The Kodaira dimension of the moduli space of Prym varieties

By G. Farkas, Katharina Ludwig. 2008

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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 of Ag−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 Prym map 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a8a4c2050f0a342d3b2074e5e58024a2cb64df2

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Citations in the corpus

Moduli of theta-characteristics via Nikulin surfaces (2011, methodology): “The conjecture is known to hold in bounded genus and has been used in [11] to show that Rg is of general type when g ≥ 14 is even.”
Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, background): “lasses in A1(Gr d): (4) a:= (fr d)∗ c 1(L)2 ,b:= (fr d)∗ c 1(L) ·c 1(ωfr d ) ,c:= (fr d)∗ c 1(ωfr d )2 = (σ 0)∗ (κ 1) M0 g . 6 The dependence on a,b,c on the choice of L is discussed in both [F2] and [FL]. We introduce the stack of gr d ’s on spin curves σ: Gr d(S 0 g/M 0 g) := S 0 g × M0 g Gr d → S”
The classification of universal Jacobians over the moduli space of curves (2010, background): “out marked points), are varieties of general type, except a ﬁnite number of casesoccurring for relatively small genus,whenthesevarieties tend to be unirational, or at least uniruled, see [HM], [EH1], [FL], [Log], [V] for illustrations of this fact. In all known cases, the transition from unirulednes”
Classical Algebraic Geometry: A Modern View (2012, background): “It is known to be of general type for g > 13 and g 6= 15 [217].”