Invariants and separating morphisms for algebraic group actions
The first part of this paper is a refinement of Winkelmann’s work on invariant rings and quotients of algebraic group actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient X//GX//G given by the possibly not finitely generated ring of inva...
| Главные авторы: | , |
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| Формат: | Journal article |
| Опубликовано: |
Springer Berlin Heidelberg
2015
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| Итог: | The first part of this paper is a refinement of Winkelmann’s work on invariant rings and quotients of algebraic group actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient X//GX//G given by the possibly not finitely generated ring of invariants is “almost” an algebraic variety, and that the quotient morphism π:X→X//Gπ:X→X//G has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective. These general results are then refined for actions of the additive group GaGa , where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separating variety, which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts’ famous example of a an action of GaGa on 7-dimensional affine space with a non-finitely generated ring of invariants. |
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