Codimension of singular locus of quotient V/G

Question

I guess this is really standard but I simply wasn't able to figure it out. Let $V$ be a finite-dimensional complex vector space and let $G \subset GL(V)$ be a finite group. Consider the variety $V/G$. What is the codimension of the singular locus in $V/G$? I know $V/G$ smooth if and only if G is generated by reflections. The dimension of the tangent space $T_x(V/G)$ has something to do with the stabilizer subgroup $G_x$ and its fixed space. What is the precise connection?

Answer 1

You are correct about your statement regarding regular quotients, and just for people who do not know, I think you are referring to the Chevalley-Shephard-Todd-Theorem.

Now in the general case (i.e. $G$ is not generated by reflections), you still know that $V$ is normal and $G$ is reductive (all finite groups are reductive by Maschke's Theorem), so you can conclude that $V/G$ is also a normal variety, see for example 27.5.1 in the book Lie Algebras and Algebraic Groups by Patrice Tauvel and Rupert W. T. Yu.

Hence, you know that the singularities of $V/G$ have codimension at least two and you can also not expect a better bound in general - just consider any quotient $\Bbb C^2/G$ when $G$ is not generated by reflections: The variety is singular by the result you quote and so the singular locus must have codimension at most two. From this you can construct higher-dimensional examples as well.

Unfortunately, I do not know any assumptions that would improve this bound.