Autoduality of the compactified Jacobian
Steven Kleiman | Esteves, Eduardo | Gagné, Mathieu
Singular curve | Jacobian | compactification | Abel map | autoduality
We prove the following autoduality theorem for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1 on C, form the corresponding Abel map A(L), which maps C into its compactified Jacobian K, and form its pullback map A(L)*, which carries P, the connected component of 0 in the Picard scheme of K, back to the Jacobian J. 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 special properties of points of multiplicity 2.