The Kodaira dimension of the moduli space of Prym varieties

Details The Kodaira dimension of the moduli space of Prym varieties The Kodaira dimension of the moduli space of Prym varieties

2008-04-29

arxiv.org/abs/0804.4616v4

Journal of the European Mathematical Society (JEMS) 12 (2010), 755-795

Mathematics

Algebraic Geometry

36 pages. Minor corrections. To appear in the Journal of the European Mathematical Society

Scientific paper

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.

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