Infinitesimal Torelli for elliptic surfaces revisited

TL;DR

This paper provides new proofs for the infinitesimal Torelli theorem for certain elliptic surfaces and discusses new counterexamples, enhancing understanding of the Torelli problem in algebraic geometry.

Contribution

It offers novel proofs for the infinitesimal Torelli theorem in specific cases of elliptic surfaces and introduces new counterexamples, expanding the theoretical framework.

Findings

New proof for nonconstant j-invariant elliptic surfaces with Euler number ≥ 24

New proof for constant j-invariant elliptic surfaces with Euler number ≥ 72

Identification of several new counterexamples to Torelli-type questions

Abstract

In this article we give a new proof for the infinitesimal Torelli theorem for minimal elliptic surfaces without multiple fibers with Euler number at least 24 for nonconstant $j$-invariant. In the case of constant $j$-invariant we find a new proof in the case of Euler number at least 72. We also discuss several new counterexamples.