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.     

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.

     

On the equivalence of plane curve singularities

A systematic method to calculate cleft extensions for pointed Hopf algebras is developed and applied to U_q(sl ₂) and the Frobenius-Lusztig kernel U_q(sl ₂)'.     

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.     

Moduli of plane curve singularities with a single characteristic exponent

This paper studies the moduli space corresponding to irreducible germs of plane analytic curve with a single characteristic exponent. We stratify the moduli space corresponding to such germs using an analytical invariant introduced by Zariski. Then, we compute the minimum Tjurina number on each stratum as well as the dimension of the strata.

     

Analytic types of plane curve singularities defined by weighted homogeneous polynomials

We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials , which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that either satisfies the same property as the above does or is homogeneous, then we prove easily that the weights of the above determine the topological type of and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in , which was already known. Also, as an application, it can be shown that for a given , where is a quasihomogeneous holomorphic function with an isolated singularity at the origin or with a positive integer , analytic types of isolated hypersurface singularities defined by are easily classified where is defined just as above.

     

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