Lifting Prym Divisors on a Bielliptic Loop of Loops

Abstract

Let $C$ be a bielliptic curve over an algebraically closed nonarchimedean field of odd characteristic or characteristic zero, and let $f \colon \widetilde{C} \to C$ be an étale double cover. We prove that, if $f$ specializes to a certain double cover of a loop of loops, then every Prym divisor on the loop of loops lifts to a Prym divisor of the same rank on $\widetilde{C}$. As consequences, we compute the dimension of the Prym-Brill-Noether variety $V^r (C,f)$, show that it is reduced and irreducible when the dimension is positive, and compute its cohomology class.