Autoduality of the compactified Jacobian

We prove the following autoduality theorem for an integral projective curve C in any characteristic. Given an invertible sheaf L 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, points of multiplicity 2, then A L is an isomorphism, and forming it commutes with specializing C. Much of our work is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, we use the determinant of cohomology to construct a right inverse to A L . Then we prove a scheme-theoretic version of the theorem of the cube, generalizing Mumford’s, and use it to prove that A L is independent of the choice of L. Finally, we prove our autoduality theorem: we use the presentation scheme to achieve an induction on the difference between the arithmetic and geometric genera; here, we use a few special properties of points of multiplicity 2.