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Let $X$ be a (smooth) algebriac algebraic variety (over$\mathbb{C}$). over $\mathbb{C}$). Let $G \subset Aut(X)$ \operatorname{Aut}(X)$be a subgroup of automorphisms of$X$. Is it true (when is it true ? ) that for any$x\in X$the closure$\overline{O_x}$of the orbit of$x$,$\overline{O_x}$, x$ is a (may be possibly singular) subvariety (subscheme) or subscheme of $X$?

If not, can stronger hypotheses be given to guarantee a subscheme structure? In the case I am interested in, $G$ is isomorphic to $\mathbb{Z}$.

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# closure of orbit of a group action on a variety

Let $X$ be a (smooth) algebriac variety (over$\mathbb{C}$). Let $G \subset Aut(X)$ be a subgroup of automorphisms of $X$. Is it true (when is it true? ) that for any $x\in X$ the closure of the orbit of $x$, $\overline{O_x}$, is a (may be singular) subvariety (subscheme) of $X$?

In the case I am interested in, $G$ is isomorphic to $\mathbb{Z}$.