Name: Santiago Zarzuela

Title: Betti numbers of arrangements of linear varieties 
       and local cohomology  

Abstract: 

\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

\begin{document}

\begin{center}

{\Large\sc

Betti numbers of arrangements of \\

\medskip

linear varieties and local cohomolgy }

\bigskip

{Santiago Zarzuela}

\medskip

{\footnotesize {\it Departament d'\`Algebra i Geometria
\\ Universitat de Barcelona \\ Gran Via 585 \\ Barcelona 08007, Spain}
\\ {\tt zarzuela@mat.ub.es}}

\bigskip
{Abstract for the}

\medskip

{\bf {Colloque

\smallskip
Alg\`ebre commutative Interactions avec \\

\smallskip
la g\'eom\'etrie alg\'ebrique}}

\medskip
{Grenoble, du 9 au 13 juillet 2001.}
\end{center}

\bigskip

\newcommand{\ea}{{\mathbb A}^{n}_{k}}
\newcommand{\dli}{\mbox{indlim}_{\,P}\,}
\newcommand{\dlii}{\mbox{indlim}^{(i)}_{\,P}\,}
\newcommand{\ili}{projlim}
\newcommand{\supp}{\mbox{supp}_{+}\,}
\newcommand{\cpx}{{\mathbb C}}
\newcommand{\cp}{{\mathbb C}^n}
\newcommand{\NN}{{\mathbb N}}
\newcommand{\h}{\mbox{h}}
\newcommand{\ZZ}{{\mathbb Z}}
\newcommand{\RR}{\mathbb R}
\newcommand{\Roos}{\mbox{Roos}}
\newcommand{\height}{\mbox{h}}
\newcommand{\Hom}{\mbox{Hom}}
\newcommand{\mc}{\mathcal}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\q}{\mathfrak{q}}
\newcommand{\rr}{\mathfrak{r}}
\newcommand{\poplus}{\textstyle{\bigoplus}}
\newcommand{\mm}{\mathfrak{m}}
\newcommand{\QQ}{{\mathbb Q}}
\newcommand{\Ext}{\mbox{Ext}^1_R\,}
\newcommand{\Ker}{\mbox{Ker}\,}
\newcommand{\E}[1]{\mbox{E}_{R}(R/#1)}
\newcommand{\lc}[1]{H^{#1}_{I}(R)}
\newcommand{\R}{k[x_1,\ldots,x_{n}]}
\newcommand{\chc}[2]{H^{#1}_c(#2)}
\newcommand{\vs}{\vspace{6mm}}
\renewcommand{\labelenumi}{\roman{enumi})}

\vs

Let $\ea$ denote the affine space of dimension $n$ over a field
$k$ and $X\subset \ea$ be an arrangement of linear subvarieties.
Set $R=\R$ and let $I\subset R$ denote an ideal which defines $X$.
If $k$ is a field of characteristic zero, the local cohomology
modules modules $\lc{r}$ are known to have a module structure over
the Weyl algebra $A_n(k)$, and one can therefore consider their
characteristic cycles, denoted $CC(\lc{r})$ in this talk. In
either the real or the complex case, we shall determine the Betti
numbers of the complement of the arrangement $X$ in terms of the
multiplicities of the local cohomology modules $\lc{r}$. \vs

\centerline{ (Joint work with Josep \`Alvarez-Montaner  and
Ricardo Garc\'{\i}a- L\'opez) }

\end{document}