The Kodaira dimension of the moduli space of Prym varieties

We study the enumerative geometry of the moduli space Rg of Prym varieties of dimension g–1. Our main result is that the compactication of Rg 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 Rg: 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 ≥4, pluricanonical forms extend to any desingularization of the moduli space.