Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.



> 
> Suppose there exists some point $x \in X$ whose stabiliser $G\_x$ is trivial. Does there exist an open subset $U \subset X$ such that the stabiliser $G\_u$ is trivial for all $u \in U$?
> 
> 
> 


I'm open to the fact that the answer might be no in general, however I have an application in mind where my $X$ and $G$ are nice. If it helps, I can also assume that


* $G$ is reductive.
* $X$ is smooth.
* For all $x \in X$, the stabiliser $G\_x$ is finite and the orbit $Gx$ is closed.



Stabilisers of group actions
Under some of your extra hypotheses, namely if $G$ is reductive and the orbit $Gx$ is closed, the answer is yes. This follows easily from Luna's slice theorem; see for instance [these lectures](http://webusers.imj-prg.fr/~jean-marc.drezet/papers/Wykno.pdf), Proposition 5.7.


