ℛ15 is of general type

Abstract

We prove that the moduli space R 15 of Prym curves of genus 15 is of general type. To this end we exhibit a virtual divisor D 15 on R 15 as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that K R 15 is big. This complements a 2010 result of Farkas and Ludwig: now the spaces R g are known to be of general type for g 14.

Prym enumerative geometry and a Hurwitz divisor in $\overline{\mathcal{R}}_{2i}$

“…An important problem regarding this moduli space is understanding its birational geometry. As such, we recognize the important role played by Hurwitz divisors in proving that M g is of general type when g ≥ 24, see [HM82], [Har84] , [EH87] and that R g is of general type when g ≥ 13, g = 16, see [FL10], [Bru16] and [FJP21]. Along with those, several other Hurwitz divisors appear in the literature in [Dia85], [vdGK12], [Far09] and [Bud21].…”

“…The algebraic theory of Prym curves developed by Mumford, together with the modular interpretation of R g provided by Beauville laid the foundation for an algebraic geometric study of Prym curves. Through this perspective, the associated map P g : R g → A g−1 to the moduli space of principally polarized Abelian varieties was used to provide an algebraic proof of the Schottky-Jung relations, see [Mum74], and to understand the birational geometry of the moduli of Prym varieties, see [FL10], [Bru16], [FV16], [FJP21] and the references therein.…”

A Hurwitz divisor on the moduli of Prym curves

“…and by describing explicitly the points and their multiplicity as in the proof of Proposition 4.1 we deduce Corollary 4. 6 The degree of the map…”

“…To outline its importance, we recall that R g comes equipped with a map P g : R g → A g−1 to the moduli space of principally polarized abelian varieties of dimension g − 1. This natural application relating curves to Prym varieties inside A g−1 was used to provide an algebraic proof of the Schottky-Jung relations, see [24], and, among others, to understand the birational geometry of the moduli of Prym varieties, see [6,12,16] and the references therein.…”

Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

“…It is known (see [FL10] and [Bru16]) that R g,2 is of general type for g 14 while R g,3 is known to be of general type for g 12 (see [CEFS13]). We can use the divisor B 8,3 previously constructed to prove the following: Theorem 1.2.…”