Crepant Resolutions of 3-Dimensional Quotient Singularities via Cox Rings

ABSTRACT

We study Cox rings of crepant resolutions of quotient singularities ${\mathbb{C}}^3/G$, where G is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of different resolutions, and relations between them. In particular, we treat explicitly several examples where G contains elements of age 2.

KEYWORDS:

• quotient singularity
• resolution of singularities
• crepant resolution
• Cox ring

2010 AMS Subject Classification:

• 14E15
• 14E30
• 14E16
• 14L30
• 14L24
• 14C20

Acknowledgments

The authors are very grateful to Michael Wemyss for asking questions that inspired this work. They would like to thank Jarosław Wiśniewski for discussions and Simon Keicher for his help with some useful software packages.

Funding

The first author was supported by the Polish National Science Center project 2013/08/A/ST1/00804. This project was started when she held a Dahlem Research School Postdoctoral Fellowship at Freie Universität Berlin. The second author was supported by the Polish National Science Center project 2015/17/N/ST1/02329.