  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.
