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Let $\xi_5$ be a 5-root of the unity. We consider $\mathbb{C}^4/G$, where $G=\left\langle \sigma,\tau\right\rangle$, with $\sigma$ and $\tau$ the automorphisms given, respectively, by the following two matrices: $$\begin{pmatrix} \xi_5& 0 & 0 & 0\\ 0 &\xi_5^{-1} & 0 & 0\\ 0 & 0 &\xi_5^{-1} & 0\\ 0 & 0 & 0 &\xi_5 \end{pmatrix}\ \text{and}\ \begin{pmatrix} 0& 0 & 1 & 0\\ 0 &0 & 0 & 1\\ 1 & 0 &0 & 0\\ 0 & 1 & 0 &0 \end{pmatrix}.$$ In theory, we know that we can find a crepant resolution in codimension 2 of $\mathbb{C}^4/G$. Would you know how doing it in practice?

At least, I will be happy to know the number of irreducible components of the exceptional divisor and the remaining singularities.

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  • $\begingroup$ By a crepant resolution in codimension 2, I mean a sujective morphism $r:X\rightarrow \mathbb{C}^4/G$ such that $X$ has only isolated singularities, $X$ has a trivial canonical bundle and the restriction $r^{-1}(\mathbb{C}^4/G\smallsetminus\Sigma)\rightarrow \mathbb{C}^4/G\smallsetminus\Sigma$ is an isomorphism, where $\Sigma$ is the surface of singularities in $\mathbb{C}^4/G$. $\endgroup$
    – Kovalevskaia
    Commented Apr 11, 2021 at 12:16

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