Summary: 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 ${\Acal}_g$ 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.

14K10

abelian variety, theta divisor, andreotti-Mayer loci, Schottky problem