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

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

doi:10.1090/jag/827

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

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

Department of Mathematical Sciences

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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 language	English
Pages (from-to)	757-793
Number of pages	37
Journal	Journal of Algebraic Geometry
Volume	33
Issue number	4
Early online date	1 Apr 2024
DOIs	https://doi.org/10.1090/jag/827
Publication status	Published - 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.

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

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10.1090/jag/827

https://arxiv.org/pdf/2112.09878.pdfLicence: CC BY
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Cite this

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AB - 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.

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

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