scholarly journals Extended McKay correspondence for quotient surface singularities

2018 ◽  
Vol 70 (2) ◽  
pp. 395-408
Author(s):  
Akira Ishii ◽  
Iku Nakamura
Keyword(s):  
Hilbert Scheme ◽  
Finite Subgroup ◽  
Minimal Resolution ◽  
Mckay Correspondence ◽  
Small Subgroup ◽  
Quotient Surface ◽  
Surface Singularities ◽  
The Ideal

Abstract Let G be a finite subgroup of GL(2) acting on A2/{0} freely. The G-orbit Hilbert scheme G-Hilb(A2) is a minimal resolution of the quotient A2/G as given by A. Ishii, On the McKay correspondence for a finite small subgroup of GL(2,C), J. Reine Angew. Math. 549 (2002), 221–233. We determine the generator sheaf of the ideal defining the universal G-cluster over G-Hilb(A2), which somewhat strengthens a version [10] of the well-known McKay correspondence for a finite subgroup of SL(2).

Coinvariant Algebras of Finite Subgroups of SL(3;C)

2004 ◽  
Vol 56 (3) ◽  
pp. 495-528 ◽  
Author(s):  
Yasushi Gomi ◽  
Iku Nakamura ◽  
Ken-ichi Shinoda
Keyword(s):  
Hilbert Scheme ◽  
Dual Graph ◽  
Orbit Space ◽  
Rational Curves ◽  
Finite Subgroup ◽  
Crepant Resolution ◽  
Mckay Correspondence ◽  
Finite Subgroups ◽  
Explicit Formulae ◽  
Molien Series

AbstractFor most of the finite subgroups of SL(3; C) we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae [McKay99] for subgroups of SU(2). We also study the G-orbit Hilbert scheme HilbG(C3) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C3/G. In this case the fiber over the origin of the Hilbert-Chow morphism from HilbG(C3) to C3/G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the McKay correspondence in the SU(2) case.

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

2012 ◽  
Vol 55 (1) ◽  
pp. 23-57
Author(s):  
ALVARO NOLLA DE CELIS
Keyword(s):  
Moduli Space ◽  
Cyclic Subgroup ◽  
Finite Subgroup ◽  
Explicit Description ◽  
Vector Spaces ◽  
Minimal Resolution ◽  
Mckay Correspondence ◽  
Dihedral Groups ◽  
Resolution Of Singularity ◽  
Distinguished Basis

AbstractGiven 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 obtained by calculating every possible distinguished basis for as vector spaces. These basis are the so-called G-graphs. In this paper we classify G-graphs for any small binary dihedral subgroup G in GL(2,ℂ), and in the context of the special McKay correspondence we use this classification to give a combinatorial description of special representations of G appearing in Y in terms of its maximal normal cyclic subgroup H ⊴ G.

scholarly journals Infinitesimal deformations of quotient surface singularities

1988 ◽  
Vol 20 (1) ◽  
pp. 31-66 ◽  
Author(s):  
Kurt Behnke ◽  
Constantin Kahn ◽  
Oswald Riemenschneider
Keyword(s):  
Quotient Surface ◽  
Surface Singularities ◽  
Infinitesimal Deformations

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

scholarly journals Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

2019 ◽  
Author(s):  
Martin de Borbon ◽  
Cristiano Spotti
Keyword(s):  
Finite Subgroup ◽  
Minimal Resolution ◽  
Gravitational Instantons ◽  
Normal Crossing ◽  
Three Sphere ◽  
Asymptotically Locally Euclidean ◽  
Quotient Singularities ◽  
Simple Normal Crossing

Abstract We construct Asymptotically Locally Euclidean (ALE) and, more generally, asymptotically conical Calabi–Yau metrics with cone singularities along a compact simple normal crossing divisor. In particular, this includes the case of the minimal resolution of 2D quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer’s construction of smooth ALE gravitational instantons.

scholarly journals Ulrich modules over cyclic quotient surface singularities

2017 ◽  
Vol 482 ◽  
pp. 224-247 ◽  
Author(s):  
Yusuke Nakajima ◽  
Ken-ichi Yoshida
Keyword(s):  
Quotient Surface ◽  
Surface Singularities
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