Plane Branches Research Articles

Classification of unimodal parametric plane curve singularities in positive characteristic

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating “very short” parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove and which may be of independent interest.

Dicritical foliations and semiroots of plane branches

In this work we describe dicritical foliations in (ℂ2,0) at a triple point of the resolution dual graph of an analytic plane branch C using its semiroots. In particular, we obtain a constructive method to present a one-parameter family Cu of separatrices for such foliations. As a by-product we relate the contact order between a special member of Cu and C with analytic discrete invariants of plane branches.

An explicit deformation of a plane branch with constant δ-invariant

We construct an explicit deformation of a plane branch with constant δ-invariant, where δ denotes the double point number of the singularity. The generic fibers of the deformation are the union of two irreducible curves. Our construction works over algebraically closed fields of arbitrary characteristic.

Analytic Semiroots for Plane Branches and Singular Foliations

The analytic moduli of equisingular plane branches has the semimodule of differential values as the most relevant system of discrete invariants. Focusing in the case of cusps, the minimal system of generators of this semimodule is reached by the differential values attached to the differential 1-forms of the so-called standard bases. We can complete a standard basis to an extended one by adding a last differential 1-form that has the considered cusp as invariant branch and the “correct” divisorial order. The elements of such extended standard bases have the “cuspidal” divisor as a “totally dicritical divisor” and hence they define packages of plane branches that are equisingular to the initial one. These are the analytic semiroots. In this paper we prove that the extended standard bases are well structured from this geometrical and foliated viewpoint, in the sense that the semimodules of differential values of the branches in the dicritical packages are described just by a truncation of the list of generators of the initial semimodule at the corresponding differential value. In particular they have all the same semimodule of differential values.

On the Approximate Polar Curves of Foliations

We present a decomposition theorem of the generic polar curves of a generalized curve foliation with only one separatrix and the Hamiltonian foliations defined by the approximate roots of the generatrix. This is a generalization to foliations of the decomposition theorem of approximate Jacobians given by García Barroso and Gwoździewicz for plane branches.

Complete transversal and formal normal forms of germs of vector fields

In this work, inspired by the technique of the complete transversal, used for the classification of plane branches, developed by Hefez, A. and Hernandes, M., as well as Bruce, J.W., Kirk, N.P. and du Plesis, A.A., study the singularities of applications, we establish a classification of vector fields through their normal forms. In the case of vector fields with non zero linear part in $(\mathbb{C}^{2}, 0) $ and nilpotent fields in $(\mathbb {C}^{n}, 0), n\geq 2$ we recover the classical normal forms for those fields, and we provide a formal normal form different from Takens in dimension 2. Likewise, we obtain the normal form for the vector fields in $(\mathbb{C},0)$ of any multiplicity.

To the calculation of the stability of lattice elements of steel structures

Introduction. The practical method of calculating lattice elements of steel structures for local and general stability contains an internal contradiction. The general stability check assumes the presence of an additional branch bending arrow in the lattice plane, and the local stability check negates it, which does not guarantee local stability when checking the general stability of the lattice rod. The calculated combinations of forces at the ends of any structural element, under the action of the same calculated combinations of loads, almost always have different values. Consequently, the longitudinal forces in the branches of the lattice elements, separated by the calculated length from the lattice plane, will always be variable, which is not taken into account in the current design standards.
 
 Materials and methods. To eliminate the mentioned contradiction, an analytical method is proposed for solving the problem of general stability of lattice elements of steel structures, taking into account the provision of local stability of the most loaded branch. The solution of the problem of stability of a branch from the lattice plane (two-branched elements) is carried out using the inverse numerical-analytical method: according to a given stress-strain state in the most loaded section of the elastic branch, acting and fictitious, compensating for the development of plastic deformations, deformation forces are numerically determined. Then, by the inverse analytical solution of the deformation problem, the actual loading of the branch at its ends is established.
 
 Results. The coefficient of overall stability of lattice elements is determined by solving a quadratic equation that includes an additional dependence on the coefficient of stability of the branch in the lattice plane. The stability coefficient of the branch from the lattice plane is obtained in the general case of loading by a longitudinal force with different values of end eccentricities in combination with a uniformly distributed axial load.
 
 Conclusions. An analytical solution to the problem of general stability of lattice elements is proposed, taking into account the stability of the branch. The solution of the problem of stability of a branch from the lattice plane (two-branched elements) is obtained when it is actually loaded with a variable longitudinal force acting with different values of terminal eccentricities.

Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups.

On the value set of 1-forms for plane branches

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

On the analytic invariants and semiroots of plane branches

The value semigroup of a k-semiroot Ck of a plane branch C allows us to recover part of the value semigroup Γ=〈v0,…,vg〉 of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λk of Ck, 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 Ck. In particular, we show that τ≤μ−3ng−24μg−1 where ng=gcd(v0,…,vg−1), μ and μg−1 denote the Milnor number of C and Cg−1 respectively. If ng=2, we have that τ=μ−μg−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

Moderately Discontinuous Homology

AbstractWe introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow “moderately discontinuous” chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group for any and homomorphisms for any . Here is a “discontinuity rate”. The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms are essential. For Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for ‐Homology recovers the homology of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi‐Lipschitz subanalytic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer‐Vietoris long exact sequences. Moreover, fixed a discontinuity rate b we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are b‐moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, and recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs. © 2020 Wiley Periodicals LLC.

The monodromy conjecture for a space monomial curve with a plane semigroup

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded Q-resolution of this pair and use an A’Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

H-Plane Waveguide In-Phase Power Divider/Combiner With High Isolation Over the WR-3 Band

A novel <inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula>-plane waveguide in-phase power divider (also known as combiner) with high isolation working across full WR-3 band (220-330 GHz) is realized in this paper. This distinctive architecture is designed based on the sum-difference principle of Magic-T, whose <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula>-plane branch (difference port) is matched with an <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula>-plane T-junction. Both effects of processing and assembling uncertainties on this power divider/combiner performance are discussed in detail to display error sensitivity. The physical prototype is split into three blocks and processed by the state-of-the-art computer numerical control (CNC) milling technology. The absorbing materials are housed in the loaded T-junction arms to terminate return wave as well. Compared with the most reported terahertz waveguide power couplers, the meaningful measured results of this advanced in-phase power divider are delivered. The input and output matching as well as high isolation have been obtained over full WR-3 band, when an 11 dB reference level is considered. A total insertion loss of about 1.2 dB is achieved including three straight waveguides having a total length of 30 mm, which is also in good agreement with simulations. This module, which is fully compatible with the standard CNC-milling technique, can be directly integrated to terahertz detection systems.

Computational Bifurcations Occurring on Red Fixed Components in the λ-Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map

Optimal fourth-order multiple-root finders with parameter λ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the λ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The λ -parameters for periodic orbits in the parameter plane are painted in different colors depending on their periods, and the bounded or chaotic ones are colored black to illustrate λ -dependent connected components. When a red fixed component in the parameter plane branches into a q-periodic component, we encounter geometric bifurcation phenomena whose characteristics determine the desired boundary equation and bifurcation point. Computational results along with illustrated components support the bifurcation phenomena underlying this paper.

On the Saito basis and the Tjurina number for plane branches

On the Saito basis and the Tjurina number for plane branches

Note on the monodromy conjecture for a space monomial curve with a plane semigroup

Roughly speaking, the monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy. In this note, we determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a plane branch. In particular, this yields a proof of the monodromy conjecture for such a curve.

Families of $K3$ surfaces and curves of (2,3)-torus type

We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of (2,3)-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the second part, we describe a deformation of singularities of Gorenstein $K3$ surfaces in these families.

Poles of the complex zeta function of a plane curve

We study the poles and residues of the complex zeta function fs of a plane curve. We prove that most non-rupture divisors do not contribute to poles of fs or roots of the Bernstein-Sato polynomial bf(s) of f. For plane branches we give an optimal set of candidates for the poles of fs from the rupture divisors and the characteristic sequence of f. We prove that for generic plane branches fgen all the candidates are poles of fgens. As a consequence, we prove Yano's conjecture for any number of characteristic exponents if the eigenvalues of the monodromy of f are different.

Vector flows and the analytic moduli of singular plane branches

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 $$\lambda $$-invariant— are zero (so called “short parametrizations”). This is the main step missed by Zariski in his attempt to solve the moduli problem.

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.