Andreotti-Mayer loci and the Schottky problem

TL;DR

This paper establishes a lower bound for the codimension of the Andreotti-Mayer locus and characterizes cases where this bound is achieved, linking it to hyperelliptic and Jacobian loci in specific genera.

Contribution

It provides a new lower bound for the codimension of the Andreotti-Mayer locus and identifies the special cases where this bound is attained.

Findings

Lower bound for codimension of N_{g,1} established

Equality cases identified as hyperelliptic locus in genus 4 and Jacobian locus in genus 5

Study of tangential degeneracy of Theta translates related to boundary of loci

Abstract

We prove a lower bound for the codimension of the Andreotti-Mayer locus N_{g,1} and show that the lower bound is reached only for the hyperelliptic locus in genus 4 and the Jacobian locus in genus 5. In relation with the boundary of the Andreotti-Mayer loci we study subvarieties of principally polarized abelian varieties (B,Theta) parametrizing points b such that Theta and the translate Theta_b are tangentially degenerate along a variety of a given dimension.