Singularities of theta divisors and the geometry of $\mathcal A_5$

Abstract

We study the codimension two locus H in \mathcal A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class [H]\in CH^2(\mathcal A_g) for every g . For g=4 , this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5 , via the Prym map we show that H\subset \mathcal A_5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of \overline{\mathcal A_5} and show that the component \overline{N_0'} of the Andreotti-Mayer divisor has minimal slope and the Iitaka dimension \kappa(\overline{\mathcal A_5}, \overline{N_0'}) is equal to zero.