I have an algebraic group $G$ acting on an affine variety $X$, the orbit $O(m)$ of an element $m \in X$, and an affine curve $C$ contained in the Zariski closure $\overline{O(m)}$ of $O(m)$, such that $m \in C$. 


If we define $C^\prime = C \cap O(m) $ then it's not hard to see that $C^\prime$ is open in $C$. And therefore $C^\prime$ is again affine (follows for example from an exercise in Hartshorne). 


Let $p: G \to O(m)$ defined by $p(g) = g \cdot m$ be the orbit map. I want to prove that there is an affine subset $G^\prime$ of $G$ such that $p(G^\prime)$ is dense in $C^\prime$. 


Is the preimage of $X^\prime$ affine? If so why? If not, how do I construct such a $G^\prime$? 


For reference: The statement that such an $G^\prime$ exists can be found in the article 
[A characterization of orbit closure and applications](https://www.sciencedirect.com/science/article/pii/0021869388901998) (MR0944153) on the first page of the proof of Theorem 1.2. Clearly he has more assumptions on $X$ and $G$, but I do not see anything else that could be useful here. 



When does an affine subset of an orbit have affine preimage under the orbit map?
I was able to figure out an elementary answer with some help: 
It seems like a good hint is to look at abstract varieties or even easier use that every quasi-affine variety is an abstract variety. 


The preimage is presumably not affine, but it is a locally closed subset of $G$ since $C$ is locally closed. Therefore $p^{-1}(C)$ is a quasi-affine variety and as such it has an open cover $U\_i$ where each $U\_i$ has the structure of an affine varieties such that for each intersection $U\_i \cap U\_j$ the induced variety structures are isomorphic (see for example Hartshorne - Algebraic Geometry Section I.8 page 58 where this is stated without proof). So let 


$p^{-1}(C) = U\_1 \cup \dots \cup U\_\alpha$.


Since $p$ is surjective we have 


$C = p(U\_1) \cup \dots \cup p(U\_\alpha)$.


Since $C$ is irreducible it can't be the union of two proper closed subsets and therefore some $p(U\_i)$ has to be dense which finishes the proof. 


