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Invariants and Separating Morphisms for Algebraic Group Actions

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JournalMathematische Zeitschrift
DateE-pub ahead of print - 25 Jan 2015
DatePublished (current) - Jun 2015
Issue number1-2
Volume280
Number of pages25
Pages (from-to)231–255
Early online date25/01/15
Original languageEnglish

Abstract

The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient $X/\!/\!G$ given by the possibly not finitely generated ring of invariants is "almost" an algebraic variety, and that the quotient morphism $\pi\colon X \to X/\!/\! G$ 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 $\mathbb{G}_a$, 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 $\mathbb{G}_a$ on 7-dimensional affine space with a non-finitely generated ring of invariants.

Bibliographical note

23 pages

    Research areas

  • math.AC, math.AG

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