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

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Alvaro Nolla de Celis¹ 

[1] Graduate School of Mathematics, Nagoya University

关键词: McKay correspondence;    G-Hilbert scheme;    quiver representations.;   

DOI  :  10.3792/pjaa.88.78

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

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 $(Q,R)$.

【 授权许可】

Unknown