Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is the quotient of a smooth affine variety to the algebraic action of a finite group LCI? If no, is there any condition on the action that guarantees such a property? 



Are quotient varieties local complete intersections?
As discussed in the comments, $X/G$ need not in general even be Gorenstein, let alone a local complete intersection. 


Actually, if the ground field has positive characteristic, $X/G$ may not even be Cohen-Macaulay (let alone Gorenstein, let alone LCI). For example, if $X = \mathbb{A}\_k^4$, where $k = \overline{\mathbb{F}}\_2$, and $G$ is $\mathbb{Z}/4$, with the action given by cyclically permuting the coordinate axes, then $X/G$ is not Cohen-Macaulay.


If the stabilizers all have order prime to the characteristic, then $X/G$ is Cohen-Macaulay, though, by the Hochster-Eagon theorem (Proposition 13 [here](https://www.jstor.org/stable/2373744?seq=1#page_scan_tab_contents)). Without this condition, even Cohen-Macaulayness is a very delicate question, so from this point on I will assume this condition (orders of all stabilizers are coprime to the ground field characteristic) is met. For example, perhaps you are working over a characteristic zero field.


In this situation, an overly strong sufficient condition is that the stabilizers are generated by elements acting as pseudoreflections (i.e. elements $g$ in a stabilizer $G\_x$, $x\in X$ whose fixed-point sets have codimension $1$ in the neighborhood of $x$). By the [Chevalley-Shephard-Todd theorem](https://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem), this forces $X/G$ to be *smooth*.


I am not aware of a more precise sufficient condition. However, it seems to me a *necessary* condition is that the stabilizers be generated by *bireflections*, i.e. elements $g$ in a stabilizer $G\_x$, $x\in X$ whose fixed point sets have codimension $2$ in a neighborhood of $x$. See [this](https://projecteuclid.org/download/pdf_1/euclid.bams/1183548690) paper by Kac and Watanabe, which proves this in the situation of a linear action by a finite group on affine space, but it seems to me the argument works in general.


(Stabilizers-generated-by-bireflections is *not* sufficient. For examples of finite linear groups generated by bireflections whose action on affine 3-space over $\mathbb{C}$ has a quotient that is not a complete intersection, see [here](https://link.springer.com/article/10.1007/BF01165796).)


