Prym varieties of genus four curves

TL;DR

This paper explores the geometric and algebraic properties of Prym varieties associated with genus four curves, extending classical constructions to degenerations and arbitrary fields for broader applications.

Contribution

It provides a comprehensive study of Prym varieties of genus four curves, including degenerations and extensions over arbitrary fields, enhancing understanding of their geometric structure.

Findings

Establishes a bijection between double covers and Cayley cubics containing the canonical model.

Shows Prym varieties are quadratic twists of Jacobians of genus three curves.

Extends the construction to degenerations and arbitrary fields, broadening applicability.

Abstract

Double covers of a generic genus four curve C are in bijection with Cayley cubics containing the canonical model of C. The Prym variety associated to a double cover is a quadratic twist of the Jacobian of a genus three curve X. The curve X can be obtained by intersecting the dual of the corresponding Cayley cubic with the dual of the quadric containing C. We take this construction to its limit, studying all smooth degenerations and proving that the construction, with appropriate modifications, extends to the complement of a specific divisor in moduli. We work over an arbitrary field of characteristic different from two in order to facilitate arithmetic applications.