The Kodaira Dimension of the Moduli Space of Prym Varieties

Prym varieties provide a correspondence between the moduli spaces of curves and abelian varieties Mg and Ag−1, via the Prym map Pg : Rg → Ag−1 from the moduli space Rg parameterizing pairs [C, η], where [C] ∈ Mg is a smooth curve and η ∈ Pic(C)[2] is a torsion point of order 2. When g ≤ 6 the Prym map is dominant and Rg can be used directly to determine the birational type ofAg−1. It is known that Rg is rational for g = 2, 3, 4 (see [Dol] and references therein and [Ca] for the case of genus 4) and unirational for g = 5 (cf. [IGS] and [V2]). The situation in genus 6 is strikingly beautiful because P6 : R6 → A5 is equidimensional (precisely dim(R6) = dim(A5) = 15). Donagi and Smith showed that P6 is generically finite of degree 27 (cf. [DS]) and the monodromy group equals the Weyl group WE6 describing the incidence correspondence of the 27 lines on a cubic surface (cf. [D1]). There are three different proofs that R6 is unirational (cf. [D1], [MM], [V]). Verra has very recently announced a proof of the unirationality ofR7 (see also Theorem 0.8 for a weaker result). The Prymmap Pg is generically injective for g ≥ 7 (cf. [FS]), although never injective. In this range, we may regard Rg as a partial desingularization of the moduli space Pg(Rg) ⊂ Ag−1 of Prym varieties of dimension g − 1.