Setting:
Fix some field $k$. I am not very prudent about the field - although I'd prefer to assume as little as possible, you may assume as much as you want, the case of primary interest being $k=\mathbb{C}$.
Let $G\subseteq\mathrm{Gl}_n(k)$ be a closed, reductive subgroup and $X\subseteq\mathbb{A}_k^m$ an affine variety. $G$-variety. Assume that both $G$ and $X$ are cones, i.e. cut out by homogeneous equations. Note that $\mathrm{Gl}_n(k)$ is the open affine subset of $\mathbb{A}^{n\times n}_k$ where the determinant (a homogeneous polynomial) does not vanish. This yields an action of $k^\times$ on both $G$ and $X$. Under this action, assume that the action map
$$\begin{align*}\alpha_x:G&\longrightarrow X \\
g&\longmapsto g.x\end{align*} $$
is a morphism of $k^\times$-varieties for each $x\in X$, i.e. $\lambda g.x=g.\lambda x$ for $g\in G$ and $\lambda\in k^\times$.
Question:
I have a point $x\in X$ such that $H:=G_x$ is reductive. Let $U:= G.x$ be the orbit of $x$. Now in this very friendly setting, I have some questions about the closure $Z:=\overline U$.
How does the coordinate ring $k[Z]$ of $Z$ look like? It is known that $U$ itself is affine, and its coordinate ring can be described as the $H$-invariants of $k[G]$. However, what about $k[Z]$?
The orbit $U$ is smooth, but $Z$ isn't (in general). Since $Z$ is the union of orbits, however, the singular locus of $Z$ should also be a union of orbits. Is there some nice way to characterize $\mathrm{Sing}(Z)$?
