Maximal rank for planar singularities of multiplicity two

Symmetric Determinantal Singularities I: The Multiplicity of the Polar Curve

This paper is the first part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.

splice quotient singularities的abelian covers的multiplicity

The multiplicity of abelian covers of splice quotient singularities

本文提出通过解析图和伽罗瓦群计算splice quotient奇点的abelian覆盖的multiplicity的方法。

与光滑边界具有常数相交重数的平面曲线奇点

Planar curve singularities having constant intersection multiplicity with a smooth boundary

本文研究光滑曲面中与光滑边界曲线具有常数相交重数的曲线奇点的形变，刻画了等奇点轨迹的余维数及理想包含关系，并分类了余维数不超过3的奇点。

Bounds of the multiplicity of abelian quotient complete intersection singularities

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.

The $4n^2$-inequality for complete intersection singularities

The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is higher than $4n^2μ$, where $μ$ is the multiplicity of the singular point.

Enriques diagrams and adjacency of planar curve singularities

We study adjacency of equisingularity types of planar curve singularities in terms of their Enriques diagrams. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also show an example of singular curve of type D' that can be deformed to a curve of type D without D' being adjacent to D.

Birationally rigid complete intersections with a singular point of high multiplicity

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized $4n^2$-inequality for complete intersection singularities and the technique of hypertangent divisors.

The multiplicity of pairs of modules and hypersurface singularities

This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a subm…

Macdonald formula for curves with planar singularities

We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize …

具有平面奇点的曲线平展双覆盖的紧化Prym的自对偶性

Autoduality of compactified Pryms for étale double covers of curves with planar singularities

针对具有平面奇点的整曲线的平展双覆盖，构造了紧化Prym簇上的Poincaré层，并证明了相应的Fourier-Mukai变换是其导出范畴的自等价，进而应用于证明Corti-Hanamura的动机分解猜想并构造乘性动机反常滤过。

相邻奇点、TQFT与Zariski重数猜想

Adjacent Singularities, TQFTs, and Zariski's Multiplicity Conjecture

本研究利用不动点Floer上同调的TQFT结构与相邻奇点生成的辛配边，构造Floer上链的链映射并结合McLean谱序列，给出孤立超曲面奇点下Zariski重数猜想的新证明，同时重现Varchenko定理。

Invariants of Newton non-degenerate surface singularities

We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (even a stronger version of) Zariski's Conjecture about the multiplicity for such a singularity.

On the simplification of singularities by blowing up at equimultiple centers

Resolution of singularities of varieties over fields of characteristic zero can be proved by using the multiplicity as main invariant. The proof of this result leads to new questions in positive characteristic. We discuss here results which follow by induction on the dimension of the varieties. Fix a variety $X^{(d)}$ of dimension $d$ over a {\em perfect field} $k$ or, more generally, a pure dimensional scheme of finite type over $k$. Fix a closed point $x\in X^{(d)}$ of multiplicity $e>1$. Define a local simplification of the multiplicity at $x\in X^{(d)}$ as a proper birational map, say $X^{…

拟齐次非孤立奇点超曲面的Zariski重数猜想

Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities

本文针对2≤n≤4维拟齐次非孤立奇点超曲面，利用拟齐次正规形推导出由权重和次数决定的重数显式公式，证明重数在拓扑等价下不变，从而证实Zariski重数猜想。

来自一般平面曲线奇点的广义仿射Springer纤维的纯性

Purity of generalized affine Springer fibers from generic planar curve singularities

本文通过构造显式仿射铺砌，证明了广义不可约平面曲线奇点上点状Hilbert概型的上同调纯性，由此确立了广义$GL_N$-仿射Springer纤维纯性猜想的一个新情形，并推广了关于紧化Jacobian的组合学结果。

广义Milnor数与Tjurina数的商在任意特征下 hypersurface 畸变中的上界

The Quotient of Milnor Number by Tjurina Number of Hypersurface Singularities in Arbitrary Characteristic

本文通过Hilbert-Samuel乘数等工具，建立了任意特征下孤立hypersurface奇异性的广义Milnor数与Tjurina数商的精确上界，并推导出特征零下的上界，证明了孤立表面奇点的μ/τ≤3/2，回应了Almirón猜想并验证了Durfee猜想的弱形式。

Equimultiplicity of families of map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$

In 1971, Zariski proposed some questions in Theory of Singularities. One of such problems is the so-called, nowadays, Zariski's multiplicity conjecture. In this work, we consider the version of this conjecture for families. We answer positively Zariski's multiplicity conjecture for a special class of non-isolated singularities.

Multiplicity of complex hypersurface singularities, Rouche' satellites and Zariski's problem

Soient $f,g\colon (\hbox{åC}^n,0) \to (\hbox{åC},0)$ des germes de fonctions holomorphes réduits. Nous montrons que $f$ et $g$ ont la même multiplicité en 0 si et seulement s'il existe des germes réduits $f'$ et $g'$ analytiquement équivalents à $f$ et $g$, respectivement, tels que $f'$ et $g'$ satisfassent une inégalité du type de Rouché par rapport à un `petit' cercle générique autour de~0. Comme application, nous donnons une reformulation de la question de Zariski sur la multiplicité et une réponse partielle positive à celle--ci.

Segre Numbers and Hypersurface Singularities

We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-Böger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of various equisingularity conditions, such as Verdier's condition W, and conditions $A_f$ and $W_f$.

Symmetric Determinantal Singularities II: Equisingularity and SEIDS

This paper is the second part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.

Double spaces with isolated singularities

We prove the non-rationality of a double cover of $\mathbb{P}^{n}$ branched over a hypersurface $F\subset\mathbb{P}^{n}$ of degree $2n$ having isolated singularities such that $n\ge 4$ and every singular points of the hypersurface $F$ is ordinary, i.e. the projectivization of its tangent cone is smooth, whose multiplicity does not exceed $2(n-2)$.

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.

The Multiplicity Polar Theorem and Isolated Singularities

Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pa…

The multiplicity and the number of generators of an integrally closed ideal

Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay…

平面多链和有理奇点

Planar multilinks and rational singularities

本文研究了平面多链与有理奇点的关系，证明了若表面奇点的canonical接触结构由平面多链开书支撑，则该奇点必为有理奇点，并通过平面多链的组件多重性1来特征化嵌套奇点。同时，文章还展示了平面开书的一些拓扑性质可扩展到平面多链：辛填充为负定且不包含正亏格的辛表面，以及Heegaard Floer接触不变量在HF_{red}中消失。

An asymptotic existence theorem for plane curves with prescribed singularities

Let $d,m_1,...,m_r$ be ($r+1$) positive integers, and $P_1,...,P_r$ be $r$ general points in the projective plane ; let $m$ be a positive integer. We prove that there exists a bound $d_0(m)$ such that : If $m_i < m$ ($0<i<r+1$), and $d > d_0(m)$ then the linear system $L$ of plane curves of degree $d$ having a multiplicity at least $m_i$ at each point $P_i$ has the expected dimension ; moreover, if $L$ is not empty, there exists an irreducible plane curve of degree $d$, smooth away from the $r$ points $P_i$, and having an ordinary singularity of the prescribed multiplicity $m_i$ at each point …

Singularities in positive characteristic, stratification and simplification of the singular locus

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case.

On elimination of variables in the study of singularities in positive characteristic

The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an inductive manner, namely by successive elimination of variables. When applied to hypersurface singularities they lead us to a refinement of the notion of multiplicity. The main theorem proves that, under some numerical conditions expressed by these invariants, singularities can be simplified by blowups at centers prescribed by this refinement.

从平面曲线奇点的计数几何角度看对数凹性

Log-concavity from enumerative geometry of planar curve singularities

本文提出关于平面曲线奇点计数几何中BPS不变量的对数凹性猜想，并将其扩展到Legendrian链环的规则多项式和特征簇的E多项式，证明了不可约加权齐次奇点（扭线结）和ADE奇点的猜想，以及规则多项式的乘法性质。

Base points of natural line bundles on relatively generic surface singularities

In \cite{NNM} the author with A. Némethi computed the multiplicity of generic surface singularities, the formula is purely topological computable from the resolution graph of the surface singularity. In the present paper we extend the results partly to the relative case, when there is a pair of resolution graphs $\mathcal{T}_1 \subset \mathcal{T}$, a fixed singularity $\tX_1$ with resolution graph $\mathcal{T}_1$, a relatively generic singularity $\tX$ corresponding to the subsingularity $\tX_1$ with resolution graph $\mathcal{T}$. We determine the base points of the natural line bundles (unde…

计数 $\mathbb{P}^3$ 中具有退化奇点的平面曲线

Counting planar curves in $\mathbb{P}^3$ with degenerate singularities

本文针对 $\mathbb{P}^3$ 中位于 $\mathbb{P}^2$ 内、含 $δ$ 个结点和1个余维数 $k$ 奇点的 $d$ 次曲线数量问题，在 $δ+k \leq 4$ 时给出显式公式，通过扩展拓扑方法验证了与已有结果及相关猜想的一致性。

平面慢度曲面的奇点分类

Classification of singularities of planar slowness surfaces

本文完全分类了二维弹性波慢度曲面的奇点类型，包括横截自交和同心圆与椭圆相切产生的切向奇点，并证明了横截自交时弹性波算子的主符号局部光滑可对角化。

Expected term bases for generic multivariate Hermite interpolation

The main goal of the paper is to find an effective estimation for the minimal number of generic points in $\mathbb K^2$ for which the basis for Hermite interpolation consists of the first $\ell$ terms (with respect to total degree ordering). As a result we prove that the space of plane curves of degree $d$ having generic singularities of multiplicity $\leq m$ has the expected dimension if the number of low order singularities (of multiplicity $k\leq12$) is greater then some $r(m,k)$. Additionally, the upper bounds for $r(m,k)$ are given.

Milnor numbers of projective hypersurfaces with isolated singularities

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.

Invariants of plane curve singularities and Newton diagrams

We present an intersection-theoretical approach to the invariants of plane curve singularities $μ$, $δ$, $r$ related by the Milnor formula $2δ=μ+r-1$. Using Newton transformations we give formulae for $μ$, $δ$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities.

Construction of bicuspidal rational complex projective plane curves

We give several constructions of bicuspidal rational complex projective plane curves, and list the Newton pairs and the multiplicity sequences of the singularities on the resulting curves. Although the existence of some of the listed cusp configurations is already proved in the literature, the advantage of our approach is that on one hand we explicitly state the Newton pairs and the multiplicity sequences of the singularities on the curves, on the other hand, we also give birational transformations transforming a conic or a bicuspidal curve with simple equation to these rational bicuspidal cur…

Arc spaces of cA-type singularities

We study the space of arcs on a singularity of the form xy=f(z_1,..., z_n) and prove 2 main results. (i) The number of irreducible components equals the multiplicity of f minus 1. (ii) If n>1 and the leading homogeneous term of f is not a perfect square then the Nash map from the set of irreducible components to the set of essential divisors is not surjective.

Invariants of plane curve singularities and Plücker formulas in positive characteristic

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $μ(f)=2δ(f)-r(f)+1$ and that $κ(f)=2δ(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $μ(f)\geq 2δ(f)-r(f)+1$ by showing that $μ(f)-\left( 2δ(f)-r(f)+1\right)$ measures the wild vanishing cycles. By introducing new invariants $γ,\tildeγ$, we prove in this note that $κ(f)\geq γ(f)+\mathrm{mt}(f)…

The cohomology of line bundles of splice-quotient singularities

We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of relative sections of line bundles (proving that the equivariant, divisorial multi-variable Hilbert series is topological), a combinatorial description of divisors of analytic function-germs, and an expression for the multiplicity of the singularity from its resolution graph. Additional, we establish a new formula for the Seiberg-Witten invariants of any rati…

孤立曲面奇点的motivic局部密度

Motivic local density of isolated surface singularities

本文旨在计算孤立代数曲面奇点的motivic局部密度，以解释其与代数乘数的关系，引入了与奇点bilipschitz几何相关的内部速率数据。

平面、有理曲线 over ${\mathbb F}_2$ 其唯一奇点是一个二重点

Planar, rational curves over ${\mathbb F}_2$ whose only singularity is a double point

研究了 over ${\mathbb F}_2$ 的平面有理曲线，其唯一奇点为二重点，且在特征0时仅限于度数不超过6的情况。

Codimension Two Determinantal Varieties with Isolated Singularities

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…

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.

Linear Systems of Plane Curves with Base Points of Equal Multiplicity

In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed in a previous artic…

Autoduality of compactified Jacobians for curves with plane singularities

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare line bundle. We prove that the Fourier-Mukai transform given by P is an auto-equivalence of the derived category of J'.

Degenerations of elliptic curves and cusp singularities

We give more or less explicit equations for all two-dimensional cusp singularities of embedding dimension at least 4. They are closely related to Felix Klein's equations for universal curves with level n structure. The main technical result is a description of the versal deformation of an n-gon in $P^{n-1}$. The final section contains the equations for smoothings of simple elliptic singularities (of multiplicity at most 9).

通过局部不变量研究 $\mathbb C^3$ 中的曲面奇点的扎里斯基等奇异性和

Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

本文通过定义一个新的局部不变量，研究了 $\mathbb C^3$ 中曲面奇点的扎里斯基等奇异性和性质。

The link of {f(x,y)+z^n=0} and Zariski's Conjecture

We consider suspension hypersurface singularities of type g=f(x,y)+z^n, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated h…

Sharp bounds on fake weighted projective spaces with canonical singularities

We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on the index of the sublattice generated by the vertices of a lattice simplex containing only the origin as an interior lattice point. We also completely characterise when equality occurs and discuss related questions and conjectures.

The Newton polygon of a planar singular curve and its subdivision

Let a planar algebraic curve $C$ be defined over a valuation field by an equation $F(x,y)=0$. Valuations of the coefficients of $F$ define a subdivision of the Newton polygon $Δ$ of the curve $C$. If a given point $p$ is of multiplicity $m$ for $C$, then the coefficients of $F$ are subject to certain linear constraints. These constraints can be visualized on the above subdivision of $Δ$. Namely, we find a distinguished collection of faces of the above subdivision, with total area at least $\frac{3}{8}m^2$. In a sense, the union of these faces in "the region of influence" of the singular point …

Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

We prove that if $\{f_t\}$ is a family of line singularities with constant Lê numbers and such that $f_0$ is a homogeneous polynomial, then $\{f_t\}$ is equimultiple. This extends to line singularities a well known theorem of A. M. Gabrièlov and A. G. Kušnirenko concerning isolated singularities. As an application, we show that if $\{f_t\}$ is a topologically $\mathscr{V}$-equisingular family of line singularities, with $f_0$ homogeneous, then $\{f_t\}$ is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated…

Punctual Hilbert Schemes and Certified Approximate Singularities

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f\_1, \ldots, f\_N)\in C[x\_1, \ldots, x\_n]^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of $f$ such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes th…

On curves on sandwiched surface singularities

Fixed a point O on a non-singular surface S and a complete mO-primary ideal I in its local ring, the curves on the surface X obtained by blowing-up I are studied in terms of the base points of I. Criteria for the principality of these curves are obtained. New formulas for their multiplicity, intersection numbers and order of singularity at the singularities of X are given. The semigroup of branches going through a sandwiched singularity is effectively determined, too.

Durfee's conjecture on the signature of smoothings of surface singularities

In 1978 Durfee conjectured various inequalities between the signature and the geometric genus of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases. We prove a `strong' Durfee--type inequality for any smoothing of a Gorenstein singularity, provided that the intersection form the resolution is unimodular, and the conjectured `weak' inequality for all hypersurface singularities and for sufficiently large multiplicity strict complete intersections. The proofs establish general inequalities valid for any normal sur…