Suppose we have an algebraic group $G$ acting on an algebraic variety $V$ over $Spec(k)$, with $k$ a field. Picking an element $x \in V$ then there is a bijection $G/G_x \simeq G(x)$, where $G_x$ is the stabilizer of $x$ and $G(x)$ is the orbit of $x$ under the action of $G$.

When is this bijection an isomorphism of schemes over $k$?

In Borel's book on linear algebraic groups, he writes that over an algebraically closed field the natural map $\pi: G \to G(x)$ is the quotient of $G$ by $G_x$ if and only if the differential $d\pi_e : Lie(G) \to T_x G(x)$ is surjective. Although this condition doesn't seem too hard to satisfy, are there restrictions we may impose on $V$ under which there is nothing to check?

On the other hand, what if we lift the restriction that $V$ be an algebraic variety and allow nilpotents, for example, in our structure sheaf?


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