The Gauss Map on Theta Divisors with Transversal $\mathrm{A}_1$ Singularities

TL;DR

This paper computes the degree of the Gauss map on Theta divisors with specific singularities in abelian varieties using Lagrangian specialization, linking geometric invariants to singularity multiplicities.

Contribution

It introduces a method to compute the Gauss degree on Theta divisors with transversal A1 singularities and relates it to Samuel multiplicity of the singular locus.

Findings

Computed the Gauss degree for general abelian varieties in specific loci.

Established that the first coefficient of Lagrangian specialization equals Samuel multiplicity.

Connected geometric invariants with singularity multiplicities in Theta divisors.

Abstract

We use Lagrangian specialization to compute the degree of the Gauss map on Theta divisors with transversal $\mathrm{A}_1$ singularities. This computes the Gauss degree for a general abelian variety in the loci $\mathcal{A}^\delta_{t,g-t}$ that form some of the irreducible components of the Andreotti-Mayer loci. We also prove that the first coefficient of the Lagrangian specialization is the Samuel multiplicity of the singular locus.