Categorical resolutions of cuspidal singularities
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915181091618816 |
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| author | Fietz, Céline |
| author_facet | Fietz, Céline |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19380 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorical resolutions of cuspidal singularities Fietz, Céline Algebraic Geometry 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. |
| title | Categorical resolutions of cuspidal singularities |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.19380 |