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

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 ${\mathcal{A}}_{t,g-t}^{\delta }$ 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.