Flops and mutations for crepant resolutions of polyhedral singularities

TL;DR

This paper establishes a correspondence between flops of $G$-Hilb$C^3$ and mutations of McKay quivers for polyhedral groups, providing new methods to construct and analyze crepant resolutions of certain singularities.

Contribution

It introduces a one-to-one correspondence between flops and mutations, enabling systematic construction of crepant resolutions via quiver mutations and stability conditions.

Findings

Established a bijection between flops and quiver mutations.

Provided two methods to construct crepant resolutions.

Described the chamber structure where resolutions occur.

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 $G$-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.