An Explicit Non-smoothable Component of the Compactified Jacobian

TL;DR

This paper identifies a unique non-smoothable component in the compactified Jacobian of certain non-Gorenstein curves, revealing new structure in the moduli space of sheaves and related schemes.

Contribution

It demonstrates the existence and uniqueness of a non-smoothable component in the compactified Jacobian for curves with specific singularities, extending understanding of these moduli spaces.

Findings

Existence of a generically reduced component of dimension equal to the arithmetic genus.

Uniqueness of the non-smoothable component for curves with a single finite representation type singularity.

Analogous results for Hilbert schemes and Quot schemes related to the dualizing sheaf.

Abstract

This paper studies the components of the moduli space of rank 1, torsion-free sheaves, or compactified Jacobian, of a non-Gorenstein curve. We exhibit a generically reduced component of dimension equal to the arithmetic genus and prove that this is the only non-smoothable component when the curve has a unique singularity that is of finite representation type. Analogous results are proven for the Hilbert scheme of points and the Quot scheme parameterizing quotients of the dualizing sheaf.