Infinitesimal study of a factorization of the Prym map

Abstract

The Prym map \(\mathcal{R}_{g} \rightarrow\mathcal{A}_{g-1}\) factors through a space Φ={X} of intrinsically polarized varieties, namely \((\tilde{C}/C) \mapsto{X} \mapsto(P,\Xi)\), where \(\pi: \tilde{C} \rightarrow{C}\) is a connected etale double cover of a smooth curve C of genus g, \(X \subset\tilde{C}^{(2g-2)}\) consists of the set of even precanonical effective divisors on \(\tilde{C}\), and (P,Ξ) is the principally polarized Prym variety associated to π. X is a connected, reduced local complete intersection of (pure) dimension g-1, and when C is non hyperelliptic X is normal and irreducible. By analogy with the proofs of the classical Torelli theorem for curves by Andreotti and by Andreotti-Mayer and Green, which factor the Jacobi map through a symmetric product of the curve, the present factorization may be used to attack the Torelli problem for Prym varieties. In [19] we have shown that X determines the Prym variety (P,Ξ), as the Albanese variety of X, and that X also determines the double cover \(\tilde{C}/C\), at least when C is non hyperelliptic and the codimension of singΞ in P is at least 5. The next challenge in this approach to the Torelli problem is to analyze the infinitesimal structure of these maps.