The Kodaira dimension of the moduli space of Prym varieties

TL;DR

This paper proves that the moduli space of Prym varieties becomes of general type for genus greater than 13 by computing cycle classes and exploring syzygies, advancing understanding of its geometric properties.

Contribution

It establishes the general type of the compactified Prym moduli space for g>13 and introduces a Prym-Green conjecture on syzygies of Prym-canonical curves.

Findings

R_g is of general type for g>13

Computed classes of cycles related to Prym varieties

Extended pluricanonical forms despite non-canonical singularities

Abstract

We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on 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. In the appendix we show that even though R_g has non-canonical singularities, pluricanonical forms on R_g extend to any desingularization.