A universal Torelli theorem for elliptic surfaces

2017/06/02 by C. S. Rajan, Rajan, C. S., S. Subramanian +1
Mathematics · #11G05 #11G99 #14D99 #14J27 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.AG #math.NT #math.RT #msc:11G05 #msc:11G99 #msc:14D99 #msc:14J27

paper · pdf · doi:10.48550/arxiv.1706.00564

42 pages, Revised, minor modifications, correcting typos

arxiv created 2017/07/16 · arxiv updated 2017/07/18

Abstract

Given two semistable, non potentially isotrivial 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 Néron-Severi lattices of the base changed elliptic surfaces for all finite separable maps B→ 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 A_n-1 to that of A_dn-1, indexed by natural numbers d, which are closed under composition.

Related

• Isotrivial elliptic surfaces in positive characteristic
• Derived isogenies and isogenies for abelian surfaces
• Automorphisms of Torelli groups
• Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group
• Automorphism groups of rational elliptic and quasi-elliptic surfaces in all characteristics