2009/02/01 by Lex E. Renner, Lex Renner, Renner, Lex +2
Mathematics · #14L30 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:14L30
paper · pdf · doi:10.48550/arxiv.0902.0137
16 pages; v2. some proofs improved, change order of results in sect. 3, citations improved
openalex publication_date 2009/02/01 · arxiv created 2009/02/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be an affine algebraic group and let X be an affine algebraic variety. An action G× X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f∈ K[X]G such that f(Y) =0. We characterize this condition geometrically as follows. The action G× X → X is observable if and only if (1) there is a nonempty open subset U⊆ X consisting of closed orbits, and (2) the field K(X)G of G-invariant rational functions on X is equal to the quotient field of K[X]G. In case G is reductive, we conclude that there exists a unique, maximal, G-stable, closed subset X\soc of X such that G× X\soc → X\soc is observable. Furthermore, the canonical map X\soc// G → X//G is finite and bijective.