Consider a diagonal action of $\mathbb Z\_n$ on $\mathbb C^2$ generated by $(z\_1,z\_2)\to (\mu^pz\_1,\mu^qz\_2)$, with $\mu^n=1$. 



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> **Question.** Is it always possible to find a *smooth* blow up $X\to \mathbb C^2$ such that the $\mathbb Z\_n$-action lifts to $X$ and such that $X/\mathbb Z\_n$ is *smooth* as well?
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The same question can be asked for action of any finite group $G$ on any smooth variety (but I am especially interested in the above example).



Resolving $\mathbb Z_n$ action on $\mathbb C^2$
I think the answer is no. In your notation, take $n=5$, $p=1$ and $q=2$. If you consider the blowup at the origin $X\to \mathbb C^2$, then $\mathbb Z\_5$ acts on $X$ with two isolated fixed points: at one of the points the action has $p=q=1$ and at the other the action has $p=2$, $q=4$, which is the same as $p=1$, $q=2$. So the blow up introduces an isolated fixed point of the same type as the one we started with. 


