On the Geometry of Two Dimensional Prym Varieties

If Σ is a smooth genus two curve, Σ ⊂ Pic(Σ) the Abel embedding in the degree one Picard variety, |2Σ| the projective space parametrizing divisors on Pic(Σ) linearly equivalent to 2Σ, and Pic(Σ)2 = G ∼= (Z/2Z) the subgroup of points of order two in the Jacobian variety J(Σ) = Pic(Σ), then G acts on |2Σ| and the quotient variety |2Σ|/G parametrizes two fundamental moduli spaces associated with the curve Σ. Namely, Narasimhan-Ramanan’s work implies an isomorphism of |2Σ|/G with the space M of (S-equivalence classes of semistable, even) P bundles over Σ, and Verra has defined a precise birational correspondence between |2Σ|/G and Beauville’s compactification of P−1(J(Σ)) the fiber of the classical Prym map over J(Σ). In this paper we give a new (birational) construction of the composed Narasimhan-Ramanan-Verra map α : M 99K P−1(J(Σ)), defined purely in terms of the geometry of a (generic stable) P bundle X → Σ in M, and also an explicit rational inverse map β : P−1(J(Σ)) 99K M. The map α may be viewed as an analog for Prym varieties of Andreotti’s reconstruction of a curve C of genus g from the branch locus of the canonical map on the symmetric product C(g−1). The map β assigns to an étale double cover π : C̃ → C in P−1(J(Σ)), where C̃ and C are curves of genera 5 and 3 respectively, the P bundle φ : X → Σ, where X = {divisors D in C̃ : π∗(D) ≡ ωC , and h(D) is even} and φ : X → φ(X) ∼= Σ ⊂ Pic(C̃) is the Abel map.