scholarly journals COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES

2015 ◽  
Vol 58 (2) ◽  
pp. 325-355 ◽  
Author(s):  
MARIA DONTEN-BURY
Keyword(s):  
Toric Variety ◽  
Affine Space ◽  
Explicit Description ◽  
Quotient Singularity ◽  
Coordinate Ring ◽  
Minimal Resolution ◽  
Cox Rings ◽  
Cox Ring ◽  
Quotient Singularities ◽  
Minimal Resolutions

AbstractWe investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

scholarly journals A software package for Mori dream spaces

2015 ◽  
Vol 18 (1) ◽  
pp. 647-659 ◽  
Author(s):  
Jürgen Hausen ◽  
Simon Keicher
Keyword(s):  
Software Package ◽  
Toric Varieties ◽  
Quotient Singularity ◽  
Algebraic Varieties ◽  
Cox Rings ◽  
Cox Ring ◽  
Mori Dream Spaces ◽  
The Common ◽  
Symplectic Resolutions

Mori dream spaces form a large example class of algebraic varieties, comprising the well-known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy–Schedler and Donten-Bury–Wiśniewski.

ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SLn(ℂ)

2018 ◽  
Vol 60 (3) ◽  
pp. 603-634
Author(s):  
RYO YAMAGISHI
Keyword(s):  
Minimal Model ◽  
Finite Subgroup ◽  
Quotient Singularity ◽  
Crepant Resolution ◽  
Minimal Models ◽  
Cox Ring ◽  
Quotient Singularities ◽  
Finite Subgroups ◽  
Symplectic Resolutions

AbstractWe prove that a quotient singularity ℂn/G by a finite subgroup G ⊂ SLn(ℂ) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky (Asian J. Math.4(3) (2000), 553–563). We also give a procedure to compute the Cox ring of a minimal model of a given ℂn/G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities that admit projective symplectic resolutions.

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 Quotient complete intersections of affine spaces by finite linear groups

1985 ◽  
Vol 98 ◽  
pp. 1-36 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Normal Subgroup ◽  
Affine Space ◽  
Finite Subgroup ◽  
Complete Intersections ◽  
Linear Groups ◽  
Coordinate Ring ◽  
Affine Spaces ◽  
Quotient Variety ◽  
Finite Linear ◽  
Complex Number Field

Let G be a finite subgroup of GLn(C) acting naturally on an affine space Cn of dimension n over the complex number field C and denote by Cn/G the quotient variety of Cn under this action of G. The purpose of this paper is to determine G completely such that Cn/G is a complete intersection (abbrev. CI.) i.e. its coordinate ring is a C.I. when n > 10. Our main result is (5.1). Since the subgroup N generated by all pseudo-reflections in G is a normal subgroup of G and Cn/G is obtained as the quotient variety of without loss of generality, we may assume that G is a subgroup of SLn(C) (cf. [6, 16, 24, 25]).

scholarly journals On pliability of del Pezzo fibrations and Cox rings

2017 ◽  
Vol 2017 (723) ◽  
Author(s):  
Hamid Ahmadinezhad
Keyword(s):  
Moduli Space ◽  
Toric Varieties ◽  
Projective Spaces ◽  
Low Rank ◽  
Del Pezzo Surfaces ◽  
Coordinate Ring ◽  
Birational Transformations ◽  
Cox Rings ◽  
Del Pezzo

AbstractWe develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over

scholarly journals Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

2017 ◽  
Vol 2017 (730) ◽  
Author(s):  
Marian Aprodu ◽  
Gavril Farkas ◽  
Angela Ortega
Keyword(s):  
Projective Variety ◽  
K3 Surfaces ◽  
Minimal Resolution ◽  
Chow Forms ◽  
Minimal Resolutions

AbstractThe Minimal Resolution Conjecture (MRC) for points on a projective variety

scholarly journals Gorensteinness and iteration of Cox rings for Fano type varieties

2022 ◽  
Author(s):  
Lukas Braun
Keyword(s):  
Class Group ◽  
Reductive Group ◽  
Finitely Generated ◽  
Projective Varieties ◽  
Cox Rings ◽  
Cox Ring

AbstractWe show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones X, iteration of Cox rings is finite with factorial master Cox ring. In particular, even if the class group has torsion, we can express such X as quotients of a factorial canonical quasicone by a solvable reductive group.

Functions on quotient singularities

1987 ◽  
Vol 324 (1576) ◽  
pp. 1-45 ◽  
Keyword(s):  
Invariant Theory ◽  
Quotient Singularity ◽  
Two Dimensional ◽  
Comprehensive Review ◽  
Invariant Functions ◽  
Quotient Singularities ◽  
Free Actions

There are two main results: a determination of the modality of a generic function on any given two-dimensional quotient singularity and a listing of all the zero-modal functions. To achieve this, a comprehensive review of the invariant theory for free actions on C 2 is needed. The problem is put in context by a general discussion of classification of invariant functions, and several different extensions of the main results are indicated.

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.

Sign in / Sign up

Export Citation Format

Share Document