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