Prym–Brill–Noether theory for ramified double covers

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.