On the branch curve of a general projection of a surface to a plane

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.

Approximate Roots, Toric Resolutions and Deformations of a Plane Branch

We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial $f$ defining a plane branch $(C,0)$, in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non equisingular) deformations of a plane branch $(C,0)$ supported on certain monomials in the approximate roots of $f$. As a consequence we find out a Kouchnirenko type formula for the Milnor number $(C,0)$. Our results provide a geometrical approach to Abhyankar's straight line conditions and its consequences. As an application we give an equisingularity criterion f…

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.

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.

The Fundamental Group of a CP^2 Complement of a Branch Curve as an Extension of a Solvable Group by a Symmetric Group

The main result in this paper is as follows: Let S be the branch curve (in the projective plan) of a generic projection of a Veronese surface. Then the fundamental group of the complement of S is an extension of a solvable group by a symmetric group. A group with the property mentioned above is ``almost solvable'' in the sense that it contains a solvable normal subgroup of finite index. This raises the question for which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projection to the complex plane an …

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.

Plane branches with Newton nondegenerate polars

We characterize the equisingularity classes of irreducible plane curve germs whose general members have a Newton nondegenerate general polar curve. In addition, we give explicit Zariski open sets of curves in such equisingularity classes whose general polars are Newton nondegenerate and describe their topology.

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。

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…

On projections of smooth and nodal plane curves

Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $ν\colon \hat C\to C$ is its normalization, and $π\colon \hat C\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as a projection $\mathrm{pr}_p\circ ν\colon\hat C \to\mathbb P^1$, where $p\in\mathbb P^2\setminus C$ (if $\mathrm{deg}\, C=3$, one should assume in addition that $\degπ\ne4$). We prove that the morphism $π$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to(\mathbb P^2)^*$ ramified over $C^*$, where $X$ is a smooth surface. As …

Algebraic curves admitting the same Galois closure for two projections

A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. As an application, it is proved that the Hermitian curve in positive characteristic coincides with the Galois closures of projections of some plane curve from some two non-uniform points.

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…

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 Singular Locus of a Plane Projection of a Complete Intersection

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

矩阵分解与曲线奇点的通用平面投影

Matrix factorization and the generic plane projection of curve a singularity

本文研究了$\mathbb{C}^3$中曲线奇点通用平面投影定义的矩阵分解，并反向探讨了给定平面曲线奇点$Y$对应的空间曲线奇点$X$所定义的矩阵分解族。

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。

Caustics of plane curves, their birationality and matrix projections

After recalling the notion of caustics of plane curves and basic equations, we first show the birationality of the caustic map for a general source point S in the plane. Then we prove more generally a theorem for curves D in the projective space of 3x3 symmetric matrices B. For a general 3x1 vector S the projection to the plane given by B --- BS is birational on D, unless D is not a line and D is contained in a plane of the form Delta_v = {B | Bv = 0}.

Strata of differentials of the second kind, positivity and irreducibility of certain Hurwitz spaces

We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to $\overline{\mathcal{M}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the class for all genus when the divisorial projection forgets only simple zeros and show in these cases the genus $g=0$ projections are indeed nef. Hurwitz spaces: We show the Hurwitz spaces of degree $d$, genus $g$ covers of $\mathbb{P}^1$ with pure branching (one ramified point ove…

关于正规曲面上曲线经平面投影的像

On the image of a curve in a normal surface by a plane projection

本文研究了复解析正规曲面到平面的有限解析态射下约化曲线像的拓扑结构，通过迭代线性系方法描述了像的拓扑，并推广了已有光滑情形的结论，同时给出了判别式曲线的拓扑型。

有理倍数平面的构造

Constructions for rational multiple planes

本文在适当一般条件下，分类了更简单的三重平面，且给出了无限多个非克雷莫纳等价的更高倍数平面家族的例子。

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.

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。

On the factorization of the polar of a plane branch

In this paper we present the most complete description as possible of the factorization of the general polar of the general member of an equisingularity class of irreducible germs of complex plane curves. Our result will refine the rough description of the factorization given by M. Merle in the 70's and it is based on a result given by E. Casas-Alvero in the 90's that describes the cluster of the singularities of such polars. By using our analysis, it will be possible to characterize all equisingularity classes of irreducible plane germs with r characteristic exponents having the exceptional b…

On ramified covers of the projective plane I: Segre's theory and classification in small degrees, with Appendix by Eugenii Shustin

We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise. First, what is the_geography_ of branch curves among all nodal-cuspidal curves? Second: what is the_geometry_ of branch curves? In other words, how can one distinguish a branch curve from a non-branch curve with the same numerical inva…

从分支点恢复平面曲线

Recovery of Plane Curves from Branch Points

本文研究从分支点恢复三次和四次平面曲线的问题，利用平面Hurwitz数刻画解轨道数，确定实数解个数，并基于经典代数几何给出精确恢复算法。

Braid Group techniques in Complex Geometry V: The fundamental group of the complement of a Veronese generic projection

Computation of the fundamental group of the complement in the complex plane of the branch curve S , of a generic projection of the Veronese surface to the plane is presented. This paper is a continuation of our previous papers: Braid Group Technique I - IV. In I and II we developed algorithms to compute braid monodromy of a brunch curve, provided there exist a degeneration of the surface to union of planes in a configuration where the associated branch curve is partial to a line arrangement dual to generic. In III we constructed a degeneration of the Veronese surface of order 3 to union of pla…

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.

Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to S. The problem of classifying the germs of such normal surface singularities was already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts: discrete and continous. The continous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the discrete par…

Singularities of plane rational curves via projections

We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal curve $C_n\subset \mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. In particular, we define via ${\mathbf{f}}$ certain 0-dimensional schemes $X_k\subset \mathbb{P}^k$, $2\leq k\leq (n-1)$, which encode all information on the singularities of multiplicity $\geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether …

Thomae type formula for K3 surfaces given by double covers of the projective plane branching along six lines

In this paper, we give Thomae type formula for \KK surfaces $\cS$ given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type $I_{22}$ and period integrals of $X$. Moreover, we express the period integrals by using the hypergeometric function $F_S$ of four variables. As an application of our main theorem, we define $\R^4$-valued sequences by mean iterations of four terms, and express their common limits by the hypergeometric function $F_S$.

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.

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…

平面分支的Zariski模问题的构造性解

Constructive solution of Zariski's Moduli Problem for Plane Branches

该研究以算法方式计算不可约全纯平面曲线芽的Kähler微分集合，讨论其值半模的多种生成概念并给出对应基的几何性质与算法构造，得到平面分支Zariski模问题的显式构造性解。

平面中被抛物线包围的区域及其形状的树表示

Regions surrounded by parabolas in the plane and trees representing their shapes respecting their natural projection to the line

本文研究平面中由抛物线包围的区域及其形状的树表示，探讨其与实代数几何和动量映射的关系。

平面分支的解析模的通用部分的显式计算

Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

本文提出了一种算法，用于计算平面分支等奇异类中解析模的通用部分，并提供了Genzmer公式的另一种证明。