Categorical resolutions of curves and Bridgeland stability

TL;DR

This paper develops categorical resolutions for singular curves, constructs Bridgeland stability conditions on these categories, and relates the resulting moduli spaces to classical moduli of sheaves, extending known results to singular cases.

Contribution

It provides explicit categorical resolutions for singular curves and establishes Bridgeland stability conditions, linking moduli spaces of semistable objects to classical sheaf moduli on singular and resolved curves.

Findings

Existence of Bridgeland stability conditions on categorical resolutions.

Proper moduli spaces of semistable objects are constructed.

Explicit descriptions of sheaf moduli on curves with singularities like nodes, cusps, and tacnodes.

Abstract

Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety $X$ via a triangulated category that behaves like the derived category of a smooth variety. We follow these ideas to study the bounded derived category of a singular, reduced curve $C$ (with arbitrary singularities and number of components). We start by describing an explicit categorical resolution of singularities, specializing a general construction of Kuznetsov and Lunts. We prove the existence of Bridgeland stability conditions on these categories. As a consequence, we get the existence of proper, good moduli spaces of semistable objects. If the curve $C$ is irreducible, then we relate these moduli spaces to the moduli of slope-semistable torsion-free sheaves on $C$, and to the moduli of slope-semistable vector bundles on the (geometric) resolution $\tilde{C}$. This extends classical constructions by Oda and Seshadri, Bhosle and many others. Finally, we use these results to give explicit descriptions of the moduli of torsion-free sheaves on a curve with a single node, cusp, or tacnode.