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

### Gavril Farkas

Humboldt-Universität zu Berlin, Germany### Samuel Grushevsky

Stony Brook University, USA### R. Salvati Manni

Università di Roma La Sapienza, Italy### Alessandro Verra

Università degli studi Roma Tre, Italy

## 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.

## Cite this article

Gavril Farkas, Samuel Grushevsky, R. Salvati Manni, Alessandro Verra, Singularities of theta divisors and the geometry of $\mathcal A_5$. J. Eur. Math. Soc. 16 (2014), no. 9, pp. 1817–1848

DOI 10.4171/JEMS/476