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.