Picard bundles and brill–noether loci in the compactified Jacobian of a nodal curve

Abstract

Let X_k denote an irreducible projective nodal curve with k nodes and of genus g(_k). We prove a generalization of the classical Poincaré formula to the compactified Jacobian J̅⁰_k the moduli space of torsion-free sheaves of rank 1 and degree 0 on X_k. We apply it to show that the Brill–Noether loci are nonempty if the Brill–Noether number is nonnegative. We prove that for d≥2g(X_k), the Picard bundle on J̅⁰_k is stable but not ample, unlike in the case of a smooth curve. However, for the pullback of the Picard bundle to the desingularization of J̅⁰_k, the restriction to a general complete intersection subvariety of codimension k is ample. We use this to show that the Brill–Noether loci are connected if the Brill–Noether number is bigger than k. We prove that the Picard bundle is semistable for d=2g(X_k)−1.