}$ over a smooth quadric surface, and
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some rational number $0<\lambda<1/3$. Then $\mathbb{LCS}(X,
B_{X})=\varnothing$.
\end{lemma}

\begin{proof}
Suppose that $\mathbb{LCS}(X, B_{X})\ne \varnothing$. Let
$S\subset Q$ be a general hyperplane section. Then
$$
\LCS\Big(S,\ B_{X}\Big\vert_{S}\Big)=\varnothing,
$$
because $S\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathrm{lct}(\mathbb{P}^{1}\times\mathbb{P}^{1})$ (see
Example~\ref{example:del-Pezzos}).

The inequality $|\mathrm{LCS}(X, B_{X})|<+\infty$ holds by
Remark~\ref{remark:hyperplane-reduction}. Then the~locus
$$
\LCS\Big(Q,\ B_{X}+S\Big)
$$
is disconnected, which contradicts
Theorem~\ref{theorem:connectedness}.
\end{proof}

The following result is a corollary
Theorem~\ref{theorem:Shokurov-vanishing}
(see~\cite[Theorem~4.1]{Na90}).

\begin{lemma}\label{lemma:rational-tree}
Suppose that $-(K_{X}+B_{X})$ is nef and big and
$\mathrm{dim}(\mathrm{LCS}(X, B_{X}))=1$. Then
\begin{itemize}
\item the locus $\mathrm{LCS}(X, B_X)$ is a connected union of smooth rational curves,%
\item every two irreducible components of the locus
$\mathrm{LCS}(X, B_X)$ meet in at most one
 point,%
\item every intersecting irreducible components of the locus
$\mathrm{LCS}(X, B_X)$ meet
 transversally,%
\item no three irreducible components of the locus $\mathrm{LCS}(X, B_X)$ meet in one point,%
\item the locus $\mathrm{LCS}(X, B_X)$ does not contain a cycle of smooth rational curves.%
\end{itemize}
\end{lemma}

\begin{proof}
Arguing as in the proof of Theorem~\ref{theorem:connectedness}, we
see that the locus $\mathrm{LCS}(X, B_X)$ is a connected tree of
smooth rational curves with simple normal crossings.
\end{proof}


To consider another application of
Theorem~\ref{theorem:connectedness}, we need the following result
(see \cite{Pu95}, \cite{Ch03b}).

\begin{lemma}
\label{lemma:Cheltsov-Pukhlikov}%
Suppose that $X$ is a smooth complete intersection
$\cap_{i=1}^{k}G_{i}\subset\mathbb{P}^{m}$, and
$$
B_{X}\qlineq\mathcal{O}_{\mathbb{P}^{m}}(1)\Big\vert_{X},
$$
where $G_{i}$ is a hypersurface. Let $S\subsetneq X$ be an
irreducible subvariety such that $\mathrm{dim}(S)\geqslant k$.
Then
$$
\mathrm{mult}_{S}\big(B_{X}\big)\leqslant 1.
$$
\end{lemma}

\begin{proof}
We may assume $\mathrm{dim}(S)=k\leqslant (m-1)\slash 2$. Let $P$
be a sufficiently general point in $\mathbb{P}^{m}$, and let
$C\subset\mathbb{P}^{m}$ be a cone over the subvariety $S$ with
vertex in the point $P$. Then
$$
C\cap X=S\cup R,
$$
where $R$ is a curve on $X$. Let us calculate $|R\cap S|$. Let
$$
\pi\colon X\longrightarrow\mathbb{P}^{m-1},\ \pi_{1}\colon
G_{1}\longrightarrow\mathbb{P}^{m-1},\
 \ldots,\ \pi_{k}\colon G_{k}\longrightarrow\mathbb{P}^{m-1}%
$$
be projections from the point $P$, let $D\subset X$ and
$D_{i}\subset G_{i}$ be the ramification subvarieties of the
projections  $\pi$ and $\pi_{i}$, respectively. Put $C\cap
G_{i}=S\cup R_{i}$. Then
$$
R_{i}\cap S=D_{i}\cap S
$$
by \cite[Lemma~3]{Pu02a}. Hence, it follows from
$R=\cap_{i=1}^{k}R_{i}$ and $D=\cap_{i=1}^{k}D_{i}$ that $R\cap
S=D\cap S$.

Let $(z_{0},\ldots,z_{m})$ be homogeneous coordinates on
$\mathbb{P}^{m}$ such that
$$
P=\big(p_{0},\ldots, p_{m}\big),
$$
and $G_{i}\subset\mathbb{P}^{m}$ is given by the equation
$F_{i}(z_{0},\ldots,z_{m})=0$. Then $D\subset X$ is cut out by
$$
\sum_{i=0}^{m}{\frac {\partial F_{1}\big(z_{0},\ldots,z_{m}\big)}
{\partial
 z_{i}}}p_{i}=\sum_{i=0}^{m}{\frac {\partial F_{2}\big(z_{0},\ldots,z_{m}\big)} {\partial
 z_{i}}}p_{i}=\ldots=\sum_{i=0}^{m}{\frac {\partial F_{k}\big(z_{0},\ldots,z_{m}\big)} {\partial
 z_{i}}}p_{i}=0.%
$$

Let $\mathcal{F}_{r}$ be a linear system on $\mathbb{P}^{m}$ that
contains divisors
$$
\sum_{i=0}^{m}\lambda_{i}{\frac {\partial F_{r}\big(z_{0},\ldots,z_{m}\big)} {\partial z_{i}}}=0%
$$
where $(\lambda_{0},\ldots,\lambda_{m})\in\mathbb{P}^{m}$ and
$r=1,\ldots,k$. The variety $X$ is smooth. Then
$$
\big|R\cap S\big|=\big|D\cap
 S\big|=\mathrm{deg}\big(S\big)\prod_{i=1}^{k}\Big(\mathrm{deg}\big(F_{i}\big)-1\Big),%
$$
because $\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$ do not base
points on $X$. Therefore, we have the inequality
$$
\mathrm{deg}\big(S\big)\prod_{i=1}^{k}\Big(\mathrm{deg}\big(F_{i}\big)-1\Big)=B_{X}\cdot
R\geqslant\sum_{O\in R\cap
 S}\mathrm{mult}_{S}\big(B_{X}\big)=\mathrm{mult}_{S}\big(B_{X}\big)\big|R\cap S\big|,%
$$
which implies $\mathrm{mult}_{S}(B_{X})\leqslant 1$.
\end{proof}

Using Remark~\ref{remark:hyperplane-reduction},
Theorem~\ref{theorem:connectedness} and
Lemma~\ref{lemma:Cheltsov-Pukhlikov}, we obtain the following
result.

\begin{corollary}
\label{corollary:lct-ci-big-index} Let $X$ is a smooth complete
intersection $\cap_{i=1}^{k}G_{i}\subset\mathbb{P}^{m}$ such that
$$
m+1-\sum_{i=1}^{k}\mathrm{deg}\big(G_{i}\big)\geqslant k+1,
$$
where $G_{i}$ is a hypersurface in $\mathbb{P}^{m}$. Then $X$ is a
Fano variety and
$$
\mathrm{lct}\big(X\big)=\frac{1}{m+1-\sum_{i=1}^{k}\mathrm{deg}\big(G_{i}\big)}.
$$
\end{corollary}

Let us consider another  simple application of
Theorem~\ref{theorem:connectedness} and
Lemma~\ref{lemma:Cheltsov-Pukhlikov}.

\begin{lemma}
\label{lemma:singular-cubic-threefold} Let $X$ be a cubic
hypersurface in $\mathbb{P}^{4}$ such that
$|\mathrm{Sing}(X)|<+\infty$. Suppose that
$$
B_{X}\qlineq -K_{X},
$$
and there is positive rational number $\lambda<1/2$ such that
$\mathrm{LCS}(X,\lambda B_{X})\ne\varnothing$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda B_{X}\Big)=L,
$$
where $L$ is a line in $X\subset\mathbb{P}^{4}$ such that $L\cap\mathrm{Sing}(X)\ne\varnothing$.%
\end{lemma}

\begin{proof}
Let $S$ be a general hyperplane section of $X$. Then
$$
S\cup\mathrm{LCS}\Big(X,\ \lambda B_{X}\Big)\subseteq \mathrm{LCS}\Big(X,\ \lambda B_{X}+S\Big),%
$$
which implies that $\dim(\mathrm{LCS}(X,\lambda B_{X}))\ge 1$ by
Theorem~\ref{theorem:connectedness}. Then
$$
\mathrm{LCS}\Big(S,\ \lambda B_{X}\Big\vert_{S}\Big)\ne\varnothing
$$
by Remark~\ref{remark:hyperplane-reduction}. But
$|\mathrm{LCS}(S,\ \lambda B_{X}\vert_{S})|<+\infty$ by
Lemma~\ref{lemma:Cheltsov-Pukhlikov}. There is a point $O\in S$
such that
$$
\mathrm{LCS}\Big(S,\ \lambda B_{X}\Big\vert_{S}\Big)=O%
$$
by Theorem~\ref{theorem:connectedness}. Therefore, there is a line $L\subset X$
such that $\mathrm{LCS}(X,\lambda B_{X})=L$ by
Remark~\ref{remark:hyperplane-reduction}.

Arguing as in the proof of Lemma~\ref{lemma:Cheltsov-Pukhlikov}, we see that
$L\cap\mathrm{Sing}(X)\ne\varnothing$.
\end{proof}

Similar to Lemma~\ref{lemma:singular-cubic-threefold}, one can
prove the following result.

\begin{lemma}
\label{lemma:double-covers} Suppose that there is a double cover $\tau\colon
X\to\mathbb{P}^{3}$ branched over an irreducible reduced quartic surface
$R\subset\mathbb{P}^{3}$ that has at most ordinary double points, the
equivalence
$$
B_{X}\qlineq -\lambda K_{X}
$$
holds and $\mathrm{LCS}(X,B_{X})\ne\varnothing$, where
$\lambda\in\mathbb{Q}$ is such that $\lambda<1/2$. Then
$\mathrm{Sing}(X)\ne\varnothing$ and
$$
\mathrm{LCS}\Big(X,\ B_{X}\Big)=L,
$$
where $L$ is an irreducible curve on $X$ such that $-K_{X}\cdot
L=2$ and $L\cap\mathrm{Sing}(X)\ne\varnothing$.
\end{lemma}

\begin{proof}
We have $-K_{X}\sim 2H$, where $H$ is a Cartier divisor on $X$
such that
$$
H\sim\tau^{*}\Big(\mathcal{O}_{\mathbb{P}^{3}}\big(1\big)\Big),
$$
which implies that $X$ is a Fano threefold and $H^{3}=2$. Then
$$
\mathrm{LCS}\Big(X,\ B_{X}+H\Big)
$$
must be connected by Theorem~\ref{theorem:connectedness}.
Thus, there is a curve
$$
C\in\mathbb{LCS}\Big(X,\ B_{X}\Big),
$$
which implies that $\mathrm{mult}_{C}(B_{X})\geqslant
1/\lambda>2$.

Let $S$ be a general surface in $|H|$. Put $B_{S}=B_{X}\vert_{S}$. Then
$$
-K_{S}\sim H\big\vert_{S}\qlineq \frac{1}{\lambda}B_{S},
$$
but the log pair $(S, B_{S})$ is not log canonical in every point
of the intersection $S\cap\mathrm{LCS}(X,B_{X})$.

The surface $H$ is a smooth surface in $\mathbb{P}(1,1,1,2)$ of degree $4$.

Let $P$ be any point in $S\cap\mathrm{LCS}(X,B_{X})$. Then there
is a birational morphism
$$
\rho\colon S\longrightarrow\bar{S}
$$
such that $\bar{S}$ is a cubic surface in $\mathbb{P}^{3}$ and $\rho$ is an
isomorphism in a neighborhood of $P$. Then
$$
\Big(\bar{S},\ \rho\big(B_{S}\big)\Big)
$$
is not log terminal at the point $\rho(P)$. Thus, we have
$\mathrm{LCS}(\bar{S},\rho(B_{S}))\ne\varnothing$. But
$$
\frac{1}{\lambda}\rho(B_{S})\qlineq-K_{\bar{S}}\sim\mathcal{O}_{\mathbb{P}^{3}}\big(1\big)\Big\vert_{\bar{S}},
$$
which implies that $\mathrm{LCS}(\bar{S},\rho(B_{S}))$ consists of one point by
Lemma~\ref{lemma:Cheltsov-Pukhlikov} and Theorem~\ref{theorem:connectedness}.
Then
$$
P=S\cap C=S\cap\mathrm{LCS}\Big(X,B_{X}\Big)
$$
if the point $P$ is sufficiently general. Therefore, we see that
$$
\mathrm{LCS}\Big(X,B_{X}\Big)=C,
$$
the curve $C$ is irreducible and $-K_{X}\cdot C=2$. Then
$\tau(C)\subset\mathbb{P}^{3}$ is a line.

Suppose that $C\cap\mathrm{Sing}(X)=\varnothing$. Let us derive a
contradiction.

Suppose that $\tau(C)\subset R$. Take a general point $O\in C$.
Let
$$
\tau\big(O\big)\in \Pi\subset\mathbb{P}^{3}
$$
be a plane that is tangent to $R$ at the point $\tau(O)$. Arguing as in the
proof of Lemma~\ref{lemma:Cheltsov-Pukhlikov}, we see that $S\vert_{\Pi}$ is
reduced along $\tau(C)$, because $\tau(C)\cap\mathrm{Sing}(R)=\varnothing$. Fix
a general line
$$
\Gamma\subset\Pi\subset\mathbb{P}^{3}
$$
such that $\tau(O)\in\Gamma$. Let $\bar{\Gamma}\subset X$ be an
irreducible curve such that $\tau(\bar{\Gamma})=\Gamma$. Then
$$
\bar{\Gamma}\not\subseteq\mathrm{Supp}\Big(B_{X}\Big),
$$
because $\Gamma$ spans a dense subset in $\mathbb{P}^{3}$ when we
vary the point $O\in C$ and the line $\Gamma\subset\Pi$.
Note that $H\cdot\bar{\Gamma}$ equals either $1$ or $2$, and
$\mult_O(\bar{\Gamma})=2$ in the case when $H\cdot\bar{Gamma}=2$. Hence
$$
H\cdot\bar{\Gamma}>2\lambda H\cdot\bar{\Gamma}=\bar{\Gamma}\cdot B_{X}
\geqslant\mathrm{mult}_{O}\big(\bar{\Gamma}\big)\mathrm{mult}_{C}
\big(B_{V}\big)\geqslant H\cdot\bar{\Gamma}.%
$$
%
%$H\cdot\Gamma=1$. Then
%$$
%1>2\lambda=\bar{\Gamma}\cdot B_{X}\geqslant\mathrm{mult}_{O}\big(\bar{\Gamma}\big)\mathrm{mult}_{C}\big(B_{V}\big)\geqslant 1,%
%$$
which is a contradiction. Thus, we see that $\tau(C)\not\subset R$.


There is an irreducible reduced curve $\bar{C}\subset X$ such that
$$
\tau\big(\bar{C}\big)=\tau\big(C\big)\subset\mathbb{P}^{3}
$$
and $\bar{C}\ne C$. Let $Y$ be a general surface in $|H|$ that
passes through the curves $\bar{C}$ and $C$. Then $Y$ is smooth, because
$C\cap\mathrm{Sing}(X)=\varnothing$, and
$$
\bar{C}\cdot\bar{C}=C\cdot C=-2
$$
on the surface $Y$.

By construction, we have $Y\not\subset\mathrm{Supp}(B_{X})$. Put
$B_{Y}=B_{X}\vert_{Y}$. Then
$$
B_{Y}=\mathrm{mult}_{\bar{C}}\big(B_{X}\big)\bar{C}+\mathrm{mult}_{C}\big(B_{X}\big)C+\Delta%
$$
where $\Delta$ is an effective $\mathbb{Q}$-divisor on the surface
$Y$ such that
$\bar{C}\not\subset\mathrm{Supp}(\Delta)\not\supset C$. But
$$
B_{Y}\qlineq 2\lambda\Big(\bar{C}+C\Big),
$$
which implies, in particular, that
$$
\Big(2\lambda-\mathrm{mult}_{C}\big(B_{X}\big)\Big)C\cdot C=\Big(\mathrm{mult}_{\bar{C}}\big(B_{X}\big)-2\lambda\Big)\bar{C}\cdot C+\Delta\cdot C\geqslant \Big(\mathrm{mult}_{\bar{C}}\big(B_{X}\big)-2\lambda\Big)\bar{C}\cdot C\geqslant 0,%
$$
because $\Delta\cdot C\geqslant 0$ and $\bar{C}\cdot C\geqslant 0$. Then
$\mathrm{mult}_{\bar{C}}(B_{X})\geqslant 2\lambda$, because $C\cdot C<0$. Thus,
we have
$$
-\Delta\qlineq\Big(\mathrm{mult}_{\bar{C}}\big(B_{X}\big)-2\lambda\Big)\bar{C}+\Big(\mathrm{mult}_{C}\big(B_{X}\big)-2\lambda\Big)C%
$$
which is impossible, because $\mathrm{mult}_{C}(B_{X})>2\lambda$ and $Y$ is
projective.
\end{proof}




One can generalize Theorem~\ref{theorem:connectedness} in the
following way (see \cite[Lemma~5.7]{Sho93}).

\begin{theorem}
\label{theorem:log-adjunction-connectedness-theorem} Let
$\psi\colon X\to Z$ be a morphism. Then the set
$$
\mathrm{LCS}\Big(\bar{X},\ B^{\bar{X}}\Big)
$$
is connected in a neighborhood of every fiber of the morphism
$\psi\circ\pi\colon X\to Z$ in the case when
\begin{itemize}
\item the morphism $\psi$ is surjective and has connected fibers,%
\item the divisor $-(K_{X}+B_{X})$ is nef and big with respect to $\psi$.%
\end{itemize}
\end{theorem}

Let us consider one important application of
Theorem~\ref{theorem:log-adjunction-connectedness-theorem} (see
\cite[Theorem~5.50]{KoMo98}).

\begin{theorem}
\label{theorem:adjunction} Suppose that $B_{1}$ is a Cartier
divisor, $a_{1}=1$, and $B_{1}$ has at most log terminal
singularities. Then the following assertions are equivalent:
\begin{itemize}
\item the log pair $(X, B_{X})$ is log canonical in  a neighborhood of the divisor $B_{1}$;%
\item the singularities of the log pair
$(B_{1},\sum_{i=2}^{r}a_{i}B_{i}\vert_{B_{1}})$ are log
 canonical.%
\end{itemize}
\end{theorem}

\begin{proof}
Suppose that the singularities of the log pair $(X,B_{X})$ are not
log canonical in a neighborhood of the divisor $B_{1}\subset X$.
Let us show that $(B_{1},\sum_{i=2}^{r}a_{i}B_{i}\vert_{B_{1}})$
is not log canonical.

In the case when $a_{m}>1$ and $B_{m}\cap B_{1}\ne\varnothing$ for
some $m\geqslant 2$, the log pair
$$
\Big(B_{1},\ \sum_{i=2}^{r}a_{i}B_{i}\Big\vert_{B_{1}}\Big)
$$
is not log canonical by
Definition~\ref{definition:log-canonical-singularities}. Thus, we
may assume that $a_{i}\leqslant 1$ for every $i$. Then
$$
\Big(X,\ B_{1}+\sum_{i=2}^{r}\lambda a_{i}B_{i}\Big)
$$
is not log canonical as well for some rational number $\lambda<1$.
Then
$$
K_{\bar{X}}+\bar{B}_{1}+\sum_{i=2}^{r}\lambda
a_{i}\bar{B}_{i}\qlineq\
 \pi^{*}\Big(K_{X}+B_{1}+\sum_{i=2}^{r}\lambda a_{i}B_{i}\Big)+\sum_{i=1}^{n}d_{i}E_{i}%
$$
for some rational numbers $d_{1},\ldots,d_{n}$. It follows from
Theorem~\ref{theorem:log-adjunction-connectedness-theorem} that
$$
\bar{B}_{1}\cap E_{k}\ne\varnothing
$$
and the inequality $d_{k}\leqslant -1$ holds for some $k$. But
$$
K_{\bar{B}_{1}}+\sum_{i=2}^{r}\lambda
 a_{i}\bar{B}_{i}\Big\vert_{B_{1}}\qlineq\phi^{*}\Big(K_{B_{1}}+\sum_{i=2}^{r}\lambda
 a_{i}B_{i}\Big\vert_{B_{1}}\big)+\sum_{i=1}^{n}d_{i}E_{i}\Big\vert_{B_{1}},%
$$
where $\phi\colon\bar{B}_{1}\to B_{1}$ is a birational morphism
that is induced by $\pi$.

Thus, the log pair $(B_{1},\sum_{i=2}^{r}\lambda
a_{i}B_{i}\vert_{B_{1}})$ is not log terminal. Then the log pair
$$
\Big(B_{1},\ \sum_{i=2}^{r}a_{i}B_{i}\Big\vert_{B_{1}}\Big)%
$$
is not log canonical. The rest of the proof is similar (see the
proof of~\cite[Theorem~7.5]{Ko97}).
\end{proof}

The simplest application of Theorem~\ref{theorem:adjunction} is a non-obvious
result (see \cite[Corollary~5.57]{KoMo98}).

\begin{lemma}
\label{lemma:adjunction} Suppose that $\mathrm{dim}(X)=2$ and
$a_{1}\leqslant 1$. Then
$$
\Big(\sum_{i=2}^{r}a_{i}B_{i}\Big)\cdot B_{1}>1
$$
whenever $(X,B_{X})$ is not log canonical in a point $O\in B_{1}$
such that $O\not\in\mathrm{Sing}(X)\cup\mathrm{Sing}(B_{1})$.
\end{lemma}

\begin{proof}
Suppose that  $(X,B_{X})$ is not log canonical in~a point $O\in B_{1}$. By
Theorem~\ref{theorem:adjunction}, we have
$$
\Big(\sum_{i=2}^{r}a_{i}B_{i}\Big)\cdot B_{1}\geqslant
\mathrm{mult}_{O}\Big(\sum_{i=2}^{r}a_{i}B_{i}\Big\vert_{B_{1}}\Big)>1
$$
if $O\not\in\mathrm{Sing}(X)\cup\mathrm{Sing}(B_{1})$, because $(X,
B_{1}+\sum_{i=2}^{r}a_{i}B_{i})$ is not log canonical at the point $O$.
\end{proof}

Let us consider another application of Theorem~\ref{theorem:adjunction} (cf.
Lemma~\ref{lemma:lct-product}).

\begin{lemma}
\label{lemma:lct-P1-product} Suppose that $X$ is a Fano variety
with log terminal singularities. Then
$$
\mathrm{lct}\Big(\mathbb{P}^{1}\times
X\Big)=\mathrm{min}\left(\frac{1}{2},\
 \mathrm{lct}\big(X\big)\right).%
$$
\end{lemma}

\begin{proof}
The inequalities $1/2\geqslant\mathrm{lct}(V\times
U)\leqslant\mathrm{lct}(X)$ are obvious. Suppose that
$$
\mathrm{lct}\Big(\mathbb{P}^{1}\times
X\Big)<\mathrm{min}\left(\frac{1}{2},\
 \mathrm{lct}\big(X\big)\right),%
$$
and let us show that this assumption leads to a contradiction.

There is an effective $\mathbb{Q}$-divisor $D\qlineq
-K_{\mathbb{P}^{1}\times X}$ such that the log pair
$$
\Big(\mathbb{P}^{1}\times X,\ \lambda D\Big)
$$
is not log canonical in some point $P\in \mathbb{P}^{1}\times X$,
where $\lambda\in\mathbb{Q}$ is such that
$\lambda<\min(1/2, \mathrm{lct}(X))$.

Let $F$ be a fiber of the projection $\mathbb{P}^{1}\times
X\to\mathbb{P}^{1}$ such that $P\in F$. Then
$$
D=\mu F+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on
$\mathbb{P}^{1}\times X$ such that
$F\not\subset\mathrm{Supp}(\Omega)$.

Let $L$ be a general fiber of the projection $\mathbb{P}^{1}\times
X\to X$. Then
$$
2=D\cdot L=\mu+\Omega\cdot L\geqslant\mu,
$$
which implies that the log pair $(\mathbb{P}^{1}\times X,
F+\lambda\Omega)$ is not log canonical at the point $P$. Then
$$
\Big(F,\ \lambda\Omega\Big\vert_{F}\Big)
$$
is not log canonical at the point $P$ by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{F}\qlineq D\Big\vert_{F}\qlineq -K_{F},
$$
which is impossible, because $X\cong F$ and
$\lambda<\mathrm{lct}(X)$.
\end{proof}

Let $P$ be a point in $X$. Let us consider an effective  divisor
$$
\Delta=\sum_{i=1}^{r}\eps_{i}B_{i}\qlineq B_{X},%
$$
where $\eps_{i}$ is a non-negative rational number. Suppose
that
\begin{itemize}
\item the divisor $\Delta$ is a $\mathbb{Q}$-Cartier divisor,%
\item the equivalence $\Delta\qlineq B_{X}$ holds,%
\item the log pair  $(X, \Delta)$ is log canonical in the point $P\in X$.%
\end{itemize}


\begin{remark}
\label{remark:convexity} Suppose that $(X, B_{X})$ is not log
canonical in the point $P\in X$. Put
$$
\alpha=\mathrm{min}\left\{\frac{a_{i}}{\eps_{i}}\ \Big\vert\ \eps_{i}\ne 0\right\},%
$$
where $\alpha$ is well defined, because there is $\eps_{i}\ne
0$. Then $\alpha<1$, the log pair
$$
\left(X,\ \sum_{i=1}^{r}\frac{a_{i}-\alpha\eps_{i}}{1-\alpha}B_{i}\right)%
$$
is not log canonical in the point $P\in X$, the equivalence
$$
\sum_{i=1}^{r}\frac{a_{i}-\alpha\eps_{i}}{1-\alpha}B_{i}\qlineq B_{X}\qlineq\Delta%
$$
holds, and at least one irreducible component of the divisor
$\mathrm{Supp}(\Delta)$ is not contained in
$$
\mathrm{Supp}\left(\sum_{i=1}^{r}\frac{a_{i}-\alpha\eps_{i}}{1-\alpha}B_{i}\right).
$$
\end{remark}


The assertion of Remark~\ref{remark:convexity} is obvious.
Nevertheless it is very useful.


\begin{lemma}
\label{lemma:elliptic-times-P1} Suppose that $X\cong C_1\times C_2$, where
$C_1$ and $C_{2}$ are smooth curves, suppose that
$$
B_{X}\qlineq\lambda E+\mu F
$$
where $E\cong C_1$ and $F\cong C_2$ are curves on the surface $X$ such that
$$
E\cdot E=F\cdot F=0
$$
and $E\cdot F=1$, and $\lambda$ and $\mu$ are non-negative rational numbers.
Then
\begin{itemize}
\item the pair $(X, B_X)$ is log terminal if $\lambda<1$ and $\mu<1$,%
\item the pair $(X, B_X)$ is log canonical if $\lambda\leqslant 1$
and $\mu\leqslant 1$.
\end{itemize}
\end{lemma}
\begin{proof}
Suppose that $\lambda<1>\mu$, but $(X, B_{X})$ is not log terminal at some
point $P\in X$. Then
$$
\mathrm{mult}_{P}\big(B_{X}\big)\geqslant 1,
$$
and we may assume that $E\not\subset\mathrm{Supp}(B_{X})$ or
$F\not\subset\mathrm{Supp}(B_{X})$ by
Remark~\ref{remark:convexity}. But
$$
E\cdot B_{X}=\mu,\ F\cdot B_{X}=\lambda,
$$
which immediately leads to a contradiction, because
$\mathrm{mult}_{P}(B_{X})\geqslant 1$.
\end{proof}

Let $[B_{X}]$ be a (non-empty) class of $\mathbb{Q}$-rational equivalence of
the divisor $B_{X}$. Put
$$
\mathrm{lct}\Big(X,
\big[B_{X}\big]\Big)=\mathrm{inf}\left\{\mathrm{lct}\big(X,D\big)\
\Big\vert\
 D\ \text{is an effective $\mathbb{Q}$-divisor on $X$ such that}\ D\sim_{\mathbb{Q}}
 B_{X}\right\}\geqslant 0,%
$$
and put $\mathrm{lct}(X,[B_{X}])=+\infty$ in the case when
$B_{X}\sim_{\mathbb{Q}} 0$.


\begin{remark}
\label{remark:lct-class} The equality
$\mathrm{lct}(X,[-K_{X}])=\mathrm{lct}(X)$ holds
(see Definition~\ref{definition:threshold}).
\end{remark}

Arguing as in the proof of Lemma~\ref{lemma:lct-P1-product}, we obtain the
following result.

\begin{lemma}
\label{lemma:Hwang} Suppose that there is a surjective morphism with connected
fibers
$$
\phi\colon X\longrightarrow Z
$$
such that $\dim(Z)=1$. Let $F$ be a fiber of $\phi$ that has log
terminal singularities. Then either
$$
\mathrm{lct}_{F}\Big(X,
B_{X}\Big)\geqslant\mathrm{lct}\Big(F,\Big[B_{X}\big\vert_{F}\Big]\Big),
$$
or there is a positive rational number
$\eps<\mathrm{lct}(F,[B_{X}\vert_{F}])$ such that
$F\subseteq\mathrm{LCS}(X, \eps B_{X})$.
\end{lemma}

\begin{proof}
Suppose that $\mathrm{lct}_{F}(X, B_{X})<\mathrm{lct}(F,[B_{X}\vert_{F}])$.
Then there is a rational number
$$
\eps<\mathrm{lct}\Big(F,\Big[B_{X}\big\vert_{F}\Big]\Big)
$$
such that the log pair $(X,\eps B_{X})$ is not log canonical at some point $P\in
F$. Put
$$
B_{X}=\mu F+\Omega,
$$
where $\Omega$ is an
effective $\mathbb{Q}$-divisor on $X$ such that
$F\not\subset\mathrm{Supp}(\Omega)$.

We may assume that $\eps\mu\leqslant 1$. Then $(X, F+\eps\Omega)$ is
not canonical at the point $P$. Then
$$
\Big(F,\ \eps\Omega\Big\vert_{F}\Big)
$$
is not log canonical at $P$ by Theorem~\ref{theorem:adjunction}. But
$\Omega\vert_{F}\sim_{\mathbb{Q}} B_{X}\vert_{F}$, which is a contradiction.
\end{proof}

Let us show how to apply Lemma~\ref{lemma:Hwang}.

\begin{lemma}
\label{lemma:2-29-singular} Let $Q\subset\mathbb{P}^{4}$ be a cone
over a smooth quadric surface, and let $\alpha\colon X\to Q$ be a
blow up along a smooth conic $C\subset
Q\setminus\mathrm{Sing}(Q)$. Then $\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $H$ be a general hyperplane section of
$Q\subset\mathbb{P}^{4}$ that contains $C$, and let $\bar{H}$ be a
proper transform of the surface $H$ on the threefold $X$. Then
$$
-K_{X}\sim 3\bar{H}+2E,
$$
where $E$ is the exceptional divisor of $\alpha$. In particular,
the inequality $\mathrm{lct}(X)\leqslant 1/3$ holds.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log
pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/3$.

There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[rd]^{\beta}&\\%
Q\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{1},}
$$
where $\beta$ is a morphism given by the linear system
$|\bar{H}|$, and $\psi$ is a projection from the two-dimensional
linear subspace that contains the conic $C$.

Suppose that $\mathbb{LCS}(X,\lambda D)$ contains a surface
$M\subset X$. Then
$$
D=\mu M+\Omega,
$$
where $\mu\geqslant 1/\lambda$, and $\Omega$ is an effective
$\mathbb{Q}$-divisor such that
$M\not\subset\mathrm{Supp}(\Omega)$.

Let $F$ be a general fiber of $\beta$. Then
$F\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$$
D\Big\vert_{F}=\mu M\Big\vert_{F}+\Omega\Big\vert_{F}\qlineq -K_{F},%
$$
which immediately implies that $M$ is a fiber of the morphism
$\beta$. But
$$
\alpha\big(D\big)=\mu\alpha\big(M\big)+\alpha\big(\Omega\big)\qlineq-K_{Q}\sim 3\alpha\big(M\big),%
$$
which is impossible, because $\mu\geqslant 1/\lambda>3$. Thus, the
set $\mathbb{LCS}(X,\lambda D)$ contains no surfaces.

There is a fiber $S$ of the morphism $\beta$ such that
$$
S\ne S\cap\mathrm{LCS}\Big(X,\ \lambda D\Big)\ne\varnothing,
$$
which implies that $S$ is singular by Lemma~\ref{lemma:Hwang},
because $\mathrm{lct}(\mathbb{P}^{1}\times\mathbb{P}^{1})=1/2$.

Thus, the surface $S$ is an irreducible quadric cone in
$\mathbb{P}^{3}$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq S
$$
by Theorem~\ref{theorem:connectedness}.  We may assume that either
$S\not\subset\mathrm{Supp}(D)$ or $E\not\subset\mathrm{Supp}(D)$
by  Remark~\ref{remark:convexity}, because
$$
\left(X,\ S+\frac{2}{3}E\right)
$$
has log-canonical singularities, and the equivalence $3S+2E\qlineq
D$ holds.

Put $\Gamma=E\cap S$. The curve $\Gamma$ is an irreducible conic
in $S$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq \Gamma
$$
by Lemma~\ref{lemma:singular-quadric-threefold}. Intersecting $D$
with a general ruling of the cone $S\subset\mathbb{P}^{3}$, and
intersecting $D$ with a general fiber of the projection $E\to C$,
we see that
$$
\Gamma\not\subseteq\mathrm{LCS}\Big(X,\lambda D\Big),
$$
which implies that $\mathrm{LCS}(X,\lambda D)$ consists of a
single point $O\in\Gamma$ by Theorem~\ref{theorem:connectedness}.

Let $R$ be a general (not passing through $O$) surface in
$|\alpha^{*}(H)|$. Then
$$
\mathrm{LCS}\left(X,\ \lambda D+\frac{1}{2}\Big(\bar{H}+2R\Big)\right)=R\cup O,%
$$
which is impossible by Theorem~\ref{theorem:connectedness}, since
$-K_{X}\sim\bar{H}+2R\qlineq D$ and $\lambda<1/3$.
\end{proof}



The following generalization of Lemma~\ref{lemma:Hwang} is proved in
\cite{Hw06b}.

\begin{theorem}
\label{theorem:Hwang} Suppose that there is a surjective morphism with
connected fibers
$$
\phi\colon X\longrightarrow Z
$$
such that $\phi$ is smooth in a neighborhood of a fiber $F$ of the
morphism $\phi$. Then either
$$
\mathrm{lct}_{F}\Big(X,
B_{X}\Big)\geqslant\mathrm{lct}\Big(F,\Big[B_{X}\big\vert_{F}\Big]\Big),
$$
or the equality $\mathrm{lct}_{O}(X, B_{X})=\mathrm{lct}_{Q}(X, B_{X})$ holds
for  every two points $O\in F\ni Q$.
\end{theorem}

Let us consider two elementary applications of
Theorem~\ref{theorem:Hwang}.

\begin{lemma}
\label{lemma:P1xP2} Suppose that $\mathrm{LCS}(X, B_{X})\ne
\varnothing$, where $X\cong\P^1\times\P^2$ and
$$
B_{X}\qlineq -\lambda K_{X}
$$
for some rational number $0<\lambda<1/2$. Then
$\mathbb{LCS}(X, B_{X})$ contains a surface.
\end{lemma}

\begin{proof}
Suppose that $\mathbb{LCS}(X, B_{X})$ contains no surfaces. By
Theorems~\ref{theorem:connectedness} and \ref{theorem:Hwang}, we
have
$$
\LCS\Big(X,\ B_{X}\Big)=F,
$$
where $F$ is a fiber of the natural projection $\pi_{2}\colon
X\to\P^2$. Let $S$ be a general surface in
$$
\Big|\pi_{1}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|,
$$
let $M_{1}$ and $M_{2}$ be general fibers of the natural
projection $\pi_{1}\colon X\to\P^1$. Then the locus
$$
\LCS\left(X,\ \lambda
D+\frac{1}{2}\Big(M_1+M_2+3S\Big)\right)=F\cup S
$$
is disconnected,  which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}

\begin{lemma}
\label{lemma:lct-product} Let $V$ and $U$ be smooth Fano
varieties. Then
$$
\mathrm{lct}\Big(V\times
U\Big)=\mathrm{min}\Big(\mathrm{lct}\big(V\big),\
 \mathrm{lct}\big(U\big)\Big).%
$$
\end{lemma}

\begin{proof}
The inequalities $\mathrm{lct}(U)\geqslant\mathrm{lct}(V\times
U)\leqslant\mathrm{lct}(U)$ are obvious. Suppose that
$$
\mathrm{lct}\Big(V\times
U\Big)<\mathrm{min}\Big(\mathrm{lct}\big(V\big),\
 \mathrm{lct}\big(U\big)\Big),%
$$
and let us show that this assumption leads to a contradiction.

There is an effective $\mathbb{Q}$-divisor $D\qlineq -K_{V\times
U}$ such that the log pair
$$
\Big(V\times U,\ \lambda D\Big)
$$
is not log canonical in some point $P\in V\times U$, where
$\lambda\in\mathbb{Q}$ is such that
$\lambda<\min(\lct(V), \lct(U))$.

Let us identify $V$ with a fiber of the projection $V\times U\to
U$ that contains the point $P$. Then
$$
\mathrm{lct}_{O}\Big(V\times U,\ D\Big)=\mathrm{lct}_{Q}\Big(V\times U,\ D\Big)%
$$
for every points $O\in V\ni Q$ by Theorem~\ref{theorem:Hwang},
because the inequalities
$$
\mathrm{lct}\big(V\big)>\lambda>\mathrm{lct}_{V}\Big(V\times U,\ D\Big)\geqslant\mathrm{lct}\Big(V,\Big[D\big\vert_{V}\Big]\Big)=\mathrm{lct}\Big(V,\big[-K_{V}\big]\Big)=\mathrm{lct}\big(V\big)%
$$
are inconsistent. So, the log pair $(V\times U, \lambda D)$ is not
log canonical in every point of $V\subset V\times U$.

Let us identify $U$ with a general fiber of the projection
$V\times U\to V$. Then
$$
D\Big\vert_{U}\qlineq -K_{U},
$$
and $(U,\lambda D\vert_{U})$ is not log canonical in $U\cap V$ by
Remark~\ref{remark:hyperplane-reduction} (applied $\dim(V)$ times
here), which contradicts the inequality $\lambda<\mathrm{lct}(U)$.
\end{proof}

We believe that the assertion of Lemma~\ref{lemma:lct-product}
holds for log terminal varieties (cf.
Lemma~\ref{lemma:lct-P1-product}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Mukai--Umemura threefold}
\label{section:MU}

The main purpose of this section is to compute the global log canonical
threshold of one remarkable smooth Fano threefold (cf. \cite{Do07}) to
illustrate the proof of Theorem~\ref{theorem:main}.

\begin{lemma}
\label{lemma:V22} Let $X$ be the smooth Fano threefold such
that\footnote{ The threefold $X$ satisfying these assumptions is
unique and has many interesting properties
(see~\cite{Pr90},~\cite{IlSh00},~\cite{JaRa07}). }
$$
\mathrm{Pic}\big(X\big)=\mathbb{Z}\big[-K_{X}\big],
$$
the equality $-K_{X}^{3}=22$ holds, and
$\mathrm{Aut}(X)\cong\mathrm{PSL}(2,\mathbb{C})$. Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Let $U\subset\mathbb{C}[x,y]$ be a subspace of forms of degree $12$. Consider
$U\cong\mathbb{C}^{13}$ as the affine~part~of
$$
\mathbb{P}\Big(U\oplus \mathbb{C}\Big)\cong\mathbb{P}^{13},
$$
and let us identify $\mathbb{P}(U)$ with the hyperplane at
infinity.

The natural action of $\mathrm{SL}(2,\mathbb{C})$ on
$\mathbb{C}[x,y]$ induces an action on $\mathbb{P}(U\oplus
\mathbb{C})$. Put
$$
\phi=xy\Big(x^{10}-11x^5y^5-y^{10}\Big)\in U
$$
and consider the closure
$\overline{\mathrm{SL}(2,\mathbb{C})\cdot[\phi+1]}\subset\mathbb{P}(U\oplus
\mathbb{C})$. It follows from \cite{MuuM83} that
$$
X\cong\overline{\mathrm{SL}\big(2,\mathbb{C}\big)\cdot\big[\phi+1\big]},%
$$
and the embedding $X\subset\mathbb{P}(U\oplus
\mathbb{C})\cong\mathbb{P}^{13}$ is induced by $|-K_{X}|$.

The action of $\mathrm{SL}(2,\mathbb{C})$ on $X$ has the following orbits (see
\cite[Theorem~5.2.13]{IsPr99}):
\begin{itemize}
\item the three-dimensional orbit $\Sigma_3=\mathrm{SL}(2,\mathbb{C})\cdot [\phi+1]$;%
\item the two-dimensional orbit $\Sigma_2=\mathrm{SL}(2,\mathbb{C})\cdot [xy^{11}]$;%
\item the one-dimensional orbit $\Sigma_1=\mathrm{SL}(2,\mathbb{C})\cdot [y^{12}]$.%
\end{itemize}

The orbit $\Sigma_{3}$ is open. The orbit
$\Sigma_{1}\cong\mathbb{P}^{1}$ is closed. But
$$
\overline{\Sigma}_2=\Sigma_1\cup\Sigma_2,
$$
which implies that the orbit $\Sigma_2$ is neither open nor
closed. One has
$$
X\cap \mathbb{P}(U)=\Sigma_1\cup \Sigma_2
$$
and $X=\Sigma_1 \cup \Sigma_2 \cup \Sigma_3$. Put $R=X\cap
\mathbb{P}(U)$. It follows from \cite{MuuM83} that
\begin{itemize}
\item the surface $R$ is swept out by lines on $X\subset\mathbb{P}^{13}$,%
\item the surface $R$ contains all lines on $X\subset\mathbb{P}^{13}$,%
\item for any lines $L_{1}\subset R\supset L_{2}$ such that $L_{1}\ne L_{2}$,
one has $L_{1}\cap L_{2}=\varnothing$,%
\item the surface $R$ is singular along the orbit $\Sigma_1\cong\mathbb{P}^1$,%
\item the normalization of the surface $R$ is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$,%
\item for every point $P\in \Sigma_{1}$, the surface $R$ is
locally isomorphic to
$$
x^{2}=y^{3}\subset\mathbb{C}^{3}\cong\mathrm{Spec}\Big(\mathbb{C}\big[x,y,z\big]\Big),
$$
which implies that $\mathrm{lct}(X,R)=5/6$.
\end{itemize}

The structure of the surface $R$ can be seen as follows. We see
that
$$
\Sigma_2=\Big\{\Big[\big(ax+by\big)\big(cx+dy\big)^{11}\Big]\ \Big\vert\ ad-bc=1\Big\}\subset\mathbb{P}\big(U\big),%
$$
which implies that there is a birational  morphism
$\nu\colon\mathbb{P}^1 \times \mathbb{P}^1 \to R$ that is defined
by
$$
\big[a:b\big]\times\big[c:d\big]\longrightarrow\Big[\big(ax+by\big)\big(cx+dy\big)^{11}\Big]\in R,%
$$
which is a normalization of the surface $R$.

Let $V_{5}$ be a smooth Fano threefold such that
$$
-K_{V_{5}}\sim 2H
$$
and $H^{3}=5$, where $H$ is a Cartier divisor on $V_{5}$. Then
$|H|$ induces an embedding $X\subset\mathbb{P}^{6}$.

Let $L\cong\mathbb{P}^{1}$ be a line on $X$. Then
$\mathcal{N}_{L/X}\cong\mathcal{O}_{\mathbb{P}^{1}}(-2)\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)$.
Let
$$
\alpha_{L}\colon U_{L}\longrightarrow X
$$
be a blow up of the line $L$, and let $E_{L}$ be the exceptional
divisor of $\alpha_{L}$. Then
$$
E_{L}\cong\mathbb{F}_{3},
$$
and it follows from Theorem~4.3.3 in \cite{IsPr99} (see
\cite{Cut89}, \cite{Tak89}) that there is a commutative diagram
$$
\xymatrix{
U_{L}\ar@{->}[d]_{\alpha_{L}}\ar@{-->}[rr]^{\rho_{L}}&&W_{L}\ar@{->}[d]^{\beta_{L}}\\%
X\ar@{-->}[rr]_{\psi_{L}}&&V_{5},}
$$
where $\rho_{L}$ is a flop in the exceptional section of
$E\cong\mathbb{F}_{3}$, the morphism $\beta_{L}$ contracts a
surface
$$
D_{L}\subset W_{L}
$$
to a smooth rational curve of degree $5$, and $\psi_{L}$ is a
double projection from the line $L$.


Let $\bar{D}_{L}\subset X$ be the proper transform of the surface
$D_{L}$. Then
$$
\mathrm{mult}_{L}\big(\bar{D}_{L}\big)=3
$$
and $\bar{D}_{L}\sim -K_{X}$. It follows from \cite{Fur90} that
$X\setminus\bar{D}_{L}\cong\mathbb{C}^{3}$ (cf. \cite{Pet88},
\cite{Pet89}, \cite{Fur93}).

It follows from \cite{Fur92} that there is an open subset
$\breve{D}_{L}\subset\bar{D}_{L}$ that is given by
$$
\mu_{0}x^{4}+\Big(\mu_{1}yz+\mu_{2}z^{3}\Big)x^{3}+\Big(\mu_{3}y^{3}+\mu_{4}y^{2}z^{2}+\mu_{5}yz^{4}\Big)x^{2}+\Big(\mu_{6}y^{4}z+\mu_{7}y^{3}z^{3}\Big)x+\mu_{8}y^{6}+\mu_{9}y^{5}z^{2}=0%
$$
in $\mathbb{C}^{3}\cong\mathrm{Spec}(\mathbb{C}[x,y,z])$, where
$L\cap \Sigma_{1}\in\breve{D}_{L}$ is given by the equations
$x=y=z=0$, and
$$
\left.%
\aligned
&\mu_{0}=-2^{8}5^{2},\ \mu_{1}=2^{9}3^{3}5,\ \mu_{2}=-2^{6}3^{4}5,\ \mu_{3}=-2^{8}3^{3}7,\ \mu_{4}=-2^{4}3^{4}127,\,\\%
&\mu_{5}=2^{9}3^{5},\ \mu_{6}=2^{2}3^{6}89,\ \mu_{7}=-2^{8}3^{6},\ \mu_{8}=-3^{6}5^{3},\ \mu_{9}=2^{5}3^{7}.\\%
\endaligned\right.%
$$

Put $O_{L}=\Sigma_{1}\cap L$. Then
$\mathrm{mult}_{O_{L}}(\bar{D}_{L})=4$, and it follows from
Proposition~8.14 in \cite{Ko97} that
$$
\mathrm{LCS}\left(X,\ \frac{1}{2}\bar{D}_{L}\right)=O_{L}
$$
and $\mathrm{lct}(X, \bar{D}_{L})=1/2$. Thus, we see that
$\mathrm{lct}(X)\leqslant 1/2$.

Suppose that $\lct(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor
$$
D\qlineq -K_{X}
$$
such that the log pair $(X,\lambda D)$ is not log-canonical for
some positive rational number $\lambda<1/2$.

By Remark~\ref{remark:convexity}, we may assume that
$R\not\subset\mathrm{Supp}(D)$, because $\mathrm{lct}(X,R)=5/6$.

Let $C$ be a line in $X$ such that $C\not\subset\mathrm{Supp}(D)$.
Then
$$
1=D\cdot C\geqslant\mathrm{mult}_{O_{C}}\big(D\big)\mathrm{mult}_{O_{C}}\big(C\big)=\mathrm{mult}_{O_{C}}\big(D\big),%
$$
which implies that $O_{C}\not\in\mathrm{LCS}(X,\lambda D)$. In
particular, we see that $\Sigma_{1}\not\in\mathrm{LCS}(X,\lambda
D)$.

Let $\Gamma$ be an irreducible curve in $\mathrm{Supp}(D)$ such
that $O_{C}\in\Gamma$. Then
$$
\mathrm{mult}_{\Gamma}\left(\frac{1}{2}\bar{D}_{C}+\lambda D\right)=\frac{\mathrm{mult}_{\Gamma}\big(\bar{D}_{C}\big)}{2}+\lambda\mathrm{mult}_{\Gamma}\big(D\big)\leqslant\frac{\mathrm{mult}_{\Gamma}\big(\bar{D}_{C}\big)}{2}+\lambda\mathrm{mult}_{O_{C}}\big(D\big)<1,%
$$
because $\lambda<1/2$ and $\mathrm{Sing}(\bar{D}_{C})=C$, because
$\bar{D}_{C}\ne R$. Thus, we see that
$$
\Gamma\not\subseteq\mathrm{LCS}\left(X,\ \frac{1}{2}\bar{D}_{C}+\lambda D\right)\supseteq\mathrm{LCS}\Big(X, \lambda D\Big)\cup O_{C},%
$$
which is impossible by Theorem~\ref{theorem:connectedness},
because $O_{C}\not\in\mathrm{LCS}(X,\lambda D)$ and $\lambda<1/2$.
\end{proof}

The threefold satisfying  all hypotheses of Lemma~\ref{lemma:V22} is called the
Mukai--Mumemura threefold.

\begin{remark}
\label{remark:MU-SO3} Let $X$ be the Mukai--Mumemura threefold. Then it follows
from \cite{Do07} that
$$
\mathrm{lct}\Big(X,\ \mathrm{SO}\big(3\big)\Big)=\frac{5}{6}.
$$
\end{remark}



\begin{remark}
\label{remark:MU} Let $Z\subset\mathbb{P}^{2}$ be a plane quartic
curve. Then $Z$ is given by an equation
$$
\zeta\big(x,y,z\big)=0\subset\mathbb{P}^{2}\cong\mathrm{Proj}\Big(\mathbb{C}[x,y,z]\Big),
$$
where $\zeta(x,y,z)$ is a form of degree $4$. A polar hexagon of
the curve $Z\subset\mathbb{P}^{2}$ is the union
$$
\Gamma=\bigcup_{i=1}^{6}\Big(\xi_{i}\big(x,y,z\big)=0\Big)\subset\mathbb{P}^{2}
$$
such that $\zeta(x,y,z)=\sum_{i=1}^{6}\xi_{i}^{4}(x,y,z)$, where
$\xi_{i}(x,y,z)$ is a non-zero linear form.  Put
$$
X_{\zeta}=\overline{\Big\{\Gamma \in \mathrm{Hilb}_6\big(\P^2\big)\ \Big|\ \Gamma\ \mathrm{is\ polar\ to}\ Z\subset\mathbb{P}^{2}\Big\}},%
$$
where we identify $\Gamma$ with a point in
$\mathrm{Hilb}_6(\P^2)$. Then $X_{\zeta}$ is a smooth Fano
threefold such that
$$
\mathrm{Pic}\big(X_{\zeta}\big)=\mathbb{Z}\big[-K_{X_{\zeta}}\big]
$$
and $-K_{X_{\zeta}}^{3}=22$ when $\zeta(x,y,z)$ is general (see
\cite{Shr99}). It follows from \cite{MuuM83} that $X_{\zeta}$ is
the Mukai--Mumemura threefold if
$$
\zeta\big(x,y,z\big)=\Big(x^{2}+y^{2}+z^{2}\Big)^{2}.
$$
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Cubic surfaces}
\label{section:cubic-surfaces}

Let $X$ be a cubic surface in $\mathbb{P}^{3}$ that has at most
one ordinary double point.

\begin{definition}
\label{definition:Eckardt-point} A point $O\in X$ is said to be an
Eckardt point if $O\not\in\mathrm{Sing}(X)$ and
$$
O=L_{1}\cap L_{2}\cap L_{3},
$$
where $L_{1},L_{2},L_{3}$ are lines on the surface
$X\subset\mathbb{P}^{3}$.
\end{definition}

General cubic surfaces have no Eckardt points. It follows from
Example~\ref{example:del-Pezzos} and \ref{example:singular-cubics}
that
$$
\mathrm{lct}\big(X\big)=\left\{%
\aligned
&3/4\ \text{when}\ X\ \text{has no Eckardt points and}\ \mathrm{Sing}\big(X\big)=\varnothing\,\\%
&2/3\ \text{when}\ X\ \text{has an Eckardt point or}\ \mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.%
$$

Let $D$ be an effective $\mathbb{Q}$-divisor on $X$ such that
$D\qlineq -K_{X}$, and let $\omega$ be a positive rational number
such that $\omega<3/4$. In this section we prove the following
result (cf.  \cite{Ch01b}, \cite{ChPa02}, \cite{Ch07b}).

\begin{theorem}
\label{theorem:cubic-surfaces} Suppose that $(X,\omega D)$ is not
log canonical. Then
$$
\mathrm{LCS}\Big(X,\ \omega D\Big)=O,
$$
where $O\in X$ is either a singular point or an Eckardt point.
\end{theorem}

Suppose that $(X,\omega D)$ is not log canonical. Let $P$ be a
point in $\mathrm{LCS}(X,\omega D)$. Suppose that
\begin{itemize}
\item neither $P=\mathrm{Sing}(X)$,%
\item nor $P$ is an Eckardt point.
\end{itemize}

\begin{lemma}
\label{lemma:dp3-point} The equality $\mathrm{LCS}(X,\omega D)=P$
holds.
\end{lemma}

\begin{proof}
Suppose that $\mathrm{LCS}(X,\omega D)\ne P$. Then there is a
curve $C\subset X$ such that
$$
P\in C\subseteq\mathrm{LCS}\Big(X,\omega D\Big)
$$
by Theorem~\ref{theorem:connectedness}. Then there is an effective
$\mathbb{Q}$-divisor $\Omega$ on $X$ such that
$C\not\subset\mathrm{Supp}(\Omega)$ and
$$
D=\mu C+\Omega,
$$
where $\mu\in\mathbb{Q}$ such that $\mu\geqslant 1/\omega$. Let
$H$ be a general hyperplane section of $X$. Then
$$
3=H\cdot D=\mu H\cdot C+H\cdot \Omega\geqslant
\mu\mathrm{deg}\big(C\big),
$$
which implies that either $\mathrm{deg}(C)=1$, or
$\mathrm{deg}(C)=2$.

Suppose that $\mathrm{deg}(C)=1$. Let $Z$ be a general conic on
$X$ such that $-K_{X}\sim C+Z$. Then
$$
2=Z\cdot D=\mu Z\cdot C+Z\cdot \Omega\geqslant \mu Z\cdot C=\left\{%
\aligned
&2\mu\ \text{if}\ C\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3\mu/2\ \text{if}\ C\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.%
$$
which implies that $\mu\leqslant 3/4$. But $\mu\geqslant
1/\omega>4/3$.

We see that $\mathrm{deg}(C)=2$. Let $L$ be a line on $X$ such
that $-K_{X}\sim C+L$. Then
$$
D=\mu C+\lambda L+\Upsilon,
$$
where $\lambda\in\mathbb{Q}$ such that $\lambda\geqslant 0$, and
$\Upsilon$ is an effective $\mathbb{Q}$-divisor such that
$C\not\subset\mathrm{Supp}(\Upsilon)\not\supset L$. Then
$$
1=L\cdot D=\mu L\cdot C+\lambda L\cdot
L+L\cdot\Upsilon\geqslant \mu L\cdot C+\lambda L\cdot L=\left\{%
\aligned
&2\mu-\lambda\ \text{if}\ C\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3\mu/2-\lambda/2\ \text{if}\ C\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.% ,
$$
which implies that $\mu<4/3$, because $\lambda\leqslant 4/3$ (see
the case when $\mathrm{deg}(C)=1$). But $\mu>4/3$.
\end{proof}

Let $\pi\colon U\to X$ be a blow up of $P$, and let $E$ be the
$\pi$-exceptional curve. Then
$$
\bar{D}\qlineq \pi^{*}\big(D\big)+\mathrm{mult}_{P}\big(D\big)E,
$$
where $\mathbb{Q}\ni\mathrm{mult}_{P}(D)\geqslant 1/\omega$ and
$\bar{D}$ is a proper transform of $D$ on the surface $U$. The log
pair
$$
\Big(U,\ \omega\bar{D}+\big(\omega\mathrm{mult}_{P}\big(D\big)-1\big)E\Big)%
$$
is not log canonical at some point $Q\in E$. Then either
$\mathrm{mult}_{P}(D)\geqslant 2/\omega$, or
\begin{equation}
\label{equation:blow-up}
\mathrm{mult}_{Q}\big(\bar{D}\big)+\mathrm{mult}_{P}\big(D\big)\geqslant 2\slash\omega>8\slash 3,%
\end{equation}
because the divisor
$\omega\bar{D}+(\omega\mathrm{mult}_{P}(D)-1)E$ is effective.

Let $T$ be the unique hyperplane section of $X$ that is singular
at $P$. We may assume that
$$
\mathrm{Supp}\big(T\big)\not\subseteq\mathrm{Supp}\big(D\big)
$$
by Remark~\ref{remark:convexity}, because $(X,\omega T)$ is log
canonical. The following cases are possible:
\begin{itemize}
\item the curve $T$ is irreducible;%
\item the curve $T$ is a union of a line and an irreducible conic;%
\item the curve $T$ consists of $3$ lines;%
\end{itemize}
where $T$ is reduced and $\mathrm{mult}_{P}(T)=2$ by assumption.
We exclude these cases one by one.

\begin{lemma}
\label{lemma:dp3-no-lines} The curve $T$ is reducible.
\end{lemma}

\begin{proof}
Suppose that $T$ is irreducible. Then there is a commutative
diagram
$$
\xymatrix{
&U\ar@{->}[dl]_{\pi}\ar@{->}[dr]^{\psi}&\\%
X\ar@{-->}[rr]_{\rho}&&\mathbb{P}^{2},}
$$ %
where $\psi$ is a double cover branched over a quartic curve, and
$\rho$ is the projection from $P\in X$.

Let $\bar{T}$ be the proper transform of $T$ on the surface $U$.
Suppose that $Q\in\bar{T}$. Then
$$
3-2\mathrm{mult}_{P}\big(D\big)=\bar{T}\cdot\bar{D}\geqslant\mathrm{mult}_{Q}\big(\bar{T}\big)\mathrm{mult}_{Q}\big(\bar{D}\big)>\mathrm{mult}_{Q}\big(\bar{T}\big)\Big(8/3-\mathrm{mult}_{P}\big(D\big)\Big)\geqslant 8/3-\mathrm{mult}_{P}\big(D\big),%
$$
which implies that $\mathrm{mult}_{P}(D)\leqslant 1/3$. But
$\mathrm{mult}_{P}(D)>4/3$. Thus, we see that $Q\not\in\bar{T}$.

Let $\tau\in\mathrm{Aut}(U)$ be an involution\footnote{The
involution $\tau$ induces an involution in $\mathrm{Bir}(X)$ that
is called a Geiser involution.} induced by~$\psi$.~It follows from
\cite{Ma67} that
$$
\tau^{*}\Big(\pi^{*}\big(-K_{X}\big)\Big)\sim \pi^{*}\big(-2K_{X}\big)-3E,%
$$
and  $\tau(\bar{T})=E$. Put $\breve{Q}=\pi\circ\tau(Q)$. Then
$\breve{Q}\ne P$, because $Q\not\in\bar{T}$.

Let $H$ be the hyperplane section of $X$ that is singular at
$\breve{Q}$. Then
$$
T\ne H,
$$
because $P\ne \breve{Q}$ and $T$ is smooth outside of the point
$P$. Then $P\not\in H$, because otherwise
$$
3=H\cdot T\geqslant\mathrm{mult}_{P}\big(H\big)\mathrm{mult}_{P}\big(T\big)+\mathrm{mult}_{\breve{Q}}\big(H\big)\mathrm{mult}_{\breve{Q}}\big(T\big)\geqslant 4.%
$$


Let $\bar{H}$ be the proper transform of $H$ on the surface $U$.
Put $\bar{R}=\tau(\bar{H})$ and $R=\pi(\bar{R})$. Then
$$
\bar{R}\sim \pi^{*}\big(-2K_{X}\big)-3E,%
$$
ant the curve $\bar{R}$ must be singular at the point $Q$.

Suppose that $R$ is irreducible. Taking into account the possible
singularities of $\bar{R}$, we see that
$$
\left(X,\ \frac{3}{8} R\right)
$$
is log canonical. Thus, we may assume that
$R\not\subseteq\mathrm{Supp}(D)$ by Remark~\ref{remark:convexity}.
Then
$$
6-3\mathrm{mult}_{P}\big(D\big)=\bar{R}\cdot\bar{D}\geqslant \mathrm{mult}_{Q}(\bar{R})\mathrm{mult}_{Q}\big(\bar{D}\big)>2\Big(8/3-\mathrm{mult}_{P}\big(D\big)\Big),%
$$
which implies that $\mathrm{mult}_{P}(D)<2/3$. But
$\mathrm{mult}_{P}(D)>4/3$. The curve  $R$ must be reducible.

The curves $R$ and $H$ are reducible. So, there is a line
$L\subset X$ such that $P\not\in L\ni\breve{Q}$.

Let $\bar{L}$ be the proper transform of $L$ on the surface $U$.
Put $\bar{Z}=\tau(\bar{L})$. Then $\bar{L}\cdot E=0$ and
$$
\bar{L}\cdot\bar{T}=\bar{L}\cdot\pi^{*}(-K_{X})=1,%
$$
which implies that $\bar{Z}\cdot E=1$ and
$\bar{Z}\cdot\pi^{*}\big(-K_{X}\big)=2$. We have $Q\in\bar{Z}$.
Then
$$
2-\mathrm{mult}_{P}\big(D\big)=\bar{Z}\cdot\bar{D}\geqslant\mathrm{mult}_{Q}\big(\bar{D}\big)>8/3-\mathrm{mult}_{P}\big(D\big)>2-\mathrm{mult}_{P}\big(D\big)%
$$
in the case when $\bar{Z}\not\subseteq\mathrm{Supp}(\bar{D})$.
Hence, we see that $\bar{Z}\subseteq\mathrm{Supp}(\bar{D})$.

Put $Z=\pi(\bar{Z})$. Then $Z$ is a conic such that $P\in Z$ and
$$
-K_{X}\sim L+Z,
$$
which means that $L\cup Z$ is~cut~out by the hyperplane in
$\mathbb{P}^{3}$ that passes through $Z$. Put
$$
D=\eps Z+\Upsilon,
$$
where $\eps\in\mathbb{Q}$ such that $\eps\geqslant 0$, and
$\Upsilon$ is an effective $\mathbb{Q}$-divisor such that
$Z\not\subset\mathrm{Supp}(\Upsilon)$.

We may assume that $L\not\subseteq\mathrm{Supp}(\Upsilon)$ by
Remark~\ref{remark:convexity}.~Then
$$
1=L\cdot D=\eps Z\cdot L+L\cdot\Upsilon\geqslant \eps
Z\cdot L=\left\{%
\aligned
&2\eps\ \text{if}\ Z\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3\eps/2\ \text{if}\ Z\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.% ,
$$
which implies that $\eps\leqslant 2/3$.

Let $\bar{\Upsilon}$ be the proper transform of $\Upsilon$ on the
surface $U$. Then the log pair
$$
\Big(U,\
\eps\omega\bar{Z}+\omega\bar{\Upsilon}+\big(\omega\mathrm{mult}_{P}\big(D\big)-1\big)E\Big)
$$
is not log canonical at $Q\in\bar{Z}$. Then
$$
\omega\bar{\Upsilon}\cdot\bar{Z}+\Big(\omega\mathrm{mult}_{P}\big(D\big)-1\Big)=\Big(\omega\bar{\Upsilon}+\big(\omega\mathrm{mult}_{P}\big(D\big)-1\big)E\Big)\cdot\bar{Z}>1
$$
by Lemma~\ref{lemma:adjunction}, because $\eps\leqslant 2/3$.
In particular, we see that
$$
8/3-\mathrm{mult}_{P}\big(D\big)<\bar{Z}\cdot\bar{\Upsilon}=2-\mathrm{mult}_{P}\big(D\big)-\eps\bar{Z}\cdot\bar{Z}=\left\{%
\aligned
&2-\mathrm{mult}_{P}\big(D\big)+\eps\ \text{if}\ Z\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&2-\mathrm{mult}_{P}\big(D\big)+\eps/2\ \text{if}\ Z\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.% ,
$$
which implies that $\eps>2/3$. But $\eps\leqslant 2/3$.
\end{proof}

Therefore, there is a line $L_{1}\subset X$ such that $P\in
L_{1}$. Put
$$
D=m_{1}L_{1}+\Omega,
$$
where $m_{1}\in\mathbb{Q}$ such that $m_{1}\geqslant 0$, and
$\Omega$ is an effective $\mathbb{Q}$-divisor such that
$L_{1}\not\subset\in\mathrm{Supp}(\Omega)$.~Then
$$
4\slash 3<1\slash\omega<\Omega\cdot L_{1}=1-m_{1}L_{1}\cdot L_{1}=\left\{%
\aligned
&1+m_{1}\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&1+m_{1}/2\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)\ne\varnothing.\\%
\endaligned\right.% ,
$$

\begin{corollary}
\label{corollary:cubic-m1} The following inequality holds:
$$
m_{1}>\left\{%
\aligned
&1/3\ \mathrm{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&2/3\ \mathrm{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.%
$$.
\end{corollary}

\begin{remark}
\label{remark:six-lines} Suppose that $X$ is singular. Put
$O=\mathrm{Sing}(X)$. It follows from \cite{BW79} that
$$
O=\Gamma_{1}\cap\Gamma_{2}\cap \Gamma_{3}\cap \Gamma_{4}\cap\Gamma_{5}\cap\Gamma_{6},%
$$
where $\Gamma_{1},\ldots,\Gamma_{6}$ are different lines on the
surface $X\subset\mathbb{P}^{3}$. The equivalence
$$
-2K_{X}\sim\sum_{i=1}^{6}\Gamma_{i}
$$
hold. Suppose that $L_{1}=\Gamma_{1}$. Let
$\Pi_{2},\ldots,\Pi_{6}\subset \mathbb{P}^{3}$ be hyperplanes such
that
$$
L_{1}\subset\Pi_{i}\supset\Gamma_{i},
$$
and let $\Lambda_{2},\ldots,\Lambda_{6}$ be lines on the surface
$X$ such that
$$
L_{1}\cup\Gamma_{i}\cup\Lambda_{i}=\Pi_{i}\cap X\subset X\subset\mathbb{P}^{3},%
$$
which implies that $-K_{X}\sim L_{1}+\Gamma_{i}+\Lambda_{i}$. Then
$$
-5K_{X}\sim 4L_{1}+\sum_{i=2}^{6}\Lambda_{i}+\Big(L_{1}+\sum_{i=2}^{6}\Gamma_{i}\Big)\sim 4L_{1}+\sum_{i=2}^{6}\Lambda_{i}-2K_{X},%
$$
which implies that $-3K_{X}\sim 4L_{1}+\sum_{i=2}^{6}\Lambda_{i}$.
But the log pair
$$
\left(X,\ L_{1}+\frac{\sum_{i=2}^{6}\Lambda_{i}}{3}\right)
$$
is log canonical at the point $P$. Thus, we may assume that
$$
\mathrm{Supp}\left(\sum_{i=2}^{6}\Lambda_{i}\right)\not\subseteq\mathrm{Supp}\big(D\big)
$$
thanks to Remark~\ref{remark:convexity}, because
$L_{1}\subseteq\mathrm{Supp}(D)$. Then there is $\Lambda_{k}$ such
that
$$
1=D\cdot\Lambda_{k}=\Big(m_{1}L_{1}+\Omega\Big)\cdot\Lambda_{k}=m_{1}+\Omega\cdot\Lambda_{k}\geqslant
m_{1},
$$
because $O\not\in\Lambda_{k}$. Thus, we may assume that
$m_{1}\leqslant 1$ if $L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$.
\end{remark}

Arguing as in the proof of Lemma~\ref{lemma:Cheltsov-Pukhlikov},
we see that $m_{1}\leqslant 1$ if
$L_{1}\cap\mathrm{Sing}(X)=\varnothing$.



\begin{lemma}
\label{lemma:dp3-single-line} There is a line $L_{2}\subset X$
such that $L_{1}\ne L_{2}$ and $P\in L_{2}$.
\end{lemma}

\begin{proof} Suppose that there is no line $L_{2}\subset X$ such that $L_{1}\ne L_{2}$ and
$P\in L_{2}$. Then
$$
T=L_{1}+C,
$$
where $C$ is an irreducible conic on the surface
$X\subset\mathbb{P}^{3}$ such that $P\in C$.

It follows from Remark~\ref{remark:convexity} that we may assume
that $C\not\subseteq\mathrm{Supp}(\Omega)$.

Let $\bar{L}_{1}$ and $\bar{C}$ be the proper transforms of
$L_{1}$ and $C$ on the surface $U$, respectively. Then
$$
\bar{D}\qlineq m_{1}\bar{L}_{1}+\bar{\Omega}\qlineq \pi^{*}\Big(m_{1} L_{1}+\Omega\Big)-\Big(m_{1}+\mathrm{mult}_{P}\big(\Omega\big)\Big)E\qlineq\pi^{*}\big(D\big)-\mathrm{mult}_{P}\big(D\big)E,%
$$
where $\bar{\Omega}$ is the proper transform of the divisor
$\Omega$ on the surface $U$. We have
$$
0\leqslant\bar{C}\cdot\bar{\Omega}=2-\mathrm{mult}_{P}\big(D\big)+m_{1}\bar{C}\cdot
\bar{L}<2/3-m_{1}\bar{C}\cdot \bar{L}_{1}=\left\{%
\aligned
&2/3-m_{1}\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&2/3-m_{1}/2\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.% ,
$$
which implies that $m_{1}<2/3$ if
$L_{1}\cap\mathrm{Sing}(X)=\varnothing$. It follows from
inequality~\ref{equation:blow-up} that
$$
\mathrm{mult}_{Q}\big(\bar{\Omega}\big)>8/3-\mathrm{mult}_{P}\big(\Omega\big)-m_{1}\Big(1+\mathrm{mult}_{Q}\big(\bar{L}_{1}\big)\Big).
$$

Suppose that $Q\in\bar{L}_{1}$. Then it follows from
Lemma~\ref{lemma:adjunction} that
$$
8/3<\bar{L}_{1}\cdot\Big(\bar{\Omega}+\big(\mathrm{mult}_{P}\big(\Omega\big)+m_{1}\big)E\Big)=1-m_{1}\bar{L}_{1}\cdot\bar{L}_{1}=\left\{%
\aligned
&1+2m_{1}\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&1+3m_{1}/2\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.% ,
$$
which is impossible, because $m_{1}\leqslant 1$ if
$L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$ by
Remark~\ref{remark:six-lines}.

We see that $Q\not\in\bar{L}_{1}$. Suppose that $Q\in\bar{C}$.
Then
$$
2-\mathrm{mult}_{P}\big(\Omega\big)-m_{1}-m_{1}\bar{C}\cdot\bar{L}{1}=\bar{C}\cdot\bar{\Omega}>8/3-\mathrm{mult}_{P}\big(\Omega\big)-m_{1},%
$$
which is impossible, because
$m_{1}\bar{C}\cdot\bar{L}_{1}\geqslant 0$. Hence, we see that
$Q\not\in\bar{C}$.

There is a commutative diagram
$$
\xymatrix{
&U\ar@{->}[d]_{\pi}\ar@{->}[rr]^{\zeta}&&W\ar@{->}[d]^{\psi}\\%
&X\ar@{-->}[rr]_{\rho}&&\mathbb{P}^{2},&}
$$ %
where $\zeta$ is a birational morphism that contracts the curve
$\bar{L}_{1}$, the morphism $\psi$ is a double cover branched over
a quartic curve, and $\rho$ is a linear projection from the point
$P\in X$.

Let $\tau$ be the birational involution of $U$ induced
by~$\psi$.~Then
\begin{itemize}
\item the involutions $\tau$ is biregular $\iff$ $L_{1}\cap\mathrm{Sing}(X)=\varnothing$,%
\item the involutions $\tau$ acts biregularly on $U\setminus\bar{L}_{1}$ if $L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$,%
\item it follows from the construction of $\tau$ that $\tau(E)=\bar{C}$,%
\item if $L_{1}\cap\mathrm{Sing}(X)=\varnothing$, then
$$
\tau^{*}\big(\bar{L}_{1}\big)\sim \bar{L}_{1},\ \tau^{*}\big(E\big)\sim \bar{C},\ \tau^{*}\Big(\pi^{*}\big(-K_{X}\big)\Big)\sim \pi^{*}\big(-2K_{X}\big)-3E-\bar{L}_{1}.%
$$
\end{itemize}

Let $H$ be the hyperplane section of $X$ that is singular at
$\pi\circ\tau(Q)\in C$. Then $P\not\in H$, because $C$ is smooth.
Let $\bar{H}$ be the proper transform of $H$ on the surface $U$.
Then
$$
\bar{L}_{1}\not\subseteq\mathrm{Supp}\big(\bar{H}\big)\not\supseteq\bar{R},
$$
and we can put $\bar{R}=\tau(\bar{H})$ and $R=\pi(\bar{R})$. Then
$\bar{R}$ is singular at the point $Q$, and
$$
\bar{R}\sim \pi^{*}\big(-2K_{X}\big)-3E-\bar{L}_{1},%
$$
because $R$ does not pass through a singular point of $X$ if
$\mathrm{Sing}(X)\ne\varnothing$.


Suppose that $R$ is irreducible. Then $R+L_{1}\sim -2K_{X}$, but
the log pair
$$
\left(X,\ \frac{3}{8}\Big(R+L_{1}\Big)\right)
$$
is log canonical. Thus, we may assume that
$R\not\subseteq\mathrm{Supp}(D)$ by Remark~\ref{remark:convexity}.
Then
$$
5-2\Big(m_{1}+\mathrm{mult}_{P}\big(\Omega\big)\Big)+m_{1}\Big(1+\bar{L}_{1}\cdot\bar{L}_{1}\Big)=\bar{R}\cdot\bar{\Omega}\geqslant 2\mathrm{mult}_{Q}\big(\bar{\Omega}\big)>2\Big(8/3-m_{1}-\mathrm{mult}_{P}\big(\Omega\big)\Big),%
$$
which implies that $m_{1}<0$. The curve $R$ must be reducible.

There is a line $L\subset X$ such that $P\not\in L$ and
$\pi\circ\tau(Q)\in L$. Then
$$
L\cap L_{1}=\varnothing,
$$
because $\pi\circ\tau(Q)\in C$ and $(C+L_{1})\cdot L=T\cdot L=1$.
Thus, there is unique conic $Z\subset X$ such that
$$
-K_{X}\sim L+Z
$$
and $P\in Z$. Then $Z$ is irreducible and $P=Z\cap L_{1}$, because
$(L+Z)\cdot L_{1}=1$.

Let $\bar{L}$ and $\bar{Z}$ be the proper transform of $L$ and $Z$
on the surface $U$, respectively. Then
$$
\bar{L}\cdot\bar{C}=\bar{Z}\cdot E=1,\
\bar{L}_{1}\cdot\bar{Z}=\bar{L}\cdot E=\bar{L}\cdot\bar{L}_{1}=0,\ \bar{Z}\cdot\bar{Z}=1-\bar{L}\cdot\bar{Z},\ \bar{L}\cdot\bar{Z}=\left\{%
\aligned
&2\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3/2\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)\ne\varnothing.\\%
\endaligned\right.% ,
$$

We have $\tau(\bar{Z})=\bar{L}$. Then $Q\in\bar{Z}$. Suppose that
$\bar{Z}\not\subseteq\mathrm{Supp}(\bar{\Omega})$. Then
$$
2-m_{1}-\mathrm{mult}_{P}\big(\Omega\big)=\bar{Z}\cdot\bar{\Omega}>8/3-m_{1}-\mathrm{mult}_{P}\big(\Omega\big),
$$
which is impossible. Thus, we see that
$\bar{Z}\subseteq\mathrm{Supp}(\bar{\Omega})$. But the log pair
$$
\Big(X,\ \omega (L+Z)\Big)
$$
is log canonical at the point $P$. Hence, we may assume that
$\bar{L}\not\subseteq\mathrm{Supp}(\bar{\Omega})$ by
Remark~\ref{remark:convexity}. Put
$$
D=\eps Z+m_{1}L_{1}+\Upsilon,
$$
where $\eps\in\mathbb{Q}$ such that $\eps>0$, and
$\Upsilon$ is an effective $\mathbb{Q}$-divisor such that
$Z\not\subset\mathrm{Supp}(\Upsilon)\not\supset L_{1}$.~Then
$$
1=L\cdot D=\eps L\cdot Z+m_{1}L\cdot
L_{1}+L\cdot\Upsilon=\eps L\cdot Z+L\cdot\Upsilon\geqslant \eps L\cdot Z=\left\{%
\aligned
&2\eps\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3\eps/2\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.%
$$
which implies that $\eps\leqslant 2/3$. But
$\bar{Z}\cap\bar{L}_{1}=\varnothing$. Then it follows from
Lemma~\ref{lemma:adjunction} that
$$
2-\mathrm{mult}_{P}\big(D\big)-\eps\bar{Z}\cdot\bar{Z}=\bar{Z}\cdot\bar{\Upsilon}>8/3-\mathrm{mult}_{P}\big(D\big),%
$$
where $\bar{\Upsilon}$ is a proper transform of $\Upsilon$ on the
surface $U$. We deduce that  $\eps>2/3$. But
$\eps\leqslant 2/3$.
\end{proof}

Therefore, we see that $T=L_{1}+L_{2}+L_{3}$, where $L_{3}$ is a
line such that $P\not\in L_{3}$. Put
$$
D=m_{1}L_{1}+m_{2}L_{2}+\Delta,
$$
where $0\leqslant m_{1}\in\mathbb{Q}\ni m_{2}\geqslant 0$, and
$\Delta$ is an effective $\mathbb{Q}$-divisor such that
$L_{2}\not\subseteq\mathrm{Supp}(\Delta)\not\supseteq L_{2}$.

The inequalities $m_{1}>1/3$ and $m_{2}>1/3$ hold by
Corollary~\ref{corollary:cubic-m1}. We may assume~that
$$
L_{3}\not\subseteq\mathrm{Supp}\big(\Delta\big)
$$
by Remark~\ref{remark:convexity}. Without loss of generality, we
assume~that
$$
\Big(L_{1}\cup
L_{2}\Big)\cap\mathrm{Sing}(X)\ne\varnothing\Longrightarrow L_{1}\cap\mathrm{Sing}(X)\ne\varnothing,%
$$
which implies that $L_{3}\cdot L_{2}=1$ and $L_{3}\cdot L_{1}=1/2$
in the case when $L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$, and
$$
L_{3}\cdot L_{2}=L_{3}\cdot L_{1}=1
$$
in the case when $L_{1}\cap\mathrm{Sing}(X)=\varnothing$. Then
$1-m_{1}L_{1}\cdot L_{3}-m_{2}=L_{3}\cdot\Delta\geqslant 0$.

Let $\bar{L}_{1}$ and $\bar{L}_{3}$ be the proper transforms of
$L_{1}$ and $L_{2}$ on the surface $U$, respectively. Then
$$
m_{1}\bar{L}_{1}+m_{2}\bar{L}_{2}+\bar{\Delta}\qlineq \pi^{*}\Big(m_{1}L_{1}+m_{2}L_{2}+\Delta\Big)-\Big(m_{1}+m_{2}+\mathrm{mult}_{P}\big(\Delta\big)\Big)E,%
$$
where $\bar{\Delta}$ is the proper transform of $\Delta$ on the
surface $U$. The inequality~\ref{equation:blow-up} implies that
\begin{equation}
\label{equation:two-lines}
\mathrm{mult}_{Q}\big(\bar{\Delta}\big)>8/3-\mathrm{mult}_{P}\big(\Delta\big)-m_{1}\Big(1+\mathrm{mult}_{Q}\big(\bar{L}_{1}\big)\Big)-m_{1}\Big(1+\mathrm{mult}_{Q}\big(\bar{L}_{2}\big)\Big).
\end{equation}


\begin{lemma}
\label{lemma:cubic-L2} The curve $\bar{L}_{2}$ does not contain
the point $Q$.
\end{lemma}

\begin{proof}
Suppose that $Q\in\bar{L}_{2}$. Then
$$
1-\mathrm{mult}_{P}\big(\Delta\big)-m_{1}+m_{2}=\bar{L}_{2}\cdot\bar{\Delta}>8/3-\mathrm{mult}_{P}\big(\Delta\big)-m_{1}-m_{2}%
$$
by Lemma~\ref{lemma:adjunction}. Thus, we have $m_{2}>5/6$. It
follows from Lemma~\ref{lemma:adjunction} that
$$
1-m_{2}-m_{1}L_{1}\cdot L_{1}=\Delta\cdot L_{1}>4/3-m_{2},
$$
but $L_{1}\cdot L_{1}=-1$ if
$L_{1}\cap\mathrm{Sing}(X)=\varnothing$, and $L_{1}\cdot
L_{1}=-1/2$ if $L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$. Then
$$
m_{1}>\left\{%
\aligned
&1/3\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&2/3\ \text{if}\ L_{1}\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.%
$$
by Corollary~\ref{corollary:cubic-m1}, which is impossible because
$m_{2}>5/6$ and $1>m_{1}L_{1}\cdot L_{3}+m_{2}$.
\end{proof}


\begin{lemma}
\label{lemma:cubic-L1} The curve $\bar{L}_{1}$ does not contain
the point $Q$.
\end{lemma}

\begin{proof}
Suppose that $Q\in\bar{L}_{1}$. Arguing as in the proof of
Lemma~\ref{lemma:cubic-L2}, we see that
$$
L_{1}\cap\mathrm{Sing}\big(X\big)\ne\varnothing,
$$
which implies that $\bar{L}_{1}\cdot\bar{L}_{1}=-1/2$. Then
$m_{1}>10/9$, because
$$
1+3m_{1}/2=\bar{L}_{2}\cdot\Big(\bar{\Delta}+\big(\mathrm{mult}_{P}\big(\Delta\big)-m_{1}-m_{2}\big)E\Big)=1-m_{1}\bar{L}_{1}\cdot\bar{L}_{1}>8/3
$$
by Lemma~\ref{lemma:adjunction}. But $m_{1}\leqslant 1$ by
Remark~\ref{remark:six-lines}.
\end{proof}

Therefore, we see that $\bar{L}_{1}\not\ni Q\not\in\bar{L}_{2}$.
There is a commutative diagram
$$
\xymatrix{
&U\ar@{->}[d]_{\pi}\ar@{->}[rr]^{\zeta}&&W\ar@{->}[d]^{\psi}\\%
&X\ar@{-->}[rr]_{\rho}&&\mathbb{P}^{2},&}
$$ %
where $\zeta$ is a birational morphism that contracts the curves
$\bar{L}_{1}$ and $\bar{L}_{2}$, the morphism $\psi$ is a double
cover branched over a quartic curve, and $\rho$ is the projection
from the point $P$.

Let $\tau$ be the birational involution of $U$ induced
by~$\psi$.~Then
\begin{itemize}
\item the involutions $\tau$ is biregular $\iff$ $L_{1}\cap\mathrm{Sing}(X)=\varnothing$,%
\item the involutions $\tau$ acts biregularly on $U\setminus\bar{L}_{1}$ if $L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$,%
\item the equality $\tau(\bar{L}_{2})=\bar{L}_{2}$ holds,%
\item if $L_{1}\cap\mathrm{Sing}(X)=\varnothing$, then
$\tau(\bar{L}_{1})=\bar{L}_{1}$ and
$$
\tau^{*}\Big(\pi^{*}\big(-K_{X}\big)\Big)\sim \pi^{*}\big(-2K_{X}\big)-3E-\bar{L}_{1}-\bar{L}_{2}.%
$$
\end{itemize}


Let $\bar{L}_{3}$ be a proper transform of $L_{3}$ on the surface
$U$. Then $\tau(E)=\bar{L}_{3}$ and
$$
L_{1}\cup L_{2}\not\ni\pi\circ\tau\big(Q\big)\in L_{3}.
$$

\begin{lemma}
\label{lemma:cubic-no-second-line} The line $L_{3}$ is the only
line on $X$ that passes through the point $\pi\circ\tau(Q)$.
\end{lemma}

\begin{proof}
Suppose that there is a line $L\subset X$ such that $L\ne L_{3}$
and $\pi\circ\tau(Q)\in L$.  Then
$$
L\cap L_{1}=L\cap L_{2}=\varnothing,
$$
because $\pi\circ\tau(Q)\in L_{3}$ and $(L_{1}+L_{2}+L_{3})\cdot
L=1$. Thus, there is unique conic $Z\subset X$ such that
$$
-K_{X}\sim L+Z
$$
and $P\in Z$. Then $Z$ is irreducible, because $P\not\in L$ and
$P$ is not an Eckardt point.

Let $\bar{L}$ and $\bar{Z}$ be the proper transform of $L$ and $Z$
on the surface $U$, respectively. Then
$$
\bar{L}\cdot\bar{L}_{3}=\bar{Z}\cdot E=1,\ \bar{Z}\cdot\bar{Z}=1-\bar{L}\cdot\bar{Z},\ \bar{L}\cdot\bar{Z}=\left\{%
\aligned
&2\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3/2\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.% ,
$$
and $\bar{L}_{1}\cdot\bar{Z}=\bar{L}_{2}\cdot\bar{Z}=\bar{L}\cdot
E=\bar{L}\cdot\bar{L}_{1}=\bar{L}\cdot\bar{L}_{2}=0$.

We have $\tau(\bar{Z})=\bar{L}$. Then $Q\in\bar{Z}$, which implies
that $\bar{Z}\subseteq\mathrm{Supp}(\bar{\Delta})$, because
$$
2-\mathrm{mult}_{P}\big(\Delta\big)-m_{1}-m_{2}=\bar{Z}\cdot\bar{\Omega}>8/3-\mathrm{mult}_{P}\big(\Delta\big)-m_{1}-m_{2}
$$
in the case when
$\bar{Z}\not\subseteq\mathrm{Supp}(\bar{\Delta})$. On the other
hand, the log pair
$$
\Big(X,\ \omega (L+Z)\Big)
$$
is log canonical at the point $P$. Hence, we may assume that
$\bar{L}\not\subseteq\mathrm{Supp}(\bar{\Delta})$ by
Remark~\ref{remark:convexity}. Put
$$
D=\eps Z+m_{1}L_{1}+m_{2}L_{2}+\Upsilon,
$$
where $\eps\in\mathbb{Q}$ such that $\eps>0$, and
$\Upsilon$ is an effective $\mathbb{Q}$-divisor such that
$Z\not\subseteq\mathrm{Supp}(\Upsilon)$.~Then
$$
1=L\cdot D=\eps L\cdot Z+m_{1}L\cdot
L_{1}+L\cdot\Upsilon=\eps L\cdot Z+L\cdot\Upsilon\geqslant \eps L\cdot Z=\left\{%
\aligned
&2\eps\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)=\varnothing\,\\%
&3\eps/2\ \text{if}\ L\cap\mathrm{Sing}\big(X\big)\ne\varnothing,\\%
\endaligned\right.%
$$
which implies that $\eps\leqslant 2/3$. But
$\bar{Z}\cap\bar{L}_{1}=\varnothing$. Then it follows from
Lemma~\ref{lemma:adjunction} that
$$
2-\mathrm{mult}_{P}\big(D\big)-\eps\bar{Z}\cdot\bar{Z}=\bar{Z}\cdot\bar{\Upsilon}>8/3-\mathrm{mult}_{P}\big(D\big),%
$$
where $\bar{\Upsilon}$ is a proper transform of $\Upsilon$ on the
surface $U$. We deduce that  $\eps>2/3$. But
$\eps\leqslant 2/3$.
\end{proof}

Therefore, there is an unique irreducible conic $C\subset X$ such
that
$$
-K_{X}\sim L_{3}+C
$$
and $\pi\circ\tau(Q)\in C$. Then $C+L_{3}$ is a hyperplane section
of $X$ that is singular at $\pi\circ\tau(Q)$.

Let $\bar{C}$ be the proper transform of $C$ on the surface $U$.
Put $\bar{Z}=\tau(\bar{C})$ and $Z=\pi(\bar{Z})$.


\begin{lemma}
\label{lemma:cubic-final} One has
$L_{1}\cap\mathrm{Sing}(X)\ne\varnothing$.
\end{lemma}

\begin{proof}
Suppose that $L_{1}\cap\mathrm{Sing}(X)=\varnothing$. Then
$$
C\cap L_{1}=C\cap L_{2}=\varnothing,
$$
because $(L_{1}+L_{2}+L_{3})\cdot C=L_{3}\cdot C=2$. One can
easily check that
$$
\bar{Z}\sim \pi^{*}\big(-2K_{X}\big)-4E-\bar{L}_{1}-\bar{L}_{2},
$$
and $Z$ is singular at $P$.  Then $-2K_{X}\sim Z+L_{1}+L_{2}$. But
the log pair
$$
\left(U,\ \frac{1}{2}\Big(Z+L_{1}+L_{2}\Big)\right)
$$
is log canonical at $P$. Thus, we may assume that
$Z\not\subseteq\mathrm{Supp}(D)$ by Remark~\ref{remark:convexity}.

We have $Q\in\bar{Z}$ and $\bar{Z}\cdot E=2$. Then it follows from
the inequality~\ref{equation:blow-up} that
$$
4-2\mathrm{mult}_{P}\big(D\big)=\bar{Z}\cdot\bar{D}\geqslant\mathrm{mult}_{Q}\big(\bar{D}\big)>8/3-\mathrm{mult}_{P}\big(D\big),%
$$
which implies that $\mathrm{mult}_{P}(D)<4/3$. But
$\mathrm{mult}_{P}(D)>4/3$.
\end{proof}

Thus, we see that $L_{1}\cap
L_{3}=\mathrm{Sing}(X)\ne\varnothing$. Then $L_{1}\cap L_{2}\in
C$, which implies that
$$
\bar{Z}\sim \pi^{*}\big(-2K_{X}\big)-4E-2\bar{L}_{1}-\bar{L}_{2},
$$
and $Z$ is smooth rational cubic curve.  Then $-2K_{X}\sim
Z+2L_{1}+L_{2}$. But the log pair
$$
\left(U,\ \frac{1}{2}\Big(Z+2L_{1}+L_{2}\Big)\right)
$$
is log canonical at $P$. Thus, we may assume that
$Z\not\subseteq\mathrm{Supp}(D)$ by Remark~\ref{remark:convexity}.

We have $Q\in\bar{Z}$ and $\bar{Z}\cdot E=\bar{L}_{1}=1$. Then it
follows from the inequality~\ref{equation:blow-up} that
$$
3-\mathrm{mult}_{P}\big(\Delta\big)-2m_{1}-m_{2}=\bar{Z}\cdot\bar{\Delta}\geqslant\mathrm{mult}_{Q}\big(\bar{\Delta}\big)>8/3-\mathrm{mult}_{P}\big(\Delta\big)-m_{1}-m_{2},%
$$
which implies that $m_{1}<1/3$. But $m_{1}>2/3$ by
Corollary~\ref{corollary:cubic-m1}.


The obtained contradiction completes the proof
Theorem~\ref{theorem:cubic-surfaces}.


\section{Del Pezzo surfaces}
\label{section:del-Pezzo}

Let $X$ be a del Pezzo surface that has at most canonical
singularities, let $O$ be a point of the~surface $X$, and let
$B_{X}$ be an effective $\mathbb{Q}$-divisor on the surface $X$.
Suppose that
\begin{itemize}
\item the point $O$ is either smooth or an ordinary double point of $X$,%
\item the surface $X$ is smooth outside the point $O\in X$.%
\end{itemize}


\begin{lemma}
\label{lemma:singular-Del-Pezzo-2} Suppose that $\Sing(X)=O$, $K_{X}^{2}=2$ and
the equivalence
$$
B_{X}\qlineq -\mu K_{X}
$$
holds, where $\mu\in\mathbb{Q}$ is such that $0<\mu<2/3$. Then
$\mathbb{LCS}(X,\mu B_{X})=\varnothing$.
\end{lemma}

\begin{proof}
Suppose that $\mathbb{LCS}(X,\mu B_{X})\neq\varnothing$. There is a
curve $\mathbb{P}^{1}\cong L\subset X$ such that
$$
\mathrm{LCS}\Big(X, \mu B_{X}\Big)\not\subseteq L
$$
and $L\cdot L<0$. There is a birational morphism $\pi\colon X\to
S$ that contracts the curve $L$. Then
$$
\mathbb{LCS}\Big(S,\ \mu\pi\big(B_{X}\big)\Big)\ne\varnothing
$$
due to the choice of the curve $L\subset X$. But
$-K_{S}\qlineq\pi(B_{X})$, and $S\subset\mathbb{P}^{3}$ is a
cubic surface that has at most one ordinary double point, which is
impossible (see Examples~\ref{example:singular-cubics}
and~\ref{example:del-Pezzos}).
\end{proof}

\begin{lemma}
\label{lemma:del-Pezzo-quintic} Suppose that $\mathrm{Sing}(X)=\varnothing$,
$K_{X}^{2}=5$, the equivalence
$$
B_{X}\qlineq -\mu K_{X}
$$
holds, where $\mu\in\mathbb{Q}$ is such that $0<\mu<2/3$.
Assume that $\mathbb{LCS}(X, B_X)\neq\varnothing$.
Then
\begin{itemize}
\item either the set $\mathbb{LCS}(X, B_{X})$ contains a curve,%
\item or there are a curve $\mathbb{P}^{1}\cong L\subset X$ and a point $O\in
L$ such that $L\cdot L=-1$ and
$$
\mathrm{LCS}\Big(X,\ B_{X}\Big)=O.
$$
\end{itemize}
\end{lemma}

\begin{proof}
Suppose that $\mathbb{LCS}(X, B_{X})$ contains no curves. Then it follows from
Theorem~\ref{theorem:connectedness} that
$$
\mathrm{LCS}\Big(X,\ B_{X}\Big)=O,
$$
where $O\in X$ is a point. We may assume that $O$ does not lie on any curve
$\mathbb{P}^{1}\cong L\subset X$~such~that the equality $L\cdot L=-1$ holds.
Then there is a birational morphism
$$
\phi\colon X\longrightarrow\mathbb{P}^{2}
$$
such that $\phi$ is an isomorphism in a neighborhood of the point $O$. Then
$$
\phi\big(O\big)\in\LCS\Big(\mathbb{P}^{2},\ \phi\big(B_{X}\big)\Big)\ne\varnothing,%
$$
the set $\mathbb{LCS}(\mathbb{P}^{2},\phi(B_{X}))$ contains no curves, and
$$
\phi\big(B_{X}\big)\qlineq -\mu K_{\mathbb{P}^{2}}.
$$
Since $0<\mu<2/3$, the latter is impossible by
Lemma~\ref{lemma:plane}.
\end{proof}


\begin{example}
\label{example:del-Pezzo-quintic} Suppose that $O=\mathrm{Sing}(X)$ and
$K_{X}^{2}=5$. Let $\alpha\colon V\to X$ be a blow up of $O$,~and let $E$ be
the exceptional divisor of $\alpha$. Then there is a birational morphism
$\omega\colon V\to \mathbb{P}^{2}$ such that
\begin{itemize}
\item the morphism $\omega$ contracts $4$ curves
$E_{1}$, $E_{2}$, $E_{3}$, $E_{4}$,%
\item the curve $\omega(E)$ is a line in $\mathbb{P}^{2}$ that contains
$\omega(E_{1})$, $\omega(E_{2})$, $\omega(E_{3})$, but $\omega(E)\not\ni
\omega(E_{4})$.
\end{itemize}

Let $Z$ be a line in
$\mathbb{P}^{2}$ such that $\omega(E_{1})\in Z\ni \omega(E_{4})$.
Then
$$
2E+\bar{Z}+2E_{1}+E_{2}+E_{3}\sim -K_{V}
$$
where $\bar{Z}\subset V$ is a proper transform of $Z$. One has
$$
\mathrm{lct}\Big(X,\
\alpha\big(\bar{Z}+2\alpha\big(E_{1}\big)+\alpha\big(E_{2}\big)+\alpha\big(E_{3}\big)\Big)=\frac{1}{2},%
$$
which implies $\mathrm{lct}(X)\leqslant 1/2$. Suppose that
$-K_{X}\qlineq 2B_{X}$, but $(X,B_{X})$ is not log canonical. Then
$$
K_{V}+B_{V}+mE\qlineq\alpha^{*}\Big(K_{X}+B_{X}\Big),
$$
for some $m\geqslant 0$, and $B_{V}$ is
the proper transform of $B_{X}$ on the surface $V$. Then
$$
\Big(V,\ B_{V}+mE\Big)
$$
is not log canonical at some point $P\in V$. There is a birational
morphism $\pi\colon V\to U$ such that
\begin{itemize}
\item the surface $U$ is a smooth del Pezzo surface with $K_{U}^{2}=6$,%
\item the morphism $\pi$ is an isomorphism in a neighborhood of $P\in X$,%
\end{itemize}
which implies that $(U,\pi(B_{V})+m\pi(E))$ is not log canonical
at $\pi(P)$. But
$$
\pi\big(B_{V}\big)+m\pi\big(E\big)\qlineq-\frac{1}{2}K_{U},
$$
which is impossible, because $\mathrm{lct}(U)=1/2$ (see
Example~\ref{example:del-Pezzos}). We see that
$\mathrm{lct}(X)=1/2$.
\end{example}


\begin{example}
\label{example:del-Pezzo-quartic} Suppose that $K_{X}^{2}=4$.
Arguing as in Example~\ref{example:del-Pezzo-quintic}, we see that
the equalities
$$
\mathrm{lct}\big(X\big)=\left\{%
\begin{aligned}
&1/2\ \text{when}\ O=\mathrm{Sing}\big(X\big),\\
&2/3\ \text{when}\ \mathrm{Sing}\big(X\big)=\varnothing,
\end{aligned}
\right.
$$
hold (cf. \cite{Wo06}). Take $\lambda\in\mathbb{Q}$ such that
$\lambda\leqslant 1$. Suppose that
$$
B_{X}\qlineq -K_{X},
$$
and $(X,\lambda B_{X})$ is not log canonical at some point $P\in
X\setminus O$. There is a commutative diagram
$$
\xymatrix{
&V\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
X\ar@{-->}[rr]_{\psi}&&U,}
$$
where $U$ is a cubic surface in $\mathbb{P}^{3}$ that has
canonical singularities, the morphism $\alpha$ is a blow up of the
points $P$, the morphism $\beta$ is birational, and $\psi$ is a
projection from the point $P\in X$. Then
$$
K_{V}+\lambda
B_{V}+\Big(\lambda\mathrm{mult}_{P}\big(B_{X}\big)-1\Big)E\qlineq\alpha^{*}\Big(K_{X}+\lambda
B_{X}\Big),
$$
where $E$ is the exceptional divisor of $\alpha$, and $B_{V}$ is
the proper transform of $B_{X}$. Then
$$
\Big(V,\ \lambda
B_{V}+\Big(\lambda\mathrm{mult}_{P}\big(B_{X}\big)-1\Big)E\Big)
$$
is not log canonical at some point $Q\in E$ and
$\mathrm{mult}_{P}(B_{X})>1/\lambda$. Then the log pair
$$
\Big(V,\ \lambda
B_{V}+\Big(\lambda\mathrm{mult}_{P}\big(B_{X}\big)-\lambda\Big)E\Big)
$$
is not log canonical at the point $Q\in E$ as well. But the
equivalences
$$
B_{V}+\Big(\mathrm{mult}_{P}\big(B_{X}\big)-1\Big)E\qlineq-K_{V}+\alpha^{*}\Big(K_{X}+B_{X}\Big)\qlineq -K_{V},%
$$
hold. Suppose that $P$ is not contained in any line on the surface
$X$. Then
\begin{itemize}
\item the morphism $\beta\colon V\to U$ is an isomorphism,%
\item the cubic surface $U$ is smooth outside of the point $\psi(O)$,%
\item the point $\psi(O)$ is at most ordinary double point of the surface $U$,%
\end{itemize}
which implies that $\lambda>2/3$ (see
Example~\ref{example:singular-cubics}).

Suppose that $\lambda=3/4$. Then the point
$$
\psi\big(Q\big)\in U\subset\mathbb{P}^{3}
$$
must be an Eckardt point (see
Definition~\ref{definition:Eckardt-point}) of the surface $U$ by
Theorem~\ref{theorem:cubic-surfaces}. But
$$
\beta\big(E\big)\subset U\subset\mathbb{P}^{3}
$$
is a line. So, there are two conics $C_{1}\ne C_{2}$ contained in $X$ such that
$P=C_{1}\cap C_{2}$ and
$$
C_{1}+C_{2}\sim -K_{X}.
$$
\end{example}

\begin{lemma}
\label{lemma:singular-del-Pezzo-sextic} Suppose that $O=\mathrm{Sing}(X)$ and
$K_{X}^{2}=6$ such that there is a diagram
$$
\xymatrix{
&V\ar@{->}[ld]_\alpha\ar@{->}[rd]^\beta&\\
X&&\P^2}
$$
where $\beta$ is a blow up of three points $P_1, P_2, P_3\in\P^2$ lying on a
line $L\subset\mathbb{P}^{2}$, and $\alpha$ is a birational morphism that
contracts an irreducible curve $\bar{L}$ to the point $O$ such that
$\beta(\bar{L})=L$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda B_{X}\Big)=O
$$
in the case when $\mathrm{LCS}(X,\lambda B_{X})\ne\varnothing$ and
$B_{X}\qlineq -K_{X}$, where  $\lambda<1/2$.
\end{lemma}

\begin{proof}
Suppose that $B_{X}\qlineq -K_{X}$ and
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda B_{X}\Big)\ne O,
$$
let $M\subset\P^2$ be a general line, and let $\bar{M}\subset V$ be its proper
transform. Then
$$
-K_X\sim 2\alpha\big(\bar{M}\big)
$$
and $O\in \alpha(\bar{M})$. Thus, the set $\mathbb{LCS}(X, \lambda B_{X})$
contains a curve, because otherwise the locus
$$
\LCS\Big(X,\ \lambda B_{X} +\alpha\big(\bar{M}\big)\Big)
$$
would be disconnected, which is impossible by
Theorem~\ref{theorem:connectedness}.

Let $C\subset X$ be an irreducible curve such that $C\subseteq\LCS(X, \lambda
B_{X})$. Then
$$
B_{X}=\eps C+\Omega,
$$
where $\eps>2$, and $\Omega$ is an
effective $\mathbb{Q}$-divisor such that $C\not\subset\mathrm{Supp}(\Omega)$.

Let $\Gamma_{i}\subset X$ be a proper transform of a general line in
$\mathbb{P}^{2}$ that passes through  $P_{i}$. Then
$$
O\not\in\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}
$$
and $-K_{X}\cdot\Gamma_{1}=-K_{X}\cdot\Gamma_{2}=-K_{X}\cdot\Gamma_{3}=2$. But
$$
-K_{X}\qlineq\Gamma_{1}+\Gamma_{2}+\Gamma_{3},
$$
which implies that there is $m\in\{1,2,3\}$ such that $C\cdot\Gamma_{m}\ne 0$.
Then
$$
2=B_{X}\cdot\Gamma_{m}=\Big(\eps C+\Omega\Big)\cdot\Gamma_{m}\geqslant\eps C\cdot\Gamma_{m}\geqslant\eps>2,%
$$
because $\Gamma_{m}\not\subset\mathrm{Supp}(B_{X})$. The obtained contradiction
completes the proof.
\end{proof}


\begin{remark}
\label{remark:del-Pezzo-sextic} Suppose that $O=\mathrm{Sing}(X)$
and $K_{X}^{2}=6$. Let
$$
\alpha\colon V\longrightarrow X
$$
be a blow up of the point $O\in X$, and let $E$ be the exceptional divisor of
$\alpha$. Then
$$
K_{V}+B_{V}+mE\qlineq\alpha^{*}\Big(K_{X}+B_{X}\Big)
$$
for some $m\geqslant 0$, and $B_{V}$ is the proper
transform of $B_{X}$ on  $V$. Suppose that
$$
B_{X}\qlineq -K_{X},
$$
but $(X,\lambda B_{X})$ is not log canonical for some $\lambda\in\mathbb{Q}$
such that $\lambda<1/3$. Then
$$
\Big(V,\ \lambda B_{V}+\lambda mE\Big)
$$
is not log canonical at some point $P\in V$. There is a birational morphism
$$
\pi\colon V\longrightarrow U
$$
such that $U$ is either $\mathbb{F}_{1}$ or
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and $\pi$ is an isomorphism in a
neighborhood of $P\in X$. Then
$$
\Big(U,\ \lambda\pi\big(B_{V}\big)+m\pi\big(E\big)\Big)
$$
is not log canonical
at the point $\pi(P)$. But we know that
$$
-K_{U}\qlineq\pi\big(B_{V}\big)+m\pi\big(E\big),
$$
so the latter is possible
only for $\lambda>1/3$ (see Example~\ref{example:del-Pezzos}) which implies
that $\mathrm{lct}(X)=1/3$.
\end{remark}


\begin{lemma}
\label{lemma:quadric-cone-far-from-vertex} Suppose that
$X\cong\mathbb{P}(1,1,2)$ and $B_{X}\qlineq -K_X$, but there is a point $P\in
X$~such~that
$$
O\ne P\in\mathrm{LCS}\Big(X,\ \lambda B_{X}\Big)
$$
for some $\lambda<1/2$. Take
$L\in|\mathcal{O}_{\mathbb{P}(1,1,2)}(1)|$ such that $P\in L$. Then
$L\subseteq\mathrm{LCS}(X, \lambda B_{X})$.
\end{lemma}

\begin{proof}
Suppose that there is a curve $\Gamma\in\LCS(X, \lambda B_{X})$ such that
$P\in\Gamma\ne L$. Then
$$
B_{X}=\mu\Gamma+\Omega,
$$
where $\mu>2$, and $\Omega$ is an effective
$\mathbb{Q}$-divisor such that $\Gamma\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\mu\Gamma+\Omega\qlineq 4L
$$
and $\Gamma\sim mL$, where $m\in\mathbb{N}$. But $m\geqslant 2$, because
$P\in\Gamma\ne L$, which is a contradiction.

Suppose that $L\not\subseteq\mathrm{LCS}(X, \lambda B_{X})$. Then it follows
from Theorem~\ref{theorem:connectedness} that
$$
\mathrm{LCS}\Big(X, \lambda B_{X}\Big)=P,%
$$
because we proved that $\mathrm{LCS}(X, \lambda B_{X})$ contains no curves
passing through $P\in X$.

Let $C\in|\mathcal{O}_{\mathbb{P}(1,1,2)}(1)|$ be a general curve. Then
$$
\mathrm{LCS}\Big(X, \lambda B_{X}+C\Big)=P\cup C,%
$$
which is impossible by Theorem~\ref{theorem:connectedness}.
\end{proof}


\begin{lemma}
\label{lemma:F1} Suppose that $X\cong\mathbb{F}_{1}$. Then there
are $0\leqslant\mu\in\mathbb{Q}\ni\lambda\geqslant 0$ such that
$$
B_{X}\qlineq \mu C+\lambda L,
$$
where $C$ and $L$ are irreducible curves on the surface $X$ such
that
$$
C\cdot C=-1,\ C\cdot L=1
$$
and $L\cdot L=0$. Suppose that $\mu<1$ and $\lambda<1$. Then
$\mathbb{LCS}(X, B_{X})=\varnothing$.
\end{lemma}

\begin{proof}
The set $\mathbb{LCS}(X, B_{X})$ contains no curves, because $L$
and $C$ generate the~cone of effective divisors of the surface
$X$. Suppose that $\mathbb{LCS}(X, B_{X})$ contains a point $O\in
X$. Then
$$
K_{X}+B_{X}+\Big(\big(1-\mu\big)C+\big(2-\lambda\big)L\Big)\qlineq -\big(L+C\big),%
$$
because  $-K_{X}\qlineq 2C+3L$. But it follows from
Theorem~\ref{theorem:Shokurov-vanishing} that the map
$$
0=H^{0}\Big(\mathcal{O}_{X}\big(-L-C\big)\Big)\longrightarrow
 H^{0}\Big(\mathcal{O}_{\mathcal{L}(X,\,B_{X})}\Big)\ne 0,%
$$
is surjective, because the divisor $(1-\mu)C+(2-\lambda)L$ is
ample.
\end{proof}


\begin{lemma}
\label{lemma:del-Pezzo-septic}  Suppose that
$\mathrm{Sing}(X)=\varnothing$ and $K_{X}^{2}=7$. Then
$$
L_{1}\cdot L_{1}=L_{2}\cdot L_{2}=L_{3}\cdot L_{3}=-1,\ L_{1}\cdot L_{2}=L_{2}\cdot L_{3}=1,\ L_{1}\cdot L_{3}=0%
$$
where $L_{1}$, $L_{2}$, $L_{3}$ are exceptional curves on $X$.
Suppose that $\mathrm{LCS}(X, B_{X})\ne\varnothing$ and
$$
B_{X}\qlineq -\mu K_{X},
$$
where $\mu\in\mathbb{Q}$ is such that $0<\mu<1/2$. Then
$\mathrm{LCS}(X, B_{X})=L_{2}$.
\end{lemma}

\begin{proof}
Let $P$ be a point in $\mathrm{LCS}(X, B_{X})$. Then $P\in L_{2}$,
because $\mathrm{lct}(\mathbb{P}^{1}\times\mathbb{P}^{1})=1/2$,
and there is a birational morphism
$X\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ that contracts only the
curve $L_{2}$.

Suppose that $\mathrm{LCS}(X, B_{X})\ne L_{2}$. Then
$\mathrm{LCS}(X, B_{X})=P$ by Theorem~\ref{theorem:connectedness}.

We may assume that $P\not\in L_3$. There is a birational morphism
$$
\phi\colon X\longrightarrow\mathbb{P}^{2}
$$
that contracts the curves $L_{1}$ and $L_{3}$. Let $C_{1}$ and
$C_{3}$ be proper transforms on $X$ of sufficiently general lines
in $\mathbb{P}^{2}$ that pass through the points $\phi(L_{1})$ and
$\phi(L_{3})$, respectively. Then
$$
-K_{X}\sim C_{1}+2C_{3}+L_{3},
$$
and $C_{1}\not\ni P\not\in C_{2}$. Therefore, we see that
$$
C_{2}\cup P\subseteq\mathrm{LCS}\left(X,\ \lambda
D+\frac{1}{2}\Big(C_{1}+2C_{2}+L_{3}\Big)\right)\subseteq C_{2}\cup
P\cup L_3,%
$$
which is impossible by Theorem~\ref{theorem:connectedness},
because $P\not\in L_{3}$.
\end{proof}

\begin{lemma}
\label{lemma:quadric-cone} Suppose that $O=\mathrm{Sing}(X)$, the equality
$K_{X}^{2}=7$ holds, the equivalence
$$
B_{X}\qlineq C+\frac{4}{3}L
$$
holds, where $L\cong\mathbb{P}^{1}\cong C$ are curves on the surface $X$ such
that
$$
L\cdot L=-1/2,\ C\cdot C=-1,\ C\cdot L=1,
$$
but the log pair $(X, B_{X})$ is not log canonical at some point $P\in C$.
Then~$P\in L$.
\end{lemma}

\begin{proof}
Let $S$ be a quadratic cone in $\mathbb{P}^{3}$. Then $S\cong\mathbb{P}(1,1,2)$
and there is a birational morphism
$$
\phi\colon X\longrightarrow S\subset\mathbb{P}^{3}
$$
that contracts the curve $C$ to a smooth point $Q\in S$. Then
$Q\in\phi(L)\in|\mathcal{O}_{\mathbb{P}(1,1,2)}(1)|$.

Suppose that $P\not\in L$. Then it follows from Remark~\ref{remark:convexity}
that to complete the~proof we may assume that either
$C\not\subset\mathrm{Supp}(B_{X})$ or $L\not\subset\mathrm{Supp}(B_{X})$,
because the log pair
$$
\left(X,\ C+\frac{4}{3}L\right)
$$
is log canonical in the point $P\in X$. Suppose that
$C\not\subset\mathrm{Supp}(B_{X})$. Then
$$
\frac{1}{3}=B_{X}\cdot C\ge\mathrm{mult}_{P}\big(B_{X}\big)>1,
$$
which is impossible. Therefore, we see that $C\subset\mathrm{Supp}(B_{X})$.
Then  $L\not\subset\mathrm{Supp}(B_{X})$. Put
$$
B_{X}=\eps C+\Omega,
$$
where $\Omega$ is an
effective $\mathbb{Q}$-divisor such that $C\not\subset\mathrm{Supp}(\Omega)$.
Then
$$
\frac{1}{3}=B_{X}\cdot L=\eps+\Omega\cdot L\geqslant \eps,
$$
which implies that $\eps\leqslant 1/3$. Then
$$
1<\Omega\cdot C=1/3+\eps\leqslant 2/3
$$
by Lemma~\ref{lemma:adjunction}, which
is a contradiction.
\end{proof}




\section{Toric varieties}
\label{section:toric}


The purpose of this section is to prove Lemma~\ref{lemma:toric} (cf.
\cite{BaSe99} and \cite{So05}).

Let $N=\mathbb{Z}^{n}$ be a lattice of rank $n$, and let
$M=\mathrm{Hom}(N,\mathbb{Z})$ be the dual lattice. Put
$$
M_{\mathbb{R}}=M\otimes_\mathbb{Z}\mathbb{R}
$$
and $N_\mathbb{R}=N\otimes_\mathbb{Z}\mathbb{R}$. Let $X$ be a
toric variety defined by a complete fan $\Sigma\subset
N_{\mathbb{R}}$, let
$$
\Delta_1=\big\{v_1,\ldots, v_m\big\}
$$
be a set of generators of one-dimensional cones of the fan
$\Sigma$. Put
$$
\Delta=\Big\{w\in M\ \Big\vert\ \big\langle w,
v_i\big\rangle\geqslant -1\ \text{for all}\ i=1,
 \ldots, m\Big\}.%
$$

Put $T=(\mathbb{C}^{*})^{n}\subset\Aut(X)$. Let $\mathcal{N}$ be
the normalizer of $T$ in $\mathrm{Aut}(X)$ and
$\mathcal{W}=\mathcal{N}/T$.

\begin{lemma}
\label{lemma:toric} Let $G\subset\mathcal{W}$ be a subgroup. Suppose
that $X$ is $\mathbb{Q}$-factorial. Then
$$
\mathrm{lct}\Big(X,\ G\Big)=\frac{1}{1+\mathrm{max}\Big\{\big\langle w, v\big\rangle\ \big\vert\ w\in\Delta^G,\ v\in\Delta_1\Big\}},%
$$
where $\Delta^G$ is the set of the points in $\Delta$ that are fixed by the
group $G$.
\end{lemma}

\begin{proof}
Put $\mu=1+\mathrm{max}\{\langle w, v\rangle\ \vert\ w\in\Delta^G,\
v\in\Delta_1\}$. Then $\mu\in\mathbb{Q}$ is the biggest number~such~that
$$
-K_X\sim_{\mathbb{Q}}\lambda R+H,
$$
where $R$ is an integral $T\rtimes G$-invariant effective divisor, and $H$ is
an effective $\mathbb{Q}$-divisor. Then
$$
\mathrm{lct}\big(X, G\big)\leqslant\frac{1}{\mu}.
$$

Suppose that $\mathrm{lct}(X, G)<1/\mu$. Then there is an effective
$G$-invariant $\mathbb{Q}$-divisor
$$
D\qlineq -K_{X}
$$
such~that the log pair $(X,\lambda D)$ is not log-canonical for some rational
$\lambda<1/\mu$.

There exists a family $\{D_t \mid t\in\mathbb{C}\}$ of $G$-invariant effective
$\Q$-divisors such that
\begin{itemize}
\item the equivalence $D_t\qlineq D$ holds for every $t\in\mathbb{C}$,%
\item the equality $D_1=D$ holds,%
\item for every $t\neq 0$ there is $\phi_{t}\in\mathrm{Aut}(X)$ such that
$$
D_t=\phi_{t}\big(D\big)\cong D,
$$
\item the divisor $D_0$ is $T$-invariant and $G$-invariant,%
%\item the family $\{D_t\ \vert\ t\in\mathbb{C}\}$ is flat,%
\end{itemize}
which implies that $(X, \lambda D_0)$ is not log-canonical (see \cite{DeKo01}).

The divisor $D_{0}$ does not have components with multiplicity greater then
$\mu$, which implies that the log pair $(X, \lambda D_{0})$ is log-canonical
(see \cite{Ful93}, \cite{Mat02}).
\end{proof}

\begin{corollary}\label{corollary:projectivization}
Let $X=\P_{\P^n}(\O\oplus\O(-a_1)\oplus\ldots\oplus\O(-a_k))$,
$a_i\ge 0$ for $i=1, \ldots, k$.
%Proektivizaciya: tochka = giperploskost' v sloe
Then
$$
\mathrm{lct}\Big(\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^n}\oplus\mathcal{O}_{\mathbb{P}^n}\big(-a_1\big)\oplus\ldots\oplus\mathcal{O}_{\mathbb{P}^n}\big(-a_k\big)\Big)\Big)=\frac{1}{1+\mathrm{max}\Big\{k,
 n+\sum_{i=1}^{k}a_i\Big\}}.%
$$
\end{corollary}
\begin{proof}
Note that $X$ is a toric variety, and $\Delta_1$ consists of the
following vectors:
\begin{gather*}
(\overbrace{1, 0, \ldots, 0}^k, \overbrace{0, 0, \ldots, 0}^n),\\
\ldots  \\
(0, \ldots, 0, 1, 0, 0, \ldots, 0),\\
(-1, \ldots, -1, 0, 0, \ldots, 0),\\
(0, 0, \ldots, 0, 1, 0, \ldots, 0),\\
\ldots \\
(0, 0, \ldots, 0, 0, \ldots, 0, 1),\\
(-a_1, \ldots, -a_k, -1, \ldots, -1),
\end{gather*}
which implies the required assertion.
\end{proof}

Applying Corollary~\ref{corollary:projectivization}, we obtain the
following result.

\begin{corollary}\label{corollary:2-33-2-35-2-36}
In the assumptions and notation of Theorem~\ref{theorem:main}, we
have
$$
\mathrm{lct}\big(X\big)=\left\{%
\begin{aligned}
&1/4\ \text{whenever}\ \gimel(X)\in\{2.33, 2.35\},\\
&1/5\ \text{whenever}\ \gimel(X)=2.36.
\end{aligned}
\right.
$$
\end{corollary}

On the other hand, straightforward computations using
Lemma~\ref{lemma:toric} imply the following result.

\begin{corollary}\label{corollary:other-toric}
In the assumptions and notation of Theorem~\ref{theorem:main}, we
have
$$
\mathrm{lct}\big(X\big)=\left\{%
\begin{aligned}
&1/3\ \text{whenever}\ \gimel(X)\in\{3.25, 3.31, 4.9, 4.11, 5.2\},\\
&1/4\ \text{whenever}\ \gimel(X)\in\{3.26, 3.30, 4.12\},\\
&1/5\ \text{whenever}\ \gimel(X)=3.29.
\end{aligned}
\right.
$$
\end{corollary}



\section{Del Pezzo threefolds}
\label{section:del-Pezzos}

We use the assumptions and notation of Theorem~\ref{theorem:main}.
Suppose that $-K_{X}\sim 2H$, where $H$ is a Cartier divisor that
is indivisible in $\mathrm{Pic}(X)$.

The purpose of this section is to prove the following result.

\begin{theorem}\label{theorem:del-Pezzo}
The equality $\lct(X)=1/2$ holds, unless $\gimel(X)=2.35$ when $\lct(X)=1/4$.
\end{theorem}

It follows from Theorems~3.1.14 and~3.3.1 in \cite{IsPr99} that
$$
\gimel(X)\in\Big\{1.11, 1.12, 1.13, 1.14, 1.15, 1.17, 2.32, 2.35, 3.27\Big\},%
$$
and it follows from~\cite{Ch01b} and \cite{Ch08a} (see also
Lemma~\ref{lemma:double-covers}) that $\lct(X)=1/2$ if $\gimel(X)\in\{1.12,
1.13\}$.

It follows from Lemma~\ref{lemma:lct-product} that $\lct(X)=1/2$ when
$\gimel(X)=3.27$.


\begin{lemma}
\label{lemma:2-35} Suppose that $\gimel(X)=2.35$. Then
$\mathrm{lct}(X)=1/4$.
\end{lemma}

\begin{proof}
We know that $X\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}\oplus
\mathcal{O}_{\mathbb{P}^{2}}(1))$, and there is a birational morphism
$\pi\colon X\to\mathbb{P}^{3}$ that contracts a surface $E\cong\mathbb{P}^{2}$
to a point $P\in\mathbb{P}^{3}$. Then $\mathrm{lct}(X)\leqslant 1/4$, because
$$
-K_{X}\sim 2E+4T,
$$
where $T$ is the proper transform of the hyperplane in
$\mathbb{P}^{3}$ that passes through $P\in\mathbb{P}^{3}$.

Suppose that $\mathrm{lct}(X)<1/4$. Then there is an effective
$\mathbb{Q}$-divisor $D$ on the variety $X$ such that
$$
D\qlineq \frac{1}{2}E+T,
$$
but the log pair $(X,\lambda D)$ is not log canonical for some
rational number $\lambda<1$.

Let $R$ be a proper transform on $X$ of general hyperplane in
$\mathbb{P}^{3}$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda D+R\Big)
$$
must be connected by Theorem~\ref{theorem:connectedness}, because
$-(K_{X}+\lambda D+R)$ is ample. Thus, we see that
\begin{itemize}
\item the subscheme $\mathcal{L}(X,\lambda D)$ is not zero-dimensional,%
\item the locus $\mathrm{LCS}(X,\lambda D)$ is not contained in $E$,%
\end{itemize}
which implies that $(\mathbb{P}^{3},\lambda\pi(D))$ is not log
canonical. But $\mathrm{lct}(\mathbb{P}^{3})=1/4$ and
$\pi(D)\qlineq\mathcal{O}_{\mathbb{P}^{3}}(1)$.
\end{proof}

Actually, the assertion of Lemma~\ref{lemma:2-35} follows from
Corollary~\ref{corollary:2-33-2-35-2-36}. We are left with the cases
$$
\gimel(X)\in\Big\{1.11, 1.14, 1.15, 2.32\Big\},
$$
while the inequality $\mathrm{lct}(X)\leqslant 1/2$ is obvious, because $|H|$
is not empty.

\begin{lemma}
\label{lemma:P2-P2} Suppose that $\gimel(X)=2.32$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
We may suppose that $\mathrm{lct}(X)<1/2$. Then there is an effective
$\mathbb{Q}$-divisor $D\qlineq H$  such~that the log pair $(X,\lambda D)$ is not
log canonical for some positive rational number $\lambda<1$.

The threefold  $X$ is a divisor on $\P^2\times\P^2$ of bi-degree $(2, 2)$.
There are two natural $\P^1$-bundles $\pi_1\colon X\to\P^2$ and
$\pi_2\colon X\to\P^2$,
and applying Theorem~\ref{theorem:Hwang} to $\pi_1$ and~$\pi_2$ we immediately
obtain a contradiction.
\end{proof}

\begin{remark}
\label{remark:rho-1-common} Suppose that $\mathrm{Pic}(X)=\Z[H]$, and there is
an effective $\mathbb{Q}$-divisor $D\qlineq H$  such~that the log pair
$(X,\lambda D)$ is not log canonical for some positive rational number
$\lambda<1$, but
$$
D=\eps S+\Omega\qlineq H,
$$
where $S$ is an irreducible surface, $\Omega$ is an effective
$\mathbb{Q}$-divisor such that
$$
\mathrm{Supp}\big(\Omega\big)\not\supset S,
$$
and $\eps$ is a non-negative rational number. Then $\eps\leqslant 1$, because
$\mathrm{Pic}(X)=\Z[H]$, which implies that the set $\mathbb{LCS}(X,\lambda D)$
contains no surfaces. Moreover, for any choice of $H\in |H|$ the locus
$$
\mathrm{LCS}\Big(X,\ \lambda D+H\Big)
$$
is connected by Theorem~\ref{theorem:connectedness}. Let $H$ be a general
surface in $|H|$. Then there is a curve
$$
C\in\mathbb{LCS}\Big(X,\ \lambda D\Big),
$$
because $\mathcal{L}(X,\lambda D)$ contains no isolated zero-dimensional
components. Put $D\vert_{H}=\bar{D}$. Then
$$
-K_{H}\sim H\big\vert_{H}\qlineq \bar{D},
$$
but $(H, \lambda\bar{D})$ is not log canonical in every point of the
intersection $H\cap C$. The locus
$$
\mathrm{LCS}\Big(H, \lambda\bar{D}\Big)%
$$
is connected by Theorem~\ref{theorem:connectedness}. But the scheme
$\mathcal{L}(H, \lambda\bar{D})$ is zero-dimensional. We see that
$$
H\cdot C=\big|H\cap C\big|=1,
$$
and the locus $\mathrm{LCS}(X,\lambda D)$ contains no curves besides the
irreducible curve $C$.
\end{remark}



\begin{lemma}
\label{lemma:V5} Suppose that $\gimel(X)=1.15$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
We may suppose that $\mathrm{lct}(X)<1/2$. Then there is an effective
$\mathbb{Q}$-divisor $D\qlineq H$  such~that the log pair $(X,\lambda D)$ is not
log canonical for some positive rational number $\lambda<1$.

The linear system $|H|$ induces an embedding $X\subset\mathbb{P}^{6}$. Thus, it
follows from Remark~\ref{remark:rho-1-common} that the locus
$\mathrm{LCS}(X,\lambda D)$ consists of a single line $C\subset
X\subset\mathbb{P}^{6}$.

It follows from \cite[Proposition~3.4.1]{IsPr99} (see
\cite{Is77} and \cite{FurNak89}) that there is a commutative
diagram
$$
\xymatrix{
&&V\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&X\ar@{-->}[rr]_{\psi}&&Q,&}
$$
where $Q$ is a quadric in $\mathbb{P}^{4}$, the morphism $\alpha$
is a blow up of $C$, the morphism $\beta$ is a blow up of a smooth
rational cubic curve $Z\subset Q$, and the map $\psi$ is a
projection from the line $C$.

Let $S$ be the exceptional divisor of $\beta$, and let $L$ be a
fiber of the morphism $\beta$ over a general point of the curve
$Z$. Put $\bar{S}=\alpha(S)$ and $\bar{L}=\alpha(L)$. Then
$$
\bar{S}\sim H,
$$
the curve $\bar{L}$ is a line, the surface $\bar{S}$ is singular
along $C$. Moreover, the singularity of $\bar{S}$ at a general point
of $C$ is locally isomorphic to $T\times\mathbb{A}^1$, where
$T$ is a germ of a nodal curve. In particular, the pair
$(X, \bar{S})$ is log canonical.

We may assume that $\mathrm{Supp}(D)\not\supset\bar{S}$ by
Remark~\ref{remark:convexity}. Then
$$
1=\bar{L}\cdot D\geqslant\mathrm{mult}_{C}\big(D\big)>1/\lambda>1,
$$
which is a contradiction.
\end{proof}

\begin{lemma}
\label{lemma:V4} Let $\gimel(X)=1.14$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
We may suppose that $\mathrm{lct}(X)<1/2$. Then there is an effective
$\mathbb{Q}$-divisor $D\qlineq H$  such~that the log pair $(X,\lambda D)$ is not
log canonical for some positive rational number $\lambda<1$.

The linear system $|H|$ induces an embedding $X\subset\mathbb{P}^{5}$ such that
$X$ is a complete intersection of two quadrics. Then the locus
$\mathrm{LCS}(X,\lambda D)$ consists of a single line $C\subset
X\subset\mathbb{P}^{5}$ by Remark~\ref{remark:rho-1-common}.

It follows from \cite[Proposition~3.4.1]{IsPr99} that there is a commutative
diagram
$$
\xymatrix{
&&V\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&X\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{3},&}
$$
where $\psi$ is a projection from $C$, the morphism $\alpha$ is a blow up of
the line $C$, and $\beta$ is a blow up of a smooth curve
$Z\subset\mathbb{P}^{3}$ of degree $5$ and genus $2$.

Let $S$ be the exceptional divisor of $\beta$, and let $L$ be a fiber of the
morphism $\beta$ over a general point of the curve $Z$. Put $\bar{S}=\alpha(S)$
and $\bar{L}=\alpha(L)$. Then
$$
\bar{S}\sim 2H,
$$
the curve $\bar{L}$ is a line, and $\mathrm{mult}_{C}(\bar{S})=3$. But the log
pair $(X, 1/2\bar{S})$ is log canonical, which implies that we may assume that
$\mathrm{Supp}(D)\not\supset\bar{S}$ by Remark~\ref{remark:convexity}. Then
$$
1=\bar{L}\cdot D\geqslant\mathrm{mult}_{C}\big(D\big)>1/\lambda>1,
$$
which is a contradiction.
\end{proof}

\begin{remark}
\label{remark:singular-V4} Let $V\subset\mathbb{P}^{5}$ be a complete
intersection of two quadric hypersurfaces that has at most one ordinary double
point, and let $B_{V}$ be an effective $\mathbb{Q}$-divisor on $V$ such that
$B_{V}\qlineq -K_{V}$~and
$$
\mathrm{LCS}\Big(V, \mu B_{V}\Big)\ne\varnothing,
$$
where $\mu<1/2$. Arguing as in the proof of
Lemma~\ref{lemma:V4}, we see that
$$
\mathrm{LCS}\Big(V, \mu B_{V}\Big)\subseteq L,
$$
where $L\subset V$ is a line such that $L\cap\Sing(V)\ne\varnothing$.
\end{remark}

\begin{lemma}
Suppose that $\gimel(X)=1.11$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
We may suppose that $\mathrm{lct}(X)<1/2$. Then there is an effective
$\mathbb{Q}$-divisor $D\qlineq H$  such~that the log pair $(X,\lambda D)$ is not
log canonical for some positive rational number $\lambda<1$.

Recall that the threefold $X$ can be given by an equation
$$
w^{2}=t^{3}+t^{2}f_{2}\big(x,y,z\big)+tf_{4}\big(x,y,z\big)+f_{6}\big(x,y,z\big)\subset\mathbb{P}\big(1,1,1,2,3\big)\cong\mathrm{Proj}\Big(\mathbb{C}\big[x,y,z,t,w\big]\Big),
$$
where $\mathrm{wt}(x)=\mathrm{wt}(y)=\mathrm{wt}(z)=1$, $\mathrm{wt}(t)=2$,
$\mathrm{wt}(w)=3$, and $f_{i}$ is a polynomial of degree $i$.

The locus $\mathrm{LCS}(X,\lambda D)$ consists of a single curve $C\subset X$
such that $H\cdot C=1$ by Remark~\ref{remark:rho-1-common}.


Let $\psi\colon X\dasharrow\mathbb{P}^{2}$ be the natural projection. Then
$\psi$ is not defined in a point $O$ that is cut out by $x=y=z=0$. The curve
$C$ does not contain the point $O$, because otherwise we get
$$
1=\Gamma\cdot D\geqslant\mathrm{mult}_{O}\big(D\big)\mathrm{mult}_{O}\big(\Gamma\big)\geqslant\mathrm{mult}_{C}\big(D\big)>1/\lambda>1,%
$$
where $\Gamma$ is a general fiber of the projection $\psi$. Thus, we see that
$\psi(C)\subset\P^2$ is a line.

Let $S$ be the unique surface in $|H|$ such that $C\subset S$. Let $L$ be a
sufficiently general fiber of the rational map $\psi$ that intersects the curve
$C$. Then $L\subset\Supp(D)$, since otherwise
$$1=D\cdot L\ge \mult_C(D)>1/\lambda>1.$$

We may assume that $D=S$ by Remark~\ref{remark:convexity}. Then $S$ has a
cuspidal singularity along $C$.

We may assume that the surface $S$ is cut out on $X$ by the equation $x=0$, and
we may assume that the curve $C$ is given by $w=t=x=0$. Then $S$ is given by
$$
w^{2}=t^{3}+t^{2}f_{2}\big(0,y,z\big)+tf_{4}\big(0,y,z\big)\subset\mathbb{P}\big(1,1,2,3\big)\cong\mathrm{Proj}\Big(\mathbb{C}\big[y,z,t,w\big]\Big)
$$
such that $f_{6}(x,y,z)=xf_{5}(x,y,z)$, where $f_{5}(x,y,z)$ is a homogeneous
polynomial of degree $5$.

Since the surface $S$ is singular along the curve $C$, one has
$$
f_{4}\big(x,y,z\big)=xf_{3}\big(x,y,z\big),
$$
where $f_{3}(x,y,z)$ is a homogeneous polynomial of degree $3$.
Then every point of the set
$$
x=f_{5}\big(x,y,z\big)=t=w=0\subset\mathbb{P}\big(1,1,1,2,3\big)
$$
must be singular on $X$, which is a contradiction, because $X$ is smooth.
\end{proof}

The assertion of Theorem~\ref{theorem:del-Pezzo} is completely proved.

\section{Fano threefolds with $\rho=2$}
\label{section:rho-2}

We use the assumptions and notation introduced in
section~\ref{section:intro}.


\begin{lemma}\label{lemma:2-1-2-3}
Suppose that $\gimel(X)=2.1$ or $\gimel(X)=2.3$. Then
$\lct(X)=1/2$.
\end{lemma}
\begin{proof}
There is a birational morphism $\alpha\colon X\to V$ that
contracts a surface $E\subset X$ to a smooth elliptic curve
$C\subset V$, where  $V$ is one of the following smooth Fano
threefolds:
\begin{itemize}
\item smooth hypersurface in $\mathbb{P}(1,1,1,2,3)$ of degree $6$;%
\item smooth hypersurface in $\mathbb{P}(1,1,1,1,2)$ of degree $4$.%
\end{itemize}

The curve $C$ is contained in a surface $H\subset V$ such that
$$
\mathrm{Pic}\big(V\big)=\mathbb{Z}\big[H\big]
$$
and $-K_{X}\sim 2H$. Then $C$ is a complete intersection of two
surfaces in $|H|$, and
$$
-K_{X}\sim 2\bar{H}+E,
$$
where $E$ is the exceptional divisor of the birational morphism
$\alpha$, and $\bar{H}$ is a proper transform of the surface $H$
on the threefold $X$. In particular, the inequality $\lct(X)\le
1/2$ holds.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(V)=1/2$ by Theorem~\ref{theorem:del-Pezzo}
and $\alpha(D)\qlineq 2H\qlineq
-K_{V}$.

Put $k=H\cdot C$. Then $k=H^{3}\in\{1,2\}$. Note that
$$
\mathcal{N}_{C\slash
 V}\cong\mathcal{O}_{C}\Big(H\big\vert_{C}\Big)\oplus\mathcal{O}_{C}\Big(H\big\vert_{C}\Big),%
$$
which implies that $E\cong C\times\mathbb{P}^{1}$. Let $Z\cong C$
and $L\cong\mathbb{P}^{1}$ be curves on $E$ such that
$$
Z\cdot Z=L\cdot L=0
$$
and $Z\cdot L=1$. Then $\alpha^{*}(H)\vert_{E}\sim kL$, and since
$$
-2Z\sim K_E\sim \Big(K_X+E\Big)\Big\vert_{E}\sim
 \Big(2E-2\alpha^{*}\big(H\big)\Big)\Big\vert_{E}\sim -2kL+2E\Big\vert_E,%
$$
we see that $E\vert_E\sim -Z+kL$. Put
$$
D=\mu E+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $E\not\subset\mathrm{Supp}(\Omega)$.

The pair $(X, E+\lambda\Omega)$ is not log-canonical in the
neighborhood of $E$. Hence, the log pair
$$
\Big(E,\ \lambda\Omega\Big\vert_{E}\Big)
$$
is also not log canonical by Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{E}\qlineq\Big(-K_X-\mu E\Big)\Big\vert_{E}\qlineq
\Big(2\alpha^{*}\big(H\big)-\big(1+\mu\big)E\Big)\Big\vert_{E}\qlineq
\big(1+\mu\big)Z+k\big(1-\mu\big)L,%
$$
and $0\le\lambda k(1-\mu)\le 1$, which contradicts
Lemma~\ref{lemma:elliptic-times-P1}.
\end{proof}

\begin{lemma}
\label{lemma:2-4} Suppose that $\gimel(X)=2.4$ and $X$ is general.
Then $\mathrm{lct}(X)=3/4$.
\end{lemma}
\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&\mathbb{P}^3\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{1},&}
$$
where $\psi$ is a rational map, $\alpha$ is a blow~up of a smooth
curve $C\subset\mathbb{P}^{3}$ such that
$$
C=H_{1}\cdot H_{2}
$$
for some $H_{1}, H_2\in |\mathcal{O}_{\mathbb{P}^{3}}(3)|$,
and $\beta$ is a fibration into cubic surfaces.

Let $\mathcal{P}$ be a pencil in
$|\mathcal{O}_{\mathbb{P}^{3}}(3)|$  generated by $H_{1}$ and
$H_{2}$. Then $\psi$ is given by $\mathcal{P}$.

We assume that $X$ satisfies the following generality conditions:
\begin{itemize}
\item every surface in $\mathcal{P}$ has at most one ordinary double point;%
\item the curve $C$ contains no Eckardt points (see Definition~\ref{definition:Eckardt-point}) of any surface in $\mathcal{P}$.%
\end{itemize}

Let $E$  be the exceptional divisor of the birational morphism
$\alpha$. Then
$$
\frac{4}{3}\bar{H}_{1}+\frac{1}{3}E\qlineq \frac{4}{3}\bar{H}_{2}+\frac{1}{3}E\qlineq -K_{X},%
$$
where $\bar{H}_{i}$ is a proper transform of $H_{i}$ on the
threefold $X$. We see that $\mathrm{lct}(X)\leqslant 3/4$.

We suppose that $\mathrm{lct}(X)<3/4$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the log pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<3/4$.

Suppose that the set $\mathbb{LCS}(X,\lambda D)$ contains a
(irreducible) surface $S\subset X$. Then
$$
D=\eps S+\Delta,
$$
where $\eps\geqslant
1/\lambda$, and $\Delta$ is an effective $\mathbb{Q}$-divisor such
that $S\not\subset\mathrm{Supp}(\Delta)$. Then
$$
\Big(\bar{H}_{1},\ D\Big\vert_{\bar{H}_{1}}\Big)
$$
is not log canonical by Remark~\ref{remark:hyperplane-reduction}
if $S\cap\bar{H}_1\ne\varnothing$. But
$$
D\Big\vert_{\bar{H}_{1}}\qlineq -K_{\bar{H}_{1}}.
$$
We can choose $\bar{H}_{1}$ to be a smooth cubic surface in $\mathbb{P}^{3}$.
Thus, it follows from Theorem~\ref{theorem:cubic-surfaces} that
$$
S\cap\bar{H}_{1}=\varnothing,
$$
which implies that $S\sim\bar{H}_{1}$. Thus, we see that
$\alpha(S)$ is a cubic surface in $\mathcal{P}$. Then
$$
\eps\alpha\big(S\big)+\alpha\big(\Delta\big)\qlineq\mathcal{O}_{\mathbb{P}^{3}}\big(4\big),
$$
which is impossible, because $\eps\geqslant 1/\lambda>4/3$.

Let $F$ be a fiber of $\beta$ such that
$F\cap\mathrm{LCS}(X,\lambda D)\ne\varnothing$. Put
$$
D=\mu F+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor such that
$F\not\subset\mathrm{Supp}(\Omega)$. Then the log pair
$(F,\lambda\Omega\vert_F)$ is not log canonical by
Theorem~\ref{theorem:adjunction}, because $\lambda\mu<1$.  It
follows from Theorem~\ref{theorem:cubic-surfaces} that
$$
\mathrm{LCS}\Big(F,\ \lambda\Omega\Big\vert_{F}\Big)=O,
$$
where either $O$ is an Eckardt point of the surface $O$, or
$O=\mathrm{Sing}(X)$. By Theorem~\ref{theorem:connectedness}
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)=\mathrm{LCS}\Big(X,\ \lambda\mu F+\lambda\Omega D\Big)=O,%
$$
because it follows from Theorem~\ref{theorem:adjunction} that $(X,
F+\lambda\Omega D)$ is not log canonical at $O$ and is log
canonical in a punctured neighborhood of $O$. But $O\not\in E$ by
our generality assumptions. Then
$$
\alpha\big(O\big)\subset\mathrm{LCS}\Big(\mathbb{P}^{3},\ \lambda\alpha\big(D\big)\Big)\subseteq\alpha(O)\cup C,%
$$
where $\alpha(O)\not\in C$. But $\lambda<3/4$, which contradicts
Lemma~\ref{lemma:plane}.
\end{proof}



\begin{lemma}
\label{lemma:2-5-and-2-10-and-2-14} Suppose that
$\gimel(X)\in\{2.5, 2.10, 2.14\}$ and $X$ is general. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&V\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{1}&}
$$
where $V$ is a smooth Fano threefold such that $-K_{V}\sim 2H$ for
some $H\in\mathrm{Pic}(V)$ and
$$
\gimel(V)\in\big\{1.13,\ 1.14,\ 1.15\big\},
$$
the morphism $\alpha$ is a blow~up of a smooth curve $C\subset V$
such that
$$
C=H_{1}\cdot H_{2}
$$
for some $H_{1}, H_{2}\in |H|$, $H_1\neq H_2$, the morphism $\beta$ is a del
Pezzo fibration, and $\psi$ is a linear projection.

Let $E$  be the exceptional divisor of the birational morphism
$\alpha$. Then
$$
2\bar{H}_{1}+E\sim 2\bar{H}_{2}+E\sim -K_{X}
$$
where $\bar{H}_{i}$ is a proper transform of $H_{i}$ on the
threefold $X$. We see that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
because $\alpha(D)\qlineq -K_V$ and $\mathrm{lct}(V)=1/2$ by
Theorem~\ref{theorem:del-Pezzo}.

We assume that the threefold $X$ satisfies the following mild
generality condition: every fiber of the del Pezzo fibration
$\beta$ has at most one singular point that is an ordinary double
point.

Let $F$ be a fiber of $\beta$ such that
$F\cap\mathrm{LCS}(X,\lambda D)\ne\varnothing$. Put
$$
D=\mu F+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $F\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\alpha\big(D\big)=\mu\alpha\big(F\big)+\alpha\big(\Omega\big)\qlineq 2\alpha\big(F\big)\qlineq -K_{V},%
$$
which implies that $\mu\leqslant 2$. Then
$(F,\lambda\Omega\vert_F)$ is also not log canonical by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{F}\qlineq -K_{F},
%\qlineq -K_{X}\Big\vert_{F},
$$
which implies that $\mathrm{lct}(F)\leqslant \lambda<1/2$. But $F$
has at most one ordinary double point and
$$
K_{F}^{2}=H^{3}\leqslant 5,
$$
which implies that $\mathrm{lct}(F)\geqslant 1/2$ (see
Examples~\ref{example:del-Pezzos}, \ref{example:singular-cubics},
\ref{example:del-Pezzo-quintic}~and~\ref{example:del-Pezzo-quartic}).
\end{proof}


\begin{lemma}
\label{lemma:2-8} Suppose that $\gimel(X)=2.8$ and $X$ is general.
Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Let $O\in\mathbb{P}^{3}$ be a point, and let $\alpha\colon
V_{7}\to\mathbb{P}^{3}$ be a blow up of the point $O$. Then
$$
V_{7}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big),
$$
and there is a $\mathbb{P}^{1}$-bundle
$\pi\colon V_{7}\to\mathbb{P}^{1}$. Let $E$
be the exceptional divisor of $\alpha$. Then $E$ is a section~of~$\pi$.

There is a quartic surface $R\subset\mathbb{P}^{3}$ such that
$\mathrm{Sing}(R)=O$, the point $O$ is an isolated double point of the surface
$R$, and there is a commutative diagram
$$
\xymatrix{
&&&&X\ar@{->}[dl]_{\eta}\ar@/_1pc/@{->}[dlll]_{\beta}\ar@{->}[dd]^{\phi}\\%
&V_{2}\ar@{->}[rd]_{\omega}&&V_{7}\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\pi}\\%
&&\mathbb{P}^{3}\ar@{-->}^{\psi}[rr]&&\mathbb{P}^{2}}
$$
where $\omega$ is a double cover branched in $R$,
the morphism $\eta$ is a double cover branched in the proper
transform of $R$,
the morphism
$\beta$ is a birational morphism that contracts a surface
$\bar{E}$ such that
$\eta(\bar{E})=E$ to the singular point of $V_{2}$ and
$$
\omega\Big(\mathrm{Sing}\big(V_{2}\big)\Big)=O,
$$
the map $\psi$~is~a~projection from the point
$O$, and $\phi$ is a conic bundle.

We assume that $X$ satisfies the following mild generality condition: the point
$O$ is an \emph{ordinary} double point of the surface $R$. Then $
\bar{E}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Let $\bar{H}$ be the proper transform on $X$ of the general plane
in $\mathbb{P}^{3}$ that passes through $O$. Then
$$
-K_{X}\sim 2\bar{H}+\bar{E},
$$
which implies that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

It follows from Lemma~\ref{lemma:double-covers} that
$\mathrm{LCS}(X,D)\cap\bar{E}\ne\varnothing$. Put
$$
D=\mu\bar{E}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $\bar{E}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
2=D\cdot\Gamma=\Big(\mu\bar{E}+\Omega\Big)\cdot\Gamma=2\mu+\Omega\cdot\Gamma\geqslant 2\mu,%
$$
where $\Gamma$ is a general fiber of the conic bundle $\phi$. Hence
$\mu\leqslant 1$. Thus, the log pair
$$
\Big(\bar{E},\ \lambda\Omega\Big\vert_{\bar{E}}\Big)
$$
is not log canonical by Theorem~\ref{theorem:adjunction}, because
$\mathrm{LCS}(X,D)\cap\bar{E}\ne\varnothing$. But
$$
\Omega\Big\vert_{\bar{E}}\qlineq -\frac{1+\mu}{2}K_{\bar{E}},
$$
which is impossible by Lemma~\ref{lemma:elliptic-times-P1}.
\end{proof}

\begin{lemma}
\label{lemma:2-11} Suppose that $\gimel(X)=2.11$ and $X$ is
general. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $V$ be a cubic hypersurface in $\P^4$. Then there is a commutative diagram
$$
\xymatrix{
&&V\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&X\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{2}&}
$$
such that $\alpha$ contracts a surface $E\subset X$ to a line $L\subset V$, the
map $\psi$ is a linear projection from the line $L$, the morphism $\beta$ is a
conic bundle.

We assume that $X$ satisfy the following generality condition:
there is an isomorphism
$$
\mathcal{N}_{L\slash V}\cong\mathcal{O}_{L}\oplus\mathcal{O}_{L}.
$$


Let $H$ be a hyperplane section of $V$ such that $L\subset H$.~Then
$$
-K_{X}\sim 2\bar{H}+E,
$$
where $\bar{H}\subset X$ is the proper transform of the surface $H$. In
particular, $\lct(X)\le 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log pair $(X,\lambda D)$ is
not log-canonical for some positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(V)=1/2$ and $\alpha(D)\qlineq -K_V$. Note that
$E\cong\P^1\times\mathbb{P}^{1}$ by the generality condition.

Let $F\subset E$ be a fiber of the induced projection $E\to L$,
let $Z\subset E$ be a section of this projection such that $Z\cdot
Z=0$. Then $\alpha^{*}(H)\vert_{E}\sim F$ and $E\vert_{E}\sim -Z$,
because
$$
-2Z-2F\sim K_E\sim\Big(K_X+E\Big)\Big\vert_{E}\sim
2\Big(E-\alpha^{*}\big(H\big)\Big)\Big\vert_{E}\sim -2F+2E\Big\vert_{E}.%
$$

There is a non-negative rational number $\mu$ such that
$$
D=\mu E+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such that
$E\not\subset\mathrm{Supp}(\Omega)$. Then
$$
2=D\cdot\Gamma=\mu E\cdot\Gamma+\Omega\cdot\Gamma\geqslant\mu E\cdot\Gamma=2\mu,%
$$
where $\Gamma$ is a general fiber of the conic bundle $\beta$. Thus, we see
that $\mu\leqslant 1$.

The log pair $(E,\lambda\Omega\vert_E)$ is not log canonical by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{E}\qlineq \Big(-K_X-\mu
E\Big)\Big\vert_{E}\qlineq\big(1+\mu\big)Z+2F,
$$
which contradicts Lemma~\ref{lemma:elliptic-times-P1}, because $\mu\leqslant
1$ and $\lambda<1/2$.
\end{proof}

\begin{lemma}
\label{lemma:2-15} Suppose that $\gimel(X)=2.15$ and $X$ is
general. Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to\mathbb{P}^3$
that contracts a surface $E\subset X$ to a smooth curve
$C\subset\mathbb{P}^{3}$ that is complete intersection of an
irreducible quadric $Q\subset\mathbb{P}^{3}$ and a cubic
$F\subset\mathbb{P}^{3}$.

We assume that $X$ satisfies the following generality condition:
the quadric $Q$ is smooth.

Let $\bar{Q}$ be a proper transform of $Q$ on the threefold $X$.
Then there is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
\mathbb{P}^{3}&&V\ar@{-->}^{\gamma}[ll],}
$$
where $V$ is a cubic in $\mathbb{P}^{4}$ that has one ordinary
double point $P\in V$, the morphism~$\beta$~contracts the~surface
$\bar{Q}$ to the point $P$, and $\gamma$ is a linear projection
from the point $P$.

Let $E$ be the exceptional divisor of $\alpha$. Then
$$
-K_{X}\sim 2\bar{Q}+E
$$
and $\beta(E)\subset V$ is a surface that contains all lines that
pass through $P$. We see that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

It follows from Lemma~\ref{lemma:singular-cubic-threefold} that
either
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq\bar{Q},
$$
or the set $\mathrm{LCS}(X,\lambda D)$ contains a fiber of the
natural projection $E\to C$. In both cases
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)\cap\bar{Q}\ne\varnothing.
$$

We have $\bar{Q}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. Put
$$
D=\mu\bar{Q}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $\bar{Q}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\alpha\big(D\big)\qlineq \mu Q+\alpha\big(\Omega\big)\qlineq -K_{\mathbb{P}^{3}},%
$$
which gives $\mu\leqslant 2$. The log pair
$(\bar{Q},\lambda\Omega\vert_{\bar{Q}})$ is not log canonical by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{\bar{Q}}\qlineq -\frac{1+\mu}{2}K_{\bar{Q}},
$$
which implies that $\mu>1$ by Lemma~\ref{lemma:elliptic-times-P1}.

It follows from Remark~\ref{remark:convexity} that we may assume
that $E\not\subset\mathrm{Supp}(D)$. Then
$$
1=D\cdot F=\mu \bar{Q}\cdot F+\Omega\cdot F=\mu+ \Omega\cdot F\geqslant \mu,%
$$
where $F$ is a general fiber the natural projection $E\to C$. But
$\mu>1$.
\end{proof}


\begin{lemma}
\label{lemma:2-18} Suppose that  $\gimel(X)=2.18$. Then
$\lct(X)=1/2$.
\end{lemma}

\begin{proof}
There is a smooth divisor $B\subset\P^1\times\P^2$ of bidegree $(2, 2)$ such
that the diagram
$$
\xymatrix{
&& X\ar@{->}[d]_{\pi}\ar@{->}[lld]_{\phi_1}\ar@{->}[rrd]^{\phi_2} & \\
\P^1 &&\P^1\times\P^2\ar@{->}[ll]^{\pi_1}\ar@{->}[rr]_{\pi_2}&& \P^2}
$$
commutes, where $\pi$ is a double cover that is  branched in $B$, the morphisms
$\pi_1$ and $\pi_2$ are natural projections, the morphism $\phi_1$ is a quadric
fibration, and $\phi_2$ is a conic bundle.

Let $H_1$ be a general fiber of $\pi_1$, and let $H_2$ be a general surface in
$|\pi_2^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$. Then
$$
B\sim 2H_1+2H_2.
$$

Let $\bar{H}_1$ be a general fiber of $\phi_1$, and let $\bar{H}_2$ be a
general surface in $|\phi_2^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$. Then
$$
-K_X\sim \bar{H_1}+2\bar{H_2},
$$
which implies that $\lct(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log pair $(X,\lambda D)$ is
not log-canonical for some positive rational number $\lambda<1/2$.

Applying Lemma~\ref{lemma:Hwang} to the fibration $\phi_1$ we see that
$$
\varnothing\neq\LCS\Big(X,\ \lambda D\Big)\subseteq Q,
$$
where $Q$ is a singular fiber of $\phi_1$. Moreover, applying
Theorem~\ref{theorem:Hwang} to the fibration $\phi_2$, we see that
$$
\varnothing\neq\LCS\Big(X,\ \lambda D\Big)\subseteq Q\cap R,
$$
where $R\subset X$ be an irreducible surface that is swept out  by singular
fibers of $\phi_2$. In particular, the set $\mathbb{LCS}(X, \lambda
D)$ contains no surfaces.

Suppose that $\LCS(X, \lambda D)$ is zero-dimensional. Then
$$
\LCS\left(X,\ \lambda
D+\frac{1}{2}\Big(\bar{H}_1+2\bar{H}_2\Big)\right)=\LCS(X,\ \lambda D)\cup
\bar{H}_2%
$$
which is impossible by Theorem~\ref{theorem:connectedness}.

We see that the set $\mathbb{LCS}(X, \lambda D)$ contains a curve
$\Gamma\subset Q\cap R$. Put
$$
D=\mu Q+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor such that
$Q\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\Big(Q,\ \lambda\Omega\Big\vert_Q\Big)
$$
is not log canonical along $\Gamma$ by Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{Q}\qlineq \Big(-K_X-\mu Q\Big)\Big\vert_{Q}\qlineq -K_{Q},
$$
which implies that $\Gamma$ is a ruling of the cone $Q\subset\mathbb{P}^{3}$ by
Lemma~\ref{lemma:quadric-cone-far-from-vertex}. Then
$\phi_2(\Gamma)\subset\mathbb{P}^{2}$ is a line, and
$$
\phi_2\big(\Gamma\big)\subseteq\phi_2\big(R\big).
$$
But $\phi_2(R)\subset\P^2$ is a curve of degree $4$. Thus, we see that
$$
\phi_2\big(R\big)=\phi_2\big(\Gamma\big)\cup Z,
$$
where $Z\subset\mathbb{P}^{2}$ is a reduced cubic curve. Then $\phi_{2}$
induces a double cover of
$$
\phi_2\big(\Gamma\big)\setminus\Big(\phi_2\big(\Gamma\big)\cap Z\Big)
$$
that must be unramified (see \cite{Za77}). But the quartic curve
$\phi_2(R)$ has at most ordinary double points (see \cite{Za77},
\cite{Sa80}).~Then
$$
\big|\phi_2\big(\Gamma\big)\cap Z|=3,
$$
which is impossible, because $\phi_2(\Gamma)\cong\mathbb{P}^{1}$.
\end{proof}

\begin{lemma}
\label{lemma:2-19} Suppose that $\gimel(X)=2.19$ and $X$ is
general. Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
It follows from \cite[Proposition~3.4.1]{IsPr99} that there is a
commutative diagram
$$
\xymatrix{
&&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&V\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{3}&}
$$
where $V$ is a complete intersection of two quadric fourfolds in
$\mathbb{P}^{5}$, the morphism $\alpha$ is a blow~up~of a line $L\subset V$,
the morphism $\beta$ is a blow up of a smooth curve $C\subset\mathbb{P}^{3}$ of
degree $5$ and genus~$2$, and the map $\psi$ is a linear projection from the
line $L$.

Let $E$ and $R$ be the exceptional divisors of $\alpha$ and
$\beta$, respectively. Then
\begin{itemize}
\item the surface $\beta(E)\subset\mathbb{P}^{3}$ is an irreducible quadric,%
\item the surface $\alpha(R)\subset V$ is swept out by lines that intersect the line $L$.%
\end{itemize}

We assume that $X$ satisfies the following generality condition: the surface
$\beta(E)$ is smooth.

Let $H$ be any hyperplane section of $V\subset\mathbb{P}^{5}$ such
that $L\subset H$. Then
$$
2\bar{H}+E\sim R+2E\sim -K_{X},
$$
where $\bar{H}$ is a proper transform of $H$ on the threefold $X$.
We see that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log pair $(X,\lambda D)$ is
not log-canonical for some positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E\cong\P^1\times\mathbb{P}^{1},%
$$
because $\alpha(D)\qlineq -K_V$ and $\mathrm{lct}(V)=1/2$ by
Theorem~\ref{theorem:del-Pezzo}.

Let $F$ be a fiber of the projection $E\to L$, and let $Z$ be a
section of this projection such that $Z\cdot Z=0$. Then
$\alpha^{*}(H)\vert_{E}\sim F$ and $E\vert_{E}\sim -Z$, because
$$
-2Z-2F\sim K_E\sim\Big(K_X+E\Big)\Big\vert_{E}\sim
2\Big(E-\alpha^{*}\big(H\big)\Big)\Big\vert_{E}\sim 2E\Big\vert_{E}-2F.%
$$

By Remark~\ref{remark:convexity}, we may assume that either
$E\not\subset\mathrm{Supp}(D)$, or $R\not\subset\mathrm{Supp}(D)$,
because the log-pair
$$
\Big(X, \lambda\big(R+2E\big)\Big)
$$
is log canonical and $-K_{X}\sim R+2E$. Put
$$
D=\mu E+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $E\not\subset\mathrm{Supp}(\Omega)$.

Suppose that $\mu\leqslant 1$. Then $(X, E+\lambda\Omega)$ is not
log canonical, which implies that
$$
\Big(E,\ \lambda\Omega\Big\vert_E\Big)
$$
is also not log canonical by Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{E}\qlineq \Big(-K_X-\mu
E\Big)\Big\vert_{E}\qlineq\big(1+\mu\big)Z+2F,
$$
which contradicts Lemma~\ref{lemma:elliptic-times-P1}, because $\mu\leqslant
1$ and $\lambda<1/2$.

Thus, we see that $\mu>1$. Then we may assume that
$R\not\subset\mathrm{Supp}(D)$.

Let $\Gamma$ be a general fiber of the projection $R\to C$. Then
$\Gamma\not\subset\mathrm{Supp}(D)$ and
$$
1=-K_{X}\cdot \Gamma=\mu E\cdot
\Gamma+\Omega\cdot\Gamma=\mu+\Omega\cdot\Gamma\geqslant\mu,
$$
which is a contradiction.
\end{proof}


\begin{lemma}
\label{lemma:2-23} Suppose that $\gimel(X)=2.23$ and $X$ is
general. Then $\lct(X)=1/3$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to Q$ such that
$Q\subset\mathbb{P}^{4}$ is a smooth quadric~threefold, and $\alpha$ contracts
a surface $E\subset X$ to a smooth curve $C\subset Q$ that is a complete
intersection~of a hyperplane section $H\subset Q$ and a divisor $F\in
|\O_Q(2)|$.

We assume that $X$ satisfies the following generality condition:
the quadric $H$ is smooth.

Let $\bar{H}$ be a proper transform of $H$ on the threefold $X$.
Then there is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
Q&&V\ar@{-->}^{\gamma}[ll]}
$$
where $V$ is a complete intersection of two quadrics in $\P^{5}$ such that $V$
has one ordinary double point $P\in V$, the morphism~$\beta$ contracts
$\bar{H}$ to the point $P$, and $\gamma$ is a projection from~$P$.

Let $E$ be the exceptional divisor of $\alpha$. Then
$$
-K_{X}\sim 3\bar{H}+2E
$$
and $\beta(E)\subset V$ is a surface that contains all lines that
pass through $P$. Then $\lct(X)\le 1/3$.

We suppose that $\lct(X)<1/3$. Then there exists an effective $\Q$-divisor
$D\qlineq -K_{X}$ such that the log pair $(X,\lambda D)$ is not log-canonical
for some positive rational number $\lambda<1/3$.

It follows from Remark~\ref{remark:singular-V4} that either
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq\bar{H},
$$
or the set $\mathrm{LCS}(X,\lambda D)$ contains a fiber of the
natural projection $E\to C$. In both
cases
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)\cap\bar{H}\ne\varnothing.
$$

We have $\bar{H}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. There is a
non-negative rational number $\mu$ such that
$$
D=\mu\bar{H}+\Omega,
$$
where $\Omega$ is an effective $\Q$-divisor on $X$ such that
$\bar{H}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\alpha\big(D\big)\qlineq \mu H+\alpha\big(\Omega\big)\qlineq -K_{Q},%
$$
which gives $\mu\leqslant 3$. The log pair
$(\bar{H},\lambda\Omega\vert_{\bar{H}})$ is not log canonical by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{\bar{H}}\qlineq -\frac{1+\mu}{2}K_{\bar{H}},
$$
which implies that $\mu>1$ by Lemma~\ref{lemma:elliptic-times-P1}.

It follows from Remark~\ref{remark:convexity} that we may assume that
$E\not\subset\mathrm{Supp}(D)$, because the log pair
$$
\Big(X, \lambda\big(3\bar{H}+2E\big)\Big)
$$
is log-canonical. Let $F$ be a general fiber the natural projection $E\to C$.
Then
$$
1=D\cdot F=\mu \bar{H}\cdot F+\Omega\cdot F=\mu+ \Omega\cdot F\geqslant \mu,%
$$
which is a contradiction, because $\mu>1$.
\end{proof}

\begin{lemma}
\label{lemma:2-24} Suppose that $\gimel(X)=2.24$ and  $X$ is general. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
The threefold $X$ is a divisor on $\P^2\times\P^2$ of bidegree $(1, 2)$. Let
$H_{i}$ be a surface in
$$
\Big|\pi_{i}^*\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|,
$$
where $\pi_i\colon X\to\mathbb{P}^{2}$ is a projection of $X$ onto the $i$-th
factor of $\P^2\times\P^2$, and $i\in\{1, 2\}$. Then
$$
-K_X\sim 2H_1+H_2,
$$
which implies that $\lct(X)\leqslant 1/2$. Note that $\pi_1$ is a conic bundle,
and $\pi_2$ is a $\mathbb{P}^{1}$-bundle.

Let $\Delta\subset\P^2$ be the degeneration curve of the conic bundle
$\pi_{1}$. Then $\Delta$ is a cubic curve.

We suppose that $X$ satisfies the following generality condition: the curve
$\Delta$ is irreducible.

Assume that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log pair $(X,\lambda D)$ is
not log canonical for some positive rational number $\lambda<1/2$.

Suppose that the set $\mathbb{LCS}(X,\lambda D)$ contains a surface $S\subset
X$. Then
$$
D=\mu S+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor such that
$$
S\not\subset\mathrm{Supp}\big(\Omega\big),
$$
and $\mu>1/\lambda$.
Let $F_i$ be a general fiber
of $\pi_i$, where $i\in\{1, 2\}$. Then
$$
2=D\cdot F_{i}=\mu S\cdot F_{i}+\Omega\cdot F_{i}\geqslant\mu S\cdot F_{i},
$$
but either $S\cdot F_{1}\geqslant 1$ or $S\cdot F_{2}\geqslant 1$. Thus, we see
that $\mu\leqslant 2$, which is a contradiction.

By Theorem~\ref{theorem:Hwang} and Theorem~\ref{theorem:connectedness}, there
is a fiber $\Gamma_{2}$ of the $\mathbb{P}^{1}$-bundle $\pi_{2}$ such that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)=\Gamma_{2},
$$
because the set $\mathbb{LCS}(X,\lambda D)$ contains no surfaces.

Applying Theorem~\ref{theorem:Hwang} to the conic bundle $\pi_1$, we see that
$$
\pi_1\big(\Gamma_2\big)\subset\Delta,
$$
which is impossible, because $\Delta\subset\P^2$ is an irreducible cubic curve
and $\pi_{1}(\Gamma_{2})\subset\P^2$ is a line.
\end{proof}

\begin{lemma}
\label{lemma:2-25} Suppose that $\gimel(X)=2.25$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Recall that $X$ is a blow up $\alpha\colon X\to\mathbb{P}^3$ along
a normal elliptic curve of degree~$4$.

Let $Q$ be a general quadric containing $C$, and let
$\bar{Q}\subset X$ be a proper transform of $Q$. Then
$$
-K_{X}\sim 2\bar{Q}+E,
$$
where $E$ is the exceptional divisor of $\alpha$. In particular,
$\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

Note that the linear system $|\bar{Q}|$ defines a quadric
fibration
$$
\phi\colon X\longrightarrow\mathbb{P}^1,
$$
such that every fiber of $\phi$ is irreducible. Then the log pair
$(X,\lambda D)$ is log-canonical along every nonsingular fiber
$\tilde{Q}$ of
the fibration $\phi$ by Theorem~\ref{theorem:Hwang} since
$\mathrm{lct}(\tilde{Q})=1/2$ (see
Example~\ref{example:del-Pezzos}).

The locus $\mathrm{LCS}(X, \lambda D)$ does not contain any fiber
of $\phi$, because
$$
\alpha(D)\qlineq 2Q,
$$
and every fiber of $\phi$ is irreducible. Therefore, we see that
$\mathrm{dim}(\mathrm{LCS}(X, \lambda D))\leqslant 1$.

Let $Z$ be an element in $\mathbb{LCS}(X, \lambda D)$. There is a
singular fiber $\bar{Q}_{1}$ of the fibration $\phi$ such that
$$
Z\subset \bar{Q}_{1}.
$$
Note that $\phi$ has $4$ singular fibers and each of them is a proper
transform of a quadric cone in $\mathbb{P}^3$ with vertex outside $C$.

Let $\bar{Q}_2$ be a singular fiber of $\phi$ such that
$\bar{Q}_{1}\ne\bar{Q}_{2}$, let $\bar{H}$ be a proper transform
of a general hyperplane in $\mathbb{P}^{3}$ that is tangent to the
cone $\alpha(\bar{Q}_2)\subset\mathbb{P}^{3}$ along one of its
rulings $L\subset\alpha(\bar{Q}_2)$, and let $\bar{R}$ be a proper
transform of a very general hyperplane. Put
$$
\Delta=\lambda D +\frac{1}{2}\Big(\big(1+\eps\big)\bar{Q}_2+\big(2-\eps\big)\bar{H}+3\eps R\Big)%
$$
for some positive rational number $\eps<1-2\lambda$. Then
$$
\Delta\qlineq
-\Big(\lambda+\frac{1}{2}\big(1+\eps\big)\Big)K_X\qlineq
 -\frac{1+\eps+2\lambda}{2}K_X,%
$$
which implies that $-(K_{X}+\Delta)$ is ample.

Let $\bar{L}$ be a proper transform on $X$ of the curve $L$. Then
$$
Z\cup\bar{L}\subset\mathrm{LCS}\big(X,\Delta\big)\subset\bar{Q}_{1}\cup\bar{Q}_{2},
$$
which is impossible by Theorem~\ref{theorem:connectedness},
because  $-(K_{X}+\Delta)$ is ample.
\end{proof}

\begin{lemma}
\label{lemma:2-26} Suppose that $\gimel(X)=2.26$ and $X$ is
general. Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Let $V$ be a smooth Fano threefold such that $-K_{V}\sim 2H$ and
$$
\mathrm{Pic}\big(V\big)=\mathbb{Z}\big[H\big],
$$
where $H$ is a Cartier divisor such that $H^{3}=5$. Then $|H|$
induces an embedding $X\subset\mathbb{P}^{6}$.

It follows from \cite[Proposition~3.4.1]{IsPr99} (see also
\cite{FurNak89}) that there is a line
$$
L\subset X\subset\mathbb{P}^{6}
$$
such that there is a commutative diagram
$$
\xymatrix{
&&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&&&\\%
&V\ar@{-->}[rr]_{\psi}&&Q&}
$$
where $Q$ is a smooth quadric hypersurface in $\mathbb{P}^{4}$,
the morphism $\alpha$ is a blow up of the line $L\subset V$, the
morphism $\beta$ is a blow up of a twisted cubic curve
$\mathbb{P}^{1}\cong C\subset Q$, and $\psi$ is a projection from
the~line~$L$.

Let $S$ be the exceptional divisor of the morphism $\beta$. Put
$\bar{S}=\alpha(S)$. Then $\bar{S}\sim H$, and $\bar{S}$ is
singular along the line $L$. Let $E$ be the exceptional divisor of
the morphism $\alpha$. Then
$$
\beta\big(E\big)\sim\mathcal{O}_{\mathbb{P}^{4}}\big(1\big)\Big\vert_{Q},
$$
which implies that $\beta(E)$ is an irreducible quadric surface.

We suppose that $X$ satisfies the following generality condition: the surface
$\beta(E)$ is smooth.

The equivalence $-K_{X}\sim 2S+3E$ holds. Moreover, the log pair
$$
\left(X,\ \frac{1}{3}\Big(2S+3E\Big)\right)
$$
is log canonical. Thus, we see that $\lct(X)\leqslant 1/3$, and
$E\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the log pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E,
$$
because $\alpha(D)\qlineq -K_V$ and $\mathrm{lct}(V)=1/2$ by
Theorem~\ref{theorem:del-Pezzo}.

Let $F$ be a fiber of the projection $E\to L$, and let $Z$ be a
section of this projection such that $Z\cdot Z=0$ on the surface
$E\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. Then
$\alpha^{*}(H)\vert_{E}\sim F$ and $E\vert_{E}\sim -Z$, because
$$
-2Z-2F\sim K_E\sim\Big(K_X+E\Big)\Big\vert_{E}\sim
2\Big(E-\alpha^{*}\big(H\big)\Big)\Big\vert_{E}\sim 2E\Big\vert_{E}-2F.%
$$

By Remark~\ref{remark:convexity}, we may assume that either
$E\not\subset\mathrm{Supp}(D)$, or $S\not\subset\mathrm{Supp}(D)$.
Put
$$
D=\mu E+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $E\not\subset\mathrm{Supp}(\Omega)$.

Suppose that $\mu\leqslant 2$. Then $(X, E+\lambda\Omega)$ is not
log canonical, which implies that
$$
\Big(E,\ \lambda\Omega\Big\vert_E\Big)
$$
is also not log canonical by Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{E}\qlineq \Big(-K_X-\mu E\Big)\Big\vert_{E}\qlineq\big(1+\mu\big)Z+2F,%
$$
which contradicts Lemma~\ref{lemma:elliptic-times-P1}, because $\mu\leqslant
2$ and $\lambda<1/3$.

Thus, we see that $\mu>2$. Then we may assume that
$S\not\subset\mathrm{Supp}(D)$.

Let $\Gamma$ be a general fiber of the projection $S\to C$. Then
$\Gamma\not\subset\mathrm{Supp}(D)$ and
$$
1=-K_{X}\cdot \Gamma=\mu E\cdot
\Gamma+\Omega\cdot\Gamma=\mu+\Omega\cdot\Gamma\geqslant\mu,
$$
which is a contradiction.
\end{proof}

\begin{lemma}
\label{lemma:2-27} Suppose that $\gimel(X)=2.27$. Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
There is a morphism $\alpha\colon X\to\mathbb{P}^3$ contracting a
surface $E$ to a smooth cubic curve~$C\subset \mathbb{P}^{3}$, and
$X\cong\mathbb{P}(\mathcal{E})$, where $\mathcal{E}$ is a stable
rank two vector bundle  on $\mathbb{P}^{2}$  with $c_{1}(\mathcal{E})=0$ and
$c_{1}(\mathcal{E})=2$ such that the sequence
$$
0\longrightarrow\mathcal{O}_{\mathbb{P}_{2}}\big(-1\big)\oplus\mathcal{O}_{\mathbb{P}_{2}}\big(-1\big)\longrightarrow\mathcal{O}_{\mathbb{P}_{2}}\oplus\mathcal{O}_{\mathbb{P}_{2}}\oplus \mathcal{O}_{\mathbb{P}_{2}}\oplus\mathcal{O}_{\mathbb{P}_{2}}\longrightarrow \mathcal{E}\otimes\mathcal{O}_{\mathbb{P}_{2}}\big(1\big)\longrightarrow 0%
$$
is exact (see \cite{Dem80} and \cite[Application~1]{SzWi90}).

Let $Q\subset\mathbb{P}^{3}$ be a general quadric containing $C$,
and let $\bar{Q}\subset X$ be a proper transform of~$Q$. Then
$$
-K_{X}\sim 2\bar{Q}+E,
$$
where $E$ is the exceptional divisor of $\alpha$. In particular,
we see that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\lct(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log pair
$(X,\lambda D)$ is not log-canonical for some positive rational
number $\lambda<1/2$.

Suppose that the set $\mathbb{LCS}(X,\lambda D)$ contains a
surface $S\subset X$. Then
$$
D=\mu F+\Omega,
$$
where $\mu\geqslant 1/\lambda$ and $\Omega$ is an
effective $\mathbb{Q}$-divisor such that $F\not\subset\mathrm{Supp}(\Omega)$.

Let $\phi\colon X\to \mathbb{P}^{2}$ be the
natural $\mathbb{P}^{1}$-bundle. Then
$$
2=D\cdot \Gamma=\mu F\cdot \Gamma+\Omega\cdot \Gamma=\mu F\cdot\Gamma+\Omega\cdot F\geqslant \mu F\cdot\Gamma,%
$$
where $\Gamma$ is a general fiber of $\phi$. Thus, we see that $F$ is swept
out by the fibers of $\phi$. Then
$$
\alpha\big(F\big)\sim\mathcal{O}_{\mathbb{P}^{3}}\big(d\big)
$$
and $d\geqslant 2$. But $\alpha(D)\qlineq
\mu\alpha(F)+\alpha(\Omega)\qlineq\mathcal{O}_{\mathbb{P}^{3}}(4)$, which is a
contradiction.

We wee that the locus $\mathrm{LCS}(X,\lambda D)$ contains no surfaces.
Applying Theorem~\ref{theorem:Hwang}
to $(X,\lambda D)$ and $\phi$, we see that
$$
L\subseteq\mathrm{LCS}\Big(X,\ \lambda D\Big),
$$
where $L$ is a fiber of $\phi$. Note that $\alpha(L)$ is a secant line of the
curve $C\subset\P^3$. One has
$$
\alpha\big(L\big)\subseteq\mathrm{LCS}\Big(\mathbb{P}^{3}, \lambda\alpha\big(D\big)\Big)\subseteq \alpha\Big(\mathrm{LCS}\big(X,\ \lambda D\big)\Big)\cup C,%
$$
which is impossible by Lemma~\ref{lemma:P3}.
\end{proof}



\begin{lemma}
\label{lemma:2-28} Suppose that $\gimel(X)=2.28$. Then
$\mathrm{lct}(X)=1/4$.
\end{lemma}
\begin{proof}
There is a blow up $\alpha\colon X\to \mathbb{P}^3$ along a plane cubic curve
$C\subset\mathbb{P}^{3}$. One has
$$
-K_{X}\sim 4G+3E,
$$
where $E$ is the exceptional divisor of $\alpha$ and $G$ is a
proper transform of the plane in $\mathbb{P}^{3}$ that contains
the curve $C$. In particular, we see that the inequality
$\mathrm{lct}(X)\leqslant 1/4$ holds.

We suppose that $\lct(X)<1/4$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_X$ such that the log pair $(X,
\lambda D)$ is not log-canonical for some rational number
$\lambda<1/4$. One has
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(\mathbb{P}^{4})=1/4$. But any curve
$\Gamma\in\mathbb{LCS}(X, \lambda D)$ must be a fiber of the
natural projection
$$
\psi\colon E\longrightarrow C
$$
by Lemma~\ref{lemma:rational-tree}. Therefore, we see that either
the locus $\mathrm{LCS}(X, \lambda D)$ consists of a single point,
or the locus $\mathrm{LCS}(X, \lambda D)$ consists of a single
fiber of the projection $\psi$ by
Theorem~\ref{theorem:connectedness}.

Let $R$ be a sufficiently  general cone in $\mathbb{P}^3$ over the
curve $C$, and let $H$ be a sufficiently general hyperplane in
$\mathbb{P}^{3}$ that passes through the point $\Sing(R)$.
Then
$$
\mathrm{LCS}\left(X,\ \lambda
D+\frac{3}{4}\Big(\bar{R}+\bar{H}\Big)\right)=\mathrm{LCS}\Big(X,\
 \lambda D\Big)\bigcup\Sing\big(\bar{R}\big),%
$$
where $\bar{R}$ and $\bar{H}$ are proper transforms of $R$ and $H$
on the threefold $X$. But the divisor
$$
-\left(K_{X}+\lambda
D+\frac{3}{4}\Big(\bar{R}+\bar{H}\Big)\right)\qlineq
 \big(\lambda-1/4\big)K_{X}%
$$
is ample, which contradicts Theorem~\ref{theorem:connectedness}.
\end{proof}


\begin{lemma}
\label{lemma:2-29} Suppose that $\gimel(X)=2.29$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to Q$ such that $Q$
is a smooth quadric hypersurface, and $\alpha$ is a blow up along
a smooth conic $C\subset Q$.

Let $H$ be a general hyperplane section of
$Q\subset\mathbb{P}^{4}$ that contains $C$, and let $\bar{H}$ be a
proper transform of the surface $H$ on the threefold $X$. Then
$$
-K_{X}\sim 3\bar{H}+2E,
$$
where $E$ is the exceptional divisor of $\alpha$. In particular,
the inequality $\mathrm{lct}(X)\leqslant 1/3$ holds.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the log pair $(X,\lambda D)$ is not log-canonical for some
rational $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(Q)=1/3$ (see
Example~\ref{example:hypersurface-index-big}) and $\alpha(D)\qlineq
-K_{Q}$.

The linear system $|\bar{H}|$ has no base points and defines a
morphism
$$
\beta\colon X\longrightarrow\mathbb{P}^{1},
$$
whose general fiber is a smooth quadric surface in
$\mathbb{P}^{3}$. Then the log pair $(X,\lambda D)$ is
log-canonical along the smooth fibers of $\beta$ by
Theorem~\ref{theorem:Hwang} (see
Example~\ref{example:del-Pezzos}).

It follows from Theorem~\ref{theorem:connectedness} that there is
a singular fiber $S\sim\bar{H}$ of the morphism $\beta$ such that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E\cap S,
$$
and $S\subset\mathbb{P}^3$ is a quadratic cone. Put $\Gamma=E\cap
S$. Then $\Gamma$ is an irreducible conic, the log-pair
$$
\left(X,\ S+\frac{2}{3}E\right)
$$
has log-canonical singularities, and $3S+2E\qlineq D$. Therefore,
it follows from Remark~\ref{remark:convexity} that to complete the
proof we may assume that either $S\not\subset\mathrm{Supp}(D)$ or
$E\not\subset\mathrm{Supp}(D)$.

Intersecting the divisor $D$ with a general ruling of the cone
$S\subset\mathbb{P}^{3}$ and with a general fiber of the
projection $E\to C$, we see that
$$
\Gamma\not\subseteq\mathrm{LCS}\Big(X,\lambda D\Big),
$$
which implies that $\mathrm{LCS}(X,\lambda D)$ consists of a
single point $O\in\Gamma$ by Theorem~\ref{theorem:connectedness}.

Let $R$ be a general (not passing through $O$) surface in
$|\alpha^{*}(H)|$. Then
$$
\mathrm{LCS}\left(X,\ \lambda D+\frac{1}{2}\Big(\bar{H}+2R\Big)\right)=R\cup O,%
$$
which is impossible by Theorem~\ref{theorem:connectedness}, since
$-K_{X}\sim\bar{H}+2R\qlineq D$ and $\lambda<1/3$.
\end{proof}



\begin{lemma}
\label{lemma:2-30} Suppose that $\gimel(X)=2.30$. Then
$\mathrm{lct}(X)=1/4$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
\mathbb{P}^{3}&&Q\ar@{-->}^{\gamma}[ll],}
$$
where $Q$ is a smooth quadric threefold in $\mathbb{P}^{4}$, the
morphism $\alpha$ is a blow up of a smooth irreducible conic
$C\subset\mathbb{P}^{3}$, the morphism $\beta$ is a blow up of a
point, and $\gamma$ is a projection from a point.

Let $G$ be a proper transform on the variety $X$ of the unique
hyperplane in $\mathbb{P}^{3}$ that contains the conic $C$. Then
the surface $G$ is contracted by the morphism $\beta$, and
$$
-K_{X}\sim 4G+3E,
$$
where $E$ is the exceptional divisor of the blow up $\alpha$.
Thus, we see that $\mathrm{lct}(X)\leqslant 1/4$.

We suppose that $\mathrm{lct}(X)<1/4$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
rational $\lambda<1/4$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E\cap G,
$$
because $\mathrm{lct}(\mathbb{P}^{4})=1/4$ and
$\mathrm{lct}(Q)=1/3$.

We may assume that either $G\not\subset\mathrm{Supp}(X)$ or
$E\not\subset\mathrm{Supp}(X)$ by Remark~\ref{remark:convexity}.

Intersecting $D$ with lines in $G\cong\mathbb{P}^{2}$ and with
fibers of the projection $E\to C$, we see that
$$
\mathrm{LCS}\Big(X,\lambda D\Big)\subsetneq E\cap G
$$
which implies that there is a point $O\in E\cap G$ such that
$\mathrm{LCS}(X,\lambda D)=O$ by
Theorem~\ref{theorem:connectedness}.

Let $R$ be a general surface in $|\alpha^{*}(H)|$ and $F$ a
general surface in $|\alpha^{*}(2H)-E|$. Then
$$
\mathrm{LCS}\left(X,\ \lambda D+\frac{1}{2}\Big(F+2R\Big)\right)=R\cup O,%
$$
which is impossible by Theorem~\ref{theorem:connectedness} since
$-K_{X}\sim F+2R\qlineq D$ and $\lambda<1/4$.
\end{proof}

\begin{lemma}
\label{lemma:2-31} Suppose that $\gimel(X)=2.31$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to Q$ such that $Q$
is a smooth quadric hypersurface, and $\alpha$ is a blow up of the
quadric $Q$ along a line $L\subset Q$.

Let $H$ be a sufficiently general hyperplane section of the
quadric $Q\subset\mathbb{P}^{4}$ that passes through the line $L$,
and let $\bar{H}$ be a proper transform of the surface $H$ on the
threefold $X$. Then
$$
-K_{X}\sim 3\bar{H}+2E,
$$
where $E$ is the exceptional divisor of $\alpha$. In particular,
$\mathrm{lct}(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some rational
$\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(Q)=1/3$ and $\alpha(D)\qlineq
-K_{Q}$.

The linear system $|\bar{H}|$ defines a $\mathbb{P}^1$-bundle
$$
\phi\colon X\longrightarrow\mathbb{P}^2
$$
such that the induced morphism
$E\cong\mathbb{F}_{1}\to\mathbb{P}^{2}$ contracts an irreducible
curve $Z\subset E$. Then
$$
\mathrm{LCS}\Big(X,\lambda D\Big)=Z\subset E
$$
by Theorem~\ref{theorem:Hwang}. There is a nonnegative rational
number $\mu$ such that
$$
D=\mu E+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $E\not\subset\mathrm{Supp}(\Omega)$. Then
$$
2=D\cdot F=\mu E\cdot F+\Omega\cdot F=\mu+ \Omega\cdot F\geqslant \mu,%
$$
where $F$ is a general fiber of $\phi$. Then the log pair
$$
\Big(X,\ E+\lambda\Omega\Big)
$$
is not log canonical, because $\lambda<1/3$. Then
$(E,\lambda\Omega\vert_{E})$ is not log canonical by
Theorem~\ref{theorem:adjunction}.

Let $C$ be a fiber of the natural projection $E\to L$. Then
$$
\Omega\Big\vert_{E}\qlineq 3C+\big(1+\mu\big)Z,
$$
which implies that $(E, \lambda\Omega\vert_{E})$ is log canonical
by Lemma~\ref{lemma:F1}, which is a contradiction.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Fano threefolds with $\rho=3$}
\label{section:rho-3}

We use the assumptions and notation introduced in
section~\ref{section:intro}.

\begin{lemma}
\label{lemma:3-1} Suppose that $\gimel(X)=3.1$ and $X$ is general.
Then $\lct(X)=3/4$.
\end{lemma}

\begin{proof}
There is a double cover
$$
\omega\colon X\longrightarrow \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1%
$$
branched over a divisor of tridegree $(2, 2, 2)$. The
projection
$$
\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\longrightarrow\mathbb{P}^1
$$
onto the $i$-th factor induces a morphism $\pi_{i}\colon
X\to\mathbb{P}^1$, whose  fibers are del Pezzo surfaces of
degree~$4$.

Let $R_{1}$ be a fiber of the fibration $\pi_{1}$ such that the
surface $R_{1}$ is singular, let $Q$ be a singular point of the
surface $R_{1}$, and let $R_{2}$ and $R_{3}$ be fibers of
$\pi_{2}$ and $\pi_{3}$ such that
$$
R_{2}\ni Q\in R_{3},
$$
respectively. Then $\mathrm{mult}_{Q}(R_{1}+R_{2}+R_{3})=4$, which
implies that the log pair
$$
\left(X,\ \frac{3}{4}\Big(R_{1}+R_{2}+R_{3}\Big)\right)
$$
is not log terminal at $Q$. But $-K_{X}\sim R_{1}+R_{2}+R_{3}$.
Thus, we see that $\mathrm{lct}(X)\leqslant 3/4$.

We suppose that the threefold $X$ satisfies the~following
generality condition: for an arbitrary point $O\in X$, there is
$k\in\{1,2,3\}$~such~that
\begin{itemize}
\item the fiber $F_{k}$ of the fibration $\pi_{k}$ that contains $O$ is smooth at the point $O$,%
\item the singularities of the surface $F_{k}$ consists of at most one ordinary double point,%
\item for every smooth curve $\Gamma\subset F_{k}$ such that $-K_{F_{k}}\cdot\Gamma=1$, we have $O\not\in\Gamma$,%
\item for every smooth curves $\Delta_{1}\subset S_{k}\supset
\Delta_{2}$ such that
$$
-K_{F_{k}}\cdot \Delta_{1}=-K_{F_{k}}\cdot \Delta_{2}=2
$$
and $\Delta_{1}+\Delta_{2}\sim -K_{F_{k}}$, we have $O\ne \Delta_{1}\cap \Delta_{2}$.%
\end{itemize}

We suppose that $\mathrm{lct}(X)<3/4$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical at some point
$P\in X$ for some rational number~$\lambda<3/4$.

Let $S_{i}$ be a fiber of $\pi_{i}$ such that $P\in S_{i}$.
Without loss of generality, we may assume that
\begin{itemize}
\item the surface $S_{1}$ is smooth at the point $P$,%
\item the singularities of the surface $S_{1}$ consists of at most one ordinary double point,%
\item for every smooth curve $L\subset S_{1}$ such that $-K_{S_{1}}\cdot L=1$, we have $P\not\in L$,%
\item for every smooth curves $C_{1}\subset S_{1}\supset C_{2}$
such that
$$
-K_{S_{1}}\cdot C_{1}=-K_{S_{1}}\cdot C_{2}=2
$$
and $C_{1}+C_{2}\sim -K_{S_{1}}$, we have $P\ne C_{1}\cap C_{2}$.%
\end{itemize}

The surface $S_{1}$ is a del Pezzo surface of degree $4$.  One has
$$
B_{X}=\mu S_{1}+\Omega,
$$
where $\Omega$
is an effective $\mathbb{Q}$-divisor on $X$ such that
$S_{1}\not\subset\mathrm{Supp}(\Omega)$.

Let $\phi\colon X\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a
natural conic bundle induced by the linear system
$$
\Big|S_{2}+S_{3}\Big|,
$$
and let $\Gamma$ be a general fiber of the conic bundle $\phi$.
Then
$$
2=B_{X}\cdot\Gamma=\mu S_{1}\cdot\Gamma+\Omega\cdot\Gamma=2\mu+\Omega\cdot \Gamma\geqslant 2\mu,%
$$
which implies that $\mu\leqslant 1$. Then $(X,
S_{1}+\lambda\Omega)$ is not canonical at the point $P$. Then
$$
\Big(S_{1},\ \lambda\Omega\Big\vert_{S_{1}}\Big)
$$
is not log canonical at the point $P$ by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{S_{1}}\qlineq B_{X}\Big\vert_{S_{1}}\qlineq
-K_{S_{k}},
$$
which is impossible (see Example~\ref{example:del-Pezzo-quartic} and mind
the generality assumption).
\end{proof}


\begin{lemma}
\label{lemma:3-2} Suppose that $\gimel(X)=3.2$ and $X$ is general.
Then $\mathrm{lct}(X)=1/2$.
\end{lemma}
\begin{proof}
The threefold $X$ is a primitive Fano threefold (see
\cite[Definition~1.3]{MoMu83}). Put
$$
U=\mathbb{P}\Big(\mathcal{O}_{\P^1\times\P^1}\oplus\mathcal{O}_{\P^1\times\P^1}\big(-1,-1\big)\oplus\mathcal{O}_{\P^1\times\P^1}\big(-1,-1\big)\Big),%
$$
let $\pi\colon U\to\P^1\times\P^1$ be a natural projection, and let $L$ be a
tautological line bundle on $U$. Then
$$
X\in\Big|2L+\pi^*\Big(\mathcal{O}_{\P^1\times\P^1}\big(2, 3\big)\Big)\Big|
$$

Let us show that $\lct(X)\le 1/2$. Let $E_1$ and $E_2$ be surfaces in $X$ such
that
$$
\pi\big(E_{1}\big)\subset\P^1\times\P^1\supset\pi\big(E_{2}\big)
$$
are divisors on $\P^1\times\P^1$ of bi-degree $(1,0)$ and $(0,1)$,
respectively. Then
$$
-K_X\sim L\Big\vert_{X}+2E_1+E_2,
$$
which implies that $\lct(X)\le 1/2$.

%$$K_{\P}\qlineq -3L-4\pi^*e_1-4\pi^*e_2,$$

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the~log-pair $(X,\lambda D)$ is
not log-canonical at some point $P\in X$ for some rational
number~$\lambda<1/2$.

It follows from \cite[Proposition~3.8]{JaPeRa07} that there is a
commutative diagram
$$
\xymatrix{
U_{1}\ar@{->}[dd]_{\psi_{1}}\ar@{->}[rr]^{\gamma_{1}}&&V&&U_{2}\ar@{->}[dd]^{\psi_{2}}\ar@{->}[ll]_{\gamma_{2}} \\
&&X\ar@{->}[d]_{\omega}\ar@{->}[lld]_{\phi_1}\ar@{->}[rrd]^{\phi_2}\ar@{->}[u]_{\alpha}\ar@{->}[ull]^{\beta_{1}}\ar@{->}[urr]_{\beta_{2}}&& \\
\P^1 &&\P^1\times\P^1\ar@{->}[ll]^{\pi_1}\ar@{->}[rr]_{\pi_2}&& \P^1,}%
$$
where $V$ is a Fano threefold that has one ordinary double point $O\in V$ such
that
$$
\mathrm{Pic}\big(V\big)=\mathbb{Z}\big[-K_{V}\big]
$$
and $-K_{V}^{3}=16$, the morphism $\alpha$ contracts a unique surface
$$
\mathbb{P}^{1}\times \mathbb{P}^{1}\cong S\subset X
$$
such that $S\sim L\vert_{X}$ to the point $O\in V$, the morphism $\beta_{i}$
contracts $S$ to a smooth rational curve, the~morphism $\gamma_{i}$ contracts
the curve $\beta_{i}(S)$ to the point $O\in V$ such that the rational~map
$$
\gamma_{2}\circ\gamma_{1}^{-1}\colon U_{1}\dasharrow U_{2}
$$
is a flop in $\beta_{1}(S)\cong\mathbb{P}^{1}$, the morphisms
$\psi_{1}$, $\psi_{2}$, $\phi_{1}$, $\phi_{2}$ are del Pezzo
fibrations such that
\begin{itemize}
\item a general fiber of $\psi_{1}$ is an intersection of two quadrics in $\mathbb{P}^{4}$,%
\item a general fiber of $\psi_{2}$ is a smooth quadric surface in $\mathbb{P}^{3}$,%
\item a general fiber of $\phi_{1}$ is a cubic surface in $\mathbb{P}^{3}$,%
\item a general fiber of $\phi_{2}$ is a del Pezzo surface of degree $6$,%
\end{itemize}
the morphisms $\pi_1$ and $\pi_2$ are natural
projections, and $\omega=\pi\vert_{X}$. Then
$$
\mathrm{Cl}\big(V\big)=\mathbb{Z}\big[\alpha\big(E_{1}\big)\big]\oplus\mathbb{Z}\big[\alpha\big(E_{2}\big)\big],
$$
and $\omega$ is a conic bundle. The curve $\beta_{1}(S)$ is a
section of $\psi_{1}$, and $\beta_{2}(S)$ is a $2$-section of
$\psi_{2}$.

We assume that the threefold $X$ satisfies the following mild
generality condition: every singular fiber of the del Pezzo
fibration $\phi_2$ has at most $\mathbb{A}_1$ singularities.

Applying Lemma~\ref{lemma:Hwang} to the fibration $\phi_1$, we see
that
$$
\varnothing\neq\LCS\Big(X,\ \lambda D\Big)\subseteq S_{1}
$$
where $S_{1}$ is a singular fiber of the del Pezzo fibration
$\phi_1$, because the global log canonical threshold of a smooth Del
Pezzo surface of degree $6$ is equal to $1/2$ by
Example~\ref{example:del-Pezzos}.

Applying Lemma~\ref{lemma:Hwang} to $\phi_2$, we obtain a
contradiction by Example~\ref{example:singular-cubic}.
\end{proof}

\begin{lemma}
\label{lemma:3-3} Suppose that $\gimel(X)=3.3$ and $X$ is general.
Then $\lct(X)=2/3$.
\end{lemma}

\begin{proof}
The threefold $X$ is a divisor on
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of tridegree
$(1, 1, 2)$. Then
$$
-K_{X}\sim
\pi_{1}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)+\pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)+\phi^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big),
$$
where $\pi_{1}\colon X\to\mathbb{P}^{1}$ and $\pi_{1}\colon
X\to\mathbb{P}^{1}$ are  fibrations into del Pezzo surfaces of
degree $4$ that are induced by the projections of the variety
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ onto its first
and second factor, respectively, and $\phi\colon X\to\mathbb{P}^2$
is conic bundle that is induced by the natural projection
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2\to
\mathbb{P}^2$.


Let $\alpha_{2}\colon X\to\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a
birational morphism induced by the linear system
$$
\Big|\pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)+\phi^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|,
$$
let $H_{i}\in|\pi_{i}^{*}(\mathcal{O}_{\mathbb{P}^{1}}(1))|$ and
$R\in|\phi^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$ be general
surfaces. Then
$$
H_{1}\sim H_{2}+2R-E_{2},
$$
where $E_{2}$ is the exceptional divisor of the birational
morphism $\alpha_{2}$. Hence
$$
-K_{X}\qlineq\sim
H_1+H_2+R\qlineq\frac{3}{2}H_{1}+\frac{1}{2}H_{2}+\frac{1}{2}E_{2},
$$
which implies that $\mathrm{lct}(X)\leqslant 2/3$.

We suppose that the threefold $X$ satisfies the~following
generality conditions: for an arbitrary point $O\in X$, there is
$k\in\{1,2\}$ such that
\begin{itemize}
\item the fiber $F_{k}$ of the fibration $\pi_{k}$ that contains
the point $O$ is smooth at the point $O$,%
\item the singularities of the surface $F_{k}$ consist of at most one ordinary double point,%
\item for every smooth curve $\Gamma\subset F_{k}$ such that $-K_{F_{k}}\cdot \Gamma=1$, we have $O\not\in \Gamma$ if $\mathrm{Sing}(F_{k})\ne\varnothing$.%
\end{itemize}


We suppose that $\mathrm{lct}(X)<2/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical at some point
$P\in X$ for some rational number~$\lambda<2/3$.

Let $S_{i}$ be a fiber of $\pi_{i}$ such that $P\in S_{i}$. Then
we may assume that
\begin{itemize}
\item the surface $S_{1}$ is smooth at the point $P$,%
\item the singularities of the surface $S_{1}$ consist of at most one ordinary double point,%
\item for every smooth curve $L\subset S_{1}$ such that $-K_{S_{1}}\cdot L=1$, we have $P\not\in L$ if $\mathrm{Sing}(S_{1})\ne\varnothing$.%
\end{itemize}

There is a non-negative rational number $\mu$ such that
$$
B_{X}=\mu S_{1}+\Omega,
$$
where $\Omega$
is an effective $\mathbb{Q}$-divisor such that
$S_{1}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\Big(H_{2},\ \lambda\mu
S_{1}\Big\vert_{H_{2}}+\lambda\Omega\Big\vert_{H_{2}}\Big)
$$
is log canonical because $\mathrm{lct}(H_{2})=2/3$. Thus, we see
that $\mu\leqslant 1/\lambda$. Hence
$$
\Big(S_{1},\ \lambda\Omega\Big\vert_{S_{1}}\Big)
$$
is not log canonical at the point $P$ by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{S_{1}}\qlineq-K_{S_{1}},
$$
which is impossible (see
Example~\ref{example:del-Pezzo-quartic}).
\end{proof}

\begin{lemma}
\label{lemma:3-4} Suppose that $\gimel(X)=3.4$. Then
$\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $O$ be a point in $\mathbb{P}^{2}$. Then there is a
commutative diagram
$$
\xymatrix{
&&X\ar@{->}[d]_{\alpha}\ar@/_1pc/@{->}[lldd]_{\eta_{1}}\ar@/_1pc/@{->}[rrd]^{\eta_{2}}\ar@/^1pc/@{->}[rrrrd]^{\phi}&&\\
&&V\ar@{->}[d]_{\omega}\ar@{->}[lld]_{\gamma_{1}}\ar@{->}[rrd]^{\gamma_{2}}&&\mathbb{F}_{1}\ar@{->}[d]^{\beta}\ar@{->}[rr]_{\upsilon}&&\mathbb{P}^1\\
\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}[ll]^{\pi_{1}}\ar@{->}[rr]_{\pi_{2}}&&\mathbb{P}^{2}&&}
$$
such that $\pi_{i}$ and $\upsilon$ are natural projections,
$\omega$ is a double cover branched over a divisor
$$
B\subset\P^1\times\P^2
$$
of bi-degree $(2, 2)$, the morphism $\gamma_{1}$ is a fibration
into quadrics, $\gamma_{2}$ and $\eta_{2}$ are conic~bundles,
the~morphism $\beta$ is a blow up of the point $O$, the morphism
$\alpha$ is a blow up of a smooth curve that is a fiber of
$\gamma_{2}$ over the point $O$, the morphism $\eta_{1}$ is a
fibration into del Pezzo surfaces~of~degree~$6$, and $\phi$ is a
fibration into del Pezzo surfaces of degree $4$.

Let $H$ be a general fiber of $\eta_{1}$, and let $S$ be a general
fiber of $\phi$. Then
$$
-K_{X}\sim H+2S+E,
$$
where $E$ is the exceptional divisor of $\alpha$. Thus, we see
that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E,
$$
because $\alpha(D)\qlineq -K_V$ and $\mathrm{lct}(V)=1/2$ by
Lemma~\ref{lemma:2-18}.

Let $\Gamma$ be a fiber of $\eta_{2}$ such that
$\Gamma\cap\mathrm{LCS}(X,\lambda D)\ne\varnothing$. Then
$$
\Gamma\subseteq\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E,
$$
by Theorem~\ref{theorem:Hwang}. Hence $(H,\lambda D\vert_{H})$ is
not log canonical at the points $H\cap\Gamma$. But
$$
D\Big\vert_{H}\qlineq -K_{X}\Big\vert_{H}\sim -K_{H}
$$
and $\mathrm{lct}(H)=1/2$, because $H$ is a del Pezzo surface of
degree $6$, which is a contradiction.
\end{proof}

\begin{lemma}
\label{lemma:3-5} Suppose that $\gimel(X)=3.5$ and $X$ is general.
Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to\P^1\times\P^2$
that contracts a  surface $E\subset X$~to~a~curve $C\subset
\P^1\times\P^2$ of bidegree $(5, 2)$. Let
$\pi_{1}\colon\P^1\times\P^2\to\P^1$ and
$\pi_{2}\colon\P^1\times\P^2\to\P^2$ be natural projections. There
is
$$
Q\in\Big|\pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(2\big)\Big)\Big|
$$
such that $C\subset Q$. Let $H_1$ be a general fiber of $\pi_1$,
let $H_2$ be a surface in
$|\pi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{1}}(1))|$. We have
$$
-K_X\sim 2\bar{H_1}+\bar{H_2}+\bar{Q},
$$
where $\bar{H_1}, \bar{H_2},\bar{Q}\subset X$ are proper
transforms of $H_1, H_2, Q$, respectively. Then $\lct(X)\le 1/2$.

We suppose that $X$ satisfies the following generality condition:
for every fiber $F$ of $\pi_1\circ\alpha$, the~surface $F$ has at
most one singular point that is an ordinary double point of $F$.

Assume that $\lct(X)<1/2$. Then there exists an effective
$\Q$-divisor $D\qlineq -K_{X}$ such that the log-pair $(X,\lambda
D)$ is not log-canonical for some positive rational number
$\lambda<1/2$.

Let $S$ be an irreducible surface on the threefold $X$. Put
$$
D=\mu S+\Omega,
$$
where $\Omega$
is an effective $\mathbb{Q}$-divisor  such that
$S\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\left(\bar{H}_{1},\ \frac{1}{2}\Big(\mu S+\Omega\Big)\Big\vert_{\bar{H}_{1}}\right)%
$$
is log canonical (see Example~\ref{example:del-Pezzos}). Thus, either
$\mu\leqslant 2$, or $S$ is a fiber of $\pi_{1}\circ\alpha$.

Let $\Gamma\cong\mathbb{P}^{1}$ be a general fiber of the conic
bundle $\pi_{2}\circ\alpha$. Then
$$
2=D\cdot\Gamma=\mu S\cdot\Gamma+\Omega\cdot\Gamma\geqslant\mu S\cdot\Gamma,%
$$
which implies that  $\mu\leqslant 2$ in the case when $S$ is a fiber of
$\pi_{1}\circ\alpha$.

We see that the set $\mathbb{LCS}(X, \lambda D)$ contains no
surfaces. Now, applying Lemma~\ref{lemma:Hwang} to
$\pi_{1}\circ\alpha$, we obtain a contradiction with
Example~\ref{example:del-Pezzo-quartic}.
\end{proof}

\begin{lemma}\label{lemma:3-6}
Suppose that $\gimel(X)=3.6$ and $X$ is general. Then
$\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $\eps\colon V\to\mathbb{P}^{3}$ be a blow up of a line
$L\subset\mathbb{P}^{3}$. Then
$$
V\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)
$$
and there is a natural $\mathbb{P}^{2}$-bundle $\eta\colon V\to\mathbb{P}^{1}$.
There is a smooth elliptic curve $C\subset\mathbb{P}^{3}$ of degree~$4$ such
that $L\cap C=\varnothing$ and there is a commutative diagram
$$
\xymatrix{
Y\ar@{->}[d]_{\eps}&&X\ar@{->}[ll]_{\gamma}\ar@{->}[d]^{\beta}\ar@{->}[rr]^{\phi}&&\P^1\\
\P^3&&V\ar@{->}[ll]^{\delta}\ar@{->}[rru]_{\eta}&&}
$$
where $\delta$ is a blow up of $C$,  the morphism $\beta$ is a
blow up of the proper transform of the line $L$, the~morphism
$\gamma$ is a blow up of the proper transform of the curve $C$,
and $\phi$ is a~del~Pezzo~fibration.

We suppose that $X$ satisfies the following generality condition:
for every fiber $F$ of $\phi$, the~surface $F$ has at most one
singular point that is an ordinary double point of the surface
$F$.

Let $E$ and $G$ be the exceptional surfaces of $\beta$ and
$\gamma$, respectively, let $H\subset\P^3$ be a general plane that
passes through $L$, and let $Q\subset\P^3$ a quadric surface that
passes through $C$. Then
$$
-K_{X}\sim 2\bar{H}+\bar{Q}+E,
$$
where $\bar{H}\subset X\supset\bar{Q}$ are proper transforms of
$H$ and $Q$, respectively. We have $\lct(X)\le 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

It follows from Lemma~\ref{lemma:2-25} that $\lct(V)=1/2$.
Therefore, we see that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq G,
$$
but every fiber of the fibration $\phi$ is a del Pezzo surface of
degree $5$ that has at most one ordinary double point. Thus,
applying Lemma~\ref{lemma:Hwang} to $\phi$, we obtain a
contradiction with Example~\ref{example:del-Pezzo-quintic}.
\end{proof}

\begin{lemma}
\label{lemma:3-7} Suppose that $\gimel(X)=3.7$ and $X$ is general.  Then
$\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $W$ be a divisor on $\P^2\times\P^2$ of bi-degree $(1, 1)$. Then
$$
-K_{W}\sim 2H,
$$
where $H$ is a Cartier divisor on $W$. There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[ld]_\beta\ar@{->}[dd]_\alpha\ar@{->}[rd]^\gamma\ar@{->}`[rr]`[ddd]^{\omega}[ddd]&&\\
\P^1\times\P^2\ar@{->}[d]_{\phi}&&\P^1\times\P^2\ar@{->}[d]^{\psi}&\\
\P^2&W\ar@{-->}[d]_{\rho}\ar@{->}[l]_{\xi}\ar@{->}[r]^{\zeta}&\P^2&\\
&\mathbb{P}^{1}&& }
$$
where $\phi$ and $\phi$ are natural projections, $\alpha$ is a blow up of a
smooth curve $C\subset W$ such that
$$
C=H_{1}\cap H_{2},
$$
where $H_{1}\ne H_{2}$ are surfaces in $|H|$, the map $\rho$ is
induced by the pencil generated by $H_{1}$~and~$H_{2}$, the
morphism $\omega$ is a del Pezzo fibration of degree $6$,  the
morphisms $\zeta$ and $\xi$ are $\P^1$-bundles, while~$\beta$~and
$\gamma$ contract surfaces $\bar{M}_1\subset X\supset\bar{M}_{2}$
such that
$\phi\circ\beta(\bar{M}_{1})=\xi(C)$~and~$\psi\circ\gamma(\bar{M}_{2})=
\zeta(C)$.

Note that $\lct(X)\le 1/2$, because
$$
-K_{X}\sim 2\bar{H}_{1}+E,
$$
where $\bar{H}_{1}\subset X$ is the proper transform of $H_{1}$, and $E$ is
the
exceptional surface of $\alpha$.



We suppose that $X$ satisfies the following generality condition: all singular
fibers of the~fibration $\omega$ satisfy the hypotheses of
Lemma~\ref{lemma:singular-del-Pezzo-sextic}.

Assume that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log-pair $(X,\lambda D)$ is
not log-canonical for some positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E,
$$
because $\lct(W)=1/2$ by Theorem~\ref{theorem:del-Pezzo}. Applying
Lemma~\ref{lemma:Hwang}, we see that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E\cap F,
$$
where $F$ is a singular fiber of $\omega$. Note that $F$ is a del Pezzo
surface of degree $6$. Put
$$
D=\mu F+\Omega,
$$
where $\Omega$ is an
effective $\mathbb{Q}$-divisor  such that $F\not\subset\mathrm{Supp}(\Omega)$.
Then
$$
\Omega\Big\vert_{F}\qlineq D\Big\vert_{F}\qlineq -K_{F},
$$
and the surface $F$ is smooth along the curve $E\cap F$. But the
log pair $(F,\lambda\Omega\vert_{F})$ is~not~log~canonical at some
point $P\in E\cap F$ by Theorem~\ref{theorem:adjunction}, which is
impossible by Lemma~\ref{lemma:singular-del-Pezzo-sextic}.
\end{proof}

\begin{remark}
\label{remark:3-7} Let us use the notation and assumptions of the
proof of Lemma~\ref{lemma:3-7}. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E\cap F,%
$$
where $F$ is a singular fiber of the fibration $\omega$. Applying
Theorem~\ref{theorem:Hwang} to $\phi$ and $\psi$, we see that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E\cap F\cap \bar{M_1}\cap\bar{M_2},%
$$
by Lemma~\ref{lemma:P1xP2}. Regardless to how singular $F$ is, if the
threefold $X$ is sufficiently general, then
$$
E\cap F\cap \bar{M_1}\cap\bar{M_2}=\varnothing,
$$
which implies that an alternative generality condition can be used
in Lemma~\ref{lemma:3-7}.
\end{remark}


\begin{lemma}
\label{lemma:3-8} Suppose that $\gimel(X)=3.8$ and $X$ is general.
Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $\pi_{1}\colon\F_1\times\P^2\to \F_1$ and
$\pi_{2}\colon\F_1\times\P^2\to\P^2$ be natural projections. Then
$$
X\in\Big|\big(\alpha\circ\pi_1\big)^*\Big(\O_{\P^2}\big(1\big)\Big)\otimes \pi_2^*\Big(\O_{\P^2}\big(2\big)\Big)\Big|,%
$$
where $\alpha\colon\F_1\to\P^2$ is a blow up of a point. Let $H$
be a surface in $|\pi_2^*(\O_{\P^2}(1))|$. Then
$$
-K_X\sim E+2L+H,
$$
where $E\subset X\supset L$ are irreducible surfaces such that
$\pi_{1}(E)\subset\mathbb{F}_{1}$ is the exceptional curve of
$\alpha$, and $\pi_{1}(L)\subset\mathbb{F}_{1}$ is a fiber of the
natural projection $\mathbb{F}_{1}\to\mathbb{P}^{2}$. We have
$\lct(X)\le 1/2$.

The projection $\pi_{1}$ induces a fibration $\phi\colon X\to\P^1$
into del Pezzo surfaces~of~degree~$5$.

We suppose that $X$ satisfies the following generality condition:
for every fiber $F$ of $\phi$, the~surface $F$ has at most one
singular point that is an ordinary double point of the surface
$F$.

Assume that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

Applying Lemma~\ref{lemma:Hwang} to the morphism $\phi$, we obtain
a contradiction with Example~\ref{example:del-Pezzo-quintic}.
\end{proof}




\begin{lemma}
\label{lemma:3-9} Suppose that $\gimel(X)=3.9$. Then $\lct(X)=1/3$.
\end{lemma}

\begin{proof}
Let $O_{1}\in V_{1}\cong V_{2}\ni O_{2}$ be singular points of $V_{1}\cong
V_{2}\cong\mathbb{P}(1,1,1,2)$, respectively, let
$$
O_{1}\not\in S_{1}\in\Big|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(2)\Big|
$$
be a smooth surface, and let $C_{1}\subset S_{1}\cong\mathbb{P}^{2}$ be a
smooth quartic curve. Then~the~diagram
$$
\xymatrix{
&&&&X\ar@{->}[dll]_{\beta_{1}}\ar@{->}[drr]^{\beta_{2}}&&&&\\
&&U_{1}\ar@{->}[d]^{\alpha_{1}}\ar@/^1pc/@{->}[ddrr]^{\gamma_{1}}&&&&U_{2}\ar@{->}[d]_{\alpha_{2}}\ar@/_1pc/@{->}[ddll]_{\gamma_{2}}&&\\
&&V_{1}\ar@{-->}[drr]_{\psi_{1}}&&&&V_{2}\ar@{-->}[dll]^{\psi_{2}}&&\\
&&&&\mathbb{P}^{2}&&&&}
$$
commutes, where $\psi_{i}$ is a natural projection, $\alpha_{i}$ is a blow up
of the point $O_{i}$ with weights $(1,1,1)$, the morphism
$\gamma_{i}$~is~a~$\mathbb{P}^{1}$-bundle, and $\beta_{i}$ is a birational
morphism that contracts a surface
$$
\mathbb{P}^{1}\times C_{1}\cong G_{i}\subset X
$$
to a smooth curve $C_{1}\cong C_{i}\subset U_{i}$.

Let $E_{i}\subset X$ be the proper transform of the exceptional divisor of
$\alpha_{i}$. Then
$$
S_{1}=\alpha_{1}\circ\beta_{1}\big(E_{2}\big)\subset V_{1}\cong\mathbb{P}\big(1,1,1,2\big)\cong V_{2}\supset\alpha_{2}\circ\beta_{2}\big(E_{1}\big)%
$$
are surfaces in $|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(2)|$ that contain the
curves $C_{1}$ and $C_{2}$, respectively. But
$$
\alpha_{1}\circ\beta_{1}\big(G_{2}\big)\subset V_{1}\cong\mathbb{P}\big(1,1,1,2\big)\cong V_{2}\supset\alpha_{2}\circ\beta_{2}\big(G_{1}\big)%
$$
are surfaces in $|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(4)|$ that contain
$O_{1}\cap C_{1}$ and $O_{2}\cap C_{2}$, respectively.

Let $\bar{H}\subset X$ be the proper transform of a general surface in
$|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(1)|$.~Then
$$
-K_{X}\sim 3\bar{H}+E_{2}+E_{1},
$$
which gives $\mathrm{lct}(X)\leqslant 1/3$.

Suppose that $\mathrm{lct}(X)<1/3$.
Then there is an effective $\mathbb{Q}$-divisor
$$
D\qlineq -K_{X}\qlineq \frac{5}{2}\Big(G_{1}+G_{2}\Big)-5\Big(E_{1}+E_{2}\Big)
$$
such that the log-pair $(X,\lambda D)$ is not log-canonical for some
$\lambda<1/3$. Put
$$
D=\mu_{1}E_{1}+\mu_{2}E_{2}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$
such that
$$
E_{1}\not\subseteq\mathrm{Supp}\big(\Omega\big)\not\supseteq E_{2}.
$$

Let $\Gamma$ be a general fiber of the conic bundle $\gamma_{1}\circ\beta_{1}$.
Then
$$
2=\Gamma\cdot D=\Gamma\cdot\Big(\mu_{1}E_{1}+\mu_{2}E_{2}+\Omega\Big)=\mu_{1}+\mu_{2}+\Gamma\cdot\Omega\geqslant\mu_{1}+\mu_{2},%
$$
and without loss of generality we may assume that $\mu_{1}\leqslant\mu_{2}$.
Then $\mu_{1}\leqslant 1$.

Suppose that there is a surface $S\in\mathbb{LCS}(X,\lambda D)$. Then $S\ne
E_{1}$ and
$$
S\ne G_{1},
$$
because
$\alpha_{2}\circ\beta_{2}(G_{1})\in|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(4)|$ and
$\alpha_{2}\circ\beta_{2}(D)\in|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(5)|$. Then
$S\cap E_{1}\ne\varnothing$. But
$$
-\frac{1}{3}K_{E_{1}}\qlineq D\Big\vert_{E_{1}}=-\frac{2\mu_{1}}{3}K_{E_{1}}+\Omega\Big\vert_{E_{1}}%
$$
and $E_{1}\cong\mathbb{P}^{2}$, which is impossible by
Theorem~\ref{theorem:adjunction}, because
$\lambda<1/3=\mathrm{lct}(\P^2)$.

We see that the set $\mathbb{LCS}(X,\lambda D)$ contains no surfaces. Let
$P\in\mathrm{LCS}(X,\lambda D)$ be a point.

Suppose that $P\not\in G_{1}$. Let $Z$ be a fiber of $\gamma_{1}$ such that
$\beta_{1}(P)\in Z$. Then
$$
Z\subseteq\mathrm{LCS}\Big(U_{1},\ \lambda\beta_{1}\big(D\big)\Big)
$$
by Theorem~\ref{theorem:Hwang}. Put $\bar{E}_{1}=\beta_{1}(E_{1})$. Then we
have
$$
Z\cap \bar{E}_{1}\in\mathrm{LCS}\Big(\bar{E}_{1},\ \lambda\Omega\Big\vert_{\bar{E}_{1}}\Big)%
$$
by Theorem~\ref{theorem:adjunction}, which is impossible by
Lemma~\ref{lemma:plane}, because $\mu_{1}\leqslant 1$. Then
$\mathrm{LCS}(X, \lambda D)\subsetneq G_{1}$.

Suppose that $\mathrm{LCS}(X, \lambda D)\subseteq G_{1}\cap G_{2}$. Then
$$
\Big|\mathrm{LCS}\Big(X,\ \lambda D\Big)\Big|=1
$$
by Lemma~\ref{lemma:rational-tree} and Theorems~\ref{theorem:connectedness}.
Then
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)\cup\bar{H}\subseteq \mathrm{LCS}\left(X,\ \lambda D+\frac{1}{3}\Big(E_{2}+E_{2}\Big)+\bar{H}\right)\subset\mathrm{LCS}\Big(X,\ \lambda D\Big)\cup\bar{H}\cup E_{1}\cup E_{1}, %
$$
which contradicts Theorems~\ref{theorem:connectedness}, because $\bar{H}$ is a
general surface in
$|(\beta_{1}\circ\gamma_{1})^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$ and
$$
\lambda D+\frac{1}{3}\Big(E_{2}+E_{2}\Big)+\bar{H}\qlineq \big(\lambda-1/3\big)K_{X}.%
$$

Thus, we see that $G_{1}\supsetneq\mathbb{LCS}(X,\lambda D)\not\subseteq
G_{1}\cap G_{2}$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(U_{2},\ \lambda\beta_{2}\big(D\big)\Big)\subsetneq\beta_{2}\big(G_{1}\big),%
$$
and it follows from Theorms~\ref{theorem:connectedness} and \ref{theorem:Hwang}
that there is a fibre $L$ of the fibration $\gamma_{2}$ such that
$$
\mathrm{LCS}\Big(U_{2},\ \lambda\beta_{2}\big(D\big)\Big)=L.
$$

Let $B$ be a general surface in
$|\alpha_{2}^{*}(\mathcal{O}_{\mathbb{P}(1,1,1,2)}(2))|$. Then
$\beta_{2}(D)\vert_{B}\qlineq\mathcal{O}_{\mathbb{P}^{2}}(5)$ and
$B\cong\mathbb{P}^{2}$.~But
$$
\mathrm{LCS}\Big(B,\ \lambda\beta_{2}\big(D\big)\Big\vert_{B}\Big)=L\cap B
$$
and $|L\cap B|=1$, which is impossible by Lemma~\ref{lemma:plane}, because
$\lambda<1/3$.
\end{proof}

\begin{lemma}
\label{lemma:3-10} Suppose that $\gimel(X)=3.10$. Then $\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Let $Q\subset\mathbb{P}^{4}$ be a smooth quadric hypersurface.
Then there are disjoint irreducible smooth conics $C_{1}\subset
Q\supset C_{2}$ such that there is a commutative diagram
$$
\xymatrix{
&&&&X\ar@{->}[lldd]^{\beta_{2}}\ar@{->}[rrdd]_{\beta_{1}}\ar@{->}[lllldd]_{\phi_{1}}\ar@{->}[rrrrdd]^{\phi_{2}}&&&&\\
&&&&&&&&\\
\P^1&&Y_{1}\ar@{->}[rr]_{\alpha_{1}}\ar@{->}[ll]^{\psi_{1}}&&Q&&Y_{2}\ar@{->}[ll]^{\alpha_{2}}\ar@{->}[rr]_{\psi_{2}}&&\P^1,}
$$
where the morphism $\alpha_{i}$ is a blow up along the conic
$C_{i}$, the morphism $\beta_{i}$ is a blow up along the~proper
transform~of the conic $C_{i}$, the morphism $\psi_{i}$ is a
natural fibration into quadric surfaces,
and~$\phi_{i}$~is~fibration, whose general fiber is isomorphic to
a smooth del Pezzo surfaces of degree $6$.

Let $E_{i}$ be the exceptional divisor of the morphism
$\beta_{i}$, and let $H_i$ be a sufficiently general hyperplane
section of the quadric $Q$ that passes through the conic $C_i$.
Then
$$
-K_{X}\sim \bar{H_1}+2\bar{H_2}+E_2,
$$
where $\bar{H}_i\subset X$ is the proper transform of the surface
$H_i$. We see that $\lct(X)\le 1/2$.


We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

Using Example~\ref{example:del-Pezzos} and applying
Lemma~\ref{theorem:Hwang}, we see that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\subseteq S_{1}\cap S_{2},%
$$
where $S_{i}$ is a singular fiber of $\phi_{i}$. Hence, the set
$\mathbb{LCS}(X, \lambda D)$ contains no surfaces.

It follows from Theorem~\ref{theorem:connectedness} that either
$\LCS(X, \lambda D)$ is a point in $E_1\cup E_{2}$, or
$$
\LCS\Big(X,\ \lambda D\Big)\cap\Big(X\setminus\big(E_1\cup E_2\big)\Big)\ne\varnothing,%
$$
which implies that we may assume that $\LCS(X, \lambda D)$ is a
point in $E_1$ by Lemma~\ref{lemma:quadric-curves}.


Since $\beta_{2}$ is an isomorphism on $X\setminus E_2$, we see that
$$
P\in\LCS\Big(Y_{1},\ \lambda\beta_{2}\big(D\big)\Big)\subset P\cup\beta_{2}\big(E_2\big)%
$$
for some point $P\in E_1$. Then $\LCS(Y_{1}, \lambda\beta_{2}(D))=P$ by
Theorem~\ref{theorem:connectedness}, because $P\not\in\beta_{2}(E_{2})$.

Let $H$ be a general hyperplane section of the quadric threefold
$Q\subset\mathbb{P}^{4}$. Then
$$
-K_{Y_{1}}\sim\tilde{H}_1+2\tilde{H}\qlineq\beta_{2}\big(D\big),
$$
where $\tilde{H}\subset Y_{1}\supset\tilde{H}_1$ are proper transforms of $H$
and $H_1$, respectively. But
$$
\LCS\left(Y_{1},\ \lambda\beta_{2}\big(D\big)+\frac{1}{2}\Big(\tilde{H}_1+2\tilde{H}\Big)\right)=P\cup\tilde{H},%
$$
which is impossible by Theorem~\ref{theorem:connectedness}, because
$\lambda<1/2$.
\end{proof}


\begin{lemma}
\label{lemma:3-11} Suppose that $\gimel(X)=3.11$. Then $\mathrm{lct}(X)=1/2$.%
\end{lemma}

\begin{proof}
Let $C\subset\mathbb{P}^{3}$ be a linearly normal quartic elliptic curve, let
$O\in C$ be a point, let
$$
\delta\colon V_{7}\longrightarrow\mathbb{P}^{3}
$$
be a blow up of the point $O$, and let $E$ be the exceptional divisor of
$\delta$. Then
$$
V_{7}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big),
$$
there is a natural $\mathbb{P}^{1}$-bundle $\eta\colon
V_{7}\to\mathbb{P}^{2}$, and $E$ is a section of $\eta$. The
diagram
$$
\xymatrix{
U\ar@/_/@{->}[ddr]_{\omega}\ar@{->}[rr]^{\gamma}&&\mathbb{P}^{3}&&V_{7}\ar@{->}[ll]_{\delta}\ar@{->}[d]^{\eta}\\%
&&X\ar@{->}[ull]^{\alpha}\ar@{->}[urr]_{\beta}\ar@{->}[dr]^{\upsilon}\ar@{->}[dl]_{\phi}&&\mathbb{P}^{2}\\%
&\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}[ll]^{\pi_{1}}\ar@/_/@{->}[ur]_{\pi_{2}}&}%
$$
commutes, where $\pi_{1}$ and $\pi_{2}$ are natural projections,
the morphism $\gamma$ contracts a surface
$$
C\times\mathbb{P}^{1}\cong G\subset U
$$
to the curve $C$, the~morphism $\alpha$ is a blow up of the fiber
of the morphism $\gamma$ over the point~$O\in\mathbb{P}^{3}$, the
morphism $\beta$ is a blow up of the proper transform of $C$, the
morphism $\omega$~is~a~fibration into quadric surfaces, the
morphism $\phi$ is a fibration into del Pezzo surfaces of degree $7$, and
$\upsilon$ contracts~a~surface
$$
C\times\mathbb{P}^{1}\cong F\subset X
$$
to an elliptic curve $Z\subset\mathbb{P}^{1}\times\mathbb{P}^{2}$
such that $-K_{\mathbb{P}^{1}\times\mathbb{P}^{2}}\cdot Z=13$ and
$Z\cong C$.

Let $H_{1}$ be a general fiber of $\phi$, and let $H_{2}$ be a
general surface in
$|(\eta\circ\beta)^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$. Then
$$
-K_{X}\sim H_{1}+2H_{2},
$$
which implies that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq\bar{E},%
$$
where $\bar{E}$ is the exceptional divisor of $\alpha$, because
$\mathrm{lct}(U)=1/2$ by Lemma~\ref{lemma:2-25}.

Let $\Gamma\cong\mathbb{P}^{2}$ be the general fiber of
$\pi_{2}\circ\upsilon$. Then
$$
2=-K_{X}\cdot\Gamma=D\cdot\Gamma=2\bar{E}\cdot\Gamma,
$$
which implies that $\bar{E}\not\subset\mathrm{LCS}(X,\lambda D)$.
Applying Theorem~\ref{lemma:Hwang} to the log pair
$$
\Big(V_{7},\ \lambda \beta\big(D\big)\Big)%
$$
we see that $\mathrm{LCS}(X,\lambda D)\subseteq\bar{E}\cap G$.
Applying Lemma~\ref{lemma:P1xP2} to the log pair
$$
\Big(\mathbb{P}^{1}\times\mathbb{P}^{2},\ \lambda \upsilon\big(D\big)\Big)%
$$
we see that $\mathrm{LCS}(X,\lambda D)=\bar{E}\cap F\cap G$, where
$|\bar{E}\cap F\cap G|=1$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda D+H_{2}\Big)=\mathrm{LCS}\Big(X,\ \lambda D\Big)\cup H_{2},%
$$
and $H_{2}\cap\mathrm{LCS}(X,\lambda D)=\varnothing$. But the
divisor
$$
-\Big(K_{X}+\lambda D+H_{2}\Big)=\left(\lambda-\frac{1}{2}\right)K_{X}+\frac{1}{2}H_{1}%
$$
is ample, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}

\begin{lemma}
\label{lemma:3-12} Suppose that $\gimel(X)=3.12$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Let $\eps\colon V\to\mathbb{P}^{3}$ be a blow up of a line
$L\subset\mathbb{P}^{3}$. There is a natural
$\mathbb{P}^{2}$-bundle $\eta\colon V\to\mathbb{P}^{1}$, there is
a smooth rational cubic curve $C\subset\mathbb{P}^{3}$ such that
$L\cap C=\varnothing$, and the diagram
$$
\xymatrix{
\P^1&&&\P^1\times\P^2\ar@{->}[rrd]^{\pi_2}\ar@{->}[lll]_{\pi_{1}}&&\\
&&X\ar@{->}[lld]^{\beta}\ar@{->}[rd]_{\gamma}\ar@{->}[llu]^{\phi}\ar@{->}[ru]_{\omega}&&&\P^2\\
V\ar@{->}[rrd]_\eps\ar@{->}[uu]^{\eta}&&&Y\ar@{->}[ld]^\alpha\ar@{->}[rru]_{\psi}&&\\
&&\P^3&&& }
$$
commutes, where $\alpha$ and $\beta$ are blow ups of the curve $C$
and its proper transform, respectively, the~morphism $\gamma$ is a
blow up of the proper transform of the line $L$, the morphism
$\psi$ is a $\P^1$-bundle, the morphism $\omega$ is a birational
contraction of a surface $F\subset X$ to a curve such that
$$
C\cup L\subset\alpha\circ\gamma\big(F\big)\subset\mathbb{P}^{3},
$$
and $\alpha\circ\gamma(F)$ consists of secant lines of $C\subset\P^3$ that
intersect $L$, the morphism $\phi$ is a fibration into del Pezzo surfaces of
degree $6$, the morphisms $\pi_1$ and $\pi_2$ are natural projections.

Let $E$ and $G$ be exceptional surfaces of $\beta$ and $\gamma$, respectively,
let $Q\subset\P^3$ be a general quadric surface that passes through $C$, let
$H\subset\P^3$ be a general plane that passes through $L$. Then
$$
-K_{X}\sim \bar{Q}+2\bar{H}+G,
$$
where $\bar{Q}\subset X\supset\bar{H}$ are proper transforms of
$Q\subset\P^3\supset H$, respectively. In particular,
$\lct(X)\le\nolinebreak 1/2$.

We suppose that $\lct(X)<1/2$. Then there exists an effective $\Q$-divisor
$D\qlineq -K_{X}$ such that the log-pair $(X,\lambda D)$ is not log-canonical
for some positive rational number $\lambda<1/2$. Then
$$
\varnothing\neq\LCS\Big(X,\ \lambda D\Big)\subset G,
$$
since $\lct(Y)=1/2$ by Lemma~\ref{lemma:2-27}. Applying
Theorem~\ref{theorem:Hwang} to $\phi$ we see that
$$
\varnothing\neq\LCS\Big(X,\ \lambda D\Big)\subset G\cap S_{\phi},
$$
where $S_{\phi}$ is a singular fiber of the del Pezzo fibration $\phi$ (see
Example~\ref{example:del-Pezzos}). Then we see that
$$
\varnothing\neq\LCS\Big(X,\ \lambda D\Big)\subset G\cap S_{\phi}\cap F,
$$
by applying Theorem~\ref{theorem:Hwang} to the log pair
$(\mathbb{P}^{1}\times\mathbb{P}^{2},\lambda\omega(D))$ and to the
$\mathbb{P}^{1}$-bundle $\pi_2$.

Let $Z_{1}\cong\P^1$ be a section of the natural projection
$$
\P^1\times\P^1\cong G\longrightarrow L\cong\P^1
$$
such that $Z_1\cdot Z_1=0$, and let $Z_2$ a fiber of this projection. Then
$$
F\Big\vert_{G}\sim Z_{1}+3Z_{2}
$$
and $S_{\phi}\vert_{G}\sim Z_{1}$. But the curve $F\cap G$ is
irreducible. Thus, we se that
$$
\Big|G\cap F\cap S_{\phi}\Big|<+\infty,
$$
which implies that the set $\LCS(X, \lambda D)$ consists of a single point
$P\in G$ by Theorem~\ref{theorem:connectedness}.

The log pair $(V, \lambda\beta(D))$ is not log-canonical. Since $\beta$ is an
isomorphism on $X\setminus E$, we see that
$$
\beta\big(P\big)\in\LCS\Big(V,\ \lambda\beta\big(D\big)\Big)\subseteq \beta\big(P\big)\cup\beta\big(E\big),%
$$
which implies that  $\LCS(V, \lambda\beta(D))=\beta(P)$ by
Theorem~\ref{theorem:connectedness}. Let $H\subset\P^3$ be a general
plane.~Then
$$
\LCS\left(V,\ \lambda\beta\big(D\big)+\frac{1}{2}\Big(\tilde{H}_1+3\tilde{H}\Big)\right)=\beta(P)\cup\bar{H},%
$$
where $\tilde{H}\subset V\supset\tilde{H}_1$ are proper transforms of
$H\subset\P^3\supset H_1$, respectively. But
$$
-K_V\sim \tilde{H}_1+3\tilde{H}\qlineq\beta\big(D\big),
$$
which contradicts Theorem~\ref{theorem:connectedness}, because $\lambda<1/2$.
\end{proof}

\begin{lemma}
\label{lemma:3-14} Suppose that $\gimel(X)=3.14$. Then
$\lct(X)=1/2$.
\end{lemma}
\begin{proof}
Let $P\in\mathbb{P}^{3}$ be a point, and let $\alpha\colon
V_{7}\to\mathbb{P}^{3}$ be a blow up of the point $P$. Then
$$
V_{7}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big),
$$
and there is a natural $\mathbb{P}^{1}$-bundle $\pi\colon
V_{7}\to\mathbb{P}^{2}$.

Let $\zeta\colon Z\to\mathbb{P}(1,1,1,2)$ be a blow up of the singular point of
$\mathbb{P}(1,1,1,2)$. Then
$$
Z\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(2\big)\Big),
$$
and there is a natural $\mathbb{P}^{1}$-bundle $\phi\colon
Z\to\mathbb{P}^{2}$.

There is a plane $\Pi\subset\mathbb{P}^{3}$ and a smooth cubic curve
$C\subset\Pi$ such that $P\not\in\Pi$ and the diagram
$$
\xymatrix{
&&&X\ar@{->}[ld]_{\gamma}\ar@{->}[rd]^{\beta}\ar@{->}[rr]^{\omega}&&Z\ar@{->}[dd]^{\phi}\ar@{->}[rrddd]^{\zeta}&&\\
&&Y\ar@{->}[rd]_{\eps}\ar@{->}[ldd]_{\eta}&&V_{7}\ar@{->}[ld]_{\alpha}\ar@{->}[rd]^{\pi}& \\
&&&\P^3\ar@{-->}[rr]_{\xi}&&\P^2\\
&U\ar@{^{(}->}[rr]&&\mathbb{P}(1,1,1,1,2)\ar@{-->}[rrrr]_{\upsilon}\ar@{-->}[u]_{\nu}&&&&\mathbb{P}(1,1,1,2)\ar@{-->}[llu]^{\psi}}
$$
commutes (see \cite[Example~3.6]{Stef96}), where we have the
following notation:
\begin{itemize}
\item the morphism $\eps$ is a blow up of the curve $C$;%
\item the threefold $U$ is a cubic hypersurface in $\mathbb{P}(1,1,1,1,2)$;%
\item the rational map $\xi$ is a projection from the point $P$;%
\item the morphism~$\gamma$~is~a~blow~up of the point that dominates $P$;%
\item the~morphism~$\beta$~is~a~blow~up of the proper transform of the curve $C$;%
\item the morphism $\eta$ contracts the proper transform of $\Pi$ to the point $\mathrm{Sing}(U)$,%
\item the~morphism $\omega$ contracts a surface $R\subset X$ to a
curve~such~that
$$
\beta\circ\alpha\big(R\big)\subset\mathbb{P}^{3}
$$
is a cone over the curve $C$ whose vertex is the point $P$;%
\item the rational maps $\psi$ and $\nu$ are natural projections;%
\item the rational map $\upsilon$ is a linear projection from a
point.
\end{itemize}

Let $E$ and $G$ be exceptional divisors of $\gamma$ and $\beta$
respectively, and let $\bar{H}\subset X$ be a proper transform of
a general plane in $\P^3$ that passes through the point $P$. Then
$$
-K_{X}\sim \bar{\Pi}+3\bar{H}+G,
$$
where $\bar{\Pi}\subset X$ is the proper transform of the plane
$\Pi$. Thus, we see that $\lct(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/3$.

Let $\bar{L}\subset X$ be a proper transform of a general line in $\P^3$
that intersects the curve $C$. Then
$$
D\cdot\bar{L}=\bar{\Pi}\cdot\bar{L}+3\bar{H}\cdot\bar{L}+G\cdot\bar{L}=3\bar{H}\cdot\bar{L}=3,
$$
which implies that $\mathbb{LCS}(X, \lambda D)$ contains no
surfaces except possibly $\bar{\Pi}$ and $E$.

Let $\Gamma$ be a general fiber of $\pi\circ\beta$. Then
$$
D\cdot\Gamma=\bar{\Pi}\cdot\Gamma+3\bar{H}\cdot\Gamma+G\cdot\Gamma=\bar{\Pi}\cdot\Gamma+G\cdot\Gamma=2,
$$
which implies that $\mathbb{LCS}(X, \lambda D)$ does not contain
$\bar{\Pi}$ and $E$. Thus, by Lemma~\ref{lemma:P3}, we have
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\subsetneq E\cup G.
$$

Suppose that $\LCS(X,\lambda D)\subseteq E$. Then
$$
\varnothing\ne\LCS\Big(V_{7},\ \lambda \beta\big(D\big)\Big)\subseteq \beta\big(E\big),%
$$
which contradicts  Theorem~\ref{theorem:Hwang}, because $\beta(E)$
is a section of $\pi$. We see that $\LCS(X,\lambda D)\subsetneq
G$.

Applying Theorem~\ref{theorem:Hwang} to $(Z, \lambda\omega(D))$
and $\phi$ and Theorem~\ref{theorem:connectedness} to $(X,\lambda
D)$, we see that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\subseteq F,
$$
where $F$ is a fiber of the natural projection $G\to \beta(G)$.
Then
$$
\varnothing\ne\LCS\Big(Y,\ \lambda\gamma\big(D\big)\Big)\subseteq\gamma(F),%
$$
where $\gamma(F)$ is a fiber of the blow up $\eps$ over a
point in the curve $C$.

Let $S\subset\P^3$ be a general cone over the curve $C$, let $O\in
C$ be an inflection point such that
$$
\eps\circ\gamma\big(F\big)\ne O
$$
let $L\subset R$ be a line that passes through the point $O$, and
let $H\subset\P^3$ be a plane that is tangent to the surface $S$
along the line $L$. Since $O$ is an inflection point of the curve
$C$, the equality
$$
\mathrm{mult}_{L}\Big(S\cdot H\Big)=3
$$
holds. Let $\breve{S}$, $\breve{H}$ and $\breve{L}$ be the proper
transforms of $S$, $H$ and $L$ on the threefold $Y$. Then
$$
\LCS\left(Y,\ \lambda\gamma(D)+\frac{2}{3}\Big(\breve{S}+\breve{H}\Big)\right)=\LCS\Big(Y,\ \lambda\gamma\big(D\big)\Big)\cup \breve{L}%
$$
due to generality in the choice of $S$. But $-K_Y\sim
\breve{S}+\breve{H}$, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}

\begin{lemma}
\label{lemma:3-15} Suppose that  $\gimel(X)=3.15$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
Let $Q\subset\mathbb{P}^{4}$ be a smooth quadric hypersurface, let $C\subset Q$
be a smooth irreducible conic, and let $\eps\colon V\to Q$ be a blow up of the
conic $C\subset Q$. Then there is a natural morphism
$$
\eta\colon V\longrightarrow\mathbb{P}^{1}
$$
induced by the projection $Q\dasharrow\mathbb{P}^{1}$ from the two-dimensional
linear subspace in $\mathbb{P}^{4}$ that contains the conic $C\subset Q$. Then
a general fiber of $\eta$ is a smooth quadric surface in $\mathbb{P}^{3}$.

There is a line $L\subset Q$ such that $L\cap C=\varnothing$ and there is a
commutative diagram
$$
\xymatrix{
\P^1&&&\P^1\times\P^2\ar@{->}[rrd]^{\pi_2}\ar@{->}[lll]_{\pi_{1}}&&\\
&&X\ar@{->}[lld]^{\beta}\ar@{->}[rd]_{\gamma}\ar@{->}[llu]^{\phi}\ar@{->}[ru]_{\omega}&&&\P^2\\
V\ar@{->}[rrd]_\eps\ar@{->}[uu]^{\eta}&&&Y\ar@{->}[ld]^\alpha\ar@{->}[rru]_{\psi}&&\\
&&Q&&& }
$$
where $\alpha$ and $\beta$ are blow ups of the line $L\subset\mathbb{P}^{3}$
and its proper transform, respectively, the~morphism $\gamma$ is a blow up of
the proper transform of the conic $C$, the morphism $\psi$ is a $\P^1$-bundle,
the morphism $\omega$ is a birational contraction of a surface $F\subset X$ to
a curve such that
$$
C\cup L\subset\alpha\circ\gamma\big(F\big)\subset Q,
$$
and $\alpha\circ\gamma(F)$ consists of all lines in $Q\subset\P^3$ that
intersect $L$ and $C$, the morphism $\phi$ is a fibration into del Pezzo
surfaces of degree $7$, the morphisms $\pi_1$ and $\pi_2$ are natural
projections.


Let $E_1$ and $E_{2}$ be exceptional surfaces of $\beta$ and $\gamma$,
respectively, let $H_{1}, H_2\subset Q$ be general hyperplane sections
that pass through the curves $L$ and $C$, respectively. We have
$$
-K_{X}\sim \bar{H}_1+2\bar{H}_2+E_2\sim \bar{H}_2+2\bar{H}_1+E_1,
$$
where $\bar{H}_1\subset X\supset\bar{H}_2$ are proper transforms
of $H_1\subset Q\supset H_2$, respectively. Then $\lct(X)\le 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log-pair $(X,\lambda D)$ is
not log-canonical for some positive rational number $\lambda<1/2$.

Let $S$ be an irreducible surface on the threefold $X$. Put
$$
D=\mu S+\Omega,
$$
where $\Omega$ is an
effective $\mathbb{Q}$-divisor  such that $S\not\subset\mathrm{Supp}(\Omega)$.
Then
$$
\mathrm{LCS}\left(\bar{H}_{2},\ \frac{1}{2}\Big(\mu
S+\Omega\Big)\Big\vert_{\bar{H}_{2}}\right)\subset E_{1}\cap\bar{H}_{2}%
$$
by Lemma~\ref{lemma:del-Pezzo-septic}. Thus, if $\mu\leqslant 2$, then either
$S=E_{1}$, or $S$ is a fiber of $\phi$.

Let $\Gamma\cong\mathbb{P}^{1}$ be a general fiber of the conic bundle
$\psi\circ\gamma$. Then
$$
2=D\cdot\Gamma=\mu S\cdot\Gamma+\Omega\cdot\Gamma\geqslant\mu S\cdot\Gamma,%
$$
which implies that  $\mu\leqslant 2$ in the case when either $S=E_{1}$, or $S$
is a fiber of $\phi$.

Therefore, we see that $\mathbb{LCS}(X, \lambda D)$ does not contain surfaces.

Applying Theorem~\ref{theorem:Hwang} to the log pair $(Y,\lambda\gamma(D))$ and
$\psi$, we see that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\subsetneq E_2\cup\bar{L},
$$
where $\mathbb{P}^1\cong\bar{L}\subset X$ is a curve such that
$\gamma(\bar{L})$ is a fiber of the conic bundle $\psi$.

Suppose that $\bar{L}\not\subset E_{1}$ and $\bar{L}\subset\mathrm{LCS}(X,
\lambda D)$. Then
$$
\alpha\circ\gamma\big(\bar{L}\big)\subseteq\LCS\Big(Q,\ \lambda \alpha\circ\gamma\big(D\big)\Big)\subseteq\alpha\circ\gamma\big(\bar{L}\big)\cup C\cup L,%
$$
which is impossible by Lemma~\ref{lemma:quadric-curves}. Hence by
Theorem~\ref{theorem:connectedness} we see that
\begin{itemize}
\item either $\LCS(X, \lambda D)\subsetneq E_2$,%
\item or $\LCS(X, \lambda D)\subseteq\bar{L}$ and $\bar{L}\subset E_{1}$.
\end{itemize}

We may assume that $\bar{L}\subset E_{1}$. But $E_{1}\cong\mathbb{F}_{1}$. Then
$\bar{L}\cdot\bar{L}=-1$ on the surface $E_{1}$.

Applying Lemma~\ref{lemma:P1xP2} to the log pair $(\P^1\times\P^2,
\lambda\omega(D))$, we see that $\LCS(X, \lambda D)\subset F$, because
$$
\omega\big(D\big)\qlineq -K_{\P^1\times\P^2}
$$
and $\lambda<1/2$. Applying Lemma~\ref{lemma:Hwang} to the log pair $(V,
\lambda\beta(D))$, we see that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\subsetneq E_1\cup S_{\phi},
$$
where $S_{\phi}$ is a singular fiber of $\phi$, because
$\mathrm{lct}(\P^1\times\P^1)=1/2$ (see Example~\ref{example:del-Pezzos}).

We have $F\cap\bar{L}=\varnothing$ and $|F\cap\bar{S_{\phi}}\cap E_2|<+\infty$.
Thus, there is point $P\in E_{2}$ such that
$$
\LCS\Big(X,\ \lambda D\Big)=P\in E_{2}
$$
by Theorem~\ref{theorem:connectedness}. But
$\beta(E_{1})\cap\beta(P)=\varnothing$. Thus, it follows from
Theorem~\ref{theorem:connectedness} that
$$
\LCS\Big(V,\ \lambda\beta\big(D\big)\Big)=\beta\big(P\big).
$$

Let $\tilde{H}_1\subset V\supset\tilde{H}_2$ be the proper transforms of
$H_1\subset Q\supset H_2$, respectively. Then
$$
-K_V\sim \tilde{H}_2+2\tilde{H}_1\qlineq\beta\big(D\big),
$$
but it follows from the generality of $H_{1}$ and $H_{2}$ that
$$
\LCS\left(V,\
\lambda\beta\big(D\big)+\frac{1}{2}\Big(\tilde{H}_2+2\tilde{H}_1\Big)\right)=\beta\big(P\big)\cup\tilde{H}_1,
$$
which is impossible by Theorem~\ref{theorem:connectedness}, because
$\lambda<1/2$.
\end{proof}


\begin{lemma}
\label{lemma:3-16} Suppose that  $\gimel(X)=3.16$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof} Let $\mathbb{P}^{1}\cong C\subset\mathbb{P}^{3}$ be a twisted
cubic curve, let $O\in C$ be a point. There is a commutative diagram
$$
\xymatrix{
\mathbb{P}\big(\mathcal{E}\big)&U\ar@{=}[l]\ar@/_/@{->}[ddr]_{\omega}\ar@{->}[rr]^{\gamma}&&\mathbb{P}^{3}&&V_{7}\ar@{->}[ll]_{\delta}\ar@{->}[d]^{\eta}\ar@{=}[r]&\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\\%
&&&X\ar@{->}[ull]^{\alpha}\ar@{->}[urr]_{\beta}\ar@{->}[dr]^{\upsilon}&&\mathbb{P}^{2}&\\%
&&\mathbb{P}^{2}&&W\ar@{->}[ll]^{\pi_{1}}\ar@/_/@{->}[ur]_{\pi_{2}},&&}%
$$
where $\mathcal{E}$ is a stable rank two vector bundle on
$\mathbb{P}^{2}$ (see the proof of Lemma~\ref{lemma:2-27}),
and we have the following notation:
\begin{itemize}
\item the morphism $\delta$ is a blow up of the point $O$;%
\item the morphism $\gamma$ contracts a surface $G\subset U$ to the curve $C\subset\mathbb{P}^{3}$;%
\item the~morphism $\alpha$ contracts a surface $E\cong\mathbb{F}_{1}$ to the fiber of $\gamma$ over the point~$O\in\mathbb{P}^{3}$;%
\item the morphism $\beta$ is a blow up of the proper transform of the curve $C$;%
\item the variety $W$ is a smooth divisor on $\mathbb{P}^{2}\times\mathbb{P}^{2}$ of bi-degree $(1,1)$;%
\item the morphisms $\pi_{1}$ and $\pi_{2}$ are natural projections;%
\item the morphisms $\omega$ and $\eta$ are natural $\mathbb{P}^{1}$-bundles;%
\item the morphism $\upsilon$ contracts~a~surface $F\subset X$ to a curve
$$
\mathbb{P}^{1}\cong Z\subset W
$$
such that $\omega\circ\alpha(E)=\pi_{1}(Z)$ and $\eta\circ\beta(G)=\pi_{2}(Z)$.%
\end{itemize}

Take general surfaces
$H_{1}\in|(\omega\circ\alpha)^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$
and
$H_{2}\in|(\eta\circ\beta)^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$.~Then
$$
-K_{X}\sim H_{1}+2H_{2},
$$
which implies that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E\cap F,%
$$
because $\mathrm{lct}(U)=1/2$ by Lemma~\ref{lemma:2-25} and
$\mathrm{lct}(W)=1/2$ by Theorem~\ref{theorem:del-Pezzo}.

Applying Lemma~\ref{lemma:V7} to the log pair $(V_{7},\lambda
\beta(D))$, we see that
$$
\mathrm{LCS}\Big(X,\lambda D\Big)=E\cap F\cap G,
$$
where $|E\cap
F\cap G|=1$. Then
$$
\mathrm{LCS}\Big(X,\ \lambda D+H_{2}\Big)=\mathrm{LCS}\Big(X,\ \lambda D\Big)\cup H_{2},%
$$
where $H_{2}\cap\mathrm{LCS}(X,\lambda D)=\varnothing$. But the
divisor
$$
-\Big(K_{X}+\lambda D+H_{2}\Big)=\left(\lambda-\frac{1}{2}\right)K_{X}+\frac{1}{2}H_{1}%
$$
is ample, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}

\begin{lemma}
\label{lemma:3-17} Suppose that $\gimel(X)=3.17$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}
\begin{proof}
The threefold $X$ is a divisor in $\P^1\times\P^1\times\P^2$ of
tri-degree $(1, 1, 1)$. Take general surfaces
$$
H_{1}\in\Big|\pi_{1}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)\Big|,\ H_{2}\in\Big|\pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)\Big|,\ H_{3}\in\Big|\pi_{3}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|,%
$$
where $\pi_i$ is a natural projection of the threefold $X$ onto
the $i$-th factor of $\P^1\times\P^1\times\P^2$. Then
$$
-K_X\sim H_1+H_2+2H_3,
$$
which implies that $\lct(X)\le 1/2$. There is a commutative
diagram
$$
\xymatrix{
\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}[ll]_{\upsilon_{1}}\ar@{->}[rr]^{\upsilon_{2}}&&\mathbb{P}^{1}\\%
&&X\ar@{->}[u]_{\zeta}\ar@{->}[dll]_{\alpha_{1}}\ar@{->}[ull]_{\pi_{1}}\ar@{->}[d]^{\pi_3}\ar@{->}[drr]^{\alpha_{2}}\ar@{->}[urr]^{\pi_{2}}&&\\%
\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}[rr]_{\omega_{1}}\ar@{->}[uu]^{\eta_{1}}&&\mathbb{P}^{2}&&\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}[ll]^{\omega_{2}}\ar@{->}[uu]_{\eta_{2}},}
$$
where $\omega_{i}$, $\eta_{i}$ and $\upsilon_{i}$ are natural
projections, $\zeta$ is a $\mathbb{P}^{1}$-bundle, and
$\alpha_{i}$~is~a~birational morphism that contracts a surface
$E_{i}\subset X$ to a smooth curve
$C_{i}\subset\mathbb{P}^{1}\times\mathbb{P}^{2}$ such that
$\omega_{1}(C_{1})=\omega_{2}(C_{2})$ is a (irreducible) conic.

Note that $E_{2}\sim H_{1}+H_{3}-H_{2}$ and $E_{1}\sim
H_{2}+H_{3}-H_{1}$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/2$.

Suppose that the set $\mathbb{LCS}(X, \lambda D)$ contains a (irreducible)
surface $S\subset X$. Put
$$
D=\mu S+\Omega,
$$
where $\mu\ge 1/\lambda$ and $\Omega$
is an effective $\mathbb{Q}$-divisor such that
$S\not\subset\mathrm{Supp}(\Omega)$. Then
$$
2=D\cdot\Gamma=\mu S\cdot\Gamma+\Omega\cdot\Gamma\geqslant\mu S\cdot\Gamma,%
$$
where $\Gamma\cong\mathbb{P}^{1}$ is a general fiber of $\zeta$.
Then $S\cdot\Gamma=0$, which implies that $E_{2}\ne S\ne E_{1}$.
But
$$
2=D\cdot\Delta=\mu S\cdot\Delta+\Omega\cdot\Delta\geqslant\mu S\cdot\Delta,%
$$
where $\Delta\cong\mathbb{P}^{1}$ is a general fiber of the conic
bundle $\pi_{2}$. Then $S\cdot\Delta=0$, which implies that
$$
S\in\Big|\pi_{3}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(m\big)\Big)\Big|
$$
for some $m\in\mathbb{Z}_{>0}$, because $E_{2}\ne S\ne E_{1}$ and $S$
is an irreducible surface. Then
$$
0=S\cdot\Gamma=m\ne 0,
$$
which is a contradiction. Thus, we see that the set
$\mathbb{LCS}(X, \lambda D)$ contains no surfaces.

Applying Theorem~\ref{theorem:Hwang} to $\zeta$ and using
Theorem~\ref{theorem:connectedness}, we see that
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)=F\cong\mathbb{P}^{1},
$$
where $F$ is a fiber of the $\mathbb{P}^{1}$-bundle $\zeta$.
Applying Theorem~\ref{theorem:Hwang} to the conic bundle
$\pi_3$,~we~see~that every fiber of the conic bundle $\pi_3$ that
intersects $F$ must be reducible. This means that
$$
\pi_3\big(F\big)\subset\omega_{1}\big(C_{1}\big)=
\omega_{2}\big(C_{2}\big)\subset\P^2,
$$
which is impossible, because $\pi_{3}(F)$ is a line, and
$\omega_{1}(C_{1})=\omega_{2}(C_{2})$~is~an irreducible conic.
\end{proof}


\begin{lemma}
\label{lemma:3-18} Suppose that $\gimel(X)=3.18$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $Q\subset\mathbb{P}^{3}$ be a smooth quadric hypersurface.
Then there is an irreducible conic $C\subset Q$, and there is a
point $O\in C$ such that there is a commutative diagram
$$
\xymatrix{
Y\ar@/^1pc/@{->}[drrrrrr]^{\eta}\ar@/_1pc/@{->}[dddrr]_{\tau}&&&&&&&&\\
&&X\ar@{->}[d]_{\gamma}\ar@{->}[ull]_{\sigma}\ar@{->}[rr]^{\beta}&&V\ar@{->}[d]^{\alpha}\ar@{->}[rr]^{\omega}&&\mathbb{P}^{1}\\
&&U\ar@{->}[d]_{\upsilon}\ar@{->}[rr]_{\zeta}&&Q\ar@{-->}[dll]^{\psi}\ar@{-->}[urr]_{\phi}&&\\
&&\P^3\ar@/_2.5pc/@{-->}[rrrruu]_{\xi}&&&&}
$$
where $\zeta$ is a blow up of the point $O$, the morphisms
$\alpha$ and $\gamma$ are blow ups of the conic $C$ and its proper
transform, respectively, $\beta$ is a blow up of the fiber of the
morphism $\alpha$ over~the~point~$O$, the map $\psi$ is a
projection from~$O$, the map $\phi$ is induced by the projection
from the two-dimen\-si\-onal linear subspace that contains the
conic $C$, the morphism $\tau$ is a blow up of the line $\psi(C)$,
the morphism $\upsilon$ is a blow up of an irreducible conic
$Z\subset\mathbb{P}^{3}$ such that
$$
\psi\big(C\big)\cap Z\ne\varnothing,
$$
and $Z$ and $\psi(C)$ are not contained in one plane, the morphism
$\sigma$ is a blow up of the proper transform of the conic $Z$,
the map $\xi$ is a projection from $\psi(C)$, the morphism $\eta$
is a $\mathbb{P}^{1}$-bundle, and $\omega$ is a fibration into
quadric surfaces.



Let $\bar{H}$ be a general fiber of $\omega\circ\beta$. Then
$\bar{H}$ is a del Pezzo surface such that $K_{\bar{H}}^{2}=7$,
and
$$
-K_{X}\sim 3\bar{H}+2E+G,
$$
where $G$ and $\bar{E}$ are the exceptional divisors of $\beta$
and $\gamma$, respectively. In particular, $\mathrm{lct}(X)\leqslant
1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some positive
rational number $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq G,
$$
since $\mathrm{lct}(V)=1/3$ by Lemma~\ref{lemma:2-29} and
$\beta(D)\qlineq -K_{V}$.

Applying Lemma~\ref{lemma:Hwang} to the del Pezzo fibration
$\omega\circ\beta$ and using Theorem~\ref{theorem:connectedness},
we see that there is a unique singular fiber $S$ of the fibration
$\omega\circ\beta$ such that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq G\cap S,
$$
because the equality $\mathrm{lct}(\bar{H})=1/3$ holds (see
Example~\ref{example:del-Pezzos}).

Let $P\in G\cap S$ be an arbitrary point of the locus
$\mathrm{LCS}(X,\lambda D)$. Put
$$
D=\mu S+\Omega,
$$
where $\Omega$
is an effective $\mathbb{Q}$-divisor such that
$S\not\subset\mathrm{Supp}(\Omega)$. Then
$$
P\in\mathrm{LCS}\Big(S,\ \lambda \Omega\Big\vert_{S}\Big)%
$$
by Theorem~\ref{theorem:adjunction}. We can identify the surface
$\beta(S)$ with an irreducible quadric cone in $\mathbb{P}^3$.

Let $L\subset S$ be a proper transform of the ruling of
$\beta(S)$ such that~$L\cap G\cap S\ne\varnothing$. Then
$$
L\cap E=\varnothing,
$$
and $|L\cap G\cap S|=1$. Hence $P=L\cap G\cap S$ by
Lemma~\ref{lemma:quadric-cone}. We see that
$\mathrm{LCS}(X,\lambda D)=P$.~Then
$$
\bar{H}\cup P\subseteq\mathrm{LCS}\left(X,\ \lambda D+\bar{H}+\frac{2}{3}E\right)\subseteq\bar{H}\cup P\cup E,%
$$
because $\bar{H}$ is a general fiber of the fibration
$\omega\circ\beta$. Therefore, the locus
$$
\varnothing\neq\mathrm{LCS}
\left(X,\ \lambda D+\bar{H}+\frac{2}{3}E\right)\subset X%
$$
must be disconnected, because $P\not\in\bar{H}$ and $P\not\in E$.
But
$$
-\left(K_{X}+\lambda
D+\bar{H}+\frac{2}{3}E\right)\qlineq\bar{H}+\frac{2}{3}\Big(E+G\Big)+\big(\lambda-1/3)K_{X}
$$
is an ample divisor, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}

The proof of Lemma~\ref{lemma:3-18} implies the following
corollary.

\begin{corollary}
\label{corollary:4-4-and-5-1} Suppose that $\gimel(X)=4.4$ or
$\gimel(X)=5.1$. Then $\mathrm{lct}(X)=1/3$.
\end{corollary}


\begin{lemma}
\label{lemma:3-19} Suppose that $\gimel(X)=3.19$. Then $\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $Q\subset\mathbb{P}^4$ be a smooth quadric, and let
$L\subset\mathbb{P}^{4}$ be a line such that
$$
L\cap Q=P_{1}\cup P_{2},
$$
where $P_{1}$ and $P_{2}$ are different points. Let $\eta\colon
Q\dasharrow\mathbb{P}^{2}$ be the projection from $L$. The diagram
$$
\xymatrix{
&&&&X\ar@{->}[ddll]^{\beta_{2}}\ar@{->}[ddrr]_{\beta_{1}}\ar@{->}[dddllll]_{\gamma_{1}}\ar@{->}[dddrrrr]^{\gamma_{2}}&&&&\\
&&&&&&&&,\\
&&U_{1}\ar@{->}[d]_{\delta_{1}}\ar@{->}[rr]^{\alpha_{1}}&&Q\ar@{-->}[dd]^{\eta}\ar@{-->}[dll]^{\xi_{1}}\ar@{-->}[drr]_{\xi_{2}}&&U_{2}\ar@{->}[d]^{\delta_{2}}\ar@{->}[ll]_{\alpha_{2}}&&\\
\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\ar@{->}[rr]^{\pi_{2}}\ar@{->}[drrrr]_{\omega_{2}}&&\mathbb{P}^{3}\ar@{-->}[drr]^{\zeta_{2}}&&&&\mathbb{P}^{3}\ar@{-->}[dll]_{\zeta_{1}}&&\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\ar@{->}[ll]_{\pi_{1}}\ar@{->}[dllll]^{\omega_{1}}\Big),\\
&&&&\mathbb{P}^{2}&&&&}
$$
commutes, where $\alpha_{i}$ is a blow up of the point $P_{i}$, the morphism
$\beta_{i}$ contracts a surface
$$
\mathbb{P}^{2}\cong E_{i}\subset X
$$
to the point that dominates $P_{i}\in Q$, the map $\xi_{i}$ is a projection
from $P_{i}$, the map $\zeta_{i}$ is a projection from the image of $P_{i}$,
the morphism $\delta_{i}$ is a contraction of a surface
$$
\mathbb{F}_{2}\cong G_{i}\subset U_{i}
$$
to a conic $C_{i}\subset\mathbb{P}^{3}$, the morphism $\pi_{i}$ is a blow up of
the image of $P_{i}$, the morphism $\gamma_{i}$ contracts the proper transform
of $G_{i}$ to the proper transform of $C_{i}$, and $\omega_{i}$ is a natural
projection.

The map $\gamma_{1}\circ\gamma_{2}^{-1}$ is an elementary transformation of a
conic bundle (see \cite{Sa80}), and
$$
\delta_{1}\circ\beta_{2}\big(E_{1}\big)\subset\mathbb{P}^{3}\supset\delta_{2}\circ\beta_{1}\big(E_{2}\big)
$$
are planes that contain the conics $C_{1}$ and $C_{2}$, respectively.

Let $H$ be a general hyperplane section of $Q$ such that $P_1\in H\ni P_{2}$.
Then
$$
-K_{X}\sim 3\bar{H}+E_{1}+E_{2},
$$
where $\bar{H}$ is the proper transform of $H$ on the threefold $X$. We see
that $\mathrm{lct}(X)\leqslant 1/3$.


We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an effective
$\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log-pair $(X,\lambda D)$ is
not log-canonical for some $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E_{1}\cup E_{2},%
$$
because $\mathrm{lct}(Q)=1/3$ (see
Example~\ref{example:hypersurface-index-big}). By
Theorem~\ref{theorem:connectedness}, we may assume that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E_{1}.%
$$

Let $\bar{G}_{2}\subset X$ be a proper transform of $G_{2}$. Then
$\bar{G}_{2}\cap E_{1}=\varnothing$, because
$\alpha_{2}(G_{2})\subset Q$~is~a~quadric cone whose vertex is the
point $P_{2}$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big),\ \lambda\gamma_{2}\big(D\big)\Big)\subseteq \gamma_{2}\big(E_{1}\big),%
$$
where $\gamma_{2}(E_{1})$ is a section of $\omega_{1}$. Applying
Theorem~\ref{theorem:Hwang} to $\omega_{1}$, we obtain a contradiction.
\end{proof}


\begin{lemma}
\label{lemma:3-20} Suppose that $\gimel(X)=3.20$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $Q\subset\mathbb{P}^{4}$ be a smooth quadric threefold, and
let
$$
W\subset\mathbb{P}^{2}\times\mathbb{P}^{2}
$$
be a smooth divisor of bi-degree $(1,1)$. Then there are lines
$L_{1}\subset Q\supset L_{2}$ such that~$L_{1}\cap
L_{2}=\varnothing$, and there is a smooth curve $C\subset W$ of
bi-degree $(1,1)$ such that there is a commutative diagram
$$
\xymatrix{
&&&&W\ar@{->}[llllddd]_{\upsilon_{1}}\ar@{->}[rrrrddd]^{\upsilon_{2}}&&&&\\
&&&&X\ar@{->}[ld]_{\beta_{2}}\ar@{->}[rd]^{\beta_{1}}\ar@{->}[u]_{\omega}&&&&\\
&&&V_{1}\ar@{->}[llld]_{\pi_{1}}\ar@{->}[rd]_{\alpha_{1}}&&V_{2}\ar@{->}[ld]^{\alpha_{2}}\ar@{->}[rrrd]^{\pi_{2}}&&&\\
\P^2&&&&Q\ar@{-->}[rrrr]_{\psi_{2}}\ar@{-->}[llll]^{\psi_{1}}&&&&\P^2}
$$
where the morphisms $\alpha_{i}$ and $\beta_{i}$ are blow ups of
the line $L_{i}$ and its proper transform, respectively, the
morphism $\omega$ is a blow up of the curve $C$, the morphisms
$\upsilon_{i}$ and $\pi_{i}$ are natural
$\mathbb{P}^{1}$-bundles, and the map $\psi_{i}$ is a linear
projection from the line $L_{i}$.

Let $\bar{H}$ be the exceptional divisor of $\omega$, and let
$E_i$ be the exceptional divisor of $\beta_{i}$. Then
$$
-K_{X}\sim 3\bar{H}+2E_1+2E_2,
$$
because $\alpha_{2}\circ\beta_{1}(\bar{H})\subset Q$ is a
hyperplane section that contains $L_{1}$ and $L_{2}$. Then
$\mathrm{lct}(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\Q$-divisor $D\qlineq -K_{X}$ such that the log-pair
$(X,\lambda D)$ is not log-canonical for some positive rational
number $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq E_1\cap E_{2}\cap\bar{H}=\varnothing,%
$$
because $\mathrm{lct}(V_{1})=\mathrm{lct}(V_{2})=1/3$ by
Lemma~\ref{lemma:2-31} and $\mathrm{lct}(W)=1/2$ by
Theorem~\ref{theorem:del-Pezzo}, which gives a contradiction.
\end{proof}




\begin{lemma}\label{lemma:3-21}
Suppose that $\gimel(X)=3.21$. Then $\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $\pi_{1}\colon \P^1\times\P^2\to\P^1$ and $\pi_{2}\colon
\P^1\times\P^2\to\P^2$ be natural projections. There is a morphism
$$
\alpha\colon X\longrightarrow\P^1\times\P^2
$$
that contracts a surface $E$ to a curve $C$ such that
$\pi_{1}^{*}(\mathcal{O}_{\mathbb{P}^{1}}(1))\cdot C=2$ and
$\pi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))\cdot C=1$.

The curve $\pi_{2}(C)\subset\mathbb{P}^{2}$ is a line. Therefore, there is a
unique surface
$$
H_{2}\in\Big|\pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|
$$
such that $C\subset H_{2}$. Let $H_{1}$ be a fiber of the
$\mathbb{P}^{2}$-bundle $\pi_{1}$. Then
$$
-K_{X}\sim 2\bar{H_1}+3\bar{H}_{2}+2E,
$$
where $\bar{H}_{i}\subset X$ is a proper transform of the surface $H_{i}$.
In particular, $\lct(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an effective
$\Q$-divisor $D\qlineq -K_{X}$ such that the log-pair $(X,\lambda D)$ is not
log-canonical for some rational $\lambda<1/3$. Then
$$
\LCS\Big(X,\ \lambda D\Big)\subseteq E,
$$
because $\mathrm{lct}(\P^1\times\P^2)=1/3$ by Lemma~\ref{lemma:lct-P1-product}.
There is a commutative diagram
$$
\xymatrix{
&&V&&\\%
U_{1}\ar@{->}[urr]^{\delta_{1}}\ar@{->}[d]_{\omega_{1}}&&X\ar@{->}[u]_{\gamma}\ar@{->}[d]_{\alpha}\ar@{->}[ll]_{\beta_{1}}\ar@{->}[rr]^{\beta_{2}}&&U_{2}\ar@{->}[d]^{\omega_{2}}\ar@{->}[ull]_{\delta_{2}}\\%
\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}[ll]_{\pi_{1}}\ar@{->}[rr]^{\pi_{2}}&&\mathbb{P}^{2}}
$$
where $V$ is a Fano threefold with one ordinary double point $O\in V$
such that
$$
-K_{V}^{3}=40,
$$
the birational morphism $\beta_{i}$ is a contraction of the surface
$\bar{H}_{2}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ to a smooth rational
curve, the~morphism $\delta_{i}$ contracts the curve $\beta_{i}(\bar{H}_{2})$
to the point $O\in V$ such that the rational~map
$$
\delta_{2}\circ\delta_{1}^{-1}\colon U_{1}\dasharrow U_{2}
$$
is a standard flop in $\beta_{1}(\bar{H}_{2})\cong\mathbb{P}^{1}$, the morphism
$\omega_{1}$ is a fibration whose general fiber is
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, the morphism $\omega_{2}$ is a
$\mathbb{P}^{1}$-bundle, and $\gamma$ is a birational morphism such that
$\gamma(\bar{H}_{2})=O\in V$.

The variety $V$ is a section of $\mathrm{Gr}(2,5)\subset\mathbb{P}^9$ by a
linear subspace of codimension $3$. Hence
$$
-K_{V}\sim 2\Big(\gamma\big(\bar{H_1}\big)+\gamma\big(E\big)\Big),
$$
and the divisor $\gamma(\bar{H_1})+\gamma(E)$ is very ample. There is a
commutative diagram
$$
\xymatrix{
X\ar@{->}[d]_{\alpha}\ar@{->}[rr]^{\gamma}&&V\ar@{^{(}->}[rr]_{\zeta}&&\mathbb{P}^{6}\ar@{-->}[d]^{\xi}\\%
\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{^{(}->}[rrrr]_{\eta}&&&&\mathbb{P}^{5}}
$$
such that the embedding $\zeta$ is given by the linear system
$|\gamma(\bar{H_1})+\gamma(E)|$, the map $\xi$ is a linear projection from the
point $O$, the embedding $\eta$ is given by the linear system $|H_{1}+H_{2}|$.

It follows from \cite[Theorem~3.6]{JaPe06} (see \cite[Theorem~3.13]{JaPeRa07}) that
$$
U_{2}\cong\mathbb{P}\big(\mathcal{E}\big),
$$
where where $\mathcal{E}$ is a stable rank two vector bundle  on
$\mathbb{P}^{2}$ such that the sequence
$$
0\longrightarrow\mathcal{O}_{\mathbb{P}_{2}}\longrightarrow\mathcal{E}\otimes\mathcal{O}_{\mathbb{P}_{2}}\big(1\big)\longrightarrow \mathcal{I}\otimes\mathcal{O}_{\mathbb{P}_{2}}\big(1\big)\longrightarrow 0%
$$
is exact, where $\mathcal{I}$ is an ideal sheaf of two general points on
$\mathbb{P}^{2}$. One has $c_{1}(\mathcal{E})=-1$ and $c_{1}(\mathcal{E})=2$, and
$\mathcal{E}$ is a Hulsbergen bundle (see \cite{Hu79}). It follows from
\cite[Theorem~3.5]{JaPe06} that
$$
U_{1}\subset\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big),
$$
and $U_{1}\in|2T-F|$, where $T$ is a tautological bundle on
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(1))$,
and $F$ is a fiber of the projection
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(1))\to\mathbb{P}^{1}$.

Either $\bar{H}_{1}$ is a smooth del Pezzo surface such that
$K_{\bar{H}_{1}}^{2}=7$, or
$$
\big|H_{1}\cap C\big|=1,
$$
because $H_{1}\cdot C=2$. Applying Lemma~\ref{lemma:Hwang} to the surface
$\bar{H}_{1}$, we see that
\begin{itemize}
\item either $|H_{1}\cap C|=1$,%
\item or $H_{1}\cap\LCS(X,\lambda D)=\varnothing$,
\end{itemize}
because $\mathrm{lct}(\bar{H}_{1})=1/3$ if $\bar{H}_{1}$ is smooth. So, there
is a fiber $L$ of the projection $E\to C$ such that
$$
\LCS\Big(X,\ \lambda D\Big)\subseteq L
$$
by Theorem~\ref{theorem:connectedness} Put $\bar{C}=\bar{H}_{2}\cap E$ and
$P=L\cap\bar{C}$. Applying Theorem~\ref{theorem:Hwang} to $\omega_{2}$ and
$$
\Big(U_{2},\ \lambda\beta_{2}\big(D\big)\Big),
$$
we see that either $\LCS(X,\lambda D)=P$ or $\LCS(X,\lambda D)=L$ by
Theorem~\ref{theorem:connectedness}.

Suppose that $\LCS(X,\lambda D)=L$. Then $-K_{V}\qlineq\gamma(D)$ and
$$
\LCS\Big(V,\ \lambda\gamma\big(D\big)\Big)=\gamma\big(L\big)\subset V\subset\mathbb{P}^{6},%
$$
where $\gamma(L)\subset V\subset\mathbb{P}^{6}$ is a line, because
$-K_{V}\cdot\gamma(L)=2$. We have $\mathrm{Sing}(V)=O\in\gamma(L)$.

Let $S\subset V$ be a general hyperplane section of
$V\subset\mathbb{P}^{6}$ such that $\gamma(L)\subset S$. Then
\begin{itemize}
\item the surface $S$ is a del Pezzo surface such that $K_{S}^{2}=5$,%
\item the point $O$ is an ordinary double point of the surface $S$,%
\item the surface $S$ is smooth outside of the point $O\in\gamma(L)$,%
\item the equivalence $K_{S}\sim\mathcal{O}_{\mathbb{P}^{6}}(1)\vert_{S}$ holds,%
\end{itemize}
which implies that $S$ contains finitely many lines that intersect the line
$\gamma(L)$.

Let $H\subset V$ be a general hyperplane section of
$V\subset\mathbb{P}^{6}$. Put $Q=\gamma(L)\cap H$. Then
$$
\LCS\Big(H,\ \lambda\gamma\big(D\big)\Big\vert_{H}\Big)=Q,%
$$
by Remark~\ref{remark:hyperplane-reduction}, which contradicts
Lemma~\ref{lemma:del-Pezzo-quintic}, because $\lambda<1/3$.

Thus, we see that $\LCS(X,\lambda D)=P\in\bar{C}$. Let $F_{1}$ be a general
fiber of $\pi_{1}$. Then
$$
F_{1}\cap C=P_{1}\cup P_{2}\not\ni\alpha\big(P\big),
$$
where $P_{1}\ne P_{2}$ are two points of the curve $C$. Then
$$
P_{1}\cup P_{2}\subset H_{2}\cap F_{1},
$$
because $C\subset H_{2}$. Let $Z$ be a general line in
$F_{1}\cong\mathbb{P}^{2}$ such that $P_{1}\in Z$. Then there is a surface
$$
F_{2}\in\Big|\pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|
$$
such that $Z\subset F_{2}$. Let $\bar{F}_{1}\subset X\supset\bar{F}_{2}$ be the
proper transforms of $F_{1}$ and $F_{2}$, respectively. Then
$$
P\not\in\bar{F}_{1}\cup\bar{F}_{2}.
$$

Let $\bar{Z}\subset X$ be the proper transforms of the curve $Z$. Then
$-K_{X}\cdot\bar{Z}=2$ and
$$
\bar{Z}\subset\bar{F}_{1}\cap\bar{F}_{2},
$$
but $\bar{Z}\cap\bar{H}_{2}=\varnothing$. Thus, the curve $\gamma(\bar{Z})$ is
a line on $V\subset\mathbb{P}^{6}$ such that
$\mathrm{Sing}(V)=O\not\in\gamma(\bar{Z})$.

Let $T$ be a general hyperplane section of the threefold
$V\subset\mathbb{P}^{6}$ such that $\gamma(\bar{Z})\subset T$. Then
$$
\bar{T}\sim 2\bar{H}_{2}+\bar{H}_{1}+E\sim 2\bar{H}_{2}+\bar{F}_{1}+E\sim 2\bar{F}_{2}+\bar{F}_{1}-E,%
$$
where $\bar{T}$ is the proper transform of the surface $T$ on the threefold
$X$. Then
$$
\bar{F}_{1}+\bar{F}_{2}+\bar{T}\sim 3\bar{F}_{2}+2\bar{F}_{1}-E\sim 2\bar{H}_{2}+2\bar{H}_{1}+2E\sim -K_{X},%
$$
and applying Theorem~\ref{theorem:connectedness}, we see that the locus
$$
P\cup\bar{Z}=\LCS\left(X,\ \lambda D+\frac{2}{3}\Big(\bar{F}_{1}+\bar{F}_{2}+\bar{T}\Big)\right)%
$$
must be connected. But $P\not\in\bar{Z}$, which is a contradiction.
\end{proof}


\begin{lemma}\label{lemma:3-22}
Suppose that $\gimel(X)=3.22$. Then $\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $\pi_{1}\colon \P^1\times\P^2\to\P^1$ and $\pi_{2}\colon
\P^1\times\P^2\to\P^2$ be natural projections. There is a morphism
$$
\alpha\colon X\longrightarrow\P^1\times\P^2
$$
that contracts a surface $E$ to a curve $C$ contained in a fiber
$H_{1}$ of $\pi_1$ such that $\pi_{2}(C)$ is a conic.

We have $E\cong\mathbb{F}_{2}$. Let $H_2$ be a general surface in
$|\pi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$. The equivalence
$$
-K_{X}\sim 2\bar{H_1}+3\bar{H_2}+E
$$
holds, where $\bar{H}_{i}\subset X$ is a proper transform of the
surface $H_{i}$. Hence $\lct(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\Q$-divisor $D\qlineq -K_{X}$ such that the log-pair
$(X,\lambda D)$ is not log-canonical for some rational
$\lambda<1/3$. Then
$$
\LCS\Big(X,\lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(\P^1\times\P^2)=1/3$ by
Lemma~\ref{lemma:lct-P1-product}.

Let $Q$ be the unique surface in
$|\pi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{2}}(2))|$ such that
$C\subset Q$, and let $\bar{Q}\subset X$ be the proper transform
of the surface $Q$. Then $\bar{Q}\cap \bar{H_1}=\varnothing$, and
there is a commutative diagram.
$$
\xymatrix{
X\ar@{->}[d]_{\alpha}\ar@{->}[rr]^{\beta}&&\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}\big(2\big)\Big)\ar@{->}[d]^{\phi}\ar@{->}[drr]^{\gamma}&&\\%
\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}^{\pi_{2}}[rr]&&\mathbb{P}_{2}&&\mathbb{P}\big(1,1,1,2\big)\ar@{-->}[ll]^{\psi}}
$$
such that $\beta$ is a contraction of $\bar{Q}$ to a curve,
$\gamma$ is a contraction of the surface
$\beta(\bar{H}_{1})$~to~a~point, the~morphism $\phi$ is a natural
$\P^1$-bundle, and the map $\psi$ is a natural projection. Then
$$
\gamma\circ\beta\big(D\big)\qlineq
\frac{5\gamma\circ\beta\big(E\big)}{2}\qlineq
 -K_{\mathbb{P}(1,1,1,2)}\qlineq\mathcal{O}_{\mathbb{P}(1,1,1,2)}\big(5\big),%
$$
which implies that $E\not\subseteq\mathrm{LCS}(X,\lambda D)$,
because $\lambda<1/3$.

Applying Theorem~\ref{theorem:Hwang} to $\phi$, we see that there
is a fiber $F$ of the projection $E\to C$ such that
$$
\varnothing\ne \LCS\Big(X,\ \lambda D\Big)\subseteq \Big(E\cap \bar{Q}\Big)\cup F,%
$$
including the possibility that  $\LCS(X, \lambda D)\subset
E\cap\bar{Q}$.

Suppose that $\LCS(X, \lambda D)\subset E\cap\bar{Q}$. Let
$M\subset\P^1\times\P^2$ be a general surface in $|H_1+H_2|$, and
let $\bar{M}\subset X$ be the proper transform of the surface $M$.
Then
$$
\bar{M}\cap\bar{H_1}=L,
$$
where $L$ is a line on $\bar{H_1}\cong\P^2$. Let $R$ be the unique
surface in $|\pi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$ such
that $\alpha(L)\subset R$, and let $\bar{R}$ be a proper transform
of the surface $R$ on the threefold $X$. Then
$$
\LCS\Big(X, \lambda D\Big)\cup L\subseteq\LCS\left(X, \lambda
 D+\frac{2}{3}\Big(\bar{M}+\bar{H_1}+\bar{R}+\bar{H_2}\Big)\right)\subseteq\LCS\Big(X, \lambda D\Big)\cup L\cup \bar{H}_{1},%
$$
but $L\cap E\cap\bar{Q}=\bar{Q}\cap\bar{H}_{1}=\varnothing$ and
$-K_{X}\sim\bar{M}+\bar{H_1}+\bar{R}+\bar{H_2}$, which contradicts
Theorem~\ref{theorem:connectedness}.

Therefore, we see that $F\subseteq\mathrm{LCS}(X,\lambda D)$. Put
$\breve{F}=\gamma\circ\beta(F)$ and
$\breve{D}=\gamma\circ\beta(D)$. Then
$$
\breve{F}\subseteq\LCS\Big(\mathbb{P}\big(1,1,1,2\big),\ \lambda
\breve{D}\Big)\subseteq
 \breve{C}\cup\breve{F},%
$$
where
$\breve{C}=\gamma\circ\beta(\bar{Q})\subset\mathbb{P}(1,1,1,2)$ is
a curve such that $\psi(\breve{C})=\pi_{2}(C)$.

Let $S$ be a general surface in
$|\mathcal{O}_{\mathbb{P}(1,1,1,2)}(2)|$. Then
$S\cong\mathbb{P}^{2}$ and
$$
\breve{F}\cap S\subseteq\LCS\Big(S,\ \lambda
\breve{D}\Big\vert_{S}\Big)\subseteq
 \Big(\breve{C}\cup\breve{F}\Big)\cap S,%
$$
but $3D\vert_{S}\qlineq -5K_{S}$, which is impossible by
Lemma~\ref{lemma:plane}.
\end{proof}

\begin{lemma}
\label{lemma:3-23} Suppose that  $\gimel(X)=3.23$. Then
$\mathrm{lct}(X)=1/4$.
\end{lemma}
\begin{proof}
Let $O\in\mathbb{P}^{3}$ be a point, let $C\subset\mathbb{P}^{3}$
be a conic such that~$O\in C$, let $\Pi\subset\mathbb{P}^{3}$ be a
unique plane such that $C\subset\Pi\cong\mathbb{P}^{2}$, and let
$Q\subset\mathbb{P}^{4}$ be a smooth quadric threefold.~Then the
diagram
$$
\xymatrix{
&&&X\ar@{->}[llldd]_{\delta}\ar@{->}[d]^{\phi}\ar@{->}[rrrdd]^{\eta}&&&\\
&&&U\ar@{->}[ld]_{\omega}\ar@{->}[rd]^{\upsilon}&&&\\
V_{7}\ar@{->}[ddrrr]_{\alpha}\ar@{->}[rr]^{\pi}&&\mathbb{P}^{2}&&Q\ar@{-->}[ll]^{\zeta}\ar@{-->}[ddl]^{\xi}&&Y\ar@{->}[ddlll]^{\beta}\ar@{->}[ll]_{\gamma}\\
&&&&&&\\
&&&\mathbb{P}^{3}\ar@{-->}[uul]^{\psi}&&&}
$$
commutes, where we have the following notation:
\begin{itemize}
\item the morphism $\alpha$ is a blow up of the point $O$ and
$$
V_{7}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big);
$$
\item the morphism $\pi$ is a natural $\mathbb{P}^{1}$-bundle;%
\item the morphisms $\delta$ and $\beta$ are blow ups of $C$ and its proper transform, respectively;%
\item the~morphism $\gamma$ contracts the proper transform of the plane $\Pi$ to a point;%
\item the~morphism $\phi$ contracts the proper transform of the plane $\Pi$ to a curve;%
\item  the morphism $\eta$ contracts the proper transform of $E$
to a curve $L\subset Y$ such~that
$$
\gamma\big(\Pi\big)\in\gamma\big(L\big)\subset Q\subset\mathbb{P}^{4}%
$$
and $\gamma(L)$ is a line in $\P^4$;%
\item  the morphism $\omega$ is a natural $\mathbb{P}^{1}$-bundle;%
\item the morphism $\upsilon$ is a blow up of the line $\gamma(L)$;%
\item the maps $\psi$, $\xi$ and $\zeta$ are projections from $O$, $\gamma(\Pi)$ and $\gamma(L)$, respectively.%
\end{itemize}

Let $E$ be the $\alpha$-exceptional surface. Then $E$ is a
section of~$\pi$.

Let $\bar{H}\subset X$ be a proper transform of $\Pi\subset\P^3$.
Then $\mathrm{lct}(X)\leqslant 1/4$, because
$$
-K_{X}\sim 4\bar{\Pi}+2\bar{E}+3G,
$$
where $\bar{E}$ and $G$ are exceptional surfaces of $\eta$ and
$\delta$, respectively.

We suppose that $\mathrm{lct}(X)<1/4$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/4$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq\bar{E}\cap\bar{\Pi}\cap G,%
$$
because $\mathrm{lct}(V_{7})=\mathrm{lct}(Y)$ and
$\mathrm{lct}(U)=1/3$ by Lemmas~\ref{lemma:2-30}, \ref{lemma:2-31}
and \ref{lemma:2-35}.

Let $R\subset\mathbb{P}^{3}$ be a general cone over $C$ whose
vertex is $P\in\mathbb{P}^{3}$, let $H_1\subset\P^3$ be a general
plane such that $O\in H_{1}\ni P$, and let $H_2\subset\P^3$ be a
general plane such that $P\in H_{2}$. Then
$$
\bar{R}\sim \big(\alpha\circ\delta\big)^*\big(R\big)-\bar{E}-G,\ \bar{H_1}\sim \big(\alpha\circ\delta\big)^*\big(H_{1}\big)-\bar{E},\ \bar{H_2}\sim\big(\alpha\circ\delta\big)^*\big(H_{2}\big),%
$$
where $\bar{R}$, $\bar{H}_1$, $\bar{H}_2$ are proper transforms of
$R$, $H_{1}$, $H_{2}$ on the threefold $X$, respectively. Then
$$
-K_{X}\sim\bar{Q}+\bar{H}_1+\bar{H}_2,
$$
but it follows from the generality of $R$, $H_{1}$, $H_{2}$ that
the locus
$$
\LCS\left(X,\ \lambda{D}+\frac{3}{4}\Big(\bar{Q}+\bar{H}_1+\bar{H}_2\Big)\right)=\mathrm{LCS}\Big(X,\ \lambda D\Big)\cup P,%
$$
is disconnected, which is impossible by
Theorem~\ref{theorem:connectedness}.
\end{proof}



\begin{lemma}
\label{lemma:3-24} Suppose that $\gimel(X)=3.24$. Then
$\lct(X)=1/3$.
\end{lemma}

\begin{proof}
Let $V$ is a divisor of bi-degree $(1,1)$ on
$\mathbb{P}^{2}\times\mathbb{P}^{2}$. There is a commutative
diagram
$$
\xymatrix{
&&X\ar@{->}[dll]_{\zeta}\ar@{->}[d]^{\pi}\ar@{->}[rr]^{\alpha}&&V\ar@{->}[d]^{\omega_{1}}\\%
\mathbb{P}^{1}&&\mathbb{F}_{1}\ar@{->}^{\xi}[ll]\ar@{->}^{\gamma}[rr]&&\mathbb{P}^{2},}
$$
where $\omega_{1}$ is a natural $\mathbb{P}^{1}$-bundle, the
morphism $\alpha$ contracts a smooth surface
$$
E\cong\mathbb{P}^{1}\times\mathbb{P}^{1}
$$
to a fiber $L$ of $\omega_{1}$, $\gamma$ is a blow up of
$\omega(L)$, the morphism $\xi$ is a $\mathbb{P}^{1}$-bundle, and
$\zeta$ is a $\mathbb{F}_{1}$-bundle.

Let $\omega_{2}\colon X\to\mathbb{P}^{2}$ be a natural
$\mathbb{P}^{1}$-bundle that is different from $\omega_{1}$. Then
there is a surface
$$
G\in\Big|\omega_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|
$$
such that $L\subset G$, because $\omega_{2}(L)$ is a line. Let
$\bar{G}\subset X$ be a proper transform of $G$. Then
$$
-K_{X}\sim 2F+2\bar{G}+3E,
$$
where $E$ is the exceptional divisor of $\alpha$, and $F$ is a
fiber of $\zeta$. We see that $\lct(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
since $\mathrm{lct}(V)=1/2$ by Theorem~\ref{theorem:del-Pezzo}. We may
assume that $F\cap\mathrm{LCS}(X,\lambda D)\ne\varnothing$. Then
$$
\mathbb{F}_{1}\cong F\subseteq\mathrm{LCS}\Big(X,\lambda
D\Big)\subseteq
 E\cong\mathbb{P}^{1}\times\mathbb{P}^{1}%
$$
by Lemma~\ref{lemma:Hwang}, because $\mathrm{lct}(F)=1/3$ (see
Example~\ref{example:del-Pezzos}), which is a contradiction.
\end{proof}

\begin{lemma}
\label{lemma:3-25} Suppose that $\gimel(X)=3.25$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
Let $L_{1}\subset\mathbb{P}^{3}\supset L_{2}$ be lines such that
$L_{1}\cap L_{2}=\varnothing$. Then there is a commutative diagram
$$
\xymatrix{
&&\mathbb{P}^{3}&&\\%
V_{1}\ar@{->}[urr]^{\alpha_{1}}\ar@{->}[d]_{\omega_{1}}&&X\ar@{->}[d]_{\omega}\ar@{->}[ll]_{\beta_{2}}\ar@{->}[rr]^{\beta_{1}}&&V_{2}\ar@{->}[d]^{\omega_{2}}\ar@{->}[ull]_{\alpha_{2}}\\%
\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}^{\gamma_{1}}[ll]\ar@{->}_{\gamma_{2}}[rr]&&\mathbb{P}^{1},}
$$
where the morphisms $\alpha_{i}$ and $\beta_{i}$ are blow up of
the line $L_{i}$ and its proper transform, respectively, the
morphism $\omega_{i}$ is a natural $\mathbb{P}^{2}$-bundle, the
morphisms $\omega$ and $\gamma_{i}$ are $\mathbb{P}^{1}$-bundles.
Note that
$$
V_{1}\cong
V_{2}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big).
$$


Let $H_{1}$ and $H_{2}$ be proper transforms on $X$ of planes in
$\mathbb{P}^{3}$ such that $L_{i}\subset\alpha(H_{i})$. Then
$$
-K_{X}\sim 2H_{1}+2H_{2}+E_{1}+E_{2}\sim 3H_{1}+H_{2}+2E_{1}\sim
H_{1}+3H_{2}+2E_{2},
$$
where $E_{i}$ is an exceptional divisors of $\beta_{i}$. Hence
$\mathrm{lct}(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/3$.

Applying Lemma~\ref{lemma:Hwang} to the $\mathbb{F}_1$-fibrations
$\omega_{2}\circ\beta_{1}$ and $\omega_{1}\circ\beta_{2}$, we
obtain a contradiction, because the equality
$\mathrm{lct}(\mathbb{F}_{1})=1/3$ holds (see
Example~\ref{example:del-Pezzos}).
\end{proof}

\begin{remark}
Actually, the result of Lemma~\ref{lemma:3-25} is contained in
Corollary~\ref{corollary:other-toric}, but we still prefer to give a
detailed proof that may have further applications.
\end{remark}





\begin{lemma}
\label{lemma:3-30} Suppose that $\gimel(X)=3.30$. Then
$\mathrm{lct}(X)=1/4$.
\end{lemma}

\begin{proof}
Let $O\in\mathbb{P}^{3}$ be a point, and let $\gamma\colon
V_{7}\to\mathbb{P}^{3}$ be a blow up of the point $O$. Then
$$
V_{7}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big),
$$
and there is a $\mathbb{P}^{1}$-bundle $\pi\colon
V_{7}\to\mathbb{P}^{1}$. There is a line $O\in
L\subset\mathbb{P}^{3}$ such that the diagram
$$
\xymatrix{
&&\mathbb{F}_{1}\ar@{->}[lld]_{\upsilon}\ar@{->}[rrd]^{\tau}&&\\
\P^1&&X\ar@{->}[ld]_{\beta}\ar@{->}[rd]^{\xi}\ar@{->}[u]_{\zeta}&&\P^2\\
&V\ar@{->}[lu]^{\eta}\ar@{->}[rd]_{\alpha}&&V_{7}\ar@{->}[ld]^{\gamma}\ar@{->}[ru]_{\pi}&\\
&&\P^3\ar@/_2.5pc/@{-->}[rruu]_{\psi}\ar@/^2.5pc/@{-->}[lluu]^{\phi}&&
}
$$
commutes, where $\alpha$ and $\xi$ are blow ups of the line $L$
and its proper transforms, respectively, the morphism $\eta$ is is
a natural $\mathbb{P}^{2}$-bundle, the~morphism $\beta$ is a blow
up of the curve
$$
\mathbb{P}^{1}\cong C\subset V\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)%
$$
such that $\beta(C)=O$, the morphisms $\zeta$ and $\upsilon$ are
$\mathbb{P}^{1}$-bundles, the maps $\phi$ and $\psi$ are linear
projections from $L$ and $O$, respectively, and $\tau$ is a blow
up of the point $\psi(L)\in\mathbb{P}^{2}$.

Let $T$  be the proper transform on $X$ of a general  hyperplane
in $\mathbb{P}^{3}$ that passes through $L\subset\mathbb{P}^{3}$,
and let $G$ be the exceptional divisor of the blow up $\beta$.
Then
$$
-K_{X}\sim 4T+3E+2G,
$$
where $E$ is the proper transform on $X$ of the exceptional
divisor of $\alpha$. In particular, $\mathrm{lct}(X)<1/4$.

We suppose that $\mathrm{lct}(X)<1/4$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the log
pair $(X,\lambda D)$ is not log canonical for some rational number
$\lambda<1/4$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq G,
$$
because $\mathrm{lct}(V)=1/4$  by
Lemma~\ref{lemma:2-35}. However, every fiber of
the morphism $\eta\circ\beta$ is isomorphic~to~$\mathbb{F}_{1}$,
which is impossible by Lemma~\ref{lemma:Hwang}, because
$\mathrm{lct}(\mathbb{F}_{1})=1/3$ by
Example~\ref{example:del-Pezzos}.
\end{proof}

The proof of Lemma~\ref{lemma:3-30} implies the following corollary (cf.
\cite[Example~3.3]{Stef96}).

\begin{corollary}
\label{corollary:4-12} Suppose that $\gimel(X)=4.12$. Then
$\mathrm{lct}(X)=1/4$.
\end{corollary}

\begin{remark}
Actually, the results of Lemma~\ref{lemma:3-30} and
Corollary~\ref{corollary:4-12} are contained in
Corollary~\ref{corollary:other-toric}, but we still prefer to give a
detailed proof that may have further applications.
\end{remark}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Fano threefolds with $\rho\ge 4$}
\label{section:rho-4}

We use the assumptions and notation introduced in
section~\ref{section:intro}.


\begin{lemma}
\label{lemma:4-1} Suppose that $\gimel(X)=4.1$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
The threefold $X$ is a divisor on
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$
of multidegree $(1, 1, 1, 1)$. Let
$$
\Big[\big(x_{1}:y_{1}\big),\ \big(x_{2}:y_{2}\big),\ \big(x_{3}:y_{3}\big),\ \big(x_{4}:y_{4}\big)\Big]%
$$
be coordinates on
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$.
Then $X$ is given by the equation
$$
F\big(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)=0,
$$
where $F$ is a multi-linear form of multidegree $(1, 1, 1, 1)$.

Let $\pi_{1}\colon
X\to\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ be a
projection induced by
$$
\Big[\big(x_{1}:y_{1}\big),\ \big(x_{2}:y_{2}\big),\
\big(x_{3}:y_{3}\big),\
 \big(x_{4}:y_{4}\big)\Big]\mapsto \Big[\big(x_{2}:y_{2}\big),\ \big(x_{3}:y_{3}\big),\
 \big(x_{4}:y_{4}\big)\Big]\in \mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1},%
$$
and let $\pi_{2}$, $\pi_{3}$ and $\pi_{4}\colon
X\to\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ be
projections defined in a similar way. Put
$$
F=x_{1}G\big(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)+y_{1}H\big(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)
$$
where $G(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4})$ and
$H(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4})$ are multi-linear forms
that do not depend on $x_{1}$ and $y_{1}$. Then $\pi_{1}$ is a
blow up of a curve
$C_{1}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
given by the equations
$$
G\big(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)=H\big(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)=0,
$$
which define a surface
$E_{1}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$
that is contracted by $\pi_{1}$. The equations
$$
x_{1}=H\big(x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)=0%
$$
define a divisor $H_{1}\subset X$ such that $-K_{X}\sim
2H_{1}+E_{1}$, which implies that $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/2$.


Let $E_{2}$, $E_{3}$, $E_{4}$ be surfaces in $X$ defined in a way
similar to $E_{1}$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq
E_{1}\cap E_{2}\cap E_{3}\cap E_{4},
$$
because
$\mathrm{lct}(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1)=1/2$
by Lemma~\ref{lemma:lct-P1-product}. But
$E_{i}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$
is given by
$$
\frac {\partial
F\big(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)}
{\partial x_{i}}=\frac
 {\partial F\big(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3},x_{4},y_{4}\big)}
 {\partial y_{i}}=0,%
$$
which implies that the intersection $E_{1}\cap E_{2}\cap E_{3}\cap
E_{4}$ is given by the equations
$$
\frac {\partial F} {\partial x_{1}}=\frac {\partial F} {\partial
y_{1}}= \frac {\partial F}
 {\partial x_{2}}=\frac {\partial F} {\partial y_{2}}=\frac {\partial F} {\partial x_{3}}=\frac
 {\partial
 F} {\partial y_{3}}=\frac {\partial F} {\partial x_{4}}=\frac {\partial F}
 {\partial y_{4}}=0.%
$$
Hence $E_{1}\cap E_{2}\cap E_{3}\cap
E_{4}=\mathrm{Sing}(X)=\varnothing$, and $\mathrm{LCS}(X,\
\lambda D)=\varnothing$.
\end{proof}



\begin{lemma}
\label{lemma:4-2} Suppose that $\gimel(X)=4.2$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $Q_{1}\subset\mathbb{P}^4\supset Q_{2}$ be quadric cones, whose vertices
are $O_{1}\in \mathbb{P}^4\ni O_{2}$, respectively. Let
$$
O_{1}\not\in S_{1}\subset Q_{1}\subset\mathbb{P}^{4}
$$
be a hyperplane section of $Q_{1}$, and let $C_{1}\subset |-K_{S_{1}}|$ be a
smooth elliptic curve. Then~the~diagram
$$
\xymatrix{
&&&&X\ar@{->}[dll]_{\beta_{1}}\ar@{->}[drr]^{\beta_{2}}\ar@/^3.5pc/@{->}[dddrrrr]^{\zeta_{2}}\ar@/_3.5pc/@{->}[dddllll]_{\zeta_{1}}&&&&\\
&&U_{1}\ar@{->}[d]^{\alpha_{1}}\ar@/^1pc/@{->}[ddrr]^{\gamma_{1}}\ar@{->}[ddll]^{\eta_{1}}&&&&U_{2}\ar@{->}[d]_{\alpha_{2}}\ar@/_1pc/@{->}[ddll]_{\gamma_{2}}\ar@{->}[ddrr]_{\eta_{2}}&&\\
&&Q_{1}\ar@{-->}[drr]_{\psi_{1}}&&&&Q_{2}\ar@{-->}[dll]^{\psi_{2}}&&\\
\mathbb{P}^{1}&&&&\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}[llll]^{\pi_{1}}\ar@{->}[rrrr]_{\pi_{2}}&&&&\mathbb{P}^{1}}
$$
commutes, where $\pi_{1}\ne\pi_{2}$ are natural projections, the map $\psi_{i}$
is a projection from $O_{i}\in Q_{i}\subset\mathbb{P}^{4}$, the~morphism
$\alpha_{i}$ is a blow up of the vertex $O_{i}$, the morphism $\beta_{i}$
contracts a surface
$$
\mathbb{P}^{1}\times C_{1}\cong G_{i}\subset X
$$
to a curve $C_{1}\cong C_{i}\subset U_{i}$, the morphism
$\eta_{i}$ is an $\mathbb{F}_{1}$-bundle,
$\gamma_{i}$~is~a~$\mathbb{P}^{1}$-bundle, and $\zeta_{i}$ is a
fibration into del Pezzo surfaces of degree $6$ that has $4$
singular fibers.

Let $E_{i}\subset X$ be the proper transform of the exceptional divisor of
$\alpha_{i}$. Then
$$
S_{1}=\alpha_{1}\circ\beta_{1}\big(E_{2}\big)\subset Q_{1}\subset\mathbb{P}^{4}\supset Q_{2}\supset\alpha_{2}\circ\beta_{2}\big(E_{1}\big)%
$$
are hyperplane sections that contain $C_{1}$ and $C_{2}$,
respectively. It is also easy to see that
$$
\alpha_{1}\circ\beta_{1}\big(G_{2}\big)\subset Q_{1}\subset\mathbb{P}^{4}\supset Q_{2}\supset\alpha_{2}\circ\beta_{2}\big(G_{1}\big)%
$$
are the cones in $\mathbb{P}^{4}$ over the curves $C_{1}$ and $C_{2}$,
respectively.

Let $\bar{H}\subset X$ be the proper transform of a hyperplane section of
$Q_{1}\subset\mathbb{P}^{4}$ that contains $O_{1}$.~Then
$$
-K_{X}\sim 2\bar{H}+E_{2}+E_{1},
$$
which gives $\mathrm{lct}(X)\leqslant 1/2$. Suppose that $\mathrm{lct}(X)<1/2$.
Then there is an effective $\mathbb{Q}$-divisor
$$
D\qlineq -K_{X}\sim E_{1}+E_{2}+G_{1}+G_{2}
$$
such that the log-pair $(X,\lambda D)$ is not log-canonical
for some $\lambda<1/2$. Put
$$
D=\mu_{1}E_{1}+\mu_{2}E_{2}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$
such that
$$
E_{1}\not\subseteq\mathrm{Supp}\big(\Omega\big)\not\supseteq E_{2}.
$$

Let $\Gamma$ be a general fiber of the conic bundle $\gamma_{1}\circ\beta_{1}$.
Then
$$
2=\Gamma\cdot D=\Gamma\cdot\Big(\mu_{1}E_{1}+\mu_{2}E_{2}+\Omega\Big)=\mu_{1}+\mu_{2}+\Gamma\cdot\Omega\geqslant\mu_{1}+\mu_{2},%
$$
and without loss of generality we may assume that $\mu_{1}\leqslant\mu_{2}$.
Then $\mu_{1}\leqslant 1$.

Suppose that there is a surface $S\in\mathbb{LCS}(X,\lambda D)$.
Then $S\ne E_{1}$. Moreover, we have $S\ne G_{1}$, because
$\alpha_{2}\circ\beta_{2}(G_{1})$ is a quadric surface and
$\lambda<1/2$. Hence $S\cap E_{1}\ne\varnothing$. But
$$
-\frac{1}{2}K_{E_{1}}\qlineq D\Big\vert_{E_{1}}\qlineq
-\frac{\mu_{1}}{2}K_{E_{1}}+\Omega\Big\vert_{E_{1}},%
$$
and $E_{1}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, which is
impossible by Theorem~\ref{theorem:adjunction} and
Lemma~\ref{lemma:elliptic-times-P1}.

We see that the set $\mathbb{LCS}(X,\lambda D)$ contains no surfaces. Let
$P\in\mathrm{LCS}(X,\lambda D)$ be a point.

Suppose that $P\not\in G_{1}$. Let $Z$ be a fiber of $\gamma_{1}$ such that
$\beta_{1}(P)\in Z$. Then
$$
Z\subseteq\mathrm{LCS}\Big(U_{1},\ \lambda\beta_{1}\big(D\big)\Big)
$$
by Theorem~\ref{theorem:Hwang}. Put $\bar{E}_{1}=\beta_{1}(E_{1})$. Then we
have
$$
Z\cap \bar{E}_{1}\in\mathrm{LCS}\Big(\bar{E}_{1},\ \lambda\Omega\Big\vert_{\bar{E}_{1}}\Big)%
$$
by Theorem~\ref{theorem:adjunction}, which is impossible by
Lemma~\ref{lemma:elliptic-times-P1}, because $\mu_{1}\leqslant 1$.

Thus, we see that $P\in G_{1}$. Let $F_{1}\subset X\supset F_{2}$ be fibers of
$\zeta_{1}$ and $\zeta_{2}$ such that
$$
F_{1}\ni P\in F_{2},
$$
respectively. Then either $F_{1}$ or $F_{2}$ is smooth, because $\alpha_1(P)\in
C_{1}$. But
$$
\mathrm{lct}\big(F_{i}\big)=1/2
$$
in the case when $F_{i}$ is smooth (see Example~\ref{example:del-Pezzos}),
which contradicts Lemma~\ref{lemma:Hwang}.
\end{proof}

\begin{lemma}
\label{lemma:4-3} Suppose that $\gimel(X)=4.3$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
Let $F_{1}\cong F_{2}\cong F_{3}\cong\P^1\times\P^1$
be fibers of three different
projections
$$
\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\longrightarrow\mathbb{P}^{1},
$$
respectively. There is a contraction $\alpha\colon X\to
\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of a surface
$E\subset X$ to a curve
$$
C\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1
$$
such that $C\cdot F_{1}=C\cdot F_{2}=1$ and $C\cdot F_{3}=2$.
There is a smooth surface
$$
\mathbb{P}^1\times\mathbb{P}^1\cong G\in \big|F_{1}+F_{2}\big|
$$
such that $C\subset G$. In particular, we see that
$$
-K_{X}\sim 2\bar{G}+E+\bar{F}_{3},
$$
where $\bar{F}_{3}$ and $\bar{G}$ are proper transforms of $F_{3}$
and $G$, respectively. Hence $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,%
$$
because
$\mathrm{lct}(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1)=1/2$
and
$\alpha(D)\qlineq-K_{\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1}$.

There is a smooth surface $H\subset |3F_{1}+F_{3}\big|$ such that
$C=G\cdot H$. Let $\bar{H}$ be a proper transform of the surface
$H$ on the threefold $X$. Then $\bar{H}\cap\bar{G}=\varnothing$
and there is a commutative diagram
$$
\xymatrix{%
&U\ar@{->}[d]_{\phi}&&X\ar@{->}[d]^{\alpha}\ar@{->}[ll]_{\gamma}\ar@{->}[rr]^{\beta}&&V\ar@{->}[d]^{\pi}&&\\
&\mathbb{P}^{1}\times\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}[ll]_{\zeta}\ar@{->}[rr]^{\xi}&&\mathbb{P}^{1}\times\mathbb{P}^{1}}
$$
such that $\beta$ and $\gamma$ are contractions of the surfaces
$\bar{G}$ and $\bar{H}$ to smooth curves,
the~morphisms $\pi$ and $\phi$ are $\mathbb{P}^{1}$-bundles, the
morphisms $\zeta$ and $\xi$ are projections that are given~by the
linear systems $|F_{1}+F_{2}|$ and $|F_{1}+F_{3}|$, respectively.

It follows from $\bar{H}\cap\bar{G}=\varnothing$ that
\begin{itemize}
\item either the log pair $(V,\lambda\beta(D))$ is not log canonical,%
\item or the log pair $(U,\lambda\gamma(D))$ is not log canonical.
\end{itemize}

Applying Theorem~\ref{theorem:Hwang} to $(V,\lambda\beta(D))$ or
$(U,\lambda\gamma(D))$ and using
Theorem~\ref{theorem:connectedness}, we see that
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)=\Gamma,
$$
where $\Gamma$ is a fiber of the~natural projection $E\to C$.

We may assume that $\alpha(\Gamma)\in F_{3}$. Let
$\bar{F}_{3}\subset X$ be the proper transform of the surface
$F_{3}$. Put
$$
D=\mu\bar{F}_{3}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $\bar{F}_{3}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\mu F_{3}+\alpha\big(\Omega\big)\qlineq 2\Big(F_{1}+F_{2}+F_{3}\Big),%
$$
which gives $\mu\leqslant 2$. Then
$(\bar{F}_{3},\lambda\Omega\vert_{\bar{F}_{3}})$ is not
log canonical along $\Gamma\subset\bar{F}_{3}$ by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{\bar{F}_{3}}\qlineq -K_{\bar{F}_{3}},
$$
and $\bar{F}_{3}$ is a del Pezzo surface such that
$K_{\bar{F}_{3}}^{2}=6$ and
\begin{itemize}
\item either $\bar{F}_{3}$ is smooth and $|C\cap F_{3}|=2$;%
\item or $\bar{F}_{3}$ has one ordinary double point and $|C\cap F_{3}|=1$.%
\end{itemize}

We have $\mathrm{lct}(\bar{F}_{3})\leqslant\lambda$. Then $\bar{F}_{3}$ is
singular by Example~\ref{example:del-Pezzos}. It follows from
Lemma~\ref{lemma:singular-del-Pezzo-sextic} that
$$
\mathrm{LCS}\Big(\bar{F}_{3},\
\lambda\Omega\vert_{\bar{F}_{3}}\Big)=\mathrm{Sing}\big(\bar{F}_{3}\big),
$$
but the log pair $(\bar{F}_{3},\lambda\Omega\vert_{\bar{F}_{3}})$
is not log canonical along the whole curve
$\Gamma\subset\bar{F}_{3}$, which is a contradiction.
\end{proof}

\begin{lemma}
\label{lemma:4-5} Suppose that $\gimel(X)=4.5$. Then $\mathrm{lct}(X)=3/7$.
\end{lemma}

\begin{proof}
Let $Q\subset\mathbb{P}^{4}$ be a quadric cone, let
$V\subset\mathbb{P}^{6}$ be a a section of
$\mathrm{Gr}(2,5)\subset\mathbb{P}^{9}$ by a linear subspace of
dimension $6$ such that $V$ has one ordinary double point.
Then the~diagram
$$
\xymatrix{
&&&&X\ar@{->}'[ddll]^(0.7){\mu}[ddddll]\ar@{->}[ddd]^{\gamma}\ar@{->}[rrd]^{\eta}\ar@{->}[rrr]^{\theta}&&&Y\ar@{->}[rr]^{\iota}\ar@{->}[rrd]^{\tau}&&Q\ar@/^0.8pc/@{-->}[d]^{\xi}&&&\\%
V\ar@/^5pc/@{-->}[urrrrrrrrr]^{\phi}&&&&&&\mathbb{P}^{1}\times\mathbb{F}_{1}\ar@/_1pc/[drr]^{\xi}\ar@{->}[ddd]_{\nu}\ar@{->}[rrr]^{\sigma}&&&\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}[rd]^{\upsilon_{2}}\ar@{->}`[rrd]`[dddd]`[lllllll]^{\upsilon_{1}}[dddlllllll]&&&\\%
&&&&&&&&\mathbb{F}_{1}\ar@/^1pc/[ddrr]^{\zeta}\ar@{->}[rr]^{\chi}&&\mathbb{P}^{1}&&\\%
U_{1}\ar@{->}[uu]^{\delta_{1}}\ar@/_1pc/@{->}[drr]_{\omega_{1}}&&&&U\ar@{->}[uullll]_(0.3){\beta}\ar@{->}[drr]^{\alpha}\ar@{->}'[ll]_{\beta_{1}}[llll]\ar@{->}'[rr]^{\beta_{2}}[rrrr]&&&&U_{2}\ar@{->}[drr]^{\omega_{2}}\ar@{->}'[ull]'[ullll]_{\delta_{2}}'[uulllll][uullllllll]&&&&\\%
&&\mathbb{P}^{1}&&&&\mathbb{P}^{1}\times\mathbb{P}^{2}\ar@{->}[llll]_{\pi_{1}}\ar@{->}[rrrr]^{\pi_{2}}&&&&\mathbb{P}^{2}\ar@{-->}[uu]_{\psi}&&\\
&&&&&&&&&&&&}%
$$
commutes (cf. \cite[Lemma~2.6]{Fur04}), where we have the
following notation:
\begin{itemize}
\item the morphisms $\pi_{i}$, $\upsilon_{i}$, $\xi$ and $\chi$ are natural projections;%
\item the morphism $\alpha$ contracts a surface
$\mathbb{F}_{3}\cong E\subset U$ to a curve $C$ such that
$$
\pi_{1}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)\cdot C=2,
\ \pi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\cdot C=1;%
$$
\item the morphism $\beta$ contracts a surface $\mathbb{P}^{1}\times\mathbb{P}^{1}\cong\bar{H}_{2}\subset U$ to the singular point of $V$;%
\item the morphism $\beta_{i}$ contracts the surface $\bar{H}_{2}$ to a smooth rational curve;%
\item the~morphism $\delta_{i}$ contracts the curve
$\beta_{i}(\bar{H}_{2})$ to the singular point of $V$ such
that the~map
$$
\delta_{2}\circ\delta_{1}^{-1}\colon U_{1}\dasharrow U_{2}
$$
is a standard flop in the curve $\beta_{1}(\bar{H}_{2})\cong\mathbb{P}^{1}$;%
\item the morphism $\omega_{1}$ is a fibration whose general fiber is $\mathbb{P}^{1}\times\mathbb{P}^{1}$;%
\item the morphisms $\omega_{2}$, $\pi_{2}$, $\xi$, $\sigma$, $\tau$ are $\mathbb{P}^{1}$-bundles;%
\item the morphism $\zeta$ is a blow up of a point $O\in\mathbb{P}^{2}$ such that $O\not\in\pi_{2}(C)$;%
\item the map $\psi$ is a linear projection from the point $O\in\mathbb{P}^{2}$;%
\item the morphism $\nu$ contracts a surface $G\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ to a curve $L$ such that $\pi_{2}(L)=O$;%
\item the morphism $\gamma$ contracts a surface $\breve{G}$ to a
curve $\bar{L}$ such~that
$$
\alpha\big(\bar{L}\big)=L\subset\mathbb{P}^{1}\times\mathbb{P}^{2}
$$
and the curve $\beta(\bar{L})$ is a line in $V\subset\mathbb{P}^{6}$ such that $\beta(\bar{L})\cap\mathrm{Sing}(V)=\varnothing$;%
\item the morphism $\eta$ contracts a surface $\breve{E}$ to a curve such~that $\nu\circ\eta(\breve{E})=C\subset\mathbb{P}^{1}\times\mathbb{P}^{2}$;%
\item the morphism $\theta$ contracts a smooth surface
$\breve{R}\subset X$ to a curve such that $\breve{R}\ne\breve{E}$
and
$$
\tau\circ\theta\big(\breve{R}\big)=\sigma\circ\eta\big(\breve{E}\big)\subset\mathbb{P}^{1}\times\mathbb{P}^{1};
$$
\item the morphism $\mu$ is a fibration into del Pezzo surfaces of degree $6$;%
\item the morphism $\iota$ contracts the surface $\theta(\breve{H}_{2})$ to the singular point of the quadric $Q$;%
\item the map $\phi$ is a linear projection from the line $\beta(\bar{L})\subset V\subset\mathbb{P}^{6}$.%
\end{itemize}


The curve $\pi_{2}(C)\subset\mathbb{P}^{2}$ is a line. Then
$\alpha(\bar{H}_{2})\in|\pi_{2}^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$
and $C\subset\alpha(\bar{H}_{2})$.

The morphism $\pi_{1}$ induces a double cover $C\to\mathbb{P}^{1}$
branched in two points $Q_{1}\in C\ni Q_{2}$.~Let
$$
T_{i}\in\Big|\pi_{1}^{*}\Big(\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big)\Big|
$$
be the unique surface such that $Q_{i}\in T_{i}$. Let
$\bar{T}_{i}\subset U$ be the proper transform of $T_{i}$. Then
\begin{itemize}
\item the surface $\bar{T}_{i}$ has one ordinary double point,%
\item the surface $\bar{T}_{i}$ is tangent to the surface $E$ along the curve $E\cap\bar{T}_{i}$,%
\item the surface $\bar{T}_{i}$ is a del Pezzo surface such that $K_{\bar{T}_{i}}^{2}=7$.%
\end{itemize}

Let $Z_{i}\subset\mathbb{P}^{2}$ be the unique line such that
$O\in Z\ni \pi_{2}\circ\alpha(Q_{i})$. Then there is a unique
surface
$$
\bar{R}_{i}\in\Big|\big(\pi_{2}\circ\alpha\big)^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|
$$
such that $Z_{i}\subset\pi_{2}\circ\alpha(\bar{R}_{i})$. Then
$\bar{L}\subset\bar{R}_{i}$ and
$$
-K_{U}\sim 2\bar{H}_{2}+\bar{R}_{i}+2\bar{T}_{i}+E
$$

Let $\Gamma_{i}$ be a fiber of the projection $E\to C$ over the
point $Q_{i}$. Then $\Gamma_{i}=E\cap\bar{T}_{i}$ and
$$
\Gamma_{i}\subset\mathrm{LCS}\left(U,\
\frac{3}{7}\Big(2\bar{H}_{2}+\bar{R}_{i}+2\bar{T}_{i}+E\Big)\right).
$$

Let $\breve{R}_{i}$ and $\breve{T}_{i}$ be the proper transforms
of $\bar{R}_{i}$ and $\bar{T}_{i}$ on the threefold $X$,
respectively. Then
$$
-K_{X}\sim 2\breve{H}_{2}+\breve{R}_{i}+2\breve{T}_{i}+\breve{E},
$$
because $\bar{L}\subset\bar{R}_{i}$. Let
$\breve{\Gamma}_{i}\subset X$ be the proper transform of the curve
$\Gamma_{i}$. Then
$$
\left(X,\
\frac{3}{7}\Big(2\breve{H}_{2}+\breve{R}_{i}+2\breve{T}_{i}+\breve{E}\Big)\right).
$$
is log canonical (and is not log terminal). Thus, we see that
$\mathrm{lct}(X)\leqslant 3/7$.

We suppose that $\mathrm{lct}(X)<3/7$. Then there exists an
effective $\Q$-divisor $D\qlineq -K_{X}$ such that the log-pair
$(X,\lambda D)$ is not log-canonical for some rational
$\lambda<3/7$.

The surfaces $\breve{T}_{1}$ and $\breve{T}_{2}$ are the only
singular fibers of the fibration $\mu\colon X\to\mathbb{P}^{1}$.
Then
$$
\breve{T}_{i}\not\subseteq\LCS\Big(X,\ \lambda D\Big)\subsetneq\breve{T}_{1}\cup\breve{T}_{2},%
$$
by Lemma~\ref{lemma:Hwang}, because $D\cdot Z=\breve{T}_{1}=2$,
where $Z$ is a general fiber of $\pi_{2}\circ\alpha\circ\gamma$.


We may assume that $\LCS(X,\lambda D)\subseteq\breve{T}_{1}$ by
Theorem~\ref{theorem:connectedness}.

Applying Theorem~\ref{theorem:Hwang} to the log pair
$(\mathbb{P}^{1}\times\mathbb{F}_{1}, \lambda\eta(D))$, we see
that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\ne \breve{T}_{1}\cap\breve{G},%
$$
because $G=\eta(\breve{G})$ is a section of the
$\mathbb{P}^{1}$-bundle $\sigma$.

Applying Theorem~\ref{theorem:Hwang} to the log pair
$(\mathbb{P}^{1}\times\mathbb{P}^{2},
\lambda\alpha\circ\gamma(D))$, we see that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)\subseteq\breve{T}_{1}\cap\breve{E}=\breve{\Gamma}_{1}%
$$
by Theorem~\ref{theorem:connectedness}, because
$\breve{G}\cap\breve{E}=\varnothing$ and $T_{1}$ is a section of
$\pi_{2}$.

Applying Theorem~\ref{theorem:Hwang} to the log pairs $(Y,
\lambda\theta(D))$ and $(U_{2}, \lambda\beta_{2}\circ\gamma(D))$,
we see that
$$
\varnothing\ne\LCS\Big(X,\ \lambda D\Big)=\breve{\Gamma}_{1},%
$$
because $\breve{R}\cap\breve{H}_{2}=\varnothing$.  Put
$\bar{D}=\gamma(D)$. Then $\varnothing\ne\LCS(U,
\lambda\bar{D})=\Gamma_{1}$. Put
$$
\bar{D}=\eps\bar{H}_{2}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor such that
$\bar{H}_{2}\not\subseteq\mathrm{Supp}(\Omega)$. Then
$$
\Omega\Big\vert_{\bar{H}_{2}}\qlineq
-\frac{\big(1+\eps\big)}{2}K_{\bar{H}_{2}},
$$
and the log pair $(\bar{H}_{2},\lambda\Omega\vert_{\bar{H}_{2}})$ is
not log canonical by Theorem~\ref{theorem:adjunction}.
The latter implies that
$$
\frac{3}{7}\frac{1+\eps}{2}>\lambda \frac{1+\eps}{2}>1/2
$$
by Lemma~\ref{lemma:elliptic-times-P1},
and hence $\eps>4/3$.

We may assume that either $E\not\subseteq\mathrm{Supp}(\bar{D})$
or $\bar{T}_{1}\not\subseteq\mathrm{Supp}(\bar{D})$ by
Remark~\ref{remark:convexity}.

Suppose that $E\not\subseteq\mathrm{Supp}(\bar{D})$. Let $Z$ be a
general fiber of the projection $E\to C$. Then
$$
1=-K_{U}\cdot Z=\bar{D}\cdot Z=\eps+\Omega\cdot Z\geqslant 1,
$$
which is a contradiction, because $\eps>4/3$. Thus, we see
that $\bar{T}_{1}\not\subseteq\mathrm{Supp}(\bar{D})$.

Let $\bar{\Delta}\subset\bar{T}_{1}$ be a proper transform of a
general line in $T_{1}\cong\mathbb{P}^{2}$ that passes through
$Q_{1}$. Then
$$
2=-K_{U}\cdot\bar{\Delta}=\bar{D}\cdot\bar{\Delta}\geqslant\mathrm{mult}_{\Gamma_{1}}\big(\bar{D}\big)\geqslant
1/\lambda>7/3,
$$
because $\bar{\Delta}\not\subset\mathrm{Supp}(\bar{D})$ and
$\bar{\Delta}\cap\Gamma_{1}\ne\varnothing$. The obtained
contradiction completes the proof.
\end{proof}

\begin{lemma}
\label{lemma:4-6} Suppose that $\gimel(X)=4.6$. Then
$\mathrm{lct}(X)=1/2$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to\mathbb{P}^{3}$
that blows up three disjoint lines $L_{1}$, $L_{2}$, $L_{3}$.

Let $H_{i}$ be the proper transform on $X$ of a general plane
in $\mathbb{P}^{3}$ such that $L_{i}\subset\alpha(H_{i})$. Then
$$
-K_{X}\sim 2H_{1}+E_{1}+H_{2}+H_{3}\sim
2H_{2}+E_{2}+H_{1}+H_{3}\sim 2H_{3}+E_{3}+H_{1}+H_{2},
$$
where $E_{i}$ is the exceptional divisor of $\alpha$ such that
$\alpha(E_{i})=L_{i}$. In particular, we see that
$\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/2$.

The surface $H_{i}$ is a smooth del Pezzo surface such that
$K_{H_{i}}^{2}=7$, the linear system $|H_{i}|$ has no base points
and induces a smooth morphism $\phi_{i}\colon X\to\mathbb{P}^{1}$,
whose fibers are isomorphic to $H_{i}$.

Suppose that $|\mathrm{LCS}(X,\lambda D)|<+\infty$. We may assume
that  $\mathrm{LCS}(X,\lambda D)\not\subseteq E_{1}$. Then the set
$$
\mathrm{LCS}\left(X,\ \lambda D+H_{1}+\frac{1}{2}E_{1}\right)
$$
is disconnected, which is impossible by
Theorem~\ref{theorem:connectedness}, because
$H_{2}+H_{3}+(\lambda-1/2)K_{X}$ is ample.

We may assume that $H_{1}\cap\mathrm{LCS}(X,\lambda
D)\ne\varnothing$. Then
$$
\varnothing\ne H_{1}\cap\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq\mathrm{LCS}\Big(H_{1},\ \lambda D\Big\vert_{H_{1}}\Big)%
$$
by Remark~\ref{remark:hyperplane-reduction}. Put
$C_{2}=E_{2}\vert_{H_{1}}$ and $C_{3}=E_{3}\vert_{H_{1}}$. Then
$$
C_{2}\cdot C_{2}=C_{3}\cdot C_{3}=-1,
$$
and there is a unique curve $\mathbb{P}^{1}\cong C\subset H_{1}$
such that $C\cdot C_{2}=C\cdot C_{3}=1$ and $C\cdot C=-1$. Then
$$
\mathrm{LCS}\Big(H_{1},\ \lambda D\Big\vert_{H_{1}}\Big)=C
$$
by Lemma~\ref{lemma:del-Pezzo-septic}.

There is a unique
smooth quadric $Q\subset\mathbb{P}^{3}$ that contains $L_{1}$,
$L_{2}$, $L_{3}$. Note that
$$
\bar{Q}\cap H_{1}=C,
$$
where $\bar{Q}\subset X$ is a proper transform of the surface $Q$.

There is a morphism $\sigma\colon
X\to\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
contracting $\bar{Q}$ to a curve of tri-degree $(1,1,1)$.
Since $\bar{Q}\cap H_{1}=C$, one obtains (see
Remark~\ref{remark:hyperplane-reduction}) that
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\supset\bar{Q},
$$
and hence $\LCS(X, \lambda D)=\bar{Q}$, because
$\mathrm{lct}(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1})=1/2$.
Then
$$
D=\mu \bar{Q}+\Omega,
$$
where $\mu\in\mathbb{Q}$ such that $\mu\geqslant 1/\lambda>2$, and
$\Omega$ is an effective $\mathbb{Q}$-divisor such that
$\bar{Q}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\alpha\big(D\big)=\mu Q+\alpha\big(\Omega\big),
$$
which is impossible, because $\alpha(D)\qlineq
2Q\sim-K_{\mathbb{P}^{3}}$ and $\mu>2$.
\end{proof}




\begin{lemma}
\label{lemma:4-7} Suppose that $\gimel(X)=4.7$. Then $\lct(X)=1/2$.
\end{lemma}

\begin{proof}
There is a birational morphism $\alpha\colon X\to W$ such that
\begin{itemize}
\item the variety $W$ is a smooth divisor of bi-degree $(1,1)$ on $\mathbb{P}^{2}\times\mathbb{P}^{2}$;%
\item the morphism $\alpha$ contracts two irreducible surfaces $E_{1}\ne E_{2}$ to two disjoint curves;%
\item the curves $\alpha(E_{1})\ne \alpha(E_{2})$ are fibers of one natural $\mathbb{P}^{1}$-bundle $W\to\P^2$.%
\end{itemize}


Put $L_{i}=\alpha(E_{i})$. There is a surface $H\subset W$ such
that $-K_{X}\sim 2H$ and $L_{1}\subset H\supset L_{2}$. Then
$$
-K_{X}\sim 2\bar{H}+E_1+E_2,
$$
where $\bar{H}$ is a proper  transform of $H$ on the threefold
$X$. In particular, $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that the
log-pair $(X,\lambda D)$ is not log-canonical for some
$\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E_1\cup
E_2,
$$
since $\mathrm{lct}(W)=1/2$ by
Theorem~\ref{theorem:del-Pezzo} and $\alpha(D)\qlineq -K_{W}$.

We may assume that $\LCS(X, \lambda D)\cap E_1\neq\varnothing$.
Let $\beta\colon X\to Y$ be a contraction of $E_2$. Then
$$
\mathbb{LCS}\Big(Y,\ \lambda\beta\big(D\big)\Big)\ne\varnothing
$$
and $\beta(D)\qlineq -K_Y$, which contradicts
Lemma~\ref{lemma:3-24}.
\end{proof}

\begin{lemma}
\label{lemma:4-8} Suppose that $\gimel(X)=4.8$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
There is blow up $\alpha\colon X\to
\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of a curve $
C\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ such
that~$C\subset F_{1}$~and
$$
C\cdot F_{2}=C\cdot F_{3}=1,
$$
where $F_{i}$ is a fiber of the projection of
$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$~to its $i$-th
factor. There is a smooth surface
$$
\mathbb{P}^1\times\mathbb{P}^1\cong G\in \big|F_{2}+F_{3}\big|
$$
such that $C\subset G$. Let $E$ be the exceptional divisor of
$\alpha$. Then
$$
-K_{X}\sim 2\bar{F}_{1}+2\bar{G}+3E,
$$
where $\bar{F}_{1}$ and $\bar{G}$ are proper transforms of $F_{1}$
and $G$, respectively. In particular, $\mathrm{lct}(X)\leqslant 1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E,
$$
because
$\mathrm{lct}(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1)=1/2$
and
$\alpha(D)\qlineq-K_{\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1}$.

Let $Q$ be an irreducible quadric cone in $\mathbb{P}^{4}$. Then
there is a commutative diagram
$$
\xymatrix{%
&&&&X\ar@{->}[dlll]_{\alpha}\ar@{->}[d]_{\beta}\ar@{->}[rrrd]^{\gamma}&&&&&\\
&\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}[drr]_{\phi}&&&V\ar@{->}[dl]_{\pi}\ar@{->}[rrd]^{\delta}&&&U\ar@{->}[dl]^{\xi}\\%
&&&\mathbb{P}^{1}\times\mathbb{P}^{1}&&&Q\ar@{-->}[lll]^{\psi}&}
$$
where we have the following notations:
\begin{itemize}
\item $V$ is a variety with $\gimel(V)=3.31$;%
\item the morphism $\beta$ is a contraction of the surface $\bar{G}$ to a curve;%
\item the morphism $\gamma$ is a contraction of $\bar{F}_{1}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ to an ordinary double point;%
\item the~morphism $\delta$ is a blow up of the vertex of the quadric cone
$Q\subset\mathbb{P}^{4}$;%
\item the morphism $\xi$~is~a~blow~up of a~smooth irreducible conic in $Q$;%
\item the map $\psi$ is a~projection from the~vertex of the cone $Q$;%
\item the morphism $\phi$ is a projection that is given by $|F_{2}+F_{3}|$,
i.\,e. the projection of $\P^1\times\P^1\times\P^1$ onto the product of the
last two factors;%
\item the morphism $\pi$ is a natural $\mathbb{P}^{1}$-bundle.%
\end{itemize}

It follows from Corollary~\ref{corollary:other-toric} that
$\mathrm{lct}(V)=1/3$. But $\mathrm{lct}(U)=1/3$ by
Lemma~\ref{lemma:2-29-singular}. Hence
$$
\varnothing\ne\mathrm{LCS}\Big(X,\ \lambda D\Big)\subseteq
E\cap\bar{G}\cap\bar{F}_{1}=\varnothing,
$$
which is a contradiction.
\end{proof}


\begin{lemma}
\label{lemma:4-9} Suppose that $\gimel(X)=4.9$. Then
$\mathrm{lct}(X)=1/3$.
\end{lemma}

\begin{proof}
There exists a point $O\in\mathbb{P}^{3}$, and there exist lines
$L_{1}\subset\mathbb{P}^{3}\supset L_{2}$ such~that~$L_{1}\cap
L_{2}=\varnothing$, the~line $L_{1}$ passes through the point $O$,
and there is a commutative diagram
$$
\xymatrix{
\mathbb{F}_{1}\ar@{->}[dddddd]_{\chi}\ar@{->}[rrrr]^{\phi}&&&&\mathbb{P}^{2}&&&&&\\%
&Z\ar@{->}[dddddd]_{\nu}\ar@{->}[ul]^{\xi}&&&&V_{7}\ar@{->}'[rrrd]'[dddddrrr]^{\sigma}[dddddd]\ar@{->}[ul]_{\pi}&&&&\mathbb{P}^{2}\times\mathbb{P}^{1}\ar@/_1pc/@{->}[ulllll]_{\iota_{1}}\ar@{->}[ddddddd]^{\iota_{2}}\\%
&&&&&&&&&\\%
&&&U_{1}\ar@{->}[uull]_{\zeta_{1}}\ar@{->}[dd]_{\delta_{1}}\ar@{->}'[ur][uurr]^{\eps_{1}}&&X\ar@/_1.8pc/@{->}[uullll]_{\rho}\ar@{->}[d]_{\tau}\ar@{->}[rr]^{\upsilon_{1}}\ar@{->}[ll]_{\upsilon_{2}}&&U_{2}\ar@{->}[uull]_{\eps_{2}}\ar@{->}[dd]^{\delta_{2}}\ar@/^0.4pc/@{->}[uurr]_(0.6){\zeta_{2}}&&\\%
&&&&&Y\ar@{->}'[ddl][dddllll]^{\eta}\ar@{->}[dll]_{\beta_{2}}\ar@{->}[drr]^{\beta_{1}}&&&&\\%
&&&V_{1}\ar@{->}[ddrr]^(0.3){\alpha_{1}}\ar@{->}'[ll]_{\omega_{1}}[dlll]&&&&V_{2}\ar@/^0.6pc/@{->}[dddrr]^(0.6){\omega_{2}}\ar@{->}[ddll]_{\alpha_{2}}&&\\%
\mathbb{P}^{1}&&&&&&&&&\\
&\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}^{\gamma_{1}}[ul]\ar@/_1pc/@{->}_{\gamma_{2}}[drrrrrrrr]&&&&\mathbb{P}^{3}&&&&\\
&&&&&&&&&\mathbb{P}^{1}}
$$
that uses the following notation:
\begin{itemize}
\item the morphism $\sigma$ is a blow up of the point $O$ and
$$
V_{7}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big);
$$
\item the morphism $\pi$ is a natural $\mathbb{P}^{1}$-bundle;%
\item the morphism $\alpha_{i}$ is a blow ups of the line $L_{i}$;%
\item the morphisms $\beta_{i}$, $\upsilon_{i}$ and $\eps_{i}$ are blow ups of the proper transforms of the line $L_{i}$;%
\item  the morphisms $\tau$ and $\delta_{1}$ are blow ups of curves that
are the preimages of the point $O$;%
\item the morphism $\delta_{2}$ is a blow up of the point that dominates the point $O$;%
\item the morphisms $\omega_{1}$ and $\omega_{2}$ are natural
$\mathbb{P}^{2}$-bundles, where
$$
V_{1}\cong V_{2}\cong\mathbb{P}\Big(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}\big(1\big)\Big);%
$$
\item the morphisms $\iota_{1}$, $\iota_{2}$, $\gamma_{1}$ and $\gamma_{2}$ are natural projections;%
\item the morphism $\phi$ is a blow up of $\pi(\bar{L}_{1})$, where $\bar{L}_{1}\subset V_{7}$ is a proper transform of $L_{1}$;%
\item the morphisms $\rho$ and $\zeta_{i}$ contracts the proper transform of the plane $\Pi\subset\mathbb{P}^{3}$ such that
$$
L_{2}\subset\Pi\ni O;
$$%
\item the morphisms $\eta$, $\nu$, $\xi$, $\chi$, $\gamma_{1}$, $\gamma_{2}$ and $\iota_{2}$ are natural $\mathbb{P}^{1}$-bundles.%
\end{itemize}

Let $H_{i}\subset X$ be the proper transform of a general
hyperplane in $\mathbb{P}^3$ that contains $L_{i}$. Then
$$
-K_{X}\sim 3H_{1}+H_{2}+2E_{1}+G.
$$
where $E_i$ and $G$ be the exceptional divisor of $\upsilon_{i}$
and $\tau$, respectively. Thus, we have $\mathrm{lct}(X)\leqslant
1/3$.

We suppose that $\mathrm{lct}(X)<1/3$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/3$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq G,
$$
since $\mathrm{lct}(Y)=1/3$ by Lemma~\ref{lemma:3-25}. But the
surface $G$ is not a fiber of the smooth morphism
$$
\omega_{1}\circ\beta_{2}\circ\tau\colon X\longrightarrow\mathbb{P}^{1},%
$$
whose fibers are del Pezzo surfaces of degree $7$, so we obtain a
contradiction applying Lemma~\ref{lemma:Hwang} to the morphism
$\omega_{1}\circ\beta_{2}\circ\tau$.
\end{proof}


The proof of Lemma~\ref{lemma:4-9} implies the following.

\begin{corollary}
\label{corollary:5-2} Suppose that $\gimel(X)=5.2$. Then
$\mathrm{lct}(X)=1/3$.
\end{corollary}


\begin{remark}
Actually, the results of Lemma~\ref{lemma:4-9} and
Corollary~\ref{corollary:5-2} are contained in
Corollary~\ref{corollary:other-toric}, but we still prefer to give a
detailed proof that may have further applications.
\end{remark}

The following result follows from
Corollaries~\ref{corollary:4-4-and-5-1} and \ref{corollary:5-2},
Lemma~\ref{lemma:lct-product} and
Example~\ref{example:del-Pezzos}.

\begin{corollary}
\label{corollary:rho-5-6-7-8-9-10} Suppose that $\rho\geqslant 5$.
Then
$$
\mathrm{lct}\big(X\big)=\left\{%
\aligned
&1/3\ \text{whenever}\ \gimel(X)\in\{5.1, 5.2\},\\%
&1/2\ \text{in the remaining cases}.\\%
\endaligned\right.%
$$
\end{corollary}

\begin{lemma}
\label{lemma:4-13} Suppose that $\gimel(X)=4.13$ and $X$ is
general. Then $\lct(X)=1/2$.
\end{lemma}
\begin{proof}
Let $F_{1}\cong F_{2}\cong F_{3}\cong\P^1\times\P^1$
be fibers of three different
projections
$$
\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\longrightarrow\mathbb{P}^{1},
$$
respectively. There is a contraction $\alpha\colon X\to
\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of a surface
$E\subset X$ to a curve
$$
C\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1
$$
such that $C\cdot F_{1}=C\cdot F_{2}=1$ and $C\cdot F_{3}=3$. Then
there is a smooth surface
$$
\mathbb{P}^1\times\mathbb{P}^1\cong G\in \big|F_{1}+F_{2}\big|
$$
such that $C\subset G$. In particular, we see that
$$
-K_{X}\sim 2\bar{G}+E+2\bar{F}_{3},
$$
where $\bar{F}_{3}$ and $\bar{G}$ are proper transforms of $F_{3}$
and $G$, respectively. Hence $\mathrm{lct}(X)\leqslant 1/2$.

We suppose that $\mathrm{lct}(X)<1/2$. Then there exists an
effective $\mathbb{Q}$-divisor $D\qlineq -K_{X}$ such that
the~log-pair $(X,\lambda D)$ is not log-canonical for some
positive rational number $\lambda<1/2$. Then
$$
\varnothing\ne\mathrm{LCS}\Big(X,\lambda D\Big)\subseteq E\cong\mathbb{F}_{4},%
$$
because
$\mathrm{lct}(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1)=1/2$
and
$\alpha(D)\qlineq-K_{\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1}$.

There are smooth surfaces $H_{1}\in |3F_{1}+F_{3}\big|$ and
$H_{2}\in |3F_{2}+F_{3}\big|$ such that
$$
C=G\cdot H_{1}=G\cdot H_{2},
$$
and $H_{1}\cong H_{2}\cong\mathbb{P}^1\times\mathbb{P}^1$. Let
$\bar{H}_{i}$ be a proper transform of $H_{i}$ on the threefold
$X$. Then
$$
\bar{H}_{1}\cap\bar{G}=\bar{H}_{2}\cap\bar{G}=\varnothing.
$$

There is a commutative diagram
$$
\xymatrix{%
&&&&&X\ar@{->}[dd]^{\alpha}\ar@{->}[dlll]_{\gamma_{1}}\ar@{->}[ddll]^{\beta}\ar@{->}[rrd]^{\gamma_{2}}&&&&\\
&&U_{1}\ar@{->}[dddr]_{\phi_{1}}&&&&&U_{2}\ar@{->}[dddl]^{\phi_{2}}\\%
&&&V\ar@/_0.37pc/[ddrr]_{\pi}&&\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\ar@{->}[dd]_{\zeta}\ar@{->}[rdd]^{\xi_{2}}\ar@{->}'[ld]^{\xi_{1}}[lldd]&&\\%
&&&&&&\\
&&&\mathbb{P}^{1}\times\mathbb{P}^{1}&&\mathbb{P}^{1}\times\mathbb{P}^{1}&\mathbb{P}^{1}\times\mathbb{P}^{1}&}
$$
such that $\beta$ and $\gamma_{i}$ are contractions of the
surfaces $\bar{G}$ and $\bar{H}_{i}$ to a smooth curves,
the~morphisms $\pi$ and $\phi_{i}$ are
$\mathbb{P}^{1}$-bundles, the morphisms $\zeta$ and $\xi_{i}$ are
projections that are given~by the linear systems $|F_{1}+F_{2}|$
and $|F_{i}+F_{3}|$, respectively.

It follows from $\bar{H}_{1}\cap\bar{G}=\varnothing$ that
\begin{itemize}
\item either the log pair $(V,\lambda\beta(D))$ is not log canonical,%
\item of the log pair $(U_{1},\lambda\gamma_{1}(D))$ is not log
canonical.
\end{itemize}

Applying Theorem~\ref{theorem:Hwang} to $(V,\lambda\beta(D))$ or
$(U_{1},\lambda\gamma_{1}(D))$ and using
Theorem~\ref{theorem:connectedness}, we see that
$$
\mathrm{LCS}\Big(X,\ \lambda D\Big)=\Gamma,
$$
where $\Gamma$ is a fiber of the~natural projection $E\to C$.

We may assume that $\alpha(\Gamma)\in F_{3}$. Let
$\bar{F}_{3}\subset X$ be the proper transform of the surface
$F_{3}$. Put
$$
D=\mu\bar{F}_{3}+\Omega,
$$
where $\Omega$ is an effective $\mathbb{Q}$-divisor on $X$ such
that $\bar{F}_{3}\not\subset\mathrm{Supp}(\Omega)$. Then
$$
\mu F_{3}+\alpha\big(\Omega\big)\qlineq 2\Big(F_{1}+F_{2}+F_{3}\Big),%
$$
which gives $\mu\leqslant 2$. Then
$(\bar{F}_{3},\lambda\Omega\vert_{\bar{F}_{3}})$ is not log
canonical along $\Gamma\subset\bar{F}_{3}$ by
Theorem~\ref{theorem:adjunction}. But
$$
\Omega\Big\vert_{\bar{F}_{3}}\qlineq -K_{\bar{F}_{3}},
$$
and $\bar{F}_{3}$ is a del Pezzo surface such that
$K_{\bar{F}_{3}}^{2}=5$. But $\bar{F}_{3}$ may be singular.
Namely, we have
$$
\mathrm{Sing}\big(\bar{F}_{3}\big)=\varnothing\iff\big|C\cap F_{3}\big|=F_{3}\cdot C=3,%
$$
and $\mathrm{Sing}(\bar{F}_{3})\subset\Gamma$. The following cases
are possible:
\begin{itemize}
\item the surface $\bar{F}_{3}$ is smooth and $|C\cap F_{3}|=3$;%
\item the surface $\bar{F}_{3}$ has one ordinary double point and $|C\cap F_{3}|=2$;%
\item the surface $\bar{F}_{3}$ has a singular point of type
$\mathbb{A}_{2}$ and $|C\cap
 F_{3}|=1$.%
\end{itemize}

We have $\mathrm{lct}(\bar{F}_{3})\leqslant\lambda<1/2$. Thus, it
follows from Examples~\ref{example:del-Pezzos} and
\ref{example:del-Pezzo-quintic} that $|C\cap F_{3}|=1$, which  is
impossible if the threefold $X$ is sufficiently general.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Upper bounds}
\label{section:bounds}

We use the assumptions and notation introduced in
section~\ref{section:intro}.
The purpose of this section is to find upper bounds for the global log
canonical thresholds of the varieties $X$ with
$$
\gimel\big(X\big)\in\Big\{1.1,1.2,\ldots,1.17,2.1,\ldots,2.36,3.1,\ldots,3.31,4.1,\ldots,4.13,5.1,\ldots,5.7,5.8\Big\}
$$
be the ordinal number of the deformation type of the threefold $X$
in the notation of Table~\ref{table:Fanos}.

\begin{lemma}
\label{lemma:1-8} Suppose that $\gimel(X)=1.8$. Then
$\mathrm{lct}(X)\leqslant 6/7$.
\end{lemma}

\begin{proof}
The linear system $|-K_{X}|$ does not have base points and induces
an embedding $X\subset\mathbb{P}^{10}$, and the threefold $X$
contains a line $L\subset X$ (see \cite{Sho79}, \cite{Tak89}).

It follows from \cite[Theorem~4.3.3]{IsPr99} (see \cite{Cut89},
\cite{Tak89})  that there is a commutative diagram
$$
\xymatrix{
U\ar@{->}[d]_{\alpha}\ar@{-->}[rr]^{\rho}&&W\ar@{->}[d]^{\beta}\\%
X\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{3},}
$$
where $\alpha$ is a blow up of the line $L$, the map $\rho$ is a
composition of flops, the morphism $\beta$ is a blow up of a
smooth curve of degree $7$ and genus $3$, and $\psi$ is a double
projection from $L$.

Let $S\subset X$ be the proper transform of the exceptional
surface of $\beta$. Then
$$
\mathrm{mult}_{L}\big(S\big)=7
$$
and $S\sim -3K_{X}$, which implies that $\mathrm{lct}(X)\leqslant
6/7$.
\end{proof}


\begin{lemma}
\label{lemma:1-9} Suppose that $\gimel(X)=1.9$. Then
$\mathrm{lct}(X)\leqslant 4/5$.
\end{lemma}

\begin{proof}
The linear system $|-K_{X}|$ does not have base points and induces
an embedding $X\subset\mathbb{P}^{11}$, and the threefold $X$
contains a line $L\subset X$ (see \cite{Sho79}, \cite{Tak89}).

It follows from \cite[Theorem~4.3.3]{IsPr99} (see \cite{Cut89},
\cite{Tak89})  that there is a commutative diagram
$$
\xymatrix{
U\ar@{->}[d]_{\alpha}\ar@{-->}[rr]^{\rho}&&W\ar@{->}[d]^{\beta}\\%
X\ar@{-->}[rr]_{\psi}&&Q,}
$$
where $Q\subset\mathbb{P}^{4}$ is a smooth quadric threefold,
$\alpha$ is a blow up of the line $L$, the map
$\rho$~is~a~composition of flops,  the morphism $\beta$ is a blow
up along a smooth curve of degree $7$ and genus $2$,
and~$\psi$~is~a~double projection from the line $L$.

Let $S\subset X$ be the proper transform of the exceptional
surface of $\beta$. Then
$$
\mathrm{mult}_{L}\big(S\big)=5
$$
and  $S\sim -2K_{X}$, which implies that $\mathrm{lct}(X)\leqslant
4/5$.
\end{proof}


\begin{lemma}
\label{lemma:1-10} Suppose that $\gimel(X)=1.10$. Then
$\mathrm{lct}(X)\leqslant 2/3$.
\end{lemma}

\begin{proof}
The linear system $|-K_{X}|$ does not have base points and induces
an embedding $X\subset\mathbb{P}^{13}$, and the threefold $X$
contains a line $L\subset X$ (see \cite{Sho79}, \cite{Tak89}).

It follows from \cite[Theorem~4.3.3]{IsPr99} (see \cite{Cut89},
\cite{Tak89})  that the diagram
$$
\xymatrix{
U\ar@{->}[d]_{\alpha}\ar@{-->}[rr]^{\rho}&&W\ar@{->}[d]^{\beta}\\%
X\ar@{-->}[rr]_{\psi}&&V_{5}}
$$
commutes, where $V_{5}$ is a smooth section of
$\mathrm{Gr}(2,5)\subset\mathbb{P}^9$ by a linear subspace of
dimension~$6$, the morphism $\alpha$ is a blow up of the line $L$,
the map $\rho$ is a flop, the morphism $\beta$ is a blow up of a
smooth rational curve of degree $5$, and $\psi$ is a double
projection from $L$.

Let $S\subset X$ be the proper transform of the exceptional
surface of $\beta$. Then
$$
\mathrm{mult}_{L}\big(S\big)=3
$$
and  $S\sim -K_{X}$, which implies that $\mathrm{lct}(X)\leqslant
2/3$.
\end{proof}

\begin{lemma}
\label{lemma:2-2}
Suppose that $\gimel(X)=2.2$. Then $\lct(X)\le 13/14$.
\end{lemma}

\begin{proof}
There is a smooth divisor $B\subset\P^1\times\P^2$  of bi-degree
$(2, 4)$ such that the diagram
$$
\xymatrix{
&& X\ar@{->}[d]_{\pi}\ar@{->}[lld]_{\phi_1}\ar@{->}[rrd]^{\phi_2} & \\
\P^1 &&\P^1\times\P^2\ar@{->}[ll]^{\pi_1}\ar@{->}[rr]_{\pi_2}&& \P^2}%
$$
commutes, where $\pi$ is a double cover branched along $B$, the
morphisms $\pi_1$ and $\pi_2$ are natural projections, $\phi_1$ is
a fibration into del Pezzo surfaces of degree $2$, and $\phi_2$ is
a conic bundle.

Let $H_1$ be a general fiber of $\phi_1$. Put
$\bar{H}_{1}=\pi(H_{1})$. Then the intersection
$$
C=\bar{H}_1\cap B\subset\bar{H}_{1}\cong\mathbb{P}^{2}
$$
is a smooth quartic curve.

There is a point $P\in C$ such that
$$
\mathrm{mult}_{P}\Big(C\cdot L\Big)\geqslant 3,
$$
where $L\subset\bar{H}_{1}\cong\mathbb{P}^{2}$ is a line that is
tangent to $C$ at the point $P$.

The curve $\pi_{2}(L)$ is a line. Thus, there is a unique surface
$$
H_{2}\in\Big|\phi_{2}^{*}\Big(\mathcal{O}_{\mathbb{P}^{2}}\big(1\big)\Big)\Big|
$$
such that $\phi_{2}(H_{2})=\pi_{2}(L)$. Then $-K_X\sim H_1+H_2$.

Let us show that $\mathrm{lct}(X,H_{1}+H_{2})\leqslant 13/14$. Put
$\bar{H}_{2}=\pi(H_{2})$. Then
$$
\mathbb{LCS}\left(X,\
\frac{13}{14}\Big(H_{1}+H_{2}\Big)\right)\ne\varnothing\iff\mathbb{LCS}\left(\P^1\times\P^2,\ \frac{1}{2}B+\frac{13}{14}\Big(\bar{H}_{1}+\bar{H}_{2}\Big)\right)\ne\varnothing%
$$
by \cite[Proposition~3.16]{Ko97}. Let $\alpha\colon
V\to\P^1\times\P^2$ be a blow up of the curve $C$. Then
$$
K_{V}+\frac{1}{2}\tilde{B}+\frac{13}{14}\Big(\tilde{H}_{1}+\tilde{H}_{2}\Big)+\frac{3}{7}E\qlineq\alpha^{*}\left(K_{\P^1\times\P^2}+\frac{1}{2}B+\frac{13}{14}\Big(\bar{H}_{1}+\bar{H}_{2}\Big)\right),
$$
where $\tilde{B},\tilde{H}_{1},\tilde{H}_{2}\subset V$ are proper
transforms of $B,\bar{H}_{1},\bar{H}_{2}$, respectively. But the
log pair
$$
\left(V,\ \frac{13}{14}\tilde{H}_{2}+\frac{3}{7}E\right)%
$$
is not log terminal along the fiber $\Gamma$ of the morphism
$\alpha$ such that $\alpha(\Gamma)=P$, because
$$
\mathrm{mult}_{\Gamma}\Big(\tilde{H}_{2}\cdot E\Big)=
\mathrm{mult}_{P}\Big(C\cdot\bar{H}_{2}\Big)
\geqslant\mathrm{mult}_{P}\Big(C\cdot L\Big)\geqslant 3%
$$
due to the choice of the curve $L$. We see that
$$
\Gamma\subseteq\mathrm{LCS}\left(V,\ \frac{13}{14}\tilde{H}_{2}+\frac{3}{7}E\right)\subseteq\mathrm{LCS}\left(V,\ \frac{1}{2}\tilde{B}+\frac{13}{14}\Big(\tilde{H}_{1}+\tilde{H}_{2}\Big)+\frac{3}{7}E\right),%
$$
which implies that $\mathrm{lct}(X,H_{1}+H_{2})\leqslant 13/14$.
Hence the inequality $\mathrm{lct}(X)\leqslant 13/14$ holds.
\end{proof}

\begin{remark}
It follows from Lemmas~\ref{lemma:Hwang} and
\ref{lemma:singular-Del-Pezzo-2} that $\mathrm{lct}(X)\geqslant
2/3$ if $\gimel(X)=2.2$ and the threefold $X$ satisfies the
following generality
condition: any fiber of $\phi_1$ satisfies the hypotheses of
Lemma~\ref{lemma:singular-Del-Pezzo-2}.
\end{remark}


\begin{lemma}
\label{lemma:2-7} Suppose that $\gimel(X)=2.7$.
Then $\mathrm{lct}(X)\leqslant 2/3$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
Q\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{1},}
$$
where $Q\subset\mathbb{P}^{4}$ is a smooth quadric threefold,
$\alpha$ is a blow up of a smooth curve $S_{1}\cap S_{2}$, where
$$
S_{1}, S_2\in\Big|\mathcal{O}_{\mathbb{P}^{4}}\big(2\big)\Big\vert_{Q}\Big|,%
$$
the morphism $\beta$ is a fibration into del Pezzo surfaces of
degree $4$, and $\psi$~is~a~rational map that is induced by the
pencil generated by the surfaces $S_{1}$ and $S_{2}$. Then
$\mathrm{lct}(X)\leqslant 2/3$, because
$$
-K_{X}\qlineq \frac{3}{2}\bar{S}_{1}+\frac{1}{2}E,
$$
where $\bar{S}_{1}\subset X$ is a proper transform of the surface
$S_{1}$, and $E$ is the exceptional divisor of $\alpha$.
\end{proof}


\begin{lemma}
\label{lemma:2-9} Suppose that $\gimel(X)=2.9$.
Then $\mathrm{lct}(X)\leqslant 3/4$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
\mathbb{P}^{3}\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{2},}
$$
where $\alpha$ is a blow up of a smooth curve
$C\subset\mathbb{P}^{3}$ of degree $7$ and genus $5$ that is a
scheme-theoretic intersection of cubic surfaces in
$\mathbb{P}^{3}$, the morphism $\beta$ is a conic bundle, and
$\psi$ is a rational map that is given by the linear system of
cubic surfaces that contain $C$. One has
$$
-K_{X}\qlineq \frac{4}{3}S+\frac{1}{3}E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 3/4$.
\end{proof}


\begin{lemma}
\label{lemma:2-12} Suppose that $\gimel(X)=2.12$.
Then $\mathrm{lct}(X)\leqslant 3/4$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
\mathbb{P}^{3}\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{3},}
$$
where $\alpha$ and $\beta$ are blow ups of smooth curves
$C\subset\mathbb{P}^{3}$ and $Z\subset\P^3$ of degree $6$ and genus $3$ that
are scheme-theoretic intersections of cubic surfaces in
$\mathbb{P}^{3}$, respectively, and $\psi$ is a birational map
that is given by the linear system of cubic surfaces that contain
$C$. Then
$$
-K_{X}\qlineq \frac{4}{3}S+\frac{1}{3}E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{3}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 3/4$.
\end{proof}

\begin{lemma}
\label{lemma:2-13} Suppose that $\gimel(X)=2.13$.
Then $\mathrm{lct}(X)\leqslant 2/3$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
Q\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{2},}
$$
where $Q\subset\mathbb{P}^{4}$ is a smooth quadric threefold,
$\alpha$ is a blow up of a smooth curve $C\subset Q$~of~degree~$6$
and genus $2$, the morphism $\beta$ is a conic bundle, and $\psi$
is a rational map that is given by the~linear system of surfaces
in $|\mathcal{O}_{\mathbb{P}^{4}}(2)\vert_{Q}|$ that contain the
curve $C$. Then
$$
-K_{X}\qlineq \frac{3}{2}S+\frac{1}{2}E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 2/3$.
\end{proof}


\begin{lemma}
\label{lemma:2-16} Suppose that $\gimel(X)=2.16$.
Then $\mathrm{lct}(X)\leqslant 1/2$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
V_{4}\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{2},}
$$
where $V_{4}\subset\mathbb{P}^{5}$ is a smooth complete
intersection of two quadric hypersurfaces, $\alpha$ is a blow up
of an irreducible conic $C\subset V_{4}$, the morphism $\beta$ is
a conic bundle, and $\psi$ is a rational map that is given by
the~linear system of surfaces in
$|\mathcal{O}_{\mathbb{P}^{5}}(1)\vert_{V_{4}}|$ that contain $C$.
Then
$$
-K_{X}\sim 2S+E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 1/2$.
\end{proof}


\begin{lemma}
\label{lemma:2-17}  Suppose that $\gimel(X)=2.17$.
Then $\mathrm{lct}(X)\leqslant 2/3$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
Q\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{3},}
$$
where $Q\subset\mathbb{P}^{4}$ is a smooth quadric threefold, the
morphisms $\alpha$ and $\beta$ are blow ups of smooth elliptic
curves $C\subset Q$ and $Z\subset\mathbb{P}^{3}$~of~degree~$5$,
respectively, and the map $\psi$ is given by the~linear system of
surfaces in $|\mathcal{O}_{\mathbb{P}^{4}}(2)\vert_{Q}|$ that
contain the curve $C$. Then
$$
-K_{X}\qlineq \frac{3}{2}S+\frac{1}{2}E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{3}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 2/3$.
\end{proof}

\begin{lemma}
\label{lemma:2-20} Suppose that $\gimel(X)=2.20$.
Then $\mathrm{lct}(X)\leqslant 1/2$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
V_{5}\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{2},}
$$
where $V_{5}\subset\mathbb{P}^{6}$ is a smooth intersection of
$\mathrm{Gr}(2,5)\subset\mathbb{P}^9$ with a linear subspace of
dimension~$6$, the morphism $\alpha$ is a blow up of a cubic curve
$\mathbb{P}^1\cong C\subset V_{5}$, the morphism $\beta$ is a
conic bundle, and $\psi$ is given by the~linear system of surfaces
in $|\mathcal{O}_{\mathbb{P}^{6}}(1)\vert_{V_{5}}|$ that contain
the curve $C$. Then
$$
-K_{X}\sim 2S+E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{2}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 1/2$.
\end{proof}


\begin{lemma}
\label{lemma:2-21}  Suppose that $\gimel(X)=2.21$.
Then $\mathrm{lct}(X)\leqslant 2/3$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
Q\ar@{-->}[rr]_{\psi}&&Q,}
$$
where $Q\subset\mathbb{P}^{4}$ is a smooth quadric threefold, the
morphisms $\alpha$ and $\beta$ are blow ups of smooth rational
curves $C\subset Q$ and $Z\subset Q$ of degree~$4$, respectively, and
$\psi$ is a birational map that is given by the~linear system of
surfaces in $|\mathcal{O}_{\mathbb{P}^{4}}(2)\vert_{Q}|$ that
contain the curve $C$. Then
$$
-K_{X}\qlineq \frac{3}{2}S+\frac{1}{2}E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{4}}(1))\vert_{Q}|$,
and $E$ is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 2/3$.
\end{proof}


\begin{lemma}
\label{lemma:2-22} Suppose that $\gimel(X)=2.22$.
Then $\mathrm{lct}(X)\leqslant 1/2$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&X\ar@{->}[dl]_{\alpha}\ar@{->}[dr]^{\beta}&\\%
V_{5}\ar@{-->}[rr]_{\psi}&&\mathbb{P}^{3},}
$$
where $V_{5}\subset\mathbb{P}^{6}$ is a smooth intersection of
$\mathrm{Gr}(2,5)\subset\mathbb{P}^9$ with a linear subspace of
dimension~$6$, the~morphisms $\alpha$ and $\beta$ are blow ups of
a conic $C\subset V_{5}$ and a quartic $\mathbb{P}^1\cong
Z\subset\mathbb{P}^{3}$,~respectively, and $\psi$ is given by
the~linear system of surfaces in
$|\mathcal{O}_{\mathbb{P}^{6}}(1)\vert_{V_{5}}|$ that contain the
curve $C$. Then
$$
-K_{X}\sim 2S+E,
$$
where $S\in|\beta^{*}(\mathcal{O}_{\mathbb{P}^{3}}(1))|$, and $E$
is the exceptional divisor of $\alpha$. We see that
$\mathrm{lct}(X)\leqslant 1/2$.
\end{proof}


\begin{lemma}
\label{lemma:3-13} Suppose that $\gimel(X)=3.13$.
Then $\mathrm{lct}(X)\leqslant 1/2$.
\end{lemma}

\begin{proof}
There is a commutative diagram
$$
\xymatrix{
&&&&\mathbb{P}^{2}&&&&\\
W_{2}\ar@{->}[dd]_{\beta_{2}}\ar@{->}[urrrr]^{\alpha_{2}}&&&&&&&&W_{3}\ar@{->}[ullll]_{\beta_{3}}\ar@{->}[dd]^{\alpha_{3}}\\
&&&&X\ar@{->}[uu]_{\phi_1}\ar@{->}[drrrr]_{\phi_2}\ar@{->}[dllll]^{\phi_3}
\ar@{->}[dd]^{\pi_{1}}\ar@{->}[ullll]^{\pi_{2}}\ar@{->}[urrrr]_{\pi_{3}}&&&&\\
\mathbb{P}^{2}&&&&&&&&\mathbb{P}^{2}\\
&&&&W_{1}\ar@{->}[ullll]^{\alpha_{1}}\ar@{->}[urrrr]_{\beta_{1}}&&&&}
$$
such that $W_{i}\subset\mathbb{P}^2\times\mathbb{P}^2$ is a
divisor of bi-degree $(1,1)$, the morphisms $\alpha_{i}$ and
$\beta_{i}$ are $\mathbb{P}^{1}$-bundles, $\pi_{i}$ is a blow
up of a smooth curve $C_{i}\subset W_{i}$ of bi-degree $(2, 2)$
such that
$$
\alpha_{i}\big(C_{i}\big)\subset\mathbb{P}^2\supset\beta_{i}\big(C_{i}\big)
$$
are irreducible conics, and $\phi_i$ is a conic bundle.
Let $E_{i}$ be the exceptional divisor of
$\pi_{i}$. Then
$$
-K_{X}\sim 2H_{1}+E_{1}\sim 2H_{2}+E_{2}\sim 2H_{3}+E_{3}\sim
E_{1}+E_{2}+E_{3},
$$
where
$H_{i}\in|\phi_{i}^*(\mathcal{O}_{\mathbb{P}^{2}}(1))|$.
We see that $\mathrm{lct}(X)\leqslant 1/2$.
\end{proof}

\begin{remark}
Let us use the notation of Lemma~\ref{lemma:3-13}
and assume that $\lct(X)<1/2$. Then there is an effective $\Q$-divisor
$D\qlineq -K_X$ such that the pair $(X, \lambda D)$ is not log canonical for
some $\lambda< 1/2$. Since $\lct(W_i)=1/2$ by
Theorem~\ref{theorem:del-Pezzo}, one has
$$\varnothing\neq\LCS(X, \lambda D)\subset E_1\cap E_2\cap E_3.$$
In particular, by Theorem~\ref{theorem:connectedness} the locus $\LCS(X,
\lambda D)$ consists of a single point $P$; note that $P$ is an
intersection
$P=F_1\cap F_2\cap F_3$ of three curves $F_i$ such that $F_2\cup F_3$
(resp., $F_1\cup F_3$, $F_1\cup F_2$) is a reducible fiber of the conic
bundle $\phi_1$ (resp., $\phi_2$, $\phi_3$).
\end{remark}

\section{The Big Table}
\label{section:appendix}


This section contains the list of nonsingular Fano threefolds. We
follow the notation and the numeration of these in~\cite{IsPr99}.
We also assume the following conventions
\begin{itemize}
\item the symbol $V_i$ denotes a smooth Fano threefold such that
$-K_{X}\sim 2H$ and
$$
\mathrm{Pic}\big(V_{i}\big)=\mathbb{Z}\big[H\big],
$$
where $H$ is a Cartier divisor on $V_{i}$, and $H^{3}=8i\in\{8,16,\ldots,40\}$,%
\item the symbol $W$ denotes a divisor on $\P^2\times\P^2$ of bidegree $(1,1)$
(or, that is the same, the variety $\P(T_{\P^2})$),%
\item the symbol $V_{7}$ denotes a blow up of $\P^3$ at a point (or, that
is the same, the variety
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$),%
\item the symbol $Q$ denotes a smooth quadric hypersurface in $\mathbb{P}^{4}$,%
\item the symbol $S_i$ denotes a smooth del Pezzo surface such that
$$
K_{S_{i}}^{2}=i\in\big\{1,\ldots,8\big\},
$$
where $S_{8}\not\cong\mathbb{P}^{1}\times\mathbb{P}^{2}$,%
\item the symbol $\mathbb{F}_{1}$ denotes
a blow up of $\P^2$ at a point (or, that is the same, the surface
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(1))$).%
\end{itemize}

The fourth column of Table~\ref{table:Fanos} contains the values of
global log canonical thresholds of the corresponding Fano varieties.
The symbol $\star$ near a number means that $\lct(X)$
is calculated for a general $X$ with a given deformation type.
If we know only the upper bound $\lct(X)\le\alpha$, we write ${}\le\alpha$
in the instead of the exact value of $\lct(X)$, and the symbol ? means that
we don't know any reasonable upper bound (apart from a trivial
$\lct(X)\le 1$).

\small{\renewcommand\arraystretch{1.6}
\begin{longtable}{|c|c|p{12cm}|c|}
\caption{Smooth Fano threefolds}\label{table:Fanos}\\
\hline $\gimel(X)$ & $-K_X^3$ &  Brief description & $\mathrm{lct}(X)$ \\
\hline $1.1$ & $2$ & a hypersurface in $\mathbb{P}(1,1,1,1,3)$ of degree $6$ & $1\star$\\
\hline $1.2$ & $4$ & a hypersurface in $\mathbb{P}^4$ of degree $4$
or\hfill\break a double cover of smooth quadric in $\mathbb{P}^{4}$
branched over a surface of degree $8$ & ? \\
\hline $1.3$ & $6$ & a complete intersection of a quadric and a cubic in
$\mathbb{P}^{5}$ & ? \\
\hline $1.4$ & $8$ & a complete intersection of three quadrics
$\mathbb{P}^{6}$ & ? \\
\hline $1.5$ & $10$ & a section of $\mathrm{Gr}(2,5)\subset\mathbb{P}^9$ by
quadric and linear subspace of dimension~$7$ & ? \\
\hline $1.6$ & $12$ & a section of the Hermitian symmetric space
$M=G/P\subset\mathbb{P}^{15}$\hfill\break of type DIII  by a linear
subspace of dimension~$8$ & ? \\
\hline $1.7$ & $14$ & a section of $\mathrm{Gr}(2,6)\subset\mathbb{P}^{14}$
by a linear subspace of codimension $5$ & ? \\
\hline $1.8$ & $16$ & a section of the Hermitian symmetric space
$M=G/P\subset \mathbb{P}^{19}$\hfill\break of type CI  by a linear subspace
of dimension~$10$ & ${}\le 6/7$\\
\hline $1.9$ & $18$ & a section of the $5$-dimensional rational
homogeneous contact\hfill\break manifold $G_2/P\subset\mathbb{P}^{13}$  by
a linear subspace of dimension~$11$ &  ${}\le 4/5$\\
\hline $1.10$ & $22$ & a zero locus of three sections of the rank $3$
vector bundle $\bigwedge^2\mathcal{Q}$,\hfill\break where $\mathcal{Q}$ is
the universal quotient bundle on $\mathrm{Gr}(7,3)$ &  ${}\le 2/3$\\
\hline $1.11$ & $8$ & $V_{1}$ that is a hypersurface in $\mathbb{P}(1,1,1,2,3)$ of degree $6$ & $1/2$\\
\hline $1.12$ & $16$ & $V_{2}$ that is a hypersurface in $\mathbb{P}(1,1,1,1,2)$ of degree $4$ & $1/2$\\
\hline $1.13$ & $24$ & $V_{3}$ that is a hypersurface in $\mathbb{P}^{4}$ of degree $3$ & $1/2$\\
\hline $1.14$ & $32$ & $V_{4}$ that is a complete intersection of two quadrics in $\mathbb{P}^{5}$ & $1/2$\\
\hline $1.15$ & $40$ & $V_{5}$ that is a section of $\mathrm{Gr}(2,5)\subset\mathbb{P}^9$ by linear subspace of  codimension $3$ & $1/2$\\
\hline $1.16$ & $54$ & $Q$ that is a hypersurface in $\mathbb{P}^{4}$ of degree $2$ & $1/3$\\
\hline $1.17$ & $64$ & $\mathbb{P}^{3}$ & $1/4$\\
\hline $2.1$ & $4$ & a blow up of the Fano threefold $V_1$ along an elliptic curve\hfill\break that is an intersection of two divisors from $|-\frac{1}{2}K_{V_1}|$   & $1/2$\\%
\hline $2.2$ & $6$ & a double cover of $\mathbb{P}^1\times\mathbb{P}^2$
whose branch locus is a divisor of bidegree $(2, 4)$   &  ${}\le 13/14$\\%
\hline $2.3$ & $8$ & the blow up of the Fano threefold $V_2$ along an elliptic curve\hfill\break that is an intersection of two divisors from $|-\frac{1}{2}K_{V_2}|$   & $1/2$\\%
\hline $2.4$ & $10$ & the blow up of $\mathbb{P}^3$ along an intersection of two cubics   & $3/4\star$ \\
\hline $2.5$ & $12$ & the blow up of the threefold $V_3\subset\mathbb{P}^4$ along a plane cubic   & $1/2\star$ \\
\hline $2.6$ & $12$ & a divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of
bidegree $(2, 2)$ or\hfill\break a double cover of $W$ whose branch locus
is a surface in $|-K_W|$  & ? \\
\hline $2.7$ & $14$ & the blow up of $Q$ along the intersection of two
divisors from $|\mathcal{O}_Q (2)|$   &  ${}\le 2/3$\\
\hline $2.8$ & $14$ & a double cover of $V_7$ whose branch locus is a surface $|-K_{V_7}|$ & $1/2\star$\\
\hline $2.9$ & $16$ & the blow up of $\mathbb{P}^3$ along a curve of degree
$7$ and genus~$5$\hfill\break which is an intersection of cubics   &
${}\le 3/4$\\
\hline $2.10$ & $16$ & the blow up of $V_4\subset\mathbb{P}^5$ along an
elliptic curve\hfill\break which is an intersection of two hyperplane
sections   &$1/2\star$  \\
\hline $2.11$ & $18$ & the blow up of $V_3$ along a line   & $1/2\star$\\
\hline $2.12$ & $20$ & the blow up of $\mathbb{P}^3$ along a curve of
degree $6$ and genus~$3$\hfill\break which is an intersection of cubics   &
${}\le 3/4$\\
\hline $2.13$ & $20$ & the blow up of $Q\subset\mathbb{P}^4$ along a curve
of degree $6$ and genus $2$   &  ${}\le 2/3$\\
\hline $2.14$ & $20$ & the blow up of $V_5\subset\mathbb{P}^6$ along an elliptic curve\hfill\break which is an intersection of two hyperplane sections   &$1/2\star$  \\
\hline $2.15$ & $22$ & the blow up of $\mathbb{P}^3$ along the intersection of a quadric and a cubic surfaces & $1/2\star$ \\
\hline $2.16$ & $22$ & the blow up of $V_4\subset\mathbb{P}^5$ along a
conic   & ${}\le 1/2$ \\
\hline $2.17$ & $24$ & the blow up of $Q\subset\mathbb{P}^4$ along an
elliptic curve of degree~$5$ &  ${}\le 2/3$\\
\hline $2.18$ & $24$ & a double cover of $\mathbb{P}^1\times\mathbb{P}^2$ whose branch locus is a divisor of bidegree $(2, 2)$   & $1/2$\\
\hline $2.19$ & $26$ & the blow up of $V_4\subset\mathbb{P}^5$ along a line   & $1/2\star$\\
\hline $2.20$ & $26$ & the blow up of $V_5\subset\mathbb{P}^6$ along a
twisted cubic   &  ${}\le 1/2$\\
\hline $2.21$ & $28$ & the blow up of $Q\subset\mathbb{P}^4$ along a
twisted quartic   &  ${}\le 2/3$\\
\hline $2.22$ & $30$ & the blow up of $V_5\subset\mathbb{P}^6$ along a
conic   &  ${}\le 1/2$\\
\hline $2.23$ & $30$ & the blow up of $Q\subset\mathbb{P}^4$ along a curve of degree $4$ that is an intersection of a surface in $|\mathcal{O}_{\mathbb{P}^{4}}(1)\vert_{Q}|$ and a surface in $|\mathcal{O}_{\mathbb{P}^{4}}(2)\vert_{Q}|$ & $1/3\star$\\
\hline $2.24$ & $30$ & a divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1, 2)$   & $1/2\star$\\
\hline $2.25$ & $32$ & the blow up of $\mathbb{P}^3$ along an elliptic curve which is an intersection of two quadrics   & $1/2$ \\
\hline $2.26$ & $34$ & the blow up of the threefold $V_5\subset\mathbb{P}^6$ along a line & $1/2\star$\\
\hline $2.27$ & $38$ & the blow up of $\mathbb{P}^3$ along a twisted cubic   & $1/2$\\
\hline $2.28$ & $40$ & the blow up of $\mathbb{P}^3$ along a plane cubic   & $1/4$\\
\hline $2.29$ & $40$ & the blow up of $Q\subset\mathbb{P}^4$ along a conic   & $1/3$\\
\hline $2.30$ & $46$ & the blow up of $\mathbb{P}^3$ along a conic   & $1/4$\\
\hline $2.31$ & $46$ & the blow up of $Q\subset\mathbb{P}^4$ along a line   & $1/3$\\
\hline $2.32$ & $48$ & $W$ that is a divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1, 1)$  & $1/2$\\
\hline $2.33$ & $54$ & the blow up of $\mathbb{P}^3$ along a line  & $1/4$\\
\hline $2.34$ & $54$ & $\mathbb{P}^1\times\mathbb{P}^2$  & $1/3$\\
\hline $2.35$ & $56$ & $V_7\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$  & $1/4$\\
\hline $2.36$ & $62$ & $\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$  & $1/5$\\
\hline $3.1$ & $12$ & a double cover of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ branched in a divisor of tridegree $(2, 2, 2)$  & $3/4\star$\\
\hline $3.2$ & $14$ & a divisor on a $\mathbb{P}^{2}$-bundle $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(-1,-1)\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(-1,-1))$\hfill\break such that $X\in|L^{{}\otimes 2}\otimes\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(2,3)|$, where $L$ is the tautological line bundle & $1/2\star$\\
\hline $3.3$ & $18$ & a divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of tridegree  $(1, 1, 2)$  & $2/3\star$\\
\hline $3.4$ & $18$ & the blow up of the Fano threefold $Y$ with $\gimel(Y)=2.18$ along a smooth fiber\hfill\break of the composition $Y\to\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ of the double cover with the projection  & $1/2$\\
\hline $3.5$ & $20$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^2$ along a curve $C$ of bidegree $(5, 2)$\hfill\break such that the composition  $C\hookrightarrow\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ is an embedding  & $1/2\star$\\
\hline $3.6$ & $22$ & the blow up of $\mathbb{P}^3$ along a disjoint union of a line and an elliptic curve of degree~$4$  & $1/2\star$\\
\hline $3.7$ & $24$ & the blow up of the threefold $W$ along an elliptic curve\hfill\break that is an intersection of two  divisors from $|-\frac{1}{2}K_W|$  & $1/2\star$\\
\hline $3.8$ & $24$ & a divisor in $|(\alpha\circ\pi_1)^*(\O_{\P^2}(1))\otimes\pi_2^*(\O_{\P^2}(2))|$, where $\pi_{1}\colon\F_1\times\P^2\to\F_1$\hfill\break and $\pi_{2}\colon\F_1\times\P^2\to\P^2$ are projections, and $\alpha\colon\F_1\to\P^2$ is a blow up of a point& $1/2\star$\\
\hline $3.9$ & $26$ & the blow up of a cone $W_4\subset\mathbb{P}^6$ over the Veronese surface  $R_4\subset\mathbb{P}^5$\hfill\break with center in a disjoint union of the vertex and a quartic on $R_4\cong\mathbb{P}^2$  & $1/3$\\
\hline $3.10$ & $26$ & the blow up of $Q\subset\mathbb{P}^4$ along a disjoint union of two conics & $1/2$ \\
\hline $3.11$ & $28$ & the blow up of the threefold $V_7$ along an elliptic curve\hfill\break that is an intersection of  two divisors from $|-\frac{1}{2}K_{V_7}|$  & $1/2$\\
\hline $3.12$ & $28$ & the blow up of $\mathbb{P}^3$ along a disjoint union of a line and a twisted cubic  & $1/2$\\
\hline $3.13$ & $30$ & the blow up of
$W\subset\mathbb{P}^2\times\mathbb{P}^2$ along a curve $C$ of bidegree $(2,
2)$\hfill\break such that $\pi_{1}(C)\subset\mathbb{P}^2$ and
$\pi_{2}(C)\subset\mathbb{P}^{2}$ are irreducible conics,\hfill\break where
$\pi_{1}\colon W\to\mathbb{P}^2$ and $\pi_{2}\colon W\to\mathbb{P}^2$ are
natural projections &  ${}\le 1/2$\\
\hline $3.14$ & $32$ & the blow up of $\mathbb{P}^3$ along a disjoint union of a plane cubic curve that\hfill\break is contained in a  plane $\Pi\subset\mathbb{P}^{3}$ and a point that is not contained in $\Pi\cong\mathbb{P}^{2}$  & $1/2$\\
\hline $3.15$ & $32$ & the blow up of $Q\subset\mathbb{P}^4$ along a disjoint union of a line and a conic  & $1/2$ \\
\hline $3.16$ & $34$ & the blow up of $V_7$ along a proper transform via the blow up $\alpha\colon V_7\to\mathbb{P}^3$\hfill\break of twisted rational cubic curve passing through the center of the blow up $\alpha$  & $1/2$\\
\hline $3.17$ & $36$ & a divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of tridegree $(1, 1, 1)$  & $1/2$\\
\hline $3.18$ & $36$ & the blow up of $\mathbb{P}^3$ along a disjoint union of a line and a conic & $1/3$\\
\hline $3.19$ & $38$ & the blow up of $Q\subset\mathbb{P}^4$ at two non-collinear points  & $1/3$\\
\hline $3.20$ & $38$ & the blow up of $Q\subset\mathbb{P}^4$ along a disjoint union of two lines & $1/3$\\
\hline $3.21$ & $38$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^2$ along a curve of bidegree  $(2, 1)$  & $1/3$\\
\hline $3.22$ & $40$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^2$ along a conic in a fiber of the projection $\mathbb{P}^{1}\times\mathbb{P}^2\to\mathbb{P}^1$  & $1/3$\\
\hline $3.23$ & $42$ & the blow up of $V_7$ along a proper transform via the blow up $\alpha\colon V_7\to\mathbb{P}^3$\hfill\break of an irreducible conic passing through the center of the blow up $\alpha$  & $1/4$\\
\hline $3.24$ & $42$ & $W\times_{\mathbb{P}^2}\mathbb{F}_1$, where $W\to\mathbb{P}^2$ is a $\mathbb{P}^1$-bundle and $\mathbb{F}_1\to\mathbb{P}^2$ is the
blow up  & $1/3$\\
\hline $3.25$ & $44$ & the blow up of $\mathbb{P}^3$ along a disjoint union of two lines & $1/3$\\
\hline $3.26$ & $46$ & the blow up of $\mathbb{P}^3$ with center in a disjoint union of a point and a line  & $1/4$\\
\hline $3.27$ & $48$ & $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$  & $1/2$\\
\hline $3.28$ & $48$ & $\mathbb{P}^1\times\mathbb{F}_1$  & $1/3$\\
\hline $3.29$ & $50$ & the blow up of the Fano threefold $V_7$ along a line
in $E\cong\P^2$,\hfill\break where $E$ is the exceptional divisor of the
blow up $V_7\to\mathbb{P}^3$  & $1/5$ \\
\hline $3.30$ & $50$ & the blow up of $V_7$ along a proper transform via the blow up $\alpha\colon V_7\to\mathbb{P}^3$\hfill\break of a line that passes through the center of the blow up $\alpha$   & $1/4$\\
\hline $3.31$ & $52$ & the blow up of a cone over a smooth quadric in $\mathbb{P}^3$ at the vertex & $1/3$\\
\hline $4.1$ & $24$ & divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of multidegree $(1, 1, 1, 1)$  & $1/2$\\
\hline $4.2$ & $28$ & the blow up of the cone over a smooth quadric $S\subset\nlb\mathbb{P}^3$\hfill\break along a disjoint union of the vertex and an elliptic curve on $S$  & $1/2$\\
\hline $4.3$ & $30$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ along a curve of tridegree $(1, 1, 2)$  & $1/2$\\
\hline $4.4$ & $32$ & the blow up of the smooth Fano threefold $Y$ with $\gimel(Y)=3.19$\hfill\break along the proper transform of a conic on the quadric $Q\subset\P^4$\hfill\break that passes through the both centers of the blow up $Y\to Q$  & $1/3$\\
\hline $4.5$ & $32$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^2$ along a disjoint union of\hfill\break two irreducible curves of bidegree $(2, 1)$ and $(1, 0)$  & $3/7$\\
\hline $4.6$ & $34$ & the blow up of $\mathbb{P}^3$ along a disjoint union of three lines & $1/2$ \\
\hline $4.7$ & $36$ & the blow up of $W\subset\mathbb{P}^2\times\mathbb{P}^2$ along a disjoint union of\hfill\break two irreducible curves of bidegree $(0, 1)$ and $(1, 0)$  & $1/2$\\
\hline $4.8$ & $38$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ along a curve of tridegree $(0, 1, 1)$  & $1/3$\\
\hline $4.9$ & $40$ & the blow up of the smooth Fano threefold $Y$ with
$\gimel(Y)=3.25$\hfill\break along a curve that is contracted by the blow
up $Y \to\mathbb{P}^3$  &  $1/3$\\
\hline $4.10$ & $42$ & $\mathbb{P}^1\times S_7$  & $1/3$\\
\hline $4.11$ & $44$ & the blow up of $\mathbb{P}^1\times\mathbb{F}_1$
along a curve $C\cong\P^1$ such that $C$ is contained\hfill\break in a
fiber $F\cong\mathbb{F}_{1}$ of the projection
$\P^1\times\mathbb{F}_{1}\to\P^1$ and $C\cdot C=-1$ on $F$ &  $1/3$\\
\hline $4.12$ & $46$ & the blow up of the smooth Fano threefold $Y$ with $\gimel(Y)=2.33$\hfill\break along two curves that are contracted by the blow up
$Y\to\mathbb{P}^3$ & $1/4$\\
\hline $4.13$ & $26$ & the blow up of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ along a curve of tridegree $(1, 1, 3)$  & $1/2\star$\\
\hline $5.1$ & $28$ & the blow up of the smooth Fano threefold $Y$ with $\gimel(Y)=2.29$\hfill\break along three curves that are contracted by the
blow up $Y\to Q$  &  $1/3$\\
\hline $5.2$ & $36$ & the blow up of the smooth Fano threefold $Y$
with $\gimel(Y)=3.25$ along\hfill\break
two curves $C_{1}\ne C_{2}$ that are contracted by the blow up
$\phi\colon Y\to\nlb\mathbb{P}^3$\hfill\break  and that
are contained in the same exceptional divisor of the blow up $\phi$  &
$1/3$\\
\hline $5.3$ & $36$ & $\mathbb{P}^1\times S_6$  & $1/2$\\
\hline $5.4$ & $30$ & $\mathbb{P}^1\times S_5$  & $1/2$\\
\hline $5.5$ & $24$ & $\mathbb{P}^1\times S_4$  & $1/2$\\
\hline $5.6$ & $18$ & $\mathbb{P}^1\times S_3$  & $1/2$\\
\hline $5.7$ & $12$ & $\mathbb{P}^1\times S_2$  & $1/2$\\
\hline $5.8$ & $6$ & $\mathbb{P}^1\times S_1$  & $1/2$\\
\hline
\end{longtable}

}

\normalsize
\appendix

\section{By Jean-Pierre Demailly. On Tian's invariant and log canonical thresholds}%
\label{section:alpha}

The goal of this appendix is to relate log canonical thresholds with
the $\alpha$-invariant introduced by G.\,Tian~\cite{Ti87} for the study of
the existence of K\"ahler--Einstein metrics. The approximation
technique of closed positive $(1,1)$-currents introduced in~\cite{De92} is
used to show that the $\alpha$-invariant actually coincides with the
log canonical threshold.

Algebraic geometers have been aware of this fact after~\cite{DeKo01}
appeared, and several papers have used it consistently in the latter
years (see e.g.~\cite{JoKo01b},~\cite{BoGaKo05}). However, it turns out that
the required result is stated only in a local analytic form
in~\cite{DeKo01},
in a language which may not be easily recognizable by algebraically
minded people. Therefore, we will repair here the lack of a proper
reference by stating and proving the statements required for the
applications to projective varieties, e.g.\ existence of
K\"ahler--Einstein metrics on Fano varieties and Fano orbifolds.

Usually, in these applications, only the case of the anticanonical
line bundle $L=-K_X$ is considered. Here we will consider more generally
the case of an arbitrary line bundle $L$ (or $\Q$-line bundle $L$) on a
complex  manifold $X$, with some additional restrictions which will
be stated later.

Assume that $L$ is equipped with a \textit{singular hermitian metric} $h$
(see e.g.~\cite{De90}). Locally, $L$ admits trivializations
$\theta:L_{|U}\simeq U\times\mathbb{C}$, and on $U$ the metric $h$ is given
by a weight function $\varphi$ such that
$$
\Vert\xi\Vert_h^2 =|\xi|^2e^{-2\varphi(z)},\qquad z\in U,~~\xi\in L_z
$$
when $\xi\in L_z$ is identified with a complex number. We are interested
in the case where $\varphi$ is (at the very least) a locally integrable
function for the Lebesgue measure, since it is then possible to compute the
curvature form
$$
\Theta_{L,h}=\frac{i}{\pi}\ddbar\varphi
$$
in the sense of distributions. We have $\Theta_{L,h}\ge 0$ as a
$(1,1)$-current
if and only if the weights $\varphi$ are plurisubharmonic functions. In
the sequel we will be interested only in that case. Let us first introduce
the concept of complex singularity exponent, following e.g.~\cite{Va82},
\cite{Va83b}, \cite{ArGV85} and~\cite{DeKo01}.

\begin{definition}
\label{definition:A1} If $K$ is a compact subset of $X$, we define
the complex singularity exponent $c_K(h)$ of the metric $h$,
written locally as $h=e^{-2\varphi}$, to be the supremum of all
positive numbers $c$ such that  $h^c=e^{-2c\varphi}$ is integrable
in a neighborhood of every point $z_0\in K$, with respect to the
Lebesgue measure in holomorphic coordinates centered at~$z_0$.
\end{definition}

Now, we introduce a generalized version of Tian's invariant $\alpha$, as
defined in~\cite{Ti87} (see also~\cite{Siu88}).

\begin{definition}
\label{definition:A2} Assume that $X$ is a compact manifold and
that $L$ is a pseudo-effective line bundle, i.\,e. $L$ admits a
singular hermitian metric $h_0$ with $\Theta_{L,h_0}\ge 0$. If $K$
is a compact subset of $X$, we put
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}}c_K(h)
$$
where $h$ runs over all singular hermitian metrics on $L$ such that
$\Theta_{L,h}\ge 0$.
\end{definition}

In algebraic geometry, it is more usual to look instead at linear systems
defined by a family of linearly independent sections
$\sigma_0,\sigma_1,\ldots\sigma_N\in
H^0(X,L^{\otimes m})$. We denote by $\Sigma$ the vector subspace generated by
these sections and by
$$
|\Sigma|:=P(\Sigma)\subset|mL|:=P(H^0(X,L^{\otimes m}))
$$
the corresponding linear system. Such an $(N+1)$-tuple of sections
$\sigma=(\sigma_j)_{0\le j\le N}$ defines a
singular hermitian metric $h$ on $L$ by putting in any trivialization
$$
\Vert \xi\Vert^2_h
=\frac{|\xi|^2}{\big(\sum_j|\sigma_j(z)|^2\big)^{1/m}}
=\frac{|\xi|^2}{|\sigma(z)|^{2/m}},\qquad \xi\in L_z,
$$
hence $h(z)=|\sigma(z)|^{-2/m}$ with
$\varphi(z)=\frac{1}{m}\log|\sigma(z)|=\frac{1}{2m}\log\sum_j|\sigma_j(z)|^2$
as the associated weight function. Therefore, we are interested in the
number $c_K(|\sigma|^{-2/m})$. In the case of a single section
$\sigma_0$ (corresponding to a one-point linear system), this is the
same as the log canonical threshold $\lct_K(X, \frac{1}{m}D)$ of the
associated divisor $D$, in the notation
of~Section~\ref{section:intro}. We will also use the formal notation
$\lct_K(X, \frac{1}{m}|\Sigma|)$ in the case of a higher dimensional
linear system $|\Sigma|\subset|mL|$.

Now, recall that the line bundle $L$ is
said to be \textit{big} if the Kodaira--Iitaka dimension
$\kappa(L)$ equals $n=\dim_\mathbb{C}(X)$. The main result of this
appendix is

\begin{theorem}
\label{theorem:A3} Let $L$ be a big line bundle on a compact
complex manifold $X$. Then for every compact set $K$ in $X$ we
have
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h)
=\inf_{m\in\Z_{>0}}\inf_{D\in|mL|}
\lct_K\Big(X, \frac{1}{m}D\Big).
$$
\end{theorem}

Observe that the inequality
$$
\inf_{m\in\Z_{>0}}\inf_{D\in|mL|}\lct_K\Big(X, \frac{1}{m}D\Big)
\geq \inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h)
$$
is trivial, since any divisor $D\in|mL|$ gives rise to a singular
hermitian metric~$h$. The converse inequality will follow from the
approximation technique of~\cite{De92} and some elementary analysis.  The
proof is parallel to the proof of \cite[Theorem~4.2]{DeKo01},
although the language used there was somewhat different. In any case,
we use again the crucial concept of multiplier ideal sheaves: if
$h$ is a singular hermitian metric with local plurisubharmonic
weights $\varphi$, the multiplier ideal sheaf $\mathcal{I}(h)\subset\O_X$
(also denoted by $\mathcal{I}(\varphi)$) is the ideal sheaf defined by
$$
\mathcal{I}(h)_z=\Big\{f\in\O_{X,z}\ \Big\vert\ \exists\;\hbox{a neighborhood $V\ni z$ such that~}\int_V|f(x)|^2e^{-2\varphi(x)}d\lambda(x)<+\infty\Big\}%
$$
where $\lambda$ is the Lebesgue measure. By Nadel
(see~\cite{Na90}), this is a coherent analytic sheaf on $X$.
Theorem~\ref{theorem:A3} has a more precise version which can be
stated as follows.

\begin{theorem}
\label{theorem:A4} Let $L$ be a line bundle on a compact complex
manifold $X$ possessing a singular hermitian metric $h$ with
$\Theta_{L,h}\ge \eps\omega$ for some $\eps>0$ and some smooth
positive definite hermitian $(1,1)$-form $\omega$ on~$X$. For
every real number $m>0$, consider the space
$\mathcal{H}_m=H^0(X,L^{\otimes m}\otimes\mathcal{I}(h^m))$ of
holomorphic sections $\sigma$ of $L^{\otimes m}$ on $X$ such that
$$
\int_X|\sigma|_{h^m}^2dV_\omega
=\int_X|\sigma|^2e^{-2m\varphi}dV_\omega<+\infty,
$$
where $dV_\omega=\frac{1}{m!}\omega^m$ is the hermitian volume form.
Then for $m\gg 1$, $\mathcal{H}_m$
is a non zero finite dimensional Hilbert space
and we consider the closed positive $(1,1)$-current
$$
T_m=\frac{i}{2\pi}\ddbar\Big(\frac{1}{2m}\log\sum_k|g_{m,k}|^2\Big)
=\frac{i}{2\pi}\ddbar\Big(\frac{1}{2m}\log\sum_k|g_{m,k}|_h^2\Big)
+\Theta_{L,h}
$$
where $(g_{m,k})_{1\le k\le N(m)}$ is an orthonormal basis of~$\mathcal{H}_m$.
Then$\;:$
\begin{list}{\labelitemi}{\leftmargin=0.6em}
\item[(i)] For every trivialization  $L_{|U}\simeq U\times\mathbb{C}$
on a cordinate open set $U$ of $X$ and every compact set $K\subset U$,
there are constants $C_1,C_2>0$ independent of $m$ and $\varphi$ such that
$$
\varphi(z)-\frac{C_1}{m}\le
\psi_m(z):=\frac{1}{2m}\log\sum_k|g_{m,k}(z)|^2
\le\sup_{|x-z|<r}\varphi(x)+\frac{1}{m}\log\frac{C_2}{r^n}
$$
for every $z\in K$ and $r\le\frac{1}{2}d(K,\partial U)$. In particular,
$\psi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$, hence $T_m$ converges weakly
to $T=\Theta_{L,h}$.
\smallskip
\item[(ii)] The Lelong numbers $\nu(T,z)=\nu(\varphi,z)$ and
$\nu(T_m,z)=\nu(\psi_m,z)$ are related by
$$
\nu(T,z)-\frac{n}{m}\le\nu(T_m,z)\le
\nu(T,z)\quad\text{ for every $z\in X$.}
$$
\item[(iii)] For every compact set $K\subset X$, the complex
singularity
exponents of the metrics given locally by $h=e^{-2\varphi}$ and
$h_m=e^{-2\psi_m}$ satisfy
$$
c_K(h)^{-1}-\frac{1}{m}\le c_K(h_m)^{-1}\le c_K(h)^{-1}.
$$
\end{list}
\end{theorem}

\begin{proof} The major part of the proof is a small variation
of the arguments already explained in~\cite{De92} (see
also~\cite[Theorem~4.2]{DeKo01}).
We give them here in some detail for the convenience of the reader.
\medskip

\noindent
(i) We note that $\sum|g_{m,k}(z)|^2$ is the square of the norm of
the evaluation linear form $\sigma\mapsto \sigma(z)$ on $\mathcal{H}_m$, hence
$$
\psi_m(z)=\sup_{\sigma\in B(1)}\frac{1}{m}\log|\sigma(z)|
$$
where $B(1)$ is the unit ball of $\mathcal{H}_m$. For
$r\le \frac{1}{2}d(K,\partial\Omega)$, the mean value inequality
applied to the plurisubharmonic function $|\sigma|^2$ implies
$$
\begin{aligned}
|\sigma(z)|^2&\le\frac{1}{\pi^nr^{2n}/n!}\int_{|x-z|<r}
|\sigma(x)|^2d\lambda(x)\\
&\le\frac{1}{\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|x-z|<r}\varphi(x)\Big)
\int_\Omega|\sigma|^2e^{-2m\varphi}d\lambda.
\end{aligned}
$$
If we take the supremum over all $\sigma\in B(1)$ we get
$$
\psi_m(z)\le\sup_{|x-z|<r}\varphi(x)+\frac{1}{2m}
\log\frac{1}{\pi^nr^{2n}/n!}
$$
and the right hand inequality in (i) is proved. Conversely, the
Ohsawa--Takegoshi extension theorem~\cite{OhT87}, \cite{Ohs88}
applied to the $0$-dimensional
subvariety $\{z\}\subset U$ shows that for any $a\in\mathbb{C}$ there is
a holomorphic function $f$ on $U$ such that $f(z)=a$ and
$$
\int_U|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},
$$
where $C_3$ only depends on $n$ and~$\diam~U$. Now, provided $a$ remains in
a compact set $K\subset U$, we can use a cut-off function $\theta$ with
support in $U$ and equal to $1$ in a neighborhood of $a$, and solve the
$\dbar$-equation $\dbar g=\dbar(\theta f)$ in the $L^2$ space associated
with the
weight $2m\varphi+2(n+1)|\log|z-a|$, that is, the singular hermitian
metric $h(z)^m|z-a|^{-2(n+1)}$ on~$L^{\otimes m}$. For this, we apply
the standard Andreotti--Vesentini--H\"ormander $L^2$ estimates
(see e.g.~\cite{De82}
for the required version). This is possible for $m\ge m_0$ thanks to the
hypothesis that $\Theta_{L,h}\ge \varepsilon\omega>0$, even if $X$
is non K\"ahler ($X$ is in any event a Moishezon variety from our
assumptions). The bound $m_0$ depends only on $\varepsilon$ and the
geometry
of a finite covering of $X$ by compact sets $K_j\subset U_j$, where
$U_j$ are coordinate balls (say); it is independent of the point $a$ and
even of the metric~$h$. It follows that $g(a)=0$ and therefore
$\sigma=\theta f-g$ is a holomorphic section of $L^{\otimes m}$ such that
$$
\int_X|\sigma|^2_{h^m}dV_\omega=
\int_X|\sigma|^2e^{-2m\varphi}dV_\omega\le C_4
\int_U|f|^2e^{-2m\varphi}dV_\omega\le C_5
|a|^2e^{-2m\varphi(z)},
$$
in particular,
$\sigma\in \mathcal{H}_m=H^0(X,L^{\otimes m}\otimes\mathcal{I}(h^m))$.
We fix $a$ such that the right hand side is~$1$. This gives
the inequality
$$
\psi_m(z)\ge\frac{1}{m}\log|a|=\varphi(z)-\frac{\log C_5}{2m}
$$
which is the left hand part of statement (i).
\medskip

\noindent
(ii) The first inequality in (i) implies $\nu(\psi_m,z)\le\nu(\varphi,z)$.
In the opposite direction, we find
$$
\sup_{|x-z|<r}\psi_m(x)\le\sup_{|x-z|<2r}\varphi(x)+
\frac{1}{m}\log\frac{C_2}{r^n}.
$$
Divide by $\log r<0$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\psi_m,x)\ge\nu(\varphi,x)-\frac{n}{m}.
$$
(iii) Again, the first inequality (in (i) immediately yields $h_m\le C_6
h$,
hence $c_K(h_m)\ge c_K(h)$. For the converse inequality, since we
have $c_{\,\cup K_j}(h)=\min_j c_{K_j}(h)$, we can assume without loss
of generality that $K$ is contained in a trivializing open patch $U$
of~$L$.
Let us take $c<c_K(\psi_m)$. Then, by definition, if
$V\subset X$ is a sufficiently small open neighborhood of $K$,
the H\"older inequality
for the conjugate exponents $p=1+mc^{-1}$ and $q=1+m^{-1}c$
implies, thanks to equality $\frac{1}{p}=\frac{c}{mq}$,
$$
\begin{aligned}
\int_V & e^{-2(m/p)\varphi}dV_\omega
=\int_V\Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}\Big)^{1/p}
\Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/mq}dV_\omega\cr
&\le
\left(\int_X\sum_{1\le k\le
N(m)}|g_{m,k}|^2e^{-2m\varphi}dV_\omega\right)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}\cr
&= N(m)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}<+\infty.
\end{aligned}
$$
From this we infer $c_K(h)\ge m/p$, i.e., $c_K(h)^{-1}\le
p/m=1/m+c^{-1}$.
As $c<c_K(\psi_m)$ was arbitrary, we get $c_K(h)^{-1}\le
1/m+c_K(h_m)^{-1}$
and the inequalities of (iii) are proved.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{theorem:A3}]
Given a big line bundle $L$ on~$X$, there exists a modification
$\mu:\tilde{X}\to X$ of $X$ such that $\tilde{X}$ is projective
and $\mu^*L=\O(A+E)$ where $A$ is an ample divisor and $E$ an
effective divisor with rational coefficients. By pushing forward
by $\mu$ a smooth metric $h_A$ with positive curvature on $A$, we
get a singular hermitian metric $h_1$ on $L$ such that
$\Theta_{L,h_1}\ge\mu_*\Theta_{A,h_A}\ge \varepsilon\omega$
on~$X$. Then For any $\delta>0$ and any singular hermitian metric
$h$ on $L$ with $\Theta_{L,h}\ge 0$, the interpolated metric
$h_\delta=h_1^\delta h^{1-\delta}$ satisfies
$\Theta_{L,h_\delta}\ge\delta\varepsilon\omega$. Since $h_1$ is
bounded away from $0$, it follows that $c_K(h)\ge
(1-\delta)c_K(h_\delta)$ by monotonicity. By
Theorem~\ref{theorem:A4}~(iii) applied to $h_\delta$, we infer
$$
c_K(h_\delta)=\lim_{m\to+\infty} c_K(h_{\delta,m}),
$$
and we also have
$$
c_K(h_{\delta,m})\ge \lct_K\Big(\frac{1}{m}D_{\delta,m}\Big)
$$
for any divisor $D_{\delta,m}$ associated with a section $\sigma\in
H^0(X,L^{\otimes m}\otimes\mathcal{I}(h_\delta^m))$, since the metric
$h_{\delta,m}$
is given by $h_{\delta,m}=(\sum_k|g_{m,k}|^2)^{-1/m}$ for an orthornormal
basis
of such sections. This clearly implies
$$
c_K(h)\ge\liminf_{\delta\to 0}\;\liminf_{m\to+\infty}\;
\lct_K\Big(\frac{1}{m}D_{\delta,m}\Big)\ge
\inf_{m\in\Z_{>0}}\inf_{D\in|mL|}
\lct_K\Big(\frac{1}{m}D\Big).\eqno\square
$$

\vskip10pt
In the applications, it is frequent to have a finite or compact group
$G$ of automorphisms of $X$ and to look at $G$-invariant objects, namely
$G$-equivariant metrics on $G$-equivariant line bundles~$L\;$; in the case
of a reductive algebraic group $G$ we simply consider a compact real form
$G^{\mathbb{R}}$ instead of $G$ itself.

One then gets an $\alpha$ invariant $\alpha_{G,K}(L)$ by
looking only at $G$-equivariant metrics in
Definition~\ref{definition:A2}. All contructions made are then
$G$-equivariant, especially $\mathcal{H}_m\subset |mL|$ is a $G$-invariant
linear system. For every $G$-invariant compact set $K$ in $X$, we thus
infer
$$
\alpha_{G,K}(L)
:=\inf_{\{h\;G~\hbox{\scriptsize equiv.},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\Z_{>0}}~~\inf_{|\Sigma|\subset|mL|,~\Sigma^G=\Sigma}~~
\lct_K\Big(\frac{1}{m}|\Sigma|\Big).
\leqno({\rm A}.5)
$$
When $G$ is a finite group, one can pick for $m$ large enough a
$G$-invariant divisor $D_{\delta,m}$ associated with a
$G$-invariant section $\sigma$, possibly after multiplying $m$ by
the order of $G$. One then gets the slightly simpler equality
$$
\alpha_{G,K}(L)
:=\inf_{\{h\;G~\hbox{\scriptsize equiv.},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\Z_{>0}}~~\inf_{D\in|mL|^G}~~\lct_K\Big(\frac{1}{m}D\Big).
\leqno({\rm A}.6)
$$
In a similar manner, one can work on an orbifold $X$ rather than on a non
singular variety. The $L^2$ techniques work in this setting with almost
no change ($L^2$ estimates are essentially insensitive  to
singularities, since one can just use an orbifold metric on
the open set of regular points).
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}
