The Moduli Space of Abelian Varieties and the Singularities of the Theta Divisor

The object of study here is the singular locus of the theta divisor Θ of a principally polarized abelian variety (X,Θ). The special case when (X,Θ) is the Jacobian of a curve C shows that this is meaningful: singularities of Θ are closely related to the existence of special linear systems on the curve C and for curves of genus g ≥ 4 the divisor Θ is always singular. But for the general principally polarized abelian variety the theta divisor Θ is smooth. In their pioneering work [A-M] Andreotti and Mayer introduced in the moduli space Ag of principally polarized abelian varieties the loci Nk of those principally polarized abelian varieties for which Θ has a k-dimensional singular locus: