Germ Of Plane Curve Research Articles

On the embedded Nash problem

Abstract The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

On a discriminant knot group problem of Brieskorn

Quite some time ago, at the singularity conference at Cargese 1972 Brieskorn asked the following question: Is the local fundamental group $\pi_1^{s}(S-D)$ of the discriminant complement inside the semi-universal unfolding $S$ of an isolated hypersurface singularity constant for $s$ in the $\mu$-constant stratum $\Sigma_E$? We review this question and give an affirmative answer in case of singular plane curve germs of multiplicity at most $3$.

On the blow-analytic equivalence of tribranched plane curves

We prove the finiteness of the number of blow-analytic equivalence classes of embedded plane curve germs for any fixed number of branches and for any fixed value of $\mu'$ —a combinatorial invariant coming from the dual graphs of good resolutions of embedded plane curve singularities. In order to do so, we develop the concept of standard form of a dual graph. We show that, fixed $\mu'$ in $\mathbb{N}$, there are only a finite number of standard forms, and to each one of them correspond a finite number of blow-analytic equivalence classes. In the tribranched case, we are able to give an explicit upper bound to the number of graph standard forms. For $\mu'\leq 2$, we also provide a complete list of standard forms.

The log-canonical threshold of a plane curve

AbstractWe give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27].

Determining plane curve singularities from its polars

This paper addresses a very classical topic that goes back at least to Plücker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.

The Milnor fiber of the singularity $$f(x,y) + zg(x,y) = 0$$ f ( x , y ) + z g ( x , y ) = 0

We give a description of the Milnor fiber and the monodromy of a singularity of the form $$f+zg = 0$$ , where f and g define germs of plane curve singularities and have no common components. In particular, this gives a description of the boundary of the Milnor fiber. The description depends only on the topological type of the two plane curve germs defined by f and g. As a corollary, we give a simple formula for the monodromy zeta function and the Euler characteristic of the fiber in terms of an embedded resolution of f and g.

Logarithmic residues along plane curves

Let (D,0)⊂(C2,0) be a plane curve germ defined by a reduced equation f. We prove that a fractional ideal I of D satisfies a symmetry property with its dual, and then apply it to study the behavior of the module of logarithmic residues of D in equisingular deformations.

Mapping class group of a plane curve germ

We prove that every topological conjugacy between two germs of singular holomorphic curves in the complex plane is homotopic to another conjugacy which extends homeomorphically to the exceptional divisors of their minimal desingularisations. As an application we give an explicit presentation of a finite index subgroup of the mapping class group of the germ of such a singularity.

Comparaison des formes de Seifert et des fonctions zêta de Denef-Loeser des germes de courbe plane à singularité isolée

We show that the topological type of a plane curve germ with isolated singularity is not determined by its integral Seifert form and Denef-Loeser zeta function. Furthermore, the integral Seifert form of such a germ is not determined by its Denef-Loeser zeta function.

Curves in a foliated plane

The paper is devoted to the classification of nonsingular and singular plane curve germs with respect to the group of local diffeomorphisms preserving the foliation of the plane by the phase curves of a fixed vector field, either nonsingular or singular. We define the multiplicity of a pair consisting of a plane curve and a vector field and prove an analog of the Tougeron theorem on finite determinacy. It leads, almost immediately, to a number of classification results; a part of them is contained in the work.

Jump of Milnor numbers

In this note we study a problem of A'Campo about the minimal non-zero difference between the Milnor numbers of a germ of plane curve and one of its deformation.

Real double-points of deformations of -simple map-germs from n to 2n

AbstractThe only stable singularities of a real map-germ $f:{\mathbb R} ^n\to{\mathbb R} ^{2n}$ are isolated transverse double-points. All ${{\cal A}}$-simple germs f have a deformation with the maximal number d(f) of real double-points (this is a partial generalization to higher n of the result of A'Campo [1] and Gusein-Zade [13] that all plane curve-germs have a deformation with δ real double points, with the extra hypothesis of ${{\cal A}}$-simplicity). The proof of this result is based on a classification of all ${{\cal A}}$-simple orbits.

Newton diagrams and equivalence of plane curve germs

We introduce an equivalence of plane curve germs which is weaker than Zariski's equisingularity and prove that the set of all Newton diagrams of a germ is an invariant of this equivalence. Then we show how to construct all Newton diagrams of a plane many-branched singularity starting with some invariants of branches and their orders of contact.

Calcul de la fonction d’Artin–Greenberg d’une branche plane

We calculate the Artin-Greenberg function of an irreducible plane curve germ. We deduce from this calculation that the Artin-Greenberg function, together with the multiplicity, determines the topological type of the curve and vice versa.

Sur la forme de Seifert entière des germes de courbe plane à deux branches

Two plane curve germs with two branches, which are isomeric, are shown to have isomorphic integral Seifert forms. The weight filtration on the integral homology of the Milnor fiber is the key ingredient of the proof. To cite this article: P. du Bois, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Topological invariants of higher order for a pair of plane curve germs

Let ( g, f) be an analytic map germ from ( C 2,0) into ( C 2,0) and denote by ( u, v) the canonical coordinates in (g,f)( C 2) ; it is ( g( x, y), f( x, y))=( u, v). In [J. London Math. Soc. (2) 59 (1999) 207–226], we showed that the set constituted of the first (not necessarily characteristic) Puiseux exponent (in the ( u, v)-coordinates) of each branch δ of the discriminant curve of ( g, f) is a topological invariant of ( g, f). Here we prove that for each branch δ there exists an integer k( δ) such that the set constituted of the first (not necessarily characteristic) k( δ) exponents of the Puiseux series in the ( u, v)-coordinates of each δ is a topological invariant of ( g, f). We give different ways to compute these invariants.

Logarithmic cohomology of the complement of a plane curve

Let D, x be a plane curve germ. We prove that the complex \( \Omega^\bullet(\log D)_x \) computes the cohomology of the complement of D, x only if D is quasihomogeneous. This is a partial converse to a theorem of [5], which asserts that this complex does compute the cohomology of the complement, whenever D is a locally weighted homogeneous free divisor (and so in particular when D is a quasihomogeneous plane curve germ). We also give an example of a free divisor \( D\subset \mathbb {C}^3 \) which is not locally weighted homogeneous, but for which this (second) assertion continues to hold.

On a class of local systems associated to plane curves

We study a class of local systems on the complement of a germ of irreducible plane curve. We exhibit local systems which by [8] give rise to regular holonomic D -modules with characteristic variety the union of the zero section with the conormal of the curve. To cite this article: P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 421–426.

Variation of the Milnor Fibration in Pencils of Hypersurface Singularities

Let Φ = (f, g):(Cn+ 1,0) → (C2, 0) be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: Φ defines an isolated complete intersection singularity; g = lN where ℓ is a generic linear form with respect to f and N> 0.) We study how the Milnor fibrations of the germs ϕ(α:β) = α g-β f are related to each other when (α:β) varies in P1. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram. 2000 Mathematical Subject Classification: 32S55, 32S15, 32S30.

Dévissage de la forme de Seifert d'un germe de courbe plane

We propose to unravel the Seifert form of a plane curve germ: from a such Seifert form, we explain how to find the topological type(s) of germs associated with this Seifert form.