Graded unipotent groups and Grosshans theory
Let U be a unipotent group which is graded in the sense that it has an extension H by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of U are strictly positive. We study embeddings of H in a general linear group G which possess Gros...
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| Format: | Journal article |
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Cambridge University Press
2017
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| _version_ | 1797082980623056896 |
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| author | Berczi, G Kirwan, F |
| author_facet | Berczi, G Kirwan, F |
| author_sort | Berczi, G |
| collection | OXFORD |
| description | Let U be a unipotent group which is graded in the sense that it has an extension H by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of U are strictly positive. We study embeddings of H in a general linear group G which possess Grosshans-like properties. More precisely, suppose H acts on a projective variety X and its action extends to an action of G which is linear with respect to an ample line bundle on X. Then, provided that we are willing to twist the linearisation of the action of H by a suitable (rational) character of H, we find that the H-invariants form a finitely generated algebra and hence define a projective variety X==H; moreover the natural morphism from the semistable locus in X to X==H is surjective, and semistable points in X are identified in X==H if and only if the closures of their H-orbits meet in the semistable locus. A similar result applies when we replace X by its product with the projective line; this gives us a projective completion of a geometric quotient of a U-invariant open subset of X by the action of the unipotent group U. |
| first_indexed | 2024-03-07T01:35:37Z |
| format | Journal article |
| id | oxford-uuid:95149b9a-610a-4b21-9e52-c25e0b3b4710 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:35:37Z |
| publishDate | 2017 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:95149b9a-610a-4b21-9e52-c25e0b3b47102022-03-26T23:43:42ZGraded unipotent groups and Grosshans theoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:95149b9a-610a-4b21-9e52-c25e0b3b4710Symplectic Elements at OxfordCambridge University Press2017Berczi, GKirwan, FLet U be a unipotent group which is graded in the sense that it has an extension H by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of U are strictly positive. We study embeddings of H in a general linear group G which possess Grosshans-like properties. More precisely, suppose H acts on a projective variety X and its action extends to an action of G which is linear with respect to an ample line bundle on X. Then, provided that we are willing to twist the linearisation of the action of H by a suitable (rational) character of H, we find that the H-invariants form a finitely generated algebra and hence define a projective variety X==H; moreover the natural morphism from the semistable locus in X to X==H is surjective, and semistable points in X are identified in X==H if and only if the closures of their H-orbits meet in the semistable locus. A similar result applies when we replace X by its product with the projective line; this gives us a projective completion of a geometric quotient of a U-invariant open subset of X by the action of the unipotent group U. |
| spellingShingle | Berczi, G Kirwan, F Graded unipotent groups and Grosshans theory |
| title | Graded unipotent groups and Grosshans theory |
| title_full | Graded unipotent groups and Grosshans theory |
| title_fullStr | Graded unipotent groups and Grosshans theory |
| title_full_unstemmed | Graded unipotent groups and Grosshans theory |
| title_short | Graded unipotent groups and Grosshans theory |
| title_sort | graded unipotent groups and grosshans theory |
| work_keys_str_mv | AT berczig gradedunipotentgroupsandgrosshanstheory AT kirwanf gradedunipotentgroupsandgrosshanstheory |