The degree of the Gauss map of the theta divisor

TL;DR

This paper investigates the degree of the Gauss map of the theta divisor in complex abelian varieties, establishing bounds on singularities, and proposing a stratification of the moduli space that relates to the Schottky problem.

Contribution

It introduces a new stratification of the moduli space based on the Gauss map degree and connects it to the Schottky problem in dimension four.

Findings

Bound on the multiplicity of the theta divisor at singular points

Stratification of the moduli space by Gauss map degree

Weak solution to the Schottky problem in dimension four

Abstract

We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppav's by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.