On the maximal number of du Val singularities for a K3 surface

Maximal Cohen-Macaulay modules over surface singularities

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss various results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we describe matrix factorizations corresponding to the indecomposable Cohen-Macaulay modules over the non-isolated singularities $A_\infty$ and $D_\infty$.

$1$-generic 矩阵的极大子式具有有理奇点

Maximal minors of $1$-generic matrices have rational singularities

本文证明 $1$-generic 矩阵极大子式理想的商环具有有理奇点，证实 Eisenbud 的正规性猜想并推广 generic Hankel 行列式环的相关结论。

Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of $k\llbracket x,y,z\rrbracket/(xyz)$ as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new …

A finiteness theorem on symplectic singularities

For positive integers N and d, there are only finite number of conical symplectic varieties of dimension 2d with maximal weights N, up to isomorphism. The maximal weight of a conical symplectic variety X is, by definition, the maximal weight of the minimal homogeneous generators of the coordinate ring R of X.

Combinatorial Aspects of Classical Resolution of Singularities

We describe combinatorial aspects of classical resolution of singularities that are free of characteristic and can be applied to singular foliations and vector fields as well as to functions and varieties. In particular, we give a combinatorial version of Hironaka's maximal contact theory in terms of characteristic polyhedra systems and we show the global existence of maximal contact in this context.

New Ideas for Resolution of Singularities in Arbitrary Characteristics

The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global s…

Maximal Modifications and Auslander-Reiten Duality for Non-isolated Singularities

We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modify…

在辛奇异性上构造极小Cohen-Macaulay层

Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities

本文研究了辛奇异性上极小Cohen-Macaulay层的构造与性质，通过反射层和上同调消失理论，推导了不可分解层的生成方法。

Polyhedral Groups and Pencils of K3-Surfaces with Maximal Picard Number

A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each pencil has Picard number 19 and each pencil contains four surfaces with Picard number 20. These surfaces are obtained as the minimal resolution of quotients X/G, where $G\subset SO(4,\R)$ is some finite subgroup and $X\subset ¶_3(\C)$ denotes a G-invariant surface. The singularities of X/G come from fix points of G on X or from singularities of X. In any case the…

Quadratic forms and singularities of genus one or two

We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves computes the discriminant of the quadratic forms of these singularities. We introduce a multiplicative branch topological invariant which determines the twisting of a non-vanishing holomorphic 1-form on the complement of the singular point.

奇点化简中的理想花朵

Idealistic Flowers in the Reduction of Singularities

本文基于“理想花朵”思想，遵循Maximal Contact Theory，为经典奇点化简提供了证明，并解决了由最大接触局部性质引发的全球化问题。

椭圆表面奇点中理想上同调

Cohomology of ideals in elliptic surface singularities

本文研究了二维正常奇点的正常还原数，证明了椭圆奇点的正常还原数为二，并探讨了非有理奇点的Gorenstein性质与$p_g$-理想的关系。

曲线奇点与δ不变量的等价性

Equivalence of Curve Singularities and delta-Invariants

本文证明了完备既约诺特曲线参数化的等价条件，以及既约不可约曲线奇点的同构性与完备化同构的关系，给出了优于Hironaka的新下界。

对数等纲平滑与最大节点退化

Logarithmic Equigeneric Smoothing and Maximal Nodal Degenerations

本文研究了曲线和曲面在代数族中的等纲变形问题，提出通过对数半正则性方法在退化情况下保持最大奇点数量的理论，并探讨了其在细化塞弗里计数中的应用。

"Weakly" Elliptic Gorenstein Singularities of Surfaces

The main message of the paper is that for Gorenstein singularities, whose (real) link is rational homology sphere, the Artin--Laufer program can be continued. Here we give the complete answer in the case of elliptic singularities. The main result of the paper says that in the case of an elliptic Gorenstein singularity whose link is rational homology sphere, the geometric genus is a topological invariant. Actually, it is exactly the length of the elliptic sequence in the minimal resolution (or, equivalently, in S. S.-T. Yau's terminology: these singularities are maximally elliptic). In the pape…

A型和D型商奇点的矩阵分解

Matrix factorization of quotient singularities of type A and D

本文扩展ADE奇点与极大库恩-麦考利模的关系至A型和D型商奇点，通过投影超曲面构造显式矩阵分解和模族，并恢复所有特殊库恩-麦考利模。

On the classification of rational surface singularities

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely certain types of graphs. This allows to extend the classification of rational singularities to multiplicity 8. We also discuss the complexity of rational resolution graphs.

Rank two Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$

We describe, by matrix factorizations, all the rank two maximal Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$.

阿米巴轮廓的奇点

Singularities of Amoeba Contours

该研究给出计算阿米巴轮廓奇点的显式实代数方程，分离对数临界轨迹退化与不同临界提升重合，针对平面曲线得到检测节点等的实用系统，还证明最大稀疏性不会最大化轮廓奇点。

Generic Hypersurface Singularities

We give a new differential proof of our result on the maximal rank of generic unions of points of multiplicity two in projective space in degrees greater than five. This simplifies somewhat our proof of the Waring conjecture.

Reduction of triangulated categories and Maximal Modification Algebras for cA_n singularities

In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi-Yau type property. If C is such a category, we say that C is Calabi-Yau with dimension at most one. We extend the notion of Calabi-Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categor…

奇异曲线的微分Hodge结构I

Infinitesimal Variations of Hodge Structure for Singular Curves I

本文研究了奇异曲线的微分Hodge结构，揭示了节点奇点与其它奇点在微分变化中的不同贡献，并在特定条件下给出了最大变化的判定标准。

A characterization of nilpotent orbit closures among symplectic singularities

We prove that a conical symplectic variety with maximal weight 1 is isomorphic to one of the following: (i) an affine space with the standard symplectic form (ii) a nilpotent orbit closure of a complex semisimple Lie algebra with the Kirillov-Kostant form.

Contact exponent and the Milnor number of plane curve singularities

We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].

Singularities of normal quartic surfaces II (char=2)

We show, in this second part, that the maximal number of singular points of a quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 14, and that, if we have 14 singularities, these are nodes and moreover the minimal resolution of $X$ is a supersingular K3 surface. We produce an irreducible component, of dimension 24, of the variety of quartics with 14 nodes. We also exhibit easy examples of quartics with 7 $A_3$-singularities.

正规四次曲面的奇点 III（特征为2，非超奇异）

Singularities of normal quartic surfaces III (char=2, non-supersingular)

该研究针对特征2的代数闭域上的非超奇异正规四次曲面，证明其奇点最多12个，并给出适用于任意特征的显式例子族，推进了特征2下四次曲面奇点计数的相关研究。

最大化曲线作为自由曲线

Maximizing curves viewed as free curves

本文建立最大化曲线与自由平面曲线的联系，并研究奇数次平面曲线最大总丘里纳数问题。

带权的光滑表面奇点与布里森完全交点奇点的同胚

Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections

本文研究了与布里森完全交点奇点同胚的带权光滑表面奇点，探讨了几何亏格最大值及极小良好解析中最大理想环与基本环重合的条件。

A short note on Hauser's Kangaroo phenomena and weak maximal contact in higher dimensions

Currently there are several approaches to resolution of singularities in positive characteristic all of which have hit some obstruction. One natural idea is to try to construct new meaningful examples at this point to gain a wider range of experience. To produce such examples we mimic the characteristic zero approach and focus on cases where it fails. In particular, this short note deals with an example-driven study of failure of maximal contact and the search for an appropriate replacement.

Hypersurface singularities in positive characteristic

The paper is motivated on the open problem of resolution of singularities in positive characteristic. The aim is to present a form of induction which is different from that used by Hironaka. In characteristic zero induction is formulated by restriction to smooth hypersurfaces (hypersurfaces of maximal contact). Our alternative approach, introduced here, replaces restrictions to smooth sub-schemes by generic projections on smooth schemes of smaller dimension. We also introduce a generalization of the discriminant, and our result makes use of the elimination theory. In the case of fields of char…

Maximal and Typical Topology of Real Polynomial Singularities

Given a polynomial map $ψ:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $ψ$ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a "singularity"). Concerning the upper estimate on the topological complexity of a polynomial singularity, we sharpen the classical bound $b(Z)\leq O(d^{m+1})$, proved by Milnor, with \begin{equation}\label{eq:abstract} b(Z)\leq O(d^{m}),\end{equation} which holds for the generic polynomial map. For what conce…

Singularities of Schubert Varieties, Tangent Cones and Bruhat Graphs

Let G be a semi-simple algebraic group over the complex numbers, B a Borel subgroup of G, T a maximal torus in B and P a parabolic in G containing B. This paper deals with singularities of T-stable subvarieties of G/P. It turns out that under the restriction that G doesn't contain any G_2-factors, the key geometric invariant determining the singular T-fixed points of X is the linear span of the reduced tangent cone to X at a T-fixed point x provided the singularity is isolated. The goal of this paper is to describe this invariant at the maximal singular T-fixed points when X is a Schubert vari…

Jumping Numbers on Algebraic Surfaces with Rational Singularities

In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are expl…

A New Proof for the Embedded Resolution of Surface Singularities in Arbitrary Characteristic

A new proof for the embedded resolution of surface singularities in a three-dimensional smooth ambient space over algebraically closed fields of arbitrary characteristic. The proof makes use of an upper semicontinuous resolution invariant which prescribes the center in each step of the resolution algorithm. The resolution invariant strictly decreases under each blowup. The definition of the resolution invariant is inspired by the proofs of resolution of singularities in arbitrary dimension over fields of characteristic zero. It was developed in an attempt to generalize these proofs to the sett…

Brill-Noether problem on splice quotient singularities and duality of topological Poincaré series

In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such t…

Hankel determinantal rings have rational singularities

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that ea…

Space curves on surfaces with ordinary singularities

We show that smooth curves in the same biliaison class on a hypersurface in $\mathbf{P}^3$ with ordinary singularities are linearly equivalent. We compute the invariants $h^0(\mathscr{I}_C(d))$, $h^1(\mathscr{I}_C(d))$ and $h^1(\mathscr{O}_C(d))$ of a curve $C$ on such a surface $X$ in terms of the cohomologies of divisors on the normalization of $X$. We then study general projections in $\mathbf{P}^3$ of curves lying on the rational normal scroll $S(a,b)\subset\mathbf{P}^{a+b+1}$. If we vary the curves in a linear system on $S(a,b)$ as well as the projections, we obtain a family of curves in …

Tropical surface singularities

In this paper, we study tropicalisations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional type, show that they can generically have only finitely many singular points, and describe all possible locations of singular points. More precisely, we show that singular points must be either vertices, or generalized midpoints and baricenters of certain faces of singular tropical surfaces, and, in some cases, there may be additional metric restrictions to faces of singular tropical surfaces.

$G$-constellations and the maximal resolution of a quotient surface singularity

For a finite subgroup $G$ of $\operatorname{GL}(2, \mathbb C)$, we consider the moduli space ${\mathcal M}_θ$ of $G$-constellations. It depends on the stability parameter $θ$ and if $θ$ is generic it is a resolution of singularities of $\mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C^2/G$ is isomorphic to ${\mathcal M}_θ$ for some generic $θ$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.

Toric G-solid Fano threefolds

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

Symmetries of Surface Singularities

The automorphism group ${\rm Aut}\: X$ of a weighted homogeneous normal surface singularity $X$ has a maximal reductive algebraic subgroup $G$ which contains every reductive algebraic subgroup of ${\rm Aut}\: X$ up to conjugation. In all cases except the cyclic quotient singularities the connected component $G_1$ of the unit equals ${\Bbb C}^*$. The induced action of $G$ on the minimal good resolution of $X$ embeds the finite group $G/G_1$ into the automorphism group of the central curve $E_0$ of the exceptional divisor. We describe $G/G_1$ as a subgroup of ${\rm Aut}\: E_0$ in case $E_0$ is r…

On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities

In earlier joint work with our collaborators Akhtar, Coates, Corti, Heuberger, Kasprzyk, Prince and Tveiten, we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces $X$ with isolated cyclic quotient singularities such that $X$ admits a $\mathbb{Q}$-Gorenstein toric degeneration correspond under Mirror Symmetry to maximally mutable Laurent polynomials $f$ in two variables, and that the quantum period of such a surface $X$, which is a generating function for Gromov-Witten invariants of $X$, coincides with th…

非交换crepant解析的三维孤立奇点上的稳定性条件

Stability conditions on noncommutative crepant resolutions of 3-dimensional isolated singularities

研究非交换crepant解析的三维孤立奇点的稳定性条件，通过构造突变锥并分析突变过程，证明突变锥的性质与最大修改模之间的联系，进而描述稳定性条件下的自同构群。

关于Hilbert方案局部最大奇点的单调性猜想

A monotonicity conjecture for the local maximal singularity of the Hilbert scheme of points

本文提出关于Hilbert方案局部最大奇点的单调性猜想，探讨了colength $l$下切空间最大维数与理想最小纯指数的关系。

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line…

The Coolidge-Nagata conjecture holds for curves with more than four cusps

Let E be a plane rational curve defined over complex numbers which has only locally irreducible singularities. The Coolidge-Nagata conjecture states that E is rectifiable, i.e. it can be transformed into a line by a birational automorphism of the plane. We show that if it is not rectifiable then the tree of the exceptional divisor for its minimal embedded resolution of singularities has at most nine maximal twigs. This settles the conjecture in case E has more than four singular points.