The Kodaira dimension of the moduli space of Prym varieties

Abstract

We study the enumerative geometry of the moduli space ${\mathcal{R}}_g$ of Prym varieties of dimension $g–1$. Our main result is that the compactication of ${\mathcal{R}}_g$ is of general type as soon as $g>13$ and $g$ is different from $15$. We achieve this by computing the class of two types of cycles on ${\mathcal{R}}_g$: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical curves. We also perform a detailed study of the singularities of the Prym moduli space, and show that for $g\ge 4$, pluricanonical forms extend to any desingularization of the moduli space.