On parametrizing exceptional tangent cones to Prym theta divisors

Abstract

The theta divisor of a Jacobian variety is parametrized by a smooth divisor variety via the Abel map, with smooth projective linear fibers. Hence the tangent cone to a Jacobian theta divisor at any singularity is parametrized by an irreducible projective linear family of linear spaces normal to the corresponding fiber. The divisor variety X X parametrizing a Prym theta divisor Ξ \Xi , on the other hand, is singular over any exceptional point. Hence although the fibers of the Abel Prym map are still smooth, the normal cone in X X parametrizing the tangent cone of Ξ \Xi can have nonlinear fibers. In this paper we highlight the diverse and interesting structure that these parametrizing maps can have for exceptional tangent cones to Prym theta divisors. We also propose an organizing framework for the various possible cases. As an illustrative example, we compute the case of a Prym variety isomorphic to the intermediate Jacobian of a cubic threefold, where the projectivized tangent cone, the threefold itself, is parametrized by the 2 parameter family of cubic surfaces cut by hyperplanes through a fixed line on the threefold.