Plumbing constructions of connected divides and the Milnor fibers of plane curve singularities

A divide is the image of a generic, relative immersion of a finite number of copies of the unit interval or the unit circle into the unit disk. N. A'Campo defined for each connected divide a link in S³ and proved that the link is fibered. In the present paper we show that the fiber surface of the fibration of a connected divide can be obtained from a disk by a successive plumbing of a finite number of positive Hopf bands. In particular, this gives us a geometric understanding of plumbing constructions of the Milnor fibers of isolated, complex plane curve singularities in terms of certain replacements of the curves of their real morsifications.     

Milnor numbers for surface singularities

An additive formula for the Milnor number of an isolated complex hypersurface singularity is shown. We apply this formula for studying surface singularities. Durfee's conjecture is proved for any absolutely isolated surface and a generalization of Yomdin singularities is given. This work was supported in part by a Spanish FPI'91 grant and by the Spanish project PB94-0291.     

Iterated inverse images of plane curve singularities

We describe the evolution of certain multiplicities and intersection numbers of plane curve singularities under the iterated action of an analytic morphism.     

On the Jacobian Newton polygon of plane curve singularities

We investigate the Jacobian Newton polygon of plane curve singularities. This invariant was introduced by Teissier in the more general context of hypersurfaces. The Jacobian Newton polygon determines the topological type of a branch (Merle’s result) but not of an arbitrary reduced curve (Eggers example). Our main result states that the Jacobian Newton Polygon determines the topological type of a non-degenerate unitangent singularity. The Milnor number, the Łojasiewicz exponent, the Hironaka exponent of maximal contact and the number of tangents are examples of invariants that can be calculated by means of the Jacobian Newton polygon. We show that the number of branches and the Newton number defined by Oka do not have this property. Dedicated to Professor Arkadiusz Płoski on his 60th birthday     

Poincaré series and zeta-function for irreducible plane curve singularities

The Poincaré series of an irreducible plane curve singularity equals the ζ-function of its monodromy, by a result of Campillo, Delgado, and Gusein-Zade. We discuss the derivation of this fact from a formula of Ebeling and Gusein-Zade relating the Poincaré series of a quasi-homogeneous complete intersection singularity to the Saito dual of a product of ζ-functions. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

On the trees associated to plane curve singularities and ideals

Si comparano diverse nozioni d’equisingolarità per le curve tracciate sopra una superficie complessa liscia, e per gli ideali di dimensione finita dell’anello locale d’uno dei suoi punti. Si vede che, in generale, questi nozioni sono diverse, ma (per le curve) queste coincidono se esse fossero membri della stessa famiglia continua.     

The boundary of the Milnor fibre of certain non-isolated singularities

Let \( \Phi : (\mathbb {C}^2, 0) \rightarrow ( \mathbb {C}^3, 0) \) be a finitely determined complex analytic germ and let \((\{f=0\},0)\) be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of f from the double-point-geometry of \(\Phi \).     

Tjurina numbers of plane curve singularities whose multiplicities are three and four

We describe Tjurina numbers of irreducible plane curve singularities whose multiplicities are three and four. Received: 4 April 2005     

On the number of rational points of a plane algebraic curve

The number of F_q -rational points of a plane non-singular algebraic curve defined over a finite field F_q is computed, provided that the generic point of is not an inflexion and that is Frobenius non-classical with respect to conics. Received: 18 March 2003     

A moduli scheme of embedded curve singularities

The main purpose of this paper is to prove the existence of the moduli space parameterizing the embedded curve singularities of with an admissible Hilbert polynomial p and to study its basic properties.     

On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.