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 $\mathbb{Z}$.
