Prym-Brill-Noether theory for ramified double covers

TL;DR

This paper extends classical Prym-Brill-Noether theory to ramified double covers, providing new dimension formulas, class computations, and demonstrating generic Prym-Brill-Noether generality for Du Val curves.

Contribution

It introduces the study of Prym-Brill-Noether theory for ramified double covers, extending key classical results and computing new invariants.

Findings

Improved dimension bounds for Prym-Brill-Noether loci

Computed classes of twisted Prym-Brill-Noether loci

Proved generic Prym-Brill-Noether generality for Du Val curves

Abstract

We initiate the study of Prym-Brill-Noether theory for ramified double covers, extending several key results from classical Prym-Brill-Noether theory to this new framework. In particular, we improve Kanev's results on the dimension of pointed Prym-Brill-Noether loci for ramified double covers. Additionally, we compute the dimension of twisted Prym-Brill-Noether loci with vanishing conditions at points, thus extending the results of Tarasca. Furthermore, we compute the class of the twisted Prym-Brill-Noether loci inside (a translation of) the Prym variety, thus extending the results of de Concini and Pragacz to ramified double covers. Finally, we prove that a generic Du Val curve is Prym-Brill-Noether general.