Flops and mutations for crepant resolutions of polyhedral singularities
Flops and mutations for crepant resolutions of polyhedral singularities
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DOI：
10.4310/ajm.2017.v21.n1.a1
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发表时间：
2011-08
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通讯作者：
Álvaro Nolla de Celis;Yuhi Sekiya
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文献类型：
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作者：
Álvaro Nolla de Celis;Yuhi Sekiya
设G是一个多面体群G\子集SO（3），其类型为$\mathbb{Z}/n\mathbb{Z}$，$D_{2n}$和$\mathbb{T}$.我们证明了G$-Hilb$\mathbb{C}^3$的触发器与不改变平凡顶点的具有势的McKay图的突变之间存在一一对应。这种对应提供了两种等价的方法来构造奇异点$\mathbb{C}^3/G$的每一个射影收缩分解，这些奇异点被构造为箭图的模空间$\mathcal{M}_C$，其中某个腔室$C$在稳定性条件空间$\Theta$中具有势。此外，我们研究了异常轨迹在$\mathcal{M}_C$与相应的Q_C$之间的关系，我们明确地描述了在$\Theta$的腔室结构的一部分，其中每个这样的决议可以找到。
Let $G$ be a polyhedral group $G\subset SO(3)$ of types $\mathbb{Z}/n\mathbb{Z}$, $D_{2n}$ and $\mathbb{T}$. We prove that there exists a one-to-one correspondence between flops of $G$-Hilb$\mathbb{C}^3$ and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities $\mathbb{C}^3/G$, which are constructed as moduli spaces $\mathcal{M}_C$ of quivers with potential for some chamber $C$ in the space $\Theta$ of stability conditions. In addition, we study the relation between the exceptional locus in $\mathcal{M}_C$ with the corresponding quiver $Q_C$, and we describe explicitly the part of the chamber structure in $\Theta$ where every such resolution can be found.