Let X be an affine/projective variety and G an algebraic group acting algebrically on X. Pick x in X, which are the properties of the orbit G.x? I've seen examples in which it can be both open or closed. I also see people writing dim G.x, does that mean G.x is some sort of algebraic subvariety of X?
Thanks!
A standard result in the theory of algebraic groups is the following.
Theorem: Let $G$ be a connected algebraic group acting on a variety $X$. Then each orbit $Gx$ is irreducible and open in its closure (hence is an irreducible variety in its own right). Its boundary is a union of orbits of strictly smaller dimension. In particular, orbits of minimal dimension are closed (so closed orbits exist).
So indeed orbits have a dimension.