closure of orbit of a group action on a variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:46:42Z http://mathoverflow.net/feeds/question/74666 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74666/closure-of-orbit-of-a-group-action-on-a-variety closure of orbit of a group action on a variety Mohammad F.Tehrani 2011-09-06T16:28:41Z 2011-09-06T18:12:11Z <p>Let $X$ be a (smooth) algebraic variety (over $\mathbb{C}$). Let $G \subset \operatorname{Aut}(X)$ be a subgroup of automorphisms of $X$. Is it true that for any $x\in X$ the closure $\overline{O_x}$ of the orbit of $x$ is a (possibly singular) subvariety or subscheme of $X$? </p> <p>If not, can stronger hypotheses be given to guarantee a subscheme structure? In the case I am interested in, $G$ is isomorphic to $\mathbb{Z}$.</p> http://mathoverflow.net/questions/74666/closure-of-orbit-of-a-group-action-on-a-variety/74668#74668 Answer by Allen Knutson for closure of orbit of a group action on a variety Allen Knutson 2011-09-06T16:47:49Z 2011-09-06T16:47:49Z <p>Yes, it can be singular. Your question is a little schizophrenic in that $X$ is algebraic, but your $G$ is not, so it becomes unclear whether you want closure in the analytic or the Zariski topology. The analytic closure is very unlikely to be algebraic, so I'm going to assume Zariski.</p> <p>Let $X = {\mathbb C}^2$, $x = (1,1)$, $G$ generated by $[{4\atop 0} {0 \atop 8}]$.</p> <p>Then the Zariski closure of the orbit $G\cdot x$ is {$(a,b) : a^3 = b^2$}.</p> http://mathoverflow.net/questions/74666/closure-of-orbit-of-a-group-action-on-a-variety/74669#74669 Answer by S. Carnahan for closure of orbit of a group action on a variety S. Carnahan 2011-09-06T16:52:05Z 2011-09-06T16:52:05Z <p>One interpretation of your question (different from Allen's) is that you are asking about the topological closure of the orbit in the analytification of $X$. In that case, let $X = \mathbb{G}_m$, and let $G$ be generated by an irrational rotation. Then the closure of any point is a circle, which is not the analytification of any subvariety.</p>