All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866912217499172864 |
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| author | Bellamy, Gwyn Craw, Alastair Rayan, Steven Schedler, Travis Weiss, Hartmut |
| author_facet | Bellamy, Gwyn Craw, Alastair Rayan, Steven Schedler, Travis Weiss, Hartmut |
| contents | 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 the quotient singularity C4/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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_09878 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces Bellamy, Gwyn Craw, Alastair Rayan, Steven Schedler, Travis Weiss, Hartmut Algebraic Geometry Representation Theory Symplectic Geometry 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 the quotient singularity C4/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. |
| title | All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces |
| topic | Algebraic Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2112.09878 |