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.



Properness of reductive group actions on smooth varieties
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.


