The locus of abelian varieties the theta divisor of which is singular at two components. (Le lieu des variétés abéliennes dont le diviseur thêta est singulier a deux composantes.) (French) Zbl 0781.14031

Debarre, Olivier

Ann. Sci. Éc. Norm. Supér. (4) 25, No. 6, 687-708 (1992).

Let \({\mathcal A}_ g\) denote the moduli space of principally polarized abelian varieties of dimension \(g\). The theta divisor of the generic element of \({\mathcal A}_ g\) is smooth. According to results of Andreotti, Mayer and Beauville the polarized abelian varieties with singular theta divisor form a divisor \({\mathcal N}_ g\) in \({\mathcal A}_ g\). It was shown by Mumford that \({\mathcal N}_ g\) decomposes as \({\mathcal N}_ g=\theta_{null,g}+2{\mathcal N}_ g'\) where \(\theta_{null,g}\) is the irreducible component of principally polarized abelian varieties the polarization of which can be represented by a symmetric theta divisor admitting a point of order 2 and even multiplicity. The article under review presents the following result:
For \(g \geq 4\) the divisor \({\mathcal N}_ g\) consists exactly of 2 irreducible components \(\theta_{null, g}\) and \({\mathcal N}_ g'\). Moreover the symmetric theta divisor of a generic element of \({\mathcal N}_ g'\) admits exactly two ordinary double points as singularities.

Reviewer: Ch.Birkenhake (Erlangen)

MSC:

14K25

Theta functions and abelian varieties

Keywords:

principally polarized abelian varieties; singular theta divisor