Brill-Noether loci and generated torsionfree sheaves over nodal and cuspidal curves

Abstract

We determine (semi)stability of kernels of evaluation maps of globally generated torsion-free sheaves on a nodal and cuspidal curve in some important cases. We use them to study Brill-Noether loci for torsion-free sheaves with slope μ less than two. The conditions for nonemptiness and singular sets of Brill-Noether loci are determined. Explicit descriptions of Brill-Noether loci are given in some cases. We also apply our results to study (semi)stability properties of the restrictions of the Picard bundle E_d (on the compactified Jacobian) to the images of Y in the compactified Jacobian for certain embeddings.