The compactified Picard scheme of the compactified Jacobian

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the corresponding Abel map AL:C → J̄ , which maps C into its compactified Jacobian, and form its pullback map A L : Pic J̄ → J , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then A L is known to be an isomorphism. We prove that A L always extends to a map between the natural compactifications, Pic J̄ → J̄, and that the extended map is an isomorphism if C has, at worst, ordinary nodes and cusps.