Algebroid plane curves whose Milnor and Tjurina numbers differ by one or two

In this paper we describe all irreducible plane algebroid curves, defined over an algebraically closed field of characteristic zero, modulo analytic equivalence, having the property that the difference between their Milnor and Tjurina numbers is 1 or 2. Our work extends a previous result of O. Zariski who described such curves when this difference is zero.Partially supported by PRONEX and CNPq.     

On resolving singularities of plane curves via a theorem attributed to Alfred Clebsch

This paper discusses a theorem in birational geometry that J.L. Coolidge attributed to Alfred Clebsch. The background is reconstructed from letters Felix Klein exchanged with Max Noether in 1894, when Noether was completing work on a lengthy report with Alexander Brill on the history of algebraic functions. Noether was deeply troubled to learn that Klein had informed him back in 1869 about relevant results that Clebsch and Leopold Kronecker had discussed in Berlin. These exchanges with Klein led to revisions in the Brill-Noether report, made in part to ensure Noether's own priority rights and larger intellectual legacy.     

Notes on pedal and contrapedal curves of fronts in the Euclidean plane

A pedal curve (a contrapedal curve) of a regular plane curve is the locus of the feet of the perpendiculars from a point to the tangents (normals) to the curve. These curves can be parametrized by using the Frenet frame of the curve. Yet provided that the curve has some singular points, the Frenet frame at these singular points is not well‐defined. Thus, we cannot use the Frenet frame to examine pedal or contrapedal curves. In this paper, pedal and contrapedal curves of plane curves, which have singular points, are investigated. By using the Legendrian Frenet frame along a front, the pedal and contrapedal curves of a front are introduced and properties of these curves are given. Then, the condition for a pedal (and a contrapedal) curve of a front to be a frontal is obtained. Furthermore, by considering the definitions of the evolute, the involute, and the offset of a front, some relationships are given. Finally, some illustrated examples are presented.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.     

On the plane section of an integral curve in positive characteristic

If is an integral curve and an algebraically closed field of characteristic 0, it is known that the points of the general plane section of are in uniform position. From this it follows easily that the general minimal curve containing is irreducible. If char, the points of may not be in uniform position. However, we prove that the general minimal curve containing is still irreducible.

     

Pedal curves of frontals in the Euclidean plane

In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.     

Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.      

Seshadri constants and the spaces of curves with prescribed singularities and positions

We will give optimal bounds for Seshadri constants of an ample line bundle at multiple points on a complex projective surface X.We also present a solution to the long-studied classical problem on the existence of curves on X with given topological singularities at r arbitrary points p1,...,pr.Namely,we obtain a universal lower bound on the degree of curves for the existence.It is independent of the position of the singular points.     

Nodal degenerations of plane curves and galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois covers.      

Euler characteristic of the Milnor fibre of plane singularities

We give a formula for the Euler characteristic of the Milnor fibre of any analytic function of two variables. This formula depends on the intersection multiplicities, the Milnor numbers and the powers of the branches of the germ of the curve defined by The goal of the formula is that it use neither the resolution nor the deformations of Moreover, it can be use for giving an algorithm to compute it.

     

Resolution of singularities of threefolds in positive characteristic II

Together with [Vincent Cossart, Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings, J. Algebra 320 (3) (2008) 1051–1082], this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields.     

On the equi-normalizable deformations of singularities of complex plane curves

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple point, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type x ^p  + y ^pk into the collections of A _k ’s. The research was constantly supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics (Mathematics Department of Ben Gurion University, Israel). Part of the work was done in Mathematische Forschungsinsitute Oberwolfach, during the author’s stay as an OWL-fellow. Some results were published in the preprint [17].     

Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes

In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at together with functions for each value in this semigroup, provided is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.

     

On holomorphic polydifferentials in positive characteristic

Let F/E be an abelian Galois extension of function fields over an algebraic closed field K of characteristic p > 0. Denote by G the Galois group of the extension F/E. In this paper, we study Ω(m), the space of holomorphic m‐(poly)differentials of the function field of F when G is cyclic or a certain elementary abelian group of order pⁿ; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the G module structure of Ω(m) in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. Finally, an application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.     

Resolution of singularities of real-analytic vector fields in dimension three

Let χ be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of χ can be made elementary by a finite number of blowing-ups in the ambient space. This work has been partially supported by the CNPq/Brasil Grant 205904/2003-5 and Fapesp Grant 02/03769-9.