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Categorical resolutions of cuspidal singularities

Céline Fietz

TL;DR

This work develops a categorical framework for projective varieties with isolated $A_2$ cuspidal singularities by constructing crepant categorical resolutions $π_*: ilde{D} o D^b(X)$ that are Verdier localizations. The kernel of the resolution is explicitly described via spinor sheaves on the exceptional nodal quadric $Y$, with two $2$-spherical generators in even dimension and a single non-$l$-spherical generator in odd dimension; the construction leverages Lefschetz decompositions and the Clifford algebra formalism to compute Ext-algebras of spinor objects. In the important case of a cubic fourfold, the resolution restricts to a crepant resolution of the Kuznetsov component $ ext{A}_X$, yielding an equivalence with $D^b(S)$ for a smooth K3 surface $S$ and identifying the kernel with two $2$-spherical objects. The results extend earlier A$_1$-type analyses to A$_2$, providing an explicit, calculable link between derived categories of singular varieties and classical K3 geometry via spinor sheaves and Morita reductions in Clifford algebras.

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 $π_*\colon \widetilde{\mathcal{D}} \to D^b(X)$, 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.

Categorical resolutions of cuspidal singularities

TL;DR

This work develops a categorical framework for projective varieties with isolated cuspidal singularities by constructing crepant categorical resolutions that are Verdier localizations. The kernel of the resolution is explicitly described via spinor sheaves on the exceptional nodal quadric , with two -spherical generators in even dimension and a single non--spherical generator in odd dimension; the construction leverages Lefschetz decompositions and the Clifford algebra formalism to compute Ext-algebras of spinor objects. In the important case of a cubic fourfold, the resolution restricts to a crepant resolution of the Kuznetsov component , yielding an equivalence with for a smooth K3 surface and identifying the kernel with two -spherical objects. The results extend earlier A-type analyses to A, providing an explicit, calculable link between derived categories of singular varieties and classical K3 geometry via spinor sheaves and Morita reductions in Clifford algebras.

Abstract

Let be a projective variety with an isolated singularity. We study its bounded derived category and prove that there exists a crepant categorical resolution , 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 singularity, we prove that this generating set is given by two -spherical objects. If is a cubic fourfold with an isolated singularity, we show that this resolution restricts to a crepant categorical resolution of the Kuznetsov component , which is equivalent to the bounded derived category of a K3 surface.

Paper Structure

This paper contains 17 sections, 49 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions.
  3. Acknowledgments.
  4. Geometric and categorical resolutions
  5. Geometric resolutions of singularities
  6. Semiorthogonal decompositions, Serre functors and mutations
  7. Categorical resolutions of singularities
  8. Preliminaries
  9. Construction of a crepant categorical resolution of $A_2$ singularities
  10. Special case of an $A_2$ cubic fourfold
  11. Spinor sheaves on singular quadrics and Clifford algebras
  12. Spinor sheaves on singular quadrics
  13. Clifford Algebras and Morita reduction
  14. The derived category of a quadric and spinor sheaves
  15. Kernels of categorical resolutions of cuspidal singularities
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X$ be a variety with an isolated $A_2$ singularity. Then there exists a crepant categorical resolution $\pi_*\colon \widetilde{\mathcal{D}} \to D^b(X)$, which is a Verdier localization. If $X$ is even dimensional, the kernel $\ker(\pi_*)$ is generated by two $2$-spherical objects $\mathcal{T}_1

Theorems & Definitions (111)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4: GW
  • Lemma 2.5
  • proof
  • ...and 101 more