# Fibre of GIT morphism

Let $$V$$ be an affine variety (over $$\mathbb C$$) with an action of a reductive group $$G$$. I would like to consider the morphism $$\pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G$$ Let $$v \in V$$. Assume that the orbit $$Gv$$ is closed in $$V$$. Assume also that the stabilizer of $$v$$ in $$G$$ is finite.

Question: Is the following true? What additional hypothesis should I place on $$v$$ in order to ensure that the following is true?

The scheme-theoretic fibre $$\pi^{-1}(\pi(v))$$ equals $$G v$$.

I looked in Mumford's book, but I could not find this.

Example: Here is an example where this does work. Suppose that $$G = SL_k$$ and $$V = Hom(\mathbb C^k, \mathbb C^n)$$ and let $$v$$ be an injective map. Then $$V // G$$ is the variety of pure $$k$$-tensors (the cone on the Grassmannian).

• That already fails for the scaling action of $\mu_n$ on $\mathbb{A}^2,$ $n>1.$ If you want a connected group, consider the action of $\mathbb{G}_m$ on $\mathbb{A}^2\times\mathbb{G}_m$ by $s\bullet((x,y),t)=(sx,sy,s^{-n}t).$ – Jason Starr Jun 5 '19 at 10:04
• You seem to be asking about smoothness of $\pi$ at the points of the orbit. The only smoothness result that I know for group quotients is the Chevalley-Shephard-Todd Theorem: en.wikipedia.org/wiki/… – Jason Starr Jun 5 '19 at 11:33
• Formally locally, your quotient is a quotient of a "slice" by the stabilizer group of a point (a more precise result is the Luna Etale Slice Theorem). Thus, your question about the scheme-theoretic fiber (your original question that I was answering) does reduce to a question about quotients by finite groups. – Jason Starr Jun 5 '19 at 12:24
• I think the answer to the set-theoretic question you asked can be answered affirmatively using results here: arxiv.org/pdf/1110.4236.pdf . – Sean Lawton Jun 5 '19 at 13:50
• Maybe I am missing something, but you are assuming your point is properly stable I believe (using the language in that paper) and so Theorem 3.8 says that properly stable is equivalent to there not being a limit to a 1-parameter subgroup. The fiber you are asking about is set-theoretically a union of orbits with a unique closed orbit. Two such orbits are in the same fiber if and only if their closures intersect which occurs if and only if there is a limit to a 1-parameter subgroup. So the fibre cannot have any other orbits since properly stable implies no limits. – Sean Lawton Jun 5 '19 at 20:48

I am just recording what was said in the comments so this question does not appear completely unanswered.

Let $$\pi_X:X\to X//G$$ be the GIT quotient of an affine variety over $$\mathbb{C}$$ by a reductive group $$G$$. WLOG assume the action is effective.

First, a point is properly stable if its orbit is closed and it has finite stabilizer. The locus of properly stable points is Zariski open. Then since each fibre $$\pi_X^{-1}(\pi_X(x))$$ is a union of orbits, this union contains a unique closed orbit, two such orbits are in the same fibre if and only if their closures intersect, and such intersections can be detected by 1-parameter subgroups, we can conclude that if $$x$$ is properly stable then $$\pi_X^{-1}(\pi_X(x))$$ is set-theoretically the orbit $$Gx$$.

Please see Stability of Affine G-varieties and Irreducibility in Reductive Groups by Casimiro and Florentino as a reference.

As pointed out by Jason Starr in the comments, the fibre is not generally scheme-theoretically the orbit however. A counter-example is the action of $$\mathbb{G}_m$$ on $$\mathbb{A}^2\times \mathbb{G}_m$$ by $$s\cdot((x,y),t)=(sx,sy,s^{-n}t)$$ for $$n>1.$$ As noted by the OP, this is apparent even for $$n=2$$.

We now refer to the Luna Slice Theorem; see Luna’s slice theorem and applications by Drézet. Let $$V$$ be a slice at a properly stable point $$x$$, and let $$\pi_V:V\to V//S$$ be the corresponding quotient where $$S$$ is the stabilizer of $$x$$ (necessarily a reductive subgroup). Then there is an isomorphism: $$G\times_S \pi^{-1}_V(\pi_V(x))\cong \pi_X^{-1}(\pi_X(x)).$$

So, the fibre is the scheme-theoretic orbit if it is smooth which, by the Chevalley-Shephard-Todd Theorem, occurs if and only if the stabilizer is generated by pseudoreflections.

• Great summary! Thanks. – Joel Kamnitzer Jun 12 '19 at 13:48