I've heard (a variant of) the following result being mentioned , but haven't been able to find a reference. I would like to know if the following is true, and if so, I'd very much appreciate a good reference (or a proof, if it is sufficiently short).
Let $X$ be a smooth quasi-projective variety, and let the finite group $G$ act on $X$. We know that the quotient $X/G$ exists as a scheme. If the set of fixed points of the action $X^G$ is a divisor (I guess we have to assume that the action is free outside $X^G$), then $X/G$ is smooth (and thus a variety).