Ivan Cheltsov, Constantin Shramov
We study exceptional quotient singularities. In particular, we prove an exceptionality criterion in terms of the $α$-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient singularities.
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In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.
We classify six-dimensional exceptional quotient singularities and show that seven-dimensional exceptional quotient singularities do not exist. Inter alia we prove that the irreducible six-dimensional projective representation of the sporadic simple Hall--Janko group gives rise to an exceptional quotient singularity.
We define and calculate the weighted multiplicities of non Gorenstein terminal singularities on threefolds and some quotient singularities. We improve freeness conditions on threefolds.
A singularity is said to be exceptional (in the sense of V. Shokurov), if for any log canonical boundary, there is at most one exceptional divisor of discrepancy -1. In our previous paper (math.AG/9805004) we found two examples of exceptional canonical singularities: these are quotients by Klein's simple group of order 168 or by its central extension of order 504. Now we classify all the three-dimensional exceptional quotient singularities.
We prove that nine-dimensional exceptional quotient singularities exist.
Deformations of isolated cyclic quotient singularities in arbitrary characteristic
该研究证明了孤立循环商奇点在任意特征下均可变形，并验证了相关猜想。
We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.
The first examples of exceptional terminal singularities are constructed.
The object of the present is a proof of the existence of functorial resolution of tame quotient singularities for quasi-projective varieties over algebraically closed fields.
We prove a result on the singularities of ball quotients $Γ\backslash\CC H^n$. More precisely, we show that a ball quotient has canonical singularities under certain restrictions on the dimension $n$ and the underlying lattice. We also extend this result to the toroidal compactification $(Γ\backslash\CC H^n)^*$.
Ordinary $r$--tuples in cyclic quotient surface singularities
本文刻画了循环商曲面奇点中哪些Weil除子构成抽象的r-元组曲线奇点，揭示了奇点结构与除子性质之间的对应关系。
A singularity is said to be weakly--exceptional if it has a unique purely log terminal blow up. In dimension $2$, V. Shokurov proved that weakly--exceptional quotient singularities are exactly those of types $D_{n}$, $E_{6}$, $E_{7}$, $E_{8}$. This paper classifies the weakly--exceptional quotient singularities in dimensions $3$ and $4$.
We give an affirmative answer to Nash Problem for quotient surface singularities, in particular for the icosahedral singularity $E_8$.
In this paper we review the classification of isolated quotient singularities over the field of complex numbers due to H. Zassenhaus, G. Vincent, and G. A. Wolf. As an application we describe Gorenstein isolated quotient singularities over C, generalizing a result of K. Kurano and S. Nishi.
This paper studies hypersurface exceptional singularities in $\mathbb C^n$ defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.
In this paper, we show a condition for two-parameter Gorenstein cyclic quotient singularities to have a crepant resolution by using the remainder polynomial in any dimension.
A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow up. This is a natural generalization of the surface singularities of types $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. Since this idea was introduced, quotient singularities of this type have been classified in dimensions up to $5$. This paper looks at such singularities in dimension $p$, where $p$ is an arbitrary prime number.
We generalize the well-known numerical criterion for projective spaces by Cho, Miyaoka and Shepherd-Barron to varieties with at worst quotient singularities. Let $X$ be a normal projective variety of dimension $n \geq 3$ with at most quotient singularities. Our result asserts that if $C \cdot (-K_X) \geq n+1$ for every curve $C \subset X$, then $X \cong \PP^n$.
Motivated by mirror symmetry for weighted projective spaces, we give a new characterization of exceptionoal unimodal singularities.
We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence needs only simple computations comparing with the results ever known.
All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the finite number of the types of exceptional ones. This subdivision is well demonstrated by the example of Du Val singularities. There are two infinite series of non-exceptional singularities: $A_n$ and $D_n$ and only three types of exceptional ones: $E_6$, $E_7$ and $E_8$. Also the importance of exceptionality phenomenon follows from the next observation: A). If …
Full exceptional collections of vector bundles on rank-two linear GIT quotients
本文在秩二线性GIT商上构造了由向量丛构成的完全强例外集合，并提出了通过装饰箭图簇构造更多此类例子的方法。
Shokurov's index conjecture for quotient singularities
本文针对商奇点情形，证明了 Shokurov 指数猜想。
A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow up. This is a natural generalization of the surface singularities of types $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. Since this idea was introduced, quotient singularities of this type have been classified in dimensions up to most $4$. This note extends that classification to dimension $5$.
In this paper the detailed classification of three-dimensional exceptional canonical hypersurface singularities which don't satisfy the condition of well-formedness is given. This result completes the classification of three-dimensional exceptional log canonical hypersurface singularities started in [4] (math.AG 0201025).
Inversion of adjunction for quotient singularities
本文研究商奇点中的伴随反转问题，建立 klt Cartier 除子的精确公式，并据此证明 klt 超商奇点最小对数偏差的半连续性。
We prove that good quotients of algebraic varieties with 1-rational singularities also have 1-rational singularities. This refines a result of Boutot on rational singularities of good quotients.
We discuss some "folklore" results on categorical crepant resolutions for varieties with quotient singularities.
Among log canonical surface singularities, the ones which have a rational homology disk smoothing are the cyclic quotient singularities $\frac{1}{n^2}(1,na-1)$ with gcd$(a,n)=1$, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with $π_1=1$, $p_g=0$, and $K^2=1,2$ for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants $π_1=1$, $p_g=0$, and $K^2=1,2,3,4$.
The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to define singularities of splice type and splice-quotient singularities. The Casson Invariant Conjecture and Milnor Fiber Conjectures are discussed, as well as some open questions about hypersurface and complete intersection singularities with integral homology sphere link.
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
In this paper we are interested in quotients of Calabi-Yau threefolds with isolated singularities. In particular, we analyze the case when $X/G$ has terminal singularities. We prove that, if $G$ is cyclic of prime order and $X/G$ has terminal singularities, then $G$ has order $2,3$ or $5$.
Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated non-cyclic quotient singularities.
In this paper, we shall discuss Hilbert property of ${\rm Hilb}^{G}(\mathbb{C}^{3})$, Fujiki-Oka resolutions and iterated Fujiki-Oka resolutions for three dimensional canonical cyclic quotient singularities by using the classification shown by Ishida and Iwashita\cite{II}. Finally, we shall prove that there exists a Hilbert desingularization for any three dimensional canonical cyclic quotient singularity.
We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of homogeneous generators of the Cox ring in a more general setup, based on the multigraded Castelnuovo-Mumford regularity and Kawamata-Viehweg vanishing.
The present work is devoted to the study of motivic integration on quotient singularities. We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces.
A recent paper of Bresciani and Vistoli studies fields of moduli using residual gerbes. One of their applications is to twisted forms of quotient singularities. We discuss a direct approach to automorphisms and twisted forms, which also works in many wildly ramified cases.
Inversion of adjunction for quotient singularities II: Non-linear actions
该研究证明了商奇点的精确伴随反演公式，推导了超商奇点极小对数偏差的半连续性，作为此前研究的续篇将相关结果推广至非线性群作用场景。
Horospherical varieties with quotient singularities
研究了horospherical变种的商奇点特征，证明了所有具有商奇点的quasiprojective horospherical变种都是平滑变种对有限阿贝尔群的商。
We prove an equivalence of triangulated categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations which is obtained from the McKay quiver by removing one vertex and half of the arrows.
This paper is a complement to the work of the second author on modular quotient singularities in odd characteristic (see arXiv:1210.8006). Here we prove that if $V$ is a three-dimensional vector space over a field of characteristic $2$ and $G<GL(V)$ is a finite subgroup generated by pseudoreflections and possessing a $2$-dimensional invariant subspace $W$ such that the restriction of $G$ to $W$ is isomorphic to the group $SL_{2}(\mathbb{F}_{2^n})$, then the quotient $V/G$ is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem …
In characteristic zero, quotient singularities are log terminal. Moreover, we can check whether a quotient variety is canonical or not by using only the age of each element of the relevant finite group if the group does not have pseudo-reflections. In positive characteristic, a quotient variety is not log terminal, in general. In this paper, we give an example of the quotient variety which is not log terminal such that the quotient varieties associated to any proper subgroups are canonical. In particular, we cannot determine whether a given quotient singularity is canonical by looking at prope…
The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces
本研究引入正特征下代数群概形的极大幂幺有限商，发现其应用于皮卡概形可编码异常挠率，借助例外恩里克斯曲面理论构造出存在该异常挠率的整法诺三维簇，打破了德尔佩佐曲面挠率为零的常规认知。
In this paper we give a criterion for birational rigidity of del Pezzo fibrations of degree 1 and 2 with only quotient singularities. As an application, we prove birational rigidity of suitable del Pezzo fibrations admitting an action of the Klein simple group.
We describe the versal deformation of two-dimensional cyclic quotient singularities in terms of equations, following Arndt, Brohme and Hamm. For the reduced components the equations are determined by certain systems of dots in a triangle. The equations of the versal deformation itself are governed by a different combinatorial structure, involving rooted trees.
Wild quotient surface singularities whose dual graphs are not star-shaped
本研究利用Kato对数结构与对数正则性理论解答Lorenzini的二维野商奇点相关问题，证明奇点消解例外曲线对偶图至少含一个节点，并构造出对偶图至少含两个节点的野商奇点实例。
We generalize the result of Kawamata concerning the strong version of Fujita's freeness conjecture for smooth 3-folds to some singular cases, namely, Gorenstein terminal singularities and quotient singularities of type 1/r(1,1,1) and of type 1/r(1,1,-1). We generalize furthermore the result of that to projective threefolds with only canonical singularities for canonical and not terminal singularities. It turns out that the estimates in the first three cases are better than the one for the smooth case, which it is not in the fourth case. We also give explicit examples which show the estimate in…
On the singularities of quotients by 1-foliations
本文研究了由1-叶形结构商映射在正特征下变种的奇点，证明了商保持MMP中的奇点及F-奇点，并提出了1-叶形结构的家族概念。
We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of different resolutions and relations between them. In particular, we treat explicitly several examples where $G$ contains elements of age 2.
$K$-regularity for finite quotient singularities
本文研究有限商奇点的$K$-正则性，证明在奇点轨迹含于仿射闭子簇时$K_1$-正则性等价于光滑性，并构造反例表明一般情形不成立，通过肯定Fogarty问题获得判据。
k-Wahl chains and cyclic quotient singularities
本文研究由k-Wahl链定义的二维环形商奇点，探讨了k-Wahl链的继续分数结构与环形群特殊表示之间的关系，并研究了双侧零继续分数对奇点变形理论的影响。
In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.
The geometric genus of splice-quotient singularities
本文提出了 splice-quotient 环奇点的几何亏格计算公式，并探讨了相关同调群的维度问题。
In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the Łojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the Łojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.
On fillings of contact links of quotient singularities
该研究利用Floer理论探讨商奇点接触链环的各类填充问题，证明多类孤立终端商奇点的接触链环不存在恰当填充并解决相关猜想，同时得到部分情形下恰当轨形填充的轨形微分同胚类型唯一性。
It is known that ambient bilipschitz equivalence preserves tangent cones. This paper explores the behavior of the Nash cone and, in particular, exceptional rays under ambient bilipschitz equivalence for real surfaces in $R^3$ with isolated singularity.
We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.
We prove that a quotient singularity $\mathbb{C}^n/G $ by a finite subgroup $G\subset SL_n(\mathbb{C})$ has a crepant resolution only if $G $ is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also give a procedure to compute the Cox ring of a minimal model of a given $\mathbb{C}^n/G$ 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 which …
Classification of threefold enc cDV quotient singularities
本文通过组合方法研究三重例外非典范 cDV 商奇点，并给出其粗略分类。
We 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.
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