Observable actions of algebraic groups
| dc.creator | Renner, Lex | |
| dc.creator | Rittatore, Alvaro | |
| dc.date | 2009-02-01 | |
| dc.date | 2009-02-05 | |
| dc.date.accessioned | 2026-07-07T12:37:55Z | |
| dc.date.available | 2026-07-07T12:37:55Z | |
| dc.description | Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0. We characterize this condition geometrically as follows. The action $G\times X \to X$ is observable if and only if (1) there is a nonempty open subset $U\subseteq X$ 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 $G\times X_{\soc} \to X_{\soc}$ is observable. Furthermore, the canonical map $X_{\soc}// G \to X//G$ is finite and bijective. | |
| dc.description | 16 pages; v2. some proofs improved, change order of results in sect. 3, citations improved | |
| dc.identifier | https://arxiv.org/abs/0902.0137 | |
| dc.identifier | http://arxiv.org/abs/0902.0137 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/218578 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14L30 | |
| dc.title | Observable actions of algebraic groups | |
| dc.type | text |