# Dihedral groups and G-Hilbert schemes

@inproceedings{Celis2008DihedralGA, title={Dihedral groups and G-Hilbert schemes}, author={{\'A}lvaro Nolla de Celis}, year={2008} }

Let G ⊂ GL(2,C) be a finite subgroup acting on the complex plane C2, and consider the following diagram
C2 -> X <- π:Y
where π is the minimal resolution of singularities. Since Du Val in the 1930s the explicit calculation of Y was made from X by blowing up the singularity at the origin, where we lose any information about the group G in the process. But, is there a direct relation between the resolution Y and the group G?
McKay [McK80] in the late 1970s was the first to realise the link…

## Figures and Tables from this paper

## 11 Citations

Hilbert schemes of points on some classes of surface singularities

- Mathematics
- 2016

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or…

G-GRAPHS AND SPECIAL REPRESENTATIONS FOR BINARY DIHEDRAL GROUPS IN GL(2,ℂ)

- MathematicsGlasgow Mathematical Journal
- 2012

Abstract Given a finite subgroup G⊂GL(2,ℂ), it is known that the minimal resolution of singularity ℂ2/G is the moduli space Y=G-Hilb(ℂ2) of G-clusters ⊂ℂ2. The explicit description of Y can be…

Euler characteristics of Hilbert schemes of points on simple surface singularities

- Mathematics
- 2015

We study the geometry and topology of Hilbert schemes of points on the orbifold surface , respectively the singular quotient surface , where is a finite subgroup of type A or D. We give a…

On generating series of classes of equivariant Hilbert schemes of fat points

- Mathematics
- 2009

In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on…

The classification of special Cohen–Macaulay modules

- Mathematics
- 2008

In this paper we completely classify all the special Cohen–Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient…

Quivers of sections on toric Deligne-Mumford stacks

- Mathematics
- 2011

Starting from a collection of line bundles on a projective toric DM stack X, we introduce a stacky analogue of the classical linear series. Our first main result extends work of King by building…

Higher representation infinite algebras from McKay quivers of metacyclic groups

- MathematicsCommunications in Algebra
- 2019

Abstract For each prime number s we introduce examples of - and s-representation infinite algebras in the sense of Herschend, Iyama and Oppermann, which arise from skew group algebras of some…

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

- Mathematics
- 2009

For a given small binary dihedral group G = BD2n(a) we use the classification of G-graphs to describe explicitly G-Hilb(C) by giving an affine open cover of Mθ(Q,R), the moduli space of θ-stable…

Divisors computing minimal log discrepancies on lc surfaces

- Mathematics
- 2021

Let (X ∋ x,B) be an lc surface germ. If X ∋ x is klt, we show that there exists a divisor computing the minimal log discrepancy of (X ∋ x,B) that is a Kollár component of X ∋ x. If B 6= 0 or X ∋ x is…

G-graphs and special representations for binary dihedral groups in GL(2,C)

- Mathematics
- 2009

We give the classification of all possible G-graphs for any small binary dihedral subgroup G in GL(2,C) and use this classification to give the combinatorial description of the special…

## References

SHOWING 1-10 OF 50 REFERENCES

La correspondance de McKay

- Mathematics
- 1999

Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite automorphism group of M acting trivially on the canonical class K_M; for example, a subgroup G of SL(n,C) acting on C^n in the…

Surface cyclic quotient singularities and Hirzebruch–Jung resolutions

- Mathematics
- 2003

If V is an affine algebraic variety and G ⊂ AutV a finite group of automorphism of V , the quotient variety is an affine algebraic variety V/G with a quotient morphism V → X = V/G. A point of X is an…

Moduli of McKay quiver representations I: the coherent component

- Mathematics
- 2005

For a finite abelian group G ⊂ GL (n, k ), we describe the coherent component Yθ of the moduli space ℳθ of θ‐stable McKay quiver representations. This is a not‐necessarily‐normal toric variety that…

Mukai implies McKay: the McKay correspondence as an equivalence of derived categories

- Mathematics
- 1999

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M…

How to calculate A-Hilb(C^3)

- Mathematics
- 2002

Iku Nakamura [Hilbert schemes of Abelian group orbits, J. Alg. Geom. 10 (2001), 757--779] introduced the G-Hilbert scheme for a finite subgroup G in SL(3,C), and conjectured that it is a crepant…

MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS

- Mathematics
- 1994

IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian…

The special McKay correspondence as an equivalence of derived categories

- Mathematics
- 2007

We give a new moduli construction of the minimal resolution of the singularity of type 1/r(1,a) by introducing the Special McKay quiver. To demonstrate that our construction trumps that of the…

Quiver representations in toric geometry

- Mathematics
- 2008

This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of…

The Classification of Special CM Modules

- Mathematics
- 2008

In this paper we completely classify all the special CM modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In…