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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).

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    $\begingroup$ 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 $s\bullet((x,y),t)=(sx,sy,s^{-n}t).$ $\endgroup$
    – Jason Starr
    Commented Jun 5, 2019 at 10:04
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    $\begingroup$ 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/… $\endgroup$
    – Jason Starr
    Commented Jun 5, 2019 at 11:33
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    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented Jun 5, 2019 at 12:24
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    $\begingroup$ I think the answer to the set-theoretic question you asked can be answered affirmatively using results here: arxiv.org/pdf/1110.4236.pdf . $\endgroup$
    – Sean Lawton
    Commented Jun 5, 2019 at 13:50
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    $\begingroup$ 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. $\endgroup$
    – Sean Lawton
    Commented Jun 5, 2019 at 20:48

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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.

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    $\begingroup$ Great summary! Thanks. $\endgroup$
    – Joel Kamnitzer
    Commented Jun 12, 2019 at 13:48

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