Proceedings of the Japan Academy, Series A, Mathematical Sciences

Dihedral $G$-Hilb via representations of the McKay quiver

Alvaro Nolla de Celis

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Abstract

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 $(Q,R)$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 5 (2012), 78-83.

Dates
First available in Project Euclid: 7 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.pja/1336394842

Digital Object Identifier
doi:10.3792/pjaa.88.78

Mathematical Reviews number (MathSciNet)
MR2925287

Zentralblatt MATH identifier
1361.14011

Subjects
Primary: 14E16: McKay correspondence
Secondary: 16G20: Representations of quivers and partially ordered sets 14C05: Parametrization (Chow and Hilbert schemes) 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]

Keywords
McKay correspondence $G$-Hilbert scheme quiver representations

Citation

Nolla de Celis, Alvaro. Dihedral $G$-Hilb via representations of the McKay quiver. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 5, 78--83. doi:10.3792/pjaa.88.78. https://projecteuclid.org/euclid.pja/1336394842


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