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 equisingularity classes for which one has the complete explicit analytic classification, and in particular for all branches of multiplicity less or equal than four.
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.
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.
\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}.
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.
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 $x=0$.
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.
We provide a geometric elementary proof of the fact that an analytic plane branch is analytically equivalent to one whose terms corresponding to contacts with holomorphic one-forms -- except for Zariski's $λ$-invariant -- are zero (so called "short parametrizations"). This is the main step missed by Zariski in his attempt to solve the moduli problem.
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.
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 $C$.
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.
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.
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…
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.
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…
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…
Consider a plane branch, that is, an irreducible germ of curve on a smooth complex analytic surface. We define its blow-up complexity as the number of blow-ups of points necessary to achieve its minimal embedded resolution. We show that there are $F_{2n-4}$ topological types of blow-up complexity $n$, where $F_{n}$ is the $n$-th Fibonacci number. We introduce complexity-preserving operations on topological types which increase the multiplicity and we deduce that the maximal multiplicity for a plane branch of blow-up complexity $n$ is $F_n$. It is achieved by exactly two topological types, one …
We present a short proof of the fact that two irreducible germs of plane analytic curves are isotopic if they are equisingular, without recourse to the structure of the associated knots.
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 …
We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<α<1$ such that those germs are not $α$-Hölder homeomorphic. For germs of complex analytic plane curves with several irreducible components we prove that if any two of them are $α$-Hölder homeomorphic, for all $0<α<1$, then there is a correspondence between their branches preserving sequence of characteristic exponents and intersection multiplicity of pair of branches. In particular, we recovery the sequence of characteristic exponents…
We study the poles and residues of the complex zeta function $ f^s $ of a plane curve. We prove that most non-rupture divisors do not contribute to poles of $ f^s $ or roots of the Bernstein-Sato polynomial $ b_f(s) $ of $ f $. For plane branches we give an optimal set of candidates for the poles of $ f^s $ from the rupture divisors and the characteristic sequence of $ f $. We prove that for generic plane branches $ f_{gen} $ all the candidates are poles of $ f_{gen}^s $. As a consequence, we prove Yano's conjecture for any number of characteristic exponents if the eigenvalues of the monodromy…
For a local singular plane curve germ $f(X,Y)=0$ we characterize all nonsingular $λ\in\bbC\{X,Y\}$ such that the Łojasiewicz exponent of $\grad\,f$ is not attained on the polar curve $\bJ(λ,f)=0$. When $f$ is not Morse we prove that for the same $λ$'s the maximal polar quotient $q_0(f,λ)$ is strictly less than its generic value $q_0(f)$. Our main tool is the Eggers tree of singularity constructed as a decorated graph of relations between balls in the space of branches defined by using a logarithmic distance.
This paper addresses a very classical topic that goes back at least to Plücker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.
