Suppose $ X$ is a singular variety over a field $ k$ , which admits an action by a finite group $ G$ . Suppose the quotient $ X/G$ is also a variety over $ k$ . If $ Y_G$ is a resolution of $ X/G$ , does there exist a variety $ Y$ which is a resolution of $ X$ such that the action on $ X$ could be extended to an action on $ Y$ and
$ $ Y/G \simeq Y_G$ $
Put it in another way, is the resolution of the quotient “the same thing” as the quotient of (equivariant) resolution?
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