All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

Alastair Craw, Gwyn Bellamy, Travis Schedler, Steven Rayan, Hartmut Weiss

Research output: Contribution to journalArticlepeer-review

1 Citation (SciVal)

Abstract

We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4, C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C 4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n − 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n ≥ 6.

Original languageEnglish
Pages (from-to)757-793
Number of pages37
JournalJournal of Algebraic Geometry
Volume33
Issue number4
Early online date1 Apr 2024
DOIs
Publication statusPublished - 31 Dec 2024

Funding

The first and second authors were partially supported by Research Project Grant RPG-2021-149 from the Leverhulme Trust. The third author was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant. The fifth author was supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP 2026 \u201CGeometry at infinity\u201D. The authors thank Amihay Hanany for pointing out [54] and for multiple discussions, and Alastair King for the observation that forms Remark 3.9. The authors also thank the referees for their comments and corrections. The third author thanks Hiraku Nakajima and Laura Schaposnik for helpful conversations. The third and fifth authors thank Laura Fredrickson, Rafe Mazzeo, and Jan Swoboda for useful discussions.

FundersFunder number
Leverhulme Trust
Deutsche Forschungsgemeinschaft
Laura Fredrickson
Amihay Hanany
Natural Sciences and Engineering Research Council of Canada

    Fingerprint

    Dive into the research topics of 'All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces'. Together they form a unique fingerprint.

    Cite this