2014/02/18 by Emilie Dufresne, Hanspeter Kraft · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Geometry #Affine transformation #Affine variety #Algebraic Geometry and Number Theory #Algebraic cycle #Algebraic group #Algebraic number #Algebraic variety #Geometric invariant theory #Invariant (physics) #Morphism #Polynomial and algebraic computation #Quotient #math.AC #math.AG
paper · pdf · doi:10.1007/s00209-015-1420-0
23 pages
arxiv created 2014/02/18 · openalex publication_date 2015/01/24 · arxiv updated 2016/02/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 π\colon 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 \mathbbGa, 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 \mathbbGa on 7-dimensional affine space with a non-finitely generated ring of invariants.