Categorical resolutions of cuspidal singularities

TL;DR

This paper constructs explicit crepant categorical resolutions for varieties with isolated A2 singularities, describing their kernels and linking them to K3 surfaces in specific cases.

Contribution

It provides an explicit description of crepant categorical resolutions for varieties with A2 singularities, including generators and their properties, and relates these resolutions to K3 surfaces.

Findings

Existence of crepant categorical resolutions as Verdier localizations.

Explicit generators for the kernels of these resolutions.

Connection of resolutions to derived categories of K3 surfaces.

Abstract

Let $X$ be a projective variety with an isolated $A_2$ singularity. We study its bounded derived category and prove that there exists a crepant categorical resolution $\pi_*\colon \widetilde{\mathcal{D}} \to D^b(X)$, which is a Verdier localization. More importantly, we give an explicit description of a generating set for its kernel. In the case of an even dimensional variety with a single $A_2$ singularity, we prove that this generating set is given by two $2$-spherical objects. If $X$ is a cubic fourfold with an isolated $A_2$ singularity, we show that this resolution restricts to a crepant categorical resolution $\widetilde{\mathcal{A}}_X$ of the Kuznetsov component $\mathcal{A}_X \subset D^b(X)$, which is equivalent to the bounded derived category of a K3 surface.