Hyperplane sections of abelian surfaces

TL;DR

This paper investigates the properties of hyperplane sections of abelian surfaces, revealing that their second Wahl map has a corank of at least 2, contrasting with K3 surfaces where the Wahl map is non-surjective.

Contribution

It demonstrates that hyperplane sections of abelian surfaces can be distinguished by the surjectivity of their second Wahl map, providing a new criterion different from K3 surfaces.

Findings

Wahl map of these curves is tendentially surjective

Second Wahl map has corank at least 2

Provides a new distinguishing criterion for hyperplane sections of abelian surfaces

Abstract

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of abelian surfaces. The somewhat surprising result is that the Wahl map of such curves is (tendentially) surjective, but their second Wahl map has corank at least 2 (in fact a more precise result is proved).