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 multiplica...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2017-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509417000196/type/journal_article |
| _version_ | 1811156206327169024 |
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| author | GERGELY BÉRCZI FRANCES KIRWAN |
| author_facet | GERGELY BÉRCZI FRANCES KIRWAN |
| author_sort | GERGELY BÉRCZI |
| collection | DOAJ |
| 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 linearization 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 | 2024-04-10T04:47:38Z |
| format | Article |
| id | doaj.art-db71eb37d6e54e62bb0f9966fe7dd2f5 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:38Z |
| publishDate | 2017-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-db71eb37d6e54e62bb0f9966fe7dd2f52023-03-09T12:34:44ZengCambridge University PressForum of Mathematics, Sigma2050-50942017-01-01510.1017/fms.2017.19GRADED UNIPOTENT GROUPS AND GROSSHANS THEORYGERGELY BÉRCZIFRANCES KIRWANLet $U$ 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 linearization 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 $U$ .https://www.cambridge.org/core/product/identifier/S2050509417000196/type/journal_article14L24 (primary)13A50 (secondary) |
| spellingShingle | GERGELY BÉRCZI FRANCES KIRWAN GRADED UNIPOTENT GROUPS AND GROSSHANS THEORY Forum of Mathematics, Sigma 14L24 (primary) 13A50 (secondary) |
| 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 |
| topic | 14L24 (primary) 13A50 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509417000196/type/journal_article |
| work_keys_str_mv | AT gergelyberczi gradedunipotentgroupsandgrosshanstheory AT franceskirwan gradedunipotentgroupsandgrosshanstheory |