Andreotti-Mayer loci and the Schottky problem

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 intersection of the Andreotti-Mayer loci with the boundary of the moduli space ${\mathcal{A}}_g$ we study subvarieties of principally polarized abelian varieties $(B,\Xi )$ parametrizing points $b$ such that $\Xi$ and the translate ${\Xi }_b$ are tangentially degenerate along a variety of a given dimension.

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