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?




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




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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).



Fibre of GIT morphism
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](https://arxiv.org/abs/1110.4236)* 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](https://hal.archives-ouvertes.fr/hal-00742479/document) 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](https://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem), occurs if and only if the stabilizer is generated by pseudoreflections.


