Cohomology of line bundles on compactified Jacobians

TL;DR

This paper proves the autoduality conjecture for line bundles on the compactified Jacobian of a singular curve and computes their cohomology, also establishing the full faithfulness of a Fourier-Mukai functor between related derived categories.

Contribution

It establishes the autoduality correspondence for line bundles on the compactified Jacobian and computes their cohomology, advancing understanding of their geometric and categorical properties.

Findings

Topologically trivial line bundles on the compactified Jacobian correspond to line bundles on the curve.

Cohomology of these line bundles is explicitly computed.

The Fourier-Mukai functor between derived categories is shown to be fully faithful.

Abstract

Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful.