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A universal Torelli-Tate theorem for elliptic surfaces

Posted in
Speaker: 
C. S. Rajan
Affiliation: 
TIFR/MPIM
Date: 
Wed, 2018-05-30 14:30 - 15:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Given two semistable elliptic surfaces over a smooth, projective 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.

This is joint work with S. Subramanian.

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