Autoduality of the compactified Jacobian

TL;DR

This paper proves an autoduality theorem for the compactified Jacobian of certain algebraic curves, establishing an isomorphism via the Abel map and extending classical results to more singular curves.

Contribution

It generalizes the autoduality theorem to integral projective curves with mild singularities, using determinant of cohomology and scheme-theoretic methods.

Findings

A_L^* is an isomorphism for curves with points of multiplicity 2.

The autoduality theorem holds in arbitrary characteristic.

The approach extends to families of curves with embedding dimension at most 2.

Abstract

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 A_L: C->J, which maps C into its compactified Jacobian, and form its pullback map A_L^*: Pic^0_J to 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.