arXiv subjects
Publications and source records attributed to Marcelo Escudeiro Hernandes.
In this paper, we consider Newton-Puiseux parametrizations of irreducible curves in (n+1)-space, n greater or equal to 1, within a fixed semigroup under the action of Mather's group A. We establish criteria for eliminating parameters while preserving the Newton-Puiseux form, extending known results for plane curves.
In this work, we study the relation between the roots of the Bernstein polynomial $\widetilde{b}$ and the value set of differentials $Λ$ for plane branches. For plane branches defined by semiquasihomogeneous polynomial we describe the set of common roots of $\widetilde{b}$ sharing by every branch with a fixed $Λ$ set.
We present some results concerning the Saito module and the torsion submodule of an analytic plane curve, and we provide a method for computing them. Using this algorithm, we compute analytic invariants for plane curves with multiplicity less than or equal to three.
We show how to obtain the Zariski invariant of a plane branch employing the contact order or the intersection multiplicity with elements in a particular family of curves and we present some consequences of this result.
In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $\mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a conjecture proposed by A. Dimca in [7]. Our results are obtained by establishing more manageable formulas to compute the colengths of fractional ideals of the local ring associated with the algebroid (not necessarily a complete intersection) curve with several branches. We then apply these results to the Jacobian ideal of a plane curve over $\mathbb C$ to get a new formula for its Tjurina number and a proof of Dimca's conjecture. We end the paper by establishing a connection between the module of Kähler differentials on the curve modulo its torsion, seen as a fractional ideal, and its Jacobian ideal, explaining the relation between the present approach and that of [1].
We introduced an $\tilde{\mathcal{A}}$-invariant for quasi-ordinary parameterizations and we consider it to describe quasi-ordinary surfaces with one generalized characteristic exponent admitting a countable moduli.
In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allow us to recover some particular cases found in the literature.
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.
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 $n_g=gcd(v_0,\ldots ,v_{g-1})$, $μ$ and $μ_{g-1}$ denote the Milnor number of $C$ and $C_{g-1}$ respectively. If $n_g=2$, we have that $τ=μ-μ_{g-1}$ for any curve in the topological class determined by $Γ$ that is a generalization of a result obtained by Luengo and Pfister.
Differently from the Milnor number, no formula relating the Tjurina number of a reducible algebroid curve to invariants of its branches was known. The aim of this work is to provide such a formula for a complete intersection algebroid curve with several branches in terms of invariants of its branches and the maximal points of the set of values of Kähler differentials on components of the curve.
In this paper we present an algorithm to compute a Standard Basis for a fractional ideal $\mathcal{I}$ of the local ring $\mathcal{O}$ of an $n$-space algebroid curve with several branches. This allows us to determine the semimodule of values of $\mathcal{I}$. When $\mathcal{I}=\mathcal{O}$, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module $Ω_{\mathcal{O}/\mathbb{C}}$ of a plane curve with a fractional ideal of $\mathcal{O}$ and applying our algorithm, we can compute the set of values of $Ω_{\mathcal{O}/\mathbb{C}}$, which is an important analytic invariant associated to the curve.
We introduce the semiring of values $Γ$ with respect to the tropical operations associated to an algebroid curve. As a set, $Γ$ determines and is determined by the well known semigroup of values $S$ and we prove that $Γ$ is always finitely generated in contrast to $S$. In particular, for a plane curve, we present a straightforward way to obtain $Γ$ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.
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 behavior that the general polar of a general curve in this equisingularity class has only irreducible components with less than r characteristic exponents, generalizing a result previously obtained for r=2 by the authors.
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.
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.