Asian Journal of Mathematics

Volume 21 (2017)

Number 1

Flops and mutations for crepant resolutions of polyhedral singularities

Pages: 1 – 46

DOI: http://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

Full Text (PDF format)

Published 16 March 2017