A new characterization of exceptional unimodal singularities

来自具有一个例外单峰点的曲面的I-曲面

I-surfaces from surfaces with one exceptional unimodal point

本研究补充了已有工作，证明特定稳定曲面是仅有的由例外单峰二重点生成的对应曲面，并明确了三类奇点所构曲面的双有理型。

在正特征下对单模孤立完全交点奇点的分类

Classification of unimodal isolated complete intersection singularities in positive characteristic

本文在接触等价下对任意特征下的单模孤立完全交点奇点进行分类，扩展了Dimca和Gibson在复数域上的工作，并将完全横截方法推广到正特征域，适用于其他分类问题。

Lipschitz单模函数芽的分类

Classification of Lipschitz unimodal function germs

本文为孤立奇点引入Lipschitz模概念，完整分类了余秩2且4-jet非零的Lipschitz单模奇点，并证明6-jet消失时模至少为2，给出上界。

The Classification of Real Singularities Using SINGULAR. Part II: The Structure of the Equivalence Classes of the Unimodal Singularities

In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence classes of the unimodal real singularities of corank 2. In other words, we explicitly answer the question which normal forms of different type are equivalent, and how a normal form can be transformed within the same equivalence class by changing the value of the parameter. This provides new theoretical insights into these singularities and has important cons…

正特征下单模参数平面曲线奇点的分类

Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

本文旨在给出正特征代数闭域上单模平面分支的完整分类，通过独立于特征的方法并结合SINGULAR计算，证明了半群和模态的半连续性，同时为特征0的分类提供了新证明。

一类Horikawa曲面的KSBA模空间中的单模奇点与边界除子

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

本文研究一类Horikawa曲面典范模型模空间的KSBA紧化，构造了八个新的不可约边界除子，并探讨其与GIT紧化及退化曲面Hodge理论的关系。

On exceptional terminal singularities

The first examples of exceptional terminal singularities are constructed.

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.

Hypersurface exceptional singularities

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.

The Classification of Real Singularities Using Singular. Part III: Unimodal Singularities of Corank 2

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying all representatives in Arnold's list of normal forms (Arnold et al. 1985) belonging to this class, and the corresponding values of the moduli parameter. We discuss how to computationally realize the individual steps of the algorithm for all singularities in consideration, and give explicit examples. The algorithm is implemented in the Singular…

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 …

Classification of three-dimensional exceptional log canonical hypersurface singularities II

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).

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.

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.

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.

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 $5$.

Deforming nonnormal isolated surface singularities and constructing 3-folds with $\mathbb{P}^1$ as exceptional set

Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as isolated, non Cohen-Macaulay threefold singularities. They arise by a small contraction of a smooth rational curve, whose normal bundle has a sufficiently positive subbundle. We study such singularities from their nonnormal general hyperplane section.

ACC for minimal log discrepancies of exceptional singularities

We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,Γ_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.

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…

Nine-dimensional exceptional quotient singularities exist

We prove that nine-dimensional exceptional quotient singularities exist.

On exceptional quotient singularities

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.

Weakly-exceptional singularities in higher dimensions

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.

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 $4$.

A note on graded hypersurface singularities

For a weighted quasihomogeneous two dimensional hypersurface singularity, we define a smoothing with unipotent monodromy and an isolated graded normal singularity. We study the natural weighted blow up of both the smoothing and the surface. In particular, we describe our construction for the quasihomogeneous singularities of type I, the 14 unimodal exceptional singularities and we relate it to their stable replacement.

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.

关于阿贝尔奇点的 crepant 解消的例外集

On the exceptional set of crepant resolutions of abelian singularities

本文证明阿贝尔奇点的环面 crepant 解消中例外集的每个分支都有同构于其典范丛全空间的开邻域，推进了子流形邻域同构于法丛零截面邻域的经典问题的研究。

Singularities of the 2-dimensional moduli spaces of stable sheaves on K3 surfaces

We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on K3 surfaces. We show that the moduli space is normal, in particular the singuralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.

Klein's group defines an exceptional singularity of dimension 3

We construct two examples of canonical exceptional singularities.

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.

Sporadic simple groups and quotient singularities

We show that the only sporadic simple group such that some of its faithful representations or some faithful representations of its stem extensions give rise to exceptional (weakly-exceptional but not exceptional, respectively) quotient singularities is the Hall-Janko group (the Suzuki group, respectively).

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.

On a certain generalization of triangle singularities

Triangle singularities are Fuchsian singularities associated with von Dyck groups, which are index two subgroups of Schwarz triangle groups. Hypersurface triangle singularities are classified by Dolgachev, and give 14 exceptional unimodal singularities classified by Arnold. We introduce a generalization of triangle singularities to higher dimensions, show that there are only finitely many hypersurface singularities of this type in each dimension, and give a complete list in dimension 3.

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。

F吹胀的例外极子与G-希尔伯特方案

Exceptional loci of F-blowups and $G$-Hilbert schemes

研究F吹胀正常分形变的例外极子，通过组合数据给出素除子中心维数公式，并探讨F吹胀与三维终端奇点和canonical奇点的关联。

Homological Mirror Symmetry and Simple Elliptic Singularities

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.

利用加权爆破在特征0中实现奇点的对数消解

Logarithmic resolution of singularities in characteristic 0 using weighted blow-ups

本文研究特征零域上的奇点消解问题，核心方法是使用加权爆破，构造出例外因子呈简单正常交叉的对数消解。

A remark on resolution of terminal singularities

Let $(Z,o)$ be a three-dimensional terminal singularity of type $cA/r$. We prove that all exceptional divisors over $o$ with discrepancies $\le 1$ are rational.

Mirror constructions for K3 surfaces from bimodal singularities

We study lattice polarizations of five exceptional pairs of families of K3 surfaces obtained via compactifications of strange dual pairs of bimodal singularities. We show that the polarizations induced by embedding these paired families in paired toric varieties obtained from polar dual reflexive polytopes cannot be mirror lattices and identify mirror sublattices.

Mirror symmetry for exceptional unimodular singularities

In this paper, we prove the mirror symmetry conjecture between the Saito-Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches the orbifold-Grothendieck-Riemann-Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction tec…

具有 $\frac13(1,1)$ 奇点的对数 del Pezzo 曲面的典范栈的例外集合

Exceptional collections for canonical stacks of log del Pezzo surfaces with $\frac13(1,1)$ singularities

研究具有 $\frac13(1,1)$ 型奇点的对数 del Pezzo 曲面的导出范畴，证明其典范 Deligne-Mumford 栈的导出范畴具有完全例外集合，并应用于一般次数 10 的超曲面。

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. For an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert-Riemenschneider type vanishing theorem for foliated surfaces with good log canonical foliation singularities.

Fano三重积分的Landau-Ginzburg模型及例外集合

Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections

本文研究Fano三重积分Picard秩为一的Landau-Ginzburg模型，探讨其与例外集合的关系，并验证Homological Mirror Symmetry的预测。

"Del Pezzo surfaces as Springer fibres for exceptional groups"

We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces from the corresponding exceptional simple algebraic groups.

Numerical properties of exceptional divisors of birational morphisms of smooth surfaces

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.

Rational homology disk smoothings of surface singularities; the exceptional cases

It is known (Stipsicz-Szabó-Wahl) that there are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a rational homology disk ($\mathbb{Q}$HD) smoothing, i.e., having a Milnor fibre with Milnor number zero. Some examples are found by an explicit "quotient construction", while others require the "Pinkham method". The fundamental group of the Milnor fibre has been known for all except the three exceptional families $\mathcal B_2^3(p), \mathcal C^3_2(p),$ and $\mathcal C^3_3(p)$. In this paper, we settle these cases. We p…

具有一维纤维的三重极值收缩的一般大象：例外情形

General elephants for threefold extremal contractions with one-dimensional fibers: exceptional case

本文研究具有一维纤维的三重极值收缩，在每分支至多含一个指数大于2点的条件下，证明一般成员是具有 Du Val 奇点的正规曲面。

Contraction of excess fibres between the McKay correspondences in dimensions two and three

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3-surfaces arising as minimal resolutions of quotients of nodal surfaces with high symmetries.

A propos de la conjecture de Nash

This paper deals with the Nash problem, which claims that there are as many families of arcs on a singular germ of surface $U$ as there are essential components of the exceptional divisor in the desingularisation of this singularity. Let $\mathcal{H}=\bigcup \bar{N_α}$ be a particular decomposition of the set of arcs on $U$, described later on. We give two new conditions to insure that $\bar{N_α}\not \subset \bar{N_β}$, $α\not = β$; more precisely,for the first one, we claim that if there exists $f \in {\mathcal{O}}_{U}$ such that $ord_{E_α}(f)<ord_{E_β}(f)$, where $E_α, E_β$ are exceptional d…

On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the exis…

Exceptional bundles associated to degenerations of surfaces

In 1981 J. Wahl described smoothings of surface quotient singularities with no vanishing cycles. Given a smoothing of a projective surface X of this type, we construct an associated exceptional vector bundle on the nearby fiber Y in the case H^{2,0}(Y)=H^1(Y)=0. If Y is the projective plane we show that our construction establishes a bijective correspondence between the possible degenerate surfaces X and exceptional bundles on Y modulo dualizing and tensoring by line bundles. If Y is of general type then our construction establishes a connection between components of the boundary of the moduli…

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, …

On tautness of two-dimensional $F$-regular and $F$-pure rational singularities

The weighted dual graph of a two-dimensional normal singularity $(X, x)$ represents the topological nature of the exceptional locus of its minimal log resolution. $(X, x)$ and its graph are said to be taut if the singularity can be uniquely determined by the graph. Laufer gave a complete list of taut singularities over $\mathbb{C}$. In positive characteristics, taut graphs over $\mathbb{C}$ are not necessarily taut and tautness have been studied only for special cases. In this paper, we prove the tautness of $F$-regular singularities. We also discuss the tautness of $F$-pure rational singulari…

完全交的Landau-Ginzburg模型的奇点与导出范畴

Singularities of Landau-Ginzburg models for complete intersections and derived categories

本文在数值层面验证了镜像对称中关于Fano完全交的Landau-Ginzburg模型纤维奇点与例外集合对应的预期。

A supplement to Fujino's paper: On isolated log canonical singularities with index one

Let $E$ be the essential part of the exceptional locus of a good resolution of an isolated, log canonical singularity of index one. We describe the dimension of the dual complex of $E$ in terms of the Hodge type of $H^{n-1}(E, O_E)$, which is one of the main results of the paper [1] of Fujino. Our proof uses only an elementary classical method, while Fujino's argument depends on the recent development in minimal model theory.