User:TakuyaMurata/Linear algebraic group action

Throughout the article, G is a linear algebraic group and X a smooth scheme (or a stack) on which G acts.

If X is a quasi-affine variety and if G is a unipotent group, then its orbits on X are closed.

Let $$f: G \to X, g \mapsto gx$$ be the orbit map at x. The differential at the identity element $$df_1$$ is surjective if and only if $$G_x$$ and $$\operatorname{ker} df_1$$ have the same dimension.

The action of G on X is called principal if. A principal action is free, but the converse does not hold in general.