On the geometry of two dimensional Prym varieties (Q1305097)

From authors' abstract: If \(\Sigma\) is a smooth genus two curve, \(\Sigma\subset\text{Pic}^1(\Sigma)\) the Abel embedding in the degree one Picard variety, \(|2\Sigma|\) the projective space parametrizing divisors on \(\text{Pic}^1(\Sigma)\) linearly equivalent to \(2\Sigma\), and \(\text{Pic}^0(\Sigma)_2=G\cong(\mathbb Z/2\mathbb Z)^4\) the subgroup of points of order two in the Jacobian variety \(J(\Sigma)=\text{Pic}^0(\Sigma)\), then \(G\) acts on \(|2\Sigma|\) and the quotient variety \(|2\Sigma|/G\) parametrizes two fundamental moduli spaces associated with the curve \(\Sigma\). Namely, Narasimhan-Ramanan's work implies an isomorphism of \(|2\Sigma|/G\) with the space \(\mathcal M\) of (\(S\)-equivalence classes of semistable, even) \(\mathbb P^1\) bundles over \(\Sigma\), and Verra has defined a precise birational correspondence between \(|2\Sigma|/G\) and Beauville's compactification of \(\mathcal P^{-1}(J(\Sigma))\), the fiber of the classical Prym map over \(J(\Sigma)\). In this paper, we give a new (birational) construction of the composed Narasimhan-Ramanan-Verra map \(\alpha:\mathcal M\dasharrow\mathcal P^{-1}(J(\Sigma))\), defined purely in terms of the geometry of a (generic stable) \(\mathbb P^1\) bundle \(X\to\Sigma\) in \(\mathcal M\), and also an explicit rational inverse map \(\beta:\mathcal P^{-1}(J(\Sigma))\dasharrow\mathcal M\). The map \(\alpha\) 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 \(C^{(g-1)}\).