关于平面分支的Zariski不变量

Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics

A simple sextic is a reduced complex projective plane curve of degree 6 with only simple singularities. We introduce a notion of Z-splitting curves for the double covering of the projective plane branching along a simple sextic, and investigate lattice Zariski k-ples of simple sextics by Z-splitting curves. We define specialization of lattice types, and classify all lattice types of Z-splitting curves of degree less than or equal to 3 up to specializations.

Torsion divisors of plane curves and Zariski pairs

In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the smooth component was implicitly considered. We formulate this relation clearly and give a criterion for distinguishing the embedded topology in terms of torsion classes. Furthermore, we give a method of systematically constructing examples of curves where our criterion is applicable, and give new examples of Zariski tuples.

Zariski multiples associated with quartic curves

We investigate Zariski multiples of plane curves $Z_1, \dots, Z_N$ such that each $Z_i$ is a union of a smooth quartic curve, some of its bitangents, and some of its 4-tangent conics. We show that, for plane curves of this type, the deformation types are equal to the homeomorphism types, and that the number of deformation types grows as $O(d^{62})$ when the degree $d$ of the plane curves tends to infinity.

Classical Zariski pairs

We compute the fundamental groups of all irreducible plane sextics constituting classical Zariski pairs

The irregularity of cyclic multiple planes after Zariski

A formula for the irregularity of a cyclic multiple plane associated to a branch curve that has arbitrary singularities and is transverse to the line at infinity is established. The irregularity is expressed as a sum of superabundances of linear systems associated to some multiplier ideals of the branch curve and the proof rests on the theory of standard cyclic coverings. Explicit computations of multiplier ideals are performed and some applications are presented.

A note on splitting numbers for Galois covers and $π_1$-equivalent Zariski $k$-plets

In this paper, we introduce \textit{splitting numbers} of subvarieties in a smooth variety for a Galois cover, and prove that the splitting numbers are invariant under certain homeomorphisms. By splitting numbers, we give a necessary and sufficient condition for two plane curves of type $(b,m)$ to be topologically equivalent as pairs of the complex projective plane and plane curves, where a plane curve of type $(b,m)$ is an arrangement of two smooth plane curves of degree $3$ and $b$ defined by I.~Shimada. Consequently, we prove that there are $π_1$-equivalent Zariski $k$-plets for any $k\geq2…

On arithmetic Zariski pairs in degree 6

We define a topological invariant of complex projective plane curves. As an application, we present new examples of arithmetic Zariski pairs.

The matroid structure of vectors of the Mordell-Weil lattice and the topology of plane quartics and bitangent lines

In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski $N$-ples of relatively low degree. Namely we show that a Zariski 102-ple of degree 18 exists.

Nodal curves with a contact-conic and Zariski pairs

In this present paper, we study the splitting of nodal plane curves with respect to contact conics. We define the notion of splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.

A weighted version of Zariski's hyperplane section theorem and fundamental groups of complements of plane curves

We formulate and prove a weighted version of Zariski's hyperplane section theorem on the topological fundamental groups of the complements of hypersurfaces in a projective space. As an application, we calculate fundamental groups of the complements of certain singular projective plane curves.

Gal($\overline{\mathbb{Q}}/\mathbb{Q}$) 在 Zariski 多重组上的忠实作用

A faithful action of Gal($\overline{\mathbb{Q}}/\mathbb{Q}$) on Zariski multiplets

本文研究了 Gal($\overline{\mathbb{Q}}/\mathbb{Q}$) 在 Zariski 多重组上的忠实作用，构造了新的算术 Zariski 对和多重组，并探讨了伽罗华表示与奇异性平面曲线几何的相互作用。

反Zariski对

Anti-Zariski pairs

提出反Zariski对概念，即同胚但属于等奇异族不同分支的平面曲线对，并给出例子及相关讨论。

计算仿射平面中轨道 Zariski 闭包的算法

Algorithm to Compute Orbit Zariski Closure in Affine Plane

本文提出一种算法，通过判断有限生成群是否包含于代数群，并计算单个三角自同构的不变子簇，进而计算仿射平面上有限生成群轨道的 Zariski 闭包。

A note on Zariski pairs

A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present a method to construct certain type of Zariski pairs, and make infinite series of examples.

On Zariski's pairs of m-th canonical discriminant curves

In this note we give examples of Zariski's pairs $B_{1,m}, B_{2,m}$ ($m \in N$ and $m \geq 5$) of plane cuspidal curves such that (i) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}:S_i \to P^2$, $i=1, 2$, (ii) $S_1$ and $S_2$ are homeomorphic surfaces of general type, (iii) $f_{i,m}$ is given by linear three-dimensional subsystem of the mth canonical class of $S_i$.

Non-homotopicity of the linking set of algebraic plane curves

The linking set is an invariant of algebraic plane curves introduced by Meilhan and the first author. It has been successfully used to detect several examples of Zariski pairs, i.e. curves with the same combinatorics and different embedding in $\mathds{CP}^2$. Differentiating Shimada's $π_1$-equivalent Zariski pair by the linking set, we prove, in the present paper, that this invariant is not determined by the fundamental group of the curve.

The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.

Uniform Zariski's Theorem On Fundamental Groups

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates isomorphisms of the fundamental groups of the complements to each hypersurface from this family in the p…

Cremona transformations of weighted projective planes, Zariski pairs, and rational cuspidal curves

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study …

Torsion divisors of plane curves with maximal flexes and Zariski pairs

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves which include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper we consider curve arrangements whi…

Analytic classification of plane branches up to multiplicity 4

We perform the analytic classification of plane branches of multiplicity less or equal than four.

On the intersection multiplicity of plane branches

We prove an intersection formula for two plane branches in terms of their semigroups and key polynomials. Then we provide a strong version of Bayer's theorem on the set of intersection numbers of two branches and apply it to the logarithmic distance in the space of branches.

An approach to plane algebroid branches

Our aim is to reprove the basic results of the theory of branches of plane algebraic curves over algebraically closed fields of arbitrary characteristic. We do not use the Hamburger-Noether expansions. Our basic tool is the logarithmic distance on the set of branches satisfying the strong triangle inequality which permits to make calculations directly on the equations of branches.

Families of affine planes: the existence of a cylinder

Given a family of complex affine planes, we show that it is trivial over a Zariski open subset of the base. The proof relies upon a relative version of the contraction theorem.

Semigroups corresponding to branches at infinity of coordinate lines in the affine plane

We characterize in terms of characteristic sequences the semigroups corresponding to branches at infinity of plane affine curves $Γ$ for which there exists a polynomial automorphism mapping $Γ$ onto the axis $x=0$.

Decompositions of the higher order polars of plane branches

\noindent In \cite{Casas} Casas-Alvero found decompositions of higher order polars of an irreducible plane curve generalizing the results of Merle. We improve his result giving a finer decomposition where we determine the topological type and the number of a kind of branches that we call {\em threshold semi-roots}.

The Analytic Classification of Plane Curves with Two Branches

In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent space to the orbits under analytic equivalence in a given equisingularity class to Kähler differentials on the curve.

On Polars of Plane Branches

It is well known that the equisingularity class of the general polar of a plane branch is not the same for all branches in a given equisingularity class, but it is the same for sufficiently general ones and depends upon the analytic type of the branch. The aim of this paper is to go beyond generality and show how one could describe the equisingularity classes of (general) polars of all branches in a given equisingularity class, making use of the analytic classification of branches as described in previous work by the first two authors. We will show how this works in some particular equisingula…

An explicit construction for $n$-contact curves to a smooth cubic via divisions and Zariski tuples

Let $E$ be a smooth cubic. A plane curve $D$ is said to be an $n$-contact curve to $E$ if the intersection multiplicities at each intersection point between $E$ and $D$ is $n$. In this note, we give an algorithm to produce $n$-contact curves to $E$ and consider its application.

平面分支的一般高阶极线的等奇异性类

On the equisingularity class of the general higher order polars of plane branches

本文研究了等奇异性类中一般分支的高阶极线分解，并将Casas-Alvero和Hefez-Hernandes-Hernández的结果推广至高阶极线。

Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility

In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.

平面分支的临界叶理与半根

Dicritical foliations and semiroots of plane branches

本文通过平面分支的半根描述三重点处的临界叶理，给出分隔线一参数族的构造方法，并关联接触阶与解析离散不变量。

Jet schemes of complex plane branches and equisingularity

For $m \in \mathbb{N}$, we determine the irreducible components of the $m$-th Jet Scheme of a complex branch $C$ and give formulas for their number $N(m)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.

On the value set of $1$-forms for plane branches

The value semigroup $Γ$ and the value set $Λ$ of $1$-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch $\mathcal{C}$ with semigroup $Γ$ there are a finitely number of distinct possible sets $Λ_i$ according to the analytic class of $\mathcal{C}$. In this work we show that the value set of $1$-forms $Λ$ determines the semigroup $Γ$ and we present an effective method to recover $Γ$ by $Λ$. In particular, this allows us to decide if a subset of $\mathbb{N}$ is a value set of $1$-forms for a plane branch.

A note on the topology of arrangements for a smooth plane quartic and its bitangent lines

In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points arising from three bitangent lines.

具有多个分支的平面曲线奇点的值半群

The value semigroup of a plane curve singularity with several branches

本文利用Apéry集给出平面曲线奇点与其爆炸之间值半群的相互构造方法，并通过数值刻画所有可能的重数树，完整描述了多分支平面曲线奇点的值半群。

Multi-Harnack smoothings of real plane branches

We introduce a new method for the construction of smoothings of a real plane branch $(C, 0)$ by using Viro Patchworking method. Since real plane branches are Newton degenerated in general, we cannot apply Viro Patchworking method directly. Instead we apply the Patchworking method for certain Newton non degenerate curve singularities with several branches. These singularities appear as a result of iterating deformations of the strict transforms of the branch at certain infinitely near points of the toric embedded resolution of singularities of $(C,0)$. We characterize the $M$-smoothings obtaine…

某个7次圆锥曲线-直线排列的实现空间与一个π₁等价Zariski对

The realization space of a certain conic line arrangement of degree 7 and a $π_1$-equivalent Zariski pair

本文通过构造一个π₁等价Zariski对，确定了具有特定固定组合结构的7次圆锥曲线-直线排列实现空间的连通分支数。

On the monodromy of the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$ does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is natura…

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。

Analytic Moduli of Plane Branches and Holomorphic Flows

We study the behaviour (in the infinitesimal neighbourhood of the singularity) of a singular plane branch under the action of holomorphic flows. The techniques we develop provide a new elementary, geometric and dynamical solution to Zariski's moduli problem for singular branches in $({\mathbb C}^{2},0)$. Furthermore, we study whether elements of the same class of analytic conjugacy are conjugated by a holomorphic flow; in particular we show that there exists an analytic class that is not complete: meaning that there are two elements of the class that are not analytically conjugated by a local …

7次和8次圆锥-直线构型的强Ziegler对示例

Examples of strong Ziegler pairs of conic-line arrangements of degree 7 and 8

本文证明了某些具有相同组合结构但嵌入拓扑不同的平面曲线对（Zariski对）同时也是Milnor代数不同构的强Ziegler对。

Arithmetic behaviour of Frobenius semistability of syzygy bundles for plane trinomial curves

Here we consider the set of bundles $\{V_n\}_{n\geq 1}$ associated to the plane trinomial curves $k[x,y,z]/(h)$. We prove that the Frobenius semistability behaviour of the reduction mod $p$ of $V_n$ is a function of the congruence class of $p$ modulo $2λ_h$ (an integer invariant associated to $h$). As one of the consequences of this, we prove that if $V_n$ is semistable in characteristic 0, then its reduction mod $p$ is strongly semistable, for $p$ in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles $V_n$, there is a common Zariski dense set of such …

平面分支的非退化条件

Non-degeneracy conditions for plane branches

本文研究复平面解析曲线芽的参数化与隐式方程的关系，证明Weierstrass多项式的系数多项式均可表示为参数化系数的多项式，为平面分支的非退化条件研究提供基础。

Bernstein多项式与平面分支微分的取值集

Bernstein polynomial and value set of differentials for plane branches

本文研究平面分支的Bernstein多项式根与微分取值集的关系，描述了半拟齐次多项式定义的平面分支中，具有固定微分取值集的分支共享的Bernstein多项式公共根集合。

具有两个分支和高相交重数的平面曲线的Tjurina数

The Tjurina number of a plane curve with two branches and high intersection multiplicity

本文构造了一族具有两个分支的约化平面曲线，证明其Tjurina数在等奇异性类中恒定，并给出基于拓扑数据的封闭公式。

The log-canonical threshold of a plane curve

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in a paper by Kuwata in Amer. J. Math. 121.

Morsifications of real plane curve singularities

A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real morsifications for real plane curve singularities having arbitrary real local branches and pairs of complex conjugate branches satisfying some conditions. This was known before only in the case of all local branches being real (A'Campo, Gusein-Zade). We also discuss a relation between real morsifications and the topology of singularities, extending to arbitrary real mor…

On the Analytic Invariants and Semiroots of Plane Branches

The value semigroup of a $k$-semiroot $C_k$ of a plane branch $C$ allow us to recover part of the value semigroup $Γ=\langle v_0,\ldots ,v_g\rangle$ of $C$, that is, it is related to topological invariants of $C$. In this paper we consider the set of values of differentials $Λ_k$ of $C_k$, that is an analytical invariant, and we show how it determine part of the set of values of differentials $Λ$ of $C$. As a consequence, in a fixed topological class, we relate the Tjurina number $τ$ of $C$ with the Tjurina number of $C_k$. In particular, we show that $τ\leq μ-\frac{3n_g-2}{4}μ_{g-1}$ where $n…

双有理平面上的二次点

Quadratic points on double planes

本文研究高维代数簇上Zariski稠密二次点的几何刻画，证明在双覆盖情形下Vojta猜想可排除典范丛充分正时的其他稠密来源，并给出反例说明该条件不可省略。

Number of rational branches of a plane singular curve over a finite field

Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points on a non-singular model of $\mathcal{F}$. In this note, the case of a curve with two singularities such that the sum of their multiplicities is precisely the degree of the curve is investigated in more depth. In particular, such plane models are completely characterized, and for $p > 3$, a curve of this type attaining one of the obtained bounds is presented…

On the Saito's basis and the Tjurina Number for Plane Branches

We introduce the concept of good Saito's basis for a plane curve $S$ and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Greuel relating the Tjurina number and the Milnor number for a singular irreducible plane curve.

Equisingularity of map germs from a surface to the plane

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_ε$ is a stabilization of $f$. We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular. As an application, we show that a family $(X_t,0)$ is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection are constant on $t$.

On plane algebroid curves

Two plane analytic branches are topologically equivalent if and only if they have the same multiplicity sequence. We show that having same semigroup is equivalent to having same multiplicity sequence, we calculate the semigroup from a parametrization, and we characterize semigroups for plane branches. These results are known, but the proofs are new. Furthermore we characterize multiplicity sequences of plane branches, and we prove that the associated graded ring, with respect to the values, of a plane branch is a complete intersection.