Categorical resolutions of curves and Bridgeland stability

TL;DR

The paper develops a derived categorical framework for singular curves by constructing a smooth noncommutative resolution D^b(R) and establishing Bridgeland stability on it. It proves the existence of moduli spaces of semistable objects M_σ(v) with good moduli spaces, showing these interpolate between slope semistable torsion free sheaves on the curve and vector bundles on its normalization. Through Auslander type constructions, non rational loci, and wall crossing, it provides explicit descriptions and birational relations for moduli spaces associated to nodes, cusps, and tacnodes, including higher rank phenomena. The work extends classical results of Oda Seshadri and Bhosle to a derived setting and delivers concrete moduli descriptions for simple curve singularities via categorical resolutions and stability conditions.

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.

Figures (1)

• Figure 1: Walls and chambers of $v=(2,1,2,4)$.

Theorems & Definitions (151)

• Theorem 1.1
• proof
• Theorem 1.2
• proof
• Theorem 1.3
• proof
• Theorem 1.4
• proof
• Definition 2.1: KL15*Definition 6.1
• Remark 2.2: KL15*Remark 6.2