Crepant Resolutions, Mutations, and the Space of Potentials

Abstract

The McKay correspondence has had much success in studying resolutions of 3-fold quotient singularities through a wide range of tools coming from geometry, combinatorics, and representation theory. We develop a computational perspective in this setting primarily realized through a web application to explore mutations of quivers with potential and crepant triangulations. We use this to study flops between different crepant resolutions of Gorenstein toric quotient singularities and find many situations in which the mutations of a quiver with potential classifies them. The application also implements key constructions of the McKay correspondence, including the Craw–Reid procedure and the process of associating a quiver to a toric resolution.

KEYWORDS:

• McKay correspondence
• Crepant resolutions
• quivers

Acknowledgments

The authors are grateful to Tom Ducat for many helpful conversations in the early stages of this project. They are also grateful to Michael Wemyss and Franco Rota for their input. They drew much inspiration from Bernhard Keller’s quiver mutation applet² for which they thank him.

Notes

1 Freely available for use at https://marybarker.github.io/quiver_mutations.

2 Freely available at https://webusers.imj-prg.fr/∼bernhard.keller/quivermutation/.