2025/12/04 by Vilches, Nicolás
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.04911
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 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.