A pointed Prym–Petri Theorem

Abstract

We construct pointed Prym–Brill–Noether varieties parametrizing line bundles assigned to an irreducible étale 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 [Ann. Sci. Ëcole Norm. Sup. (4) 18 (1985), pp. 671–683] and De Concini–Pragacz [Math. Ann. 302 (1995), pp. 687–697] 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 2 , extending to the Prym setting work of Ortega [Math. Ann. 356 (2013), pp. 809–817].