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 $[H]\in C{H}^2({\mathcal{A}}_g)$ 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 $\overline{N_0^{\prime }}$ of the Andreotti-Mayer divisor has minimal slope and the Iitaka dimension $\kappa (\overline{{\mathcal{A}}_5},\overline{N_0^{\prime }})$ is equal to zero.