A pointed Prym-Petri Theorem

TL;DR

This paper develops a pointed Prym-Petri theorem by constructing Prym-Brill-Noether varieties with prescribed vanishing, realizing them as degeneracy loci, and establishing their expected dimensions, extending classical Prym theory.

Contribution

It introduces a pointed Prym-Petri map and proves a new theorem ensuring the expected dimension in the general case, extending prior unpointed results.

Findings

Construction of pointed Prym-Brill-Noether varieties as degeneracy loci.

Proof of the pointed Prym-Petri theorem confirming expected dimensions.

Identification of Prym varieties as Prym-Tyurin varieties for certain curves.

Abstract

We construct pointed Prym-Brill-Noether varieties parametrizing line bundles assigned to an irreducible \'etale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym-Petri map and prove a pointed version of the Prym-Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters and De Concini-Pragacz on the unpointed case. Finally, we show that Prym varieties are Prym-Tyurin varieties for Prym-Brill-Noether curves of exponent enumerating standard shifted tableaux times a factor of $2$, extending to the Prym setting work of Ortega.