Prym varieties and cubic threefolds over Z

Abstract

We develop a theory of Prym varieties over fields of characteristic 2. Using this theory we study rationality problems for conic bundles over the projective plane. Namely, we find a criterion for non-rationality of conic bundles with smooth quintic discriminant. We apply this theory to show that cubic threefolds over arbitrary fields are non-rational.

We answer a question of Deligne in the case of cubic threefolds by constructing the intermediate Jacobian of the universal cubic threefold. We use this to construct a morphism of stacks which sends a family of cubic threefolds over an arbitrary base to its intermediate Jacobian. In doing so, we show that intermediate Jacobians of families of cubic threefolds exist and are stable under base change. We also prove the Torelli theorem for cubic threefolds over arbitrary fields.

Finally, we discuss specialization properties of intermediate Jacobians and formulate some new birational invariants which measure the failure of the existence of a universal codimension 2 cycle.

Date of Award	8 Oct 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Daniel Loughran (Supervisor) & Gregory Sankaran (Supervisor)