Picard Bundles and Brill–Noether Loci in the Compactified Jacobian of a Nodal Curve

Let Xk denote an irreducible projective nodal curve with k nodes and of genus g(Xk). We prove a generalization of the classical Poincaré formula to the compactified Jacobian $\bar J^0(X_k)$ the moduli space of torsion-free sheaves of rank 1 and degree 0 on Xk. 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(Xk), the Picard bundle on $\bar J^0(X_k)$ 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 $\bar J^d(X_k)$, 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(Xk)−1.