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This has been asked on MSE here, but has not had much traffic so I will ask a similar question here as many people here enjoy topics around resolving singularities.

Let $X$ be a complex threefold with an action by a finite group $G$. Suppose the quotient $X/G$ is singular, and admits a crepant resolution, for example $G\subset \operatorname{SL}_3(\mathbb{C})$ by Bridgeland-King-Reid. Instead of working with the quotient, can you blowup the fixed locus on $X$ to get fixed divisors so that the quotient of the blowup is smooth, and a crepant resolution of the quotient? In other words, is there some $\widetilde{X}$ such that the following diagram commutes \begin{array}{c} \widetilde{X} & \to & \widetilde{X/G} & \\ \downarrow& & \downarrow \\ X & \to & X/G \end{array} where $\widetilde{X/G}$ is a crepant resolution of $X/G$?

This is possible with transversal $A_n$ singularities, and triple Veronese singularities, but for example if $G = \langle \zeta_6\times \zeta_3\times \zeta_2\rangle$ acts on $\mathbb{C}^3$, where $\zeta_n$ is a primitive $n$-th root of unity, then the $y$-axis is a transveral $A_1$, the $z$-axis is a transversal $A_2$, and these can be blown up to fixed divisors, but the origin fixed by all of $G$, the intersection of the two transversal axes, is more difficult. I have tried different ways and every approach just gives extra points fixed by $G$ that I have not been able to get a corresponding divisor.

My motivation for asking is that I am hoping to avoid having to work with the singularities in $X/G$ (which are quite nasty for some of the groups I am interested in) and instead just work with affines instead, but I have been blowing up and contracting in many different ways an unable to find a desired resolution. After many attempts, I realized it may not be possible, though perhaps I simply need to get more creative with my approach.

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  • $\begingroup$ You can start with a crepant resolution, that is a smooth variety with a $G$-torsor over an open subset. You can take $\tilde{X}$ to be the normalization of that $G$-torsor over the crepant resolution. Then its quotient by $G$ will certainly be the crepant resolution. However I don't know whether this normalization can always be obtained by blowing up. $\endgroup$
    – Will Sawin
    Feb 21, 2016 at 15:38
  • $\begingroup$ In the quasi-projective case, every birational, projective morphism can be obtained as a blowing up. However, there is no guarantee that the ideal sheaf will be equivariant, nor that the corresponding closed subscheme will have support strictly contained in the fundamental locus of the birational morphism. $\endgroup$ Feb 21, 2016 at 15:56
  • $\begingroup$ The problem with taking a crepant resolution and then normalization is that the result will typically be singular. $\endgroup$ Feb 21, 2016 at 21:57

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