Description
Speaker : C S Rajan (TIFR Mumbai)
Title: A universal Torelli theorem for elliptic surfaces
Abstract: Given two semistable
elliptic surfaces over a curve $C$ defined over a field of
characteristic zero or finitely generated over its prime field, we
show that any compatible family of effective isometries of the
N{\'e}ron-Severi lattices of the base changed elliptic surfaces for
all finite separable maps $B\to C$ arises from an isomorphism of the
elliptic surfaces. Without the effectivity hypothesis, we show that
the two elliptic surfaces are isomorphic.
We also determine the group of universal automorphisms of a semistable
elliptic surface. In particular, this includes showing that the
Picard-Lefschetz transformations corresponding to an irreducible
component of a singular fibre, can be extended as universal
isometries. In the process, we get a family of homomorphisms of the
affine Weyl group associated to $\tilde{A}_{n-1}$ to that of
$\tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed
under composition.