Asian Journal of Mathematics

Volume 21 (2017)

Number 1

Flops and mutations for crepant resolutions of polyhedral singularities

Pages: 1 – 46

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n1.a1

Authors

Álvaro Nolla de Celis (Graduate School of Mathematics, Nagoya University, Nagoya, Japan; and Department of Applied Mathematics, Materials Science and Engineering and Electronic Technology, Rey Juan Carlos University, Madrid, Spain)

Yuhi Sekiya (Graduate School of Mathematics, Nagoya University, Nagoya, Japan)

Abstract

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 $G-\mathrm{Hilb}(\mathbb{C}^3)$ 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.

Keywords

crepant resolutions, polyhedral singularities, flops, mutations, moduli spaces of quiver representations

2010 Mathematics Subject Classification

14E16, 16G20

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Published 16 March 2017