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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 $\mu^n=1$.

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?

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).

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  • $\begingroup$ I think it would be better if you may talk about your idea first... $\endgroup$
    – Majid
    Commented Jun 20, 2017 at 12:46

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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.

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  • $\begingroup$ Thank you for this example Rita! It looks indeed that what I want is impossible.. $\endgroup$
    – aglearner
    Commented Jun 20, 2017 at 13:46

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