Properness of reductive group actions on smooth varieties

Question

Suppose that $G$ is a reductive algebraic group acting on a smooth variety $X$, and that the action has finite stabilizers. When is the action of $G$ on $X$ proper? What is an example where the action is not proper?

I am aware of a similar statement which is Proposition 0.8 in Mumford's GIT, which says that the action is proper if the geometric quotient $\phi : X \to X / G$ exists and $\phi$ is affine. I would like to know in what situations the action can be assumed proper without this assumption.

Answer 1

Actions of reductive groups with finite stabilizers on quasi-projective varieties are often not proper. The simplest example I know is given by he action of $\mathrm{PGL}_2$ on the projective space $\mathbb P^4$ of effective divisors of degree $4$ on $\mathbb P^1$. Consider the open subset $X \subseteq \mathbb P^4$ of points whose stabilizer is finite. These are of two types.

(1) four distinct points in $\mathbb P^1$, and (2) three distinct points, one of them double.

The stabilizer of a point of $X$ of type (1) is well-known to be of type $\mathbb Z/2 \times \mathbb Z/2$, while one of type (2) has $\mathbb Z/2$ as a stabilizer. Thus the generic stabilizer is larger than the stabilizer at a special point: this means that the inertia can not be finite, and the action can not be proper.