Klein's group defines an exceptional singularity of dimension 3

A remark on deformations of $1$-convex manifolds with exceptional curves

A formula for the dimension of the smoothing component of a $3$-dimensional isolated Cohen--Macaulay singularity is shown. We apply this formula for a $1$-convex threefold with a connected exceptional curve which is blown down to a terminal Gorenstein singularity.

On the exceptional locus of the birational projections of normal surface singularity into a plane

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $π: X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $π$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.

Six-dimensional exceptional quotient singularities

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.

Exceptional quotient singularities

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.

On purely log terminal blow-ups

In this paper we prove the existence of purely log terminal blow-up for Kawamata log terminal singularity and obtain the criterion for a singularity to be weakly exceptional in terms of the exceptional divisor of plt blow-up.

Combinatorial structure of exceptional sets in resolutions of singularities

The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity. The homotopy type of the dual complex does not depend on the choice of a resolution and thus can be considered as an invariant of singularity. In this preprint we show that the dual complex is homotopy trivial for resolutions of 3-dimensional terminal singularities and for resolutions of Brieskorn singularities. We also review our earlier results on resoluti…

Exceptional rays and bilipschitz geometry of real surface singularities

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.

Degenerate singularities of one dimensional Foliations

We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the exceptional line; we also find the degree of the locus defined by imposing a leaf of total contact with the exceptional line.

On graded stable derived categories of isolated Gorenstein quotient singularities

We show the existence of a full exceptional collection in the graded stable derived category of a Gorenstein isolated quotient singularity using a result of Orlov (arXiv:math/0503632). We also show that the equivariant graded stable derived category of a Gorenstein Veronese subring of a polynomial ring with respect to an action of a finite group has a full strong exceptional collection, even if the corresponding quotient singularity is neither isolated nor Gorenstein.

Weakly-exceptional quotient singularities in prime 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.

Five-Dimensional Weakly Exceptional Quotient Singularities

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 $5$.

A McKay correspondence for the Poincaré series of some finite subgroups of ${\rm SL}_3(\CC)$

A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…

Relative singularity categories II: DG models

We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which are known or generalise known results, on singularity categories of algebras with radical square zero, of non-commutative deformations of Kleinian singularities, of $SL_3(\mathbb{C})$-quotient singularities and of Gorenstein toric threefolds.

Weakly--exceptional quotient singularities

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 $4$.

The Nash problem on arc families of singularities

Nash proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that occurs on every resolution. He asked if the converse also holds: does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

Classification of three-dimensional exceptional log canonical hypersurface singularities I

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 …

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V. I. Arnold and V. I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.

Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type $A_{2n}$, this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence.

Kleinian 奇点的形变与量子化的自同构群

Automorphism groups of deformations and quantizations of Kleinian singularities

本文证明 A 或 D 型 Kleinian 奇点的量子化同构与形变 Poisson 同构本质相同，并刻画了对应自同构群的抽象结构，首次计算了 D 型形变间的所有仿射同构。

克莱因奇点的Stacky分解与中岛箭图簇

Stacky Resolutions of Kleinian Singularities and Nakajima Quiver Varieties

该研究构造克莱因奇点的非交换平展分解，证明其与典范轨形分解Morita等价，确定对应点希尔伯特概型为中岛箭图簇，并刻画其两类锥的结构。

代数网络与架构退化

Exceptional Singularities in Polynomial Neural Networks

研究具有单项式激活函数和无偏置的多项式神经网络的几何，定义代数网络框架，并证明全连接网络中奇异点位于架构退化轨迹内。

Braid groups and Kleinian singularities

We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.

类型A量子表面奇点的解析

Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities

本文通过构造几何解析类别\mathcal{X}，展示了类型A量子克莱因奇点的非交换解析与导出等价性，并推广了经典克莱因奇点的几何 McKay对应。

Kleinian 奇点：一些几何、组合与表示论

Kleinian singularities: some geometry, combinatorics and representation theory

本文综述了 Kleinian 奇点相关的几何、组合与表示论，重点介绍单位根替换演算，并讨论 A 型、D 型及 E 型情形的结果与问题。

Icosahedron, exceptional singularities and modular forms

We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, …

W-Constraints for the Total Descendant Potential of a Simple Singularity

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, int…