Invariants and separating morphisms for algebraic group actions

Dufresne, Emilie; Kraft, Hanspeter

Abstract

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//G given by the possibly not finitely generated ring of invariants is “almost” an algebraic variety, and that the quotient morphism π: 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 Ga, 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 Ga on 7-dimensional affine space with a non-finitely generated ring of invariants.

Citation

Dufresne, E., & Kraft, H. (2015). Invariants and separating morphisms for algebraic group actions. Mathematische Zeitschrift, 280(1-2), https://doi.org/10.1007/s00209-015-1420-0

• Journal Article Type: Article
• Acceptance Date: Nov 23, 2014
• Online Publication Date: Jan 25, 2015
• Publication Date: Jun 30, 2015
• Deposit Date: Oct 9, 2017
• Publicly Available Date: Oct 9, 2017
• Journal: Mathematische Zeitschrift
• Print ISSN: 0025-5874
• Electronic ISSN: 1432-1823
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 280
• Issue: 1-2
• DOI: https://doi.org/10.1007/s00209-015-1420-0
• Public URL: https://nottingham-repository.worktribe.com/output/753268
• Publisher URL: https://link.springer.com/article/10.1007%2Fs00209-015-1420-0
• Additional Information: The final publication is available at link.springer.com via http://dx.doi.org/10.1007/s00209-015-1420-0.