Dimension of Prym–Brill–Noether varieties in the low-degree elliptic regime

Determine the dimension of the Prym–Brill–Noether variety V^r(C,f) for a nonempty open subset of the k-elliptic locus in the moduli space of étale double covers when k<r.

Background

The paper discusses prior work on Prym–Brill–Noether varieties associated with étale double covers of k-elliptic curves. For a nonempty open subset of the k-elliptic locus, that work establishes an upper bound on the dimension of V^r(C,f), and proves that the bound is attained when k≥r. The case k<r was identified as unresolved. The present paper addresses the bielliptic case k=2 through a tropical lifting theorem, but the general problem for k<r remains the explicitly stated broader question.