Let $X$ be an affine algebraic variety over an algebraically closed field $k$ of characteristic zero. Let $G$ be a reductive algebraic group acting on $X$. In this setting, there exists a categorical quotient variety $X / G$. Are there nice conditions (involving $X$ and/or $G$) that imply that $\mathrm{dim}(X / G) = \mathrm{dim}(X) - \mathrm{dim}(G)$? As pointed out in the comments, at a minimum one would have to require the action to be faithful.



When does dimension behave nicely for quotients of affine algebraic varieties by the action of a group?
A sufficient condition is that there should exist a closed orbit of maximal dimension (i.e. of dimension $\dim(G)$). Indeed, when it is the case, the stable locus is nonempty, and the image of the stable locus in $X/G$ is an open subset of $X/G$ of dimension $\dim(X)-\dim(G)$. This happens for instance if $G$ acts properly on $X$ (in this case all orbits are closed of maximal dimension).


It will not be possible to give a nice necessary and sufficient condition, because the property $\dim(X/G)=\dim(X)-\dim(G)$ is not a very natural one in this context. For example, it does not imply that the action is faithful : take $G=\mathbb{G}\_m\times\mathbb{G}\_m$ acting on $X=\mathbb{A}^2$ by $(s,t)\cdot(x,y)=(sx,sy)$.


