On the geometry of two dimensional Prym varieties

If Σ is a smooth genus two curve, Σ ⊂ Pic1(Σ) the Abel em-bedding in the degree one Picard variety, |2Σ | the projective space parametrizing divisors on Pic1(Σ) linearly equivalent to 2Σ, and Pic0(Σ)2 = G ∼ = (Z/2Z)4 the subgroup of points of order two in the Jacobian variety J(Σ) = Pic0(Σ), then G acts on |2Σ | and the quotient variety |2Σ|/G parametrizes two fun-damental 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 semi-stable, even) P1 bundles over Σ, and Verra has defined a pre-cise 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) con-struction of the composed Narasimhan-Ramanan-Verra map α: M 99K P−1(J(Σ)), defined purely in terms of the ge