# Sections of del Pezzo fibrations over \(\mathbf P^1\)

## Sho Tanimoto

Mori’s Bend and Break shows that if we deform a rational curve
while fixing two points, then it breaks into the union of rational curves.
However, in general it is difficult to control the number of components of
a breaking curve and their properties. In this talk I will talk about our
recent result for sections of del Pezzo fibrations over \(\mathbf P^1\) which we call
as Movable Bend and Break, i.e., one can break a free section of high
height to the union of one free section and one vertical free curve. Then
we discuss several applications of this result, Batyrev’s conjecture on the
number of components of the space of sections, the irreducibility of the
space of sections for certain del Pezzo fibrations, and Batyrev’s heuristic
for del Pezzo fibrations. This is joint work with Brian Lehmann.