Theta divisors whose Gauss map has a fiber of positive dimension

TL;DR

This paper constructs specific families of abelian varieties with irreducible theta divisors containing abelian subvarieties, demonstrating cases where the Gauss map has positive-dimensional fibers, and provides bounds on Andreotti-Mayer loci.

Contribution

It introduces new examples of abelian varieties with theta divisors whose Gauss map fibers are positive-dimensional, advancing understanding of their geometric properties.

Findings

Existence of abelian varieties with theta divisors having positive-dimensional Gauss map fibers

Lower bounds on the dimension of Andreotti-Mayer loci

Construction of irreducible theta divisors containing abelian subvarieties

Abstract

We construct families of principally polarized abelian varieties whose theta divisor is irreducible and contains an abelian subvariety. These families are used to construct examples when the Gauss map of the theta divisor is only generically finite and not finite. That is, the Gauss map in these cases has at least one positive-dimensional fiber. We also obtain lower-bounds on the dimension of Andreotti-Mayer loci.