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arXiv · 2510.07137

Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities

Simon Crawford·Susan J. Sierra
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Abstract

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 $\Lambda$, 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 $\text{mod-}\Lambda$ 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.

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Simon Crawford, Susan J. Sierra. 2025-10-08. Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities. https://arxiv.org/abs/2510.07137

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