The semi-invariant ring as the Cox ring of a GIT quotient
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929319340670976 |
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| author | Bellamy, Gwyn Craw, Alastair Schedler, Travis |
| author_facet | Bellamy, Gwyn Craw, Alastair Schedler, Travis |
| contents | We study GIT quotients $X_θ=V\!/\!\!/\!_θG$ whose linearisation map defines an isomorphism between the group of characters of $G$ and the Picard group of $X_θ$ modulo torsion. Our main result establishes that the Cox ring of $X_θ$ is isomorphic to the semi-invariant ring of the $θ$-stable locus in $V$. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of $n$-points in the affine plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12225 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The semi-invariant ring as the Cox ring of a GIT quotient Bellamy, Gwyn Craw, Alastair Schedler, Travis Algebraic Geometry We study GIT quotients $X_θ=V\!/\!\!/\!_θG$ whose linearisation map defines an isomorphism between the group of characters of $G$ and the Picard group of $X_θ$ modulo torsion. Our main result establishes that the Cox ring of $X_θ$ is isomorphic to the semi-invariant ring of the $θ$-stable locus in $V$. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of $n$-points in the affine plane. |
| title | The semi-invariant ring as the Cox ring of a GIT quotient |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2404.12225 |