# Properness of reductive group actions on smooth varieties

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.

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.