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)—denote by θnull the locus of such ppavs. We describe the locus where this singularity is not an ordinary double point. By using theta function methods we first show (this was shown in , see below for a discussion). We then show that is contained in the intersection of the two irreducible components of the Andreotti-Mayer , and describe by using the geometry of the universal scheme of singularities of the theta divisor which components of this intersection are in .
Some of the intermediate results obtained along the way of our proof were concurrently obtained independently by C. Ciliberto and G. van der Geer in  and by R. de Jong in , version 2.