Open Access
May 2012 Dihedral $G$-Hilb via representations of the McKay quiver
Alvaro Nolla de Celis
Proc. Japan Acad. Ser. A Math. Sci. 88(5): 78-83 (May 2012). DOI: 10.3792/pjaa.88.78


For a given finite small binary dihedral group $G\subset\mathrm{GL}(2,\mathbf{C})$ we provide an explicit description of the minimal resolution $Y$ of the singularity $\mathbf{C}^{2}/G$. The minimal resolution $Y$ is known to be either the moduli space of $G$-clusters $G$-Hilb$(\mathbf{C}^{2})$, or the equivalent $\mathcal{M}_{\theta}(Q,R)$, the moduli space of $\theta$-stable quiver representations of the McKay quiver. We use both moduli approaches to give an explicit open cover of $Y$, by assigning to every distinguished $G$-graph $\Gamma$ an open set $U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)$, and calculating the explicit equation of $U_{\Gamma}$ using the McKay quiver with relations $(Q,R)$.


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Alvaro Nolla de Celis. "Dihedral $G$-Hilb via representations of the McKay quiver." Proc. Japan Acad. Ser. A Math. Sci. 88 (5) 78 - 83, May 2012.


Published: May 2012
First available in Project Euclid: 7 May 2012

zbMATH: 1361.14011
MathSciNet: MR2925287
Digital Object Identifier: 10.3792/pjaa.88.78

Primary: 14E16
Secondary: 14C05 , 14E15 , 16G20

Keywords: $G$-Hilbert scheme , McKay correspondence , quiver representations

Rights: Copyright © 2012 The Japan Academy

Vol.88 • No. 5 • May 2012
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