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On \(c_ 1=0\) resolution of quotient singularity. (English) Zbl 0856.14005
The purpose of this paper is to study minimal resolutions of quotient singularities with trivial canonical sheaf, more precisely the minimal resolution of the 3-dimensional quotient singularity by the isocahedral group and some higher dimensional abelian quotients.
Let \(\pi:V \to X\) the canonical projection from the vector space \(V=\mathbb{C}^n\) to the quotient space \(X=V/G\), where \(G\) is a finite subgroup of \(SL_n(\mathbb{C})\). The author wants to compare the Euler number \(\chi(X)\) of some minimal resolution of \(X\) with the “orbifold Euler characteristic” \(\chi(V,G)\). In a first part the author considers a double cover \(p:X\to Y\) of a smooth variety defined by some \(s \in \Gamma (Y, {\mathcal K}^2_Y)\), where \({\mathcal K}_Y\) is the canonical line bundle on \(Y\). Let \(B\) the ramification locus of \(p\) and assume \(\text{Sing} (B)=\bigcup^k_{j=1} C_j\), where each \(C_j\) is a non-singular codimension 2 subvariety of \(Y\). Then \(X\) is a normal variety with trivial canonical sheaf and \(\text{Sing} (X)=p^{-1} (\text{Sing} (B))\). – By induction we can construct a modification \(\widetilde \sigma : \widetilde Y\to Y\), as the composition of blow-up of a curve \(C_j\), such that the double cover \(\widetilde X\) of \(\widetilde Y\) ramified at the strict transform of \(B\) on \(\widetilde Y\) gives an explicit resolution \(\widetilde \tau: \widetilde X \to X\) with trivial canonical bundle.
Then the author studies the space \(X=\mathbb{C}^3/I\) defined as the quotient 3-space by the icosahedral group \(I\). If we denote by \(J\) the subgroup of \(GL_3 (\mathbb{C})\) generated by \(I\) and \(-\text{id}\), then we get a double cover \(p : X \to Y= \mathbb{C}^3/J\). The quotient 3-space \(Y\) is smooth and the singular set of the branch locus \(B\) is the union of three smooth curves. Then we can apply the previous procedure and construct a resolution \(\widetilde X\) of \(X\) and we can calculate its Euler number \(\chi (\widetilde X)\) and we get the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^3,I) = 5\).
In the same way we can consider the quotient \(n\)-space \(X=\mathbb{C}^n/SD\), where \(SD=D \cap SL_n (\mathbb{C}) \simeq (\mathbb{Z}/2)^{n-1}\) and \(D\) is the subgroup of \(GL_n (\mathbb{C})\) consisting of all the diagonal order 2 elements, as a double cover of the smooth variety \(Y=\mathbb{C}^n/D\). We get a minimal resolution \(\widetilde X\) of \(X\) with trivial canonical bundle and we have the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^n,SD)=2^{n-1}\).
Reviewer: M.Vaquie (Paris)

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
14J30 \(3\)-folds
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