Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914181347803136 |
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| author | Crawford, Simon Sierra, Susan J. |
| author_facet | Crawford, Simon Sierra, Susan J. |
| contents | Let $B = \Bbbk_q[u,v]^{C_{n+1}}$ be a Type $\mathbb{A}_n$ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution $Λ$, which is an "algebraic" resolution of $B$. We construct a category $\mathcal{X}$ which serves as a "geometric" resolution of $B$ by adapting techniques from quiver GIT and show that $\mathcal{X}$ and $\text{mod-}Λ$ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of $\mathcal{X}$ corresponds to a Type $\mathbb{A}_n$ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07137 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities Crawford, Simon Sierra, Susan J. Rings and Algebras Algebraic Geometry Primary: 14A22, 14E15. Secondary: 16S38, 14E16 Let $B = \Bbbk_q[u,v]^{C_{n+1}}$ be a Type $\mathbb{A}_n$ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution $Λ$, which is an "algebraic" resolution of $B$. We construct a category $\mathcal{X}$ which serves as a "geometric" resolution of $B$ by adapting techniques from quiver GIT and show that $\mathcal{X}$ and $\text{mod-}Λ$ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of $\mathcal{X}$ corresponds to a Type $\mathbb{A}_n$ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities. |
| title | Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities |
| topic | Rings and Algebras Algebraic Geometry Primary: 14A22, 14E15. Secondary: 16S38, 14E16 |
| url | https://arxiv.org/abs/2510.07137 |