Singularities of the theta divisor at points of order two

In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor). We describe the locus within the theta-null divisor where this singularity is not an ordinary double point. By using theta function methods we first show that this locus does not equal the entire theta-null divisor (this was shown previously by O. Debarre). We then show that this locus is contained in the intersection of the theta-null divisor with the other irreducible components of the Andreotti-Mayer divisor N_0, and describe by using the geometry of the universal scheme of singularities of the theta divisor the components of this intersection that are contained in this locus. Some of the intermediate results obtained along the way of our proof were concurrently obtained independently by C. Ciliberto and G. van der Geer, and by R. de Jong.