Abstract

If Σ is a smooth genus two curve, Σ ⊂ Pic¹(Σ) the Abel embedding 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 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 semi-stable, even) P¹ 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−−→P⁻¹(J(Σ)), defined purely in terms of the geometry of a (generic stable) P¹ bundle X → Σ in M, and also an explicit rational inverse map β : P⁻¹(J(Σ)) −−→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 : C → C in P⁻¹(J(Σ)), where C and C are curves of genera 5 and 3 respectively, the P¹ bundle ϕ : X → Σ, where X = {divisors D in C⁽⁴⁾ : π_*(D) ≡ ω_C, and h⁰(D) is even} and ϕ : X → ϕ(X)≅Σ ⊂ Pic⁴(C) is the Abel map.

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