## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Alvaro Nolla de Celis

#### 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

#### 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

#### References

• I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, 65, Cambridge Univ. Press, Cambridge, 2006.
• R. Bocklandt, T. Schedler and M. Wemyss, Superpotentials and higher order derivations, J. Pure Appl. Algebra 214 (2010), no. 9, 1501–1522.
• A. Craw, D. Maclagan and R. R. Thomas, Moduli of McKay quiver representations. I. The coherent component, Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 179–198.
• A. Ishii, On the McKay correspondence for a finite small subgroup of $\mathrm{GL}(2,\mathbf{C})$, J. Reine Angew. Math. 549 (2002), 221–233.
• Y. Ito and I. Nakamura, Hilbert schemes and simple singularities, in New trends in algebraic geometry (Warwick, 1996), 151–233, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge.
• A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530.
• Becky Leng, The McKay correspondence and orbifold Riemann-Roch, Ph.D. Thesis, University of Warwick, 2002.
• J. McKay, Graphs, singularities, and finite groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), 183–186, Proc. Sympos. Pure Math., 37 Amer. Math. Soc., Providence, RI, 1980.
• I. Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (2001), no. 4, 757–779.
• Alvaro Nolla de Celis, Dihedral groups and $G$-Hilbert schemes, Ph.D. Thesis, University of Warwick, 112 pp., 2008.
• Alvaro Nolla de Celis, $G$-graphs and special representations for binary dihedral groups in $\mathrm{GL}(2,\mathbf{C})$. (to appear in Glasgow Mathematical Journal).
• M. Reid, La correspondance de McKay, Astérisque No. 276 (2002), 53–72.
• Michael Wemyss, Reconstruction algebras of type $D$ (I). arXiv:0905.1154v2, 2009.
• Michael Wemyss, Reconstruction algebras of type $D$ (II). arXiv:0905.1155v1, 2009.
• J. Wunram, Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), no. 4, 583–598.
• Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge Univ. Press, Cambridge, 1990.