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We study linear pencils of curves on normal surface singularities. Using the minimal good resolution of the pencil, we describe the topological type of generic elements of the pencil and characterize the behaviour of special elements. Furthermore, we show that the critical locus associated to the pencil is linked to the special elements. This gives a decomposition of the critical locus through the minimal good resolution and as a consequence, some information on the topological type of the critical locus.

Let(formula here)be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.For each α ∈ A, we define a rational number dα by(formula here)where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.

We consider a finite analytic morphism $\phi = (f,g) : (X,p)\to (\C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $\pi : (Y,E_{Y})\to (X,p)$ be a good resolution of $\phi$ with exceptional divisor $E_{Y}=\pi ^{-1}(p)$. We denote $G(Y)$ the dual graph of the resolution $\pi $. We study the behaviour of the Hironaka quotients of $(f,g)$ associated to the vertices of $G(Y)$. We show that there exists maximal oriented arcs in $G(Y)$ along which the Hironaka quotients of $(f,g)$ strictly increase and they are constant on the connected components of the closure of the complement of the union of the maximal oriented arcs.

Soit Φ:= (f, g): C 2 → C 2 ou f et g sont des applications polynomiales. Nous etablissons le lien qui existe entre le polygone de Newton de la courbe reunion du discriminant et du lieu de non-proprete de Φ et la topologie des entrelacs a l'infini des courbes affines f -1 (0) et g -1 (0). Nous en deduisons alors des consequences liees a la conjecture du jacobien.

We will analyze the relationships between the special fibres of a pencil Λ of plane curve singularities and the Jacobian curve J of Λ (defined by the zero locus of the Jacobian determinant for any fixed basis ′ ∈ Λ). From the results, we find decompositions of J (and of any special fibre of the pencil) in terms of the minimal resolution of Λ. Using these decompositions and the topological type of any generic pair of curves of Λ, we obtain some topological information about J. More precise decompositions for J can be deduced from the minimal embedded resolution of any pair of fibres (not necessarily generic) or from the minimal embedded resolution of all the special fibres.

For a finite morphism $$\varphi =(f,g)$$ from the plane to the plane we describe the topology of the image of a branch in the source by the use of iterated pencils of analytic functions, constructed inductively in a natural way starting from the components of the map. In the case of the study of the topology of the discriminant curve, image by $$\varphi $$ of the critical locus of the map, we show that the special fibres of the pencil $$ \langle f,g\rangle $$ suffice to determine the topological type of each branch of the discriminant curve. This is due to the known relations that exist between the branches of the critical locus of $$\varphi $$ and the special fibres.

Resume Soit φ :=(f,g) : C 2 → C 2 ou f et g sont deux polynomes. Nous etablissons le lien qui existe entre le polygone de Newton de la courbe reunion du discriminant et du lieu de non proprete de φ et la topologie des entrelacs a l'infini des courbes affines f−1(0) et g−1(0).

Soient f et g deux germes de fonctions analytiques a l'origine dans C 2 et N ∈ N*. Nous calculons explicitement l'ensemble B des valeurs atypiques du pinceau {f a = f+ag N ; a E C}. Avec notre description de B, nous pouvons determiner les valeurs irregulieres a l'infini des applications polynomiales de C 2 dans C (voir les exemples du chapitre 5). Nous comparons les fibrations de Milnor des germes f a du pinceau et nous exhibons un facteur commun aux polynomes caracteristiques de leurs monodromies. Lorsque g = x, nous montrons que la repartition de la multiplicite par dicritique est independante du parametre a et nous decrivons la topologie des membres generiques en fonction de la resolution minimale de f . x.

Let f and g be two analytic function germs without common branches. We define the Jacobian quotients of ( g , f ), which are ‘first order invariants’ of the discriminant curve of ( g , f ), and we prove that they only depend on the topological type of ( g , f ). We compute them with the help of the topology of ( g , f ). If g is a linear form transverse to f , the Jacobian quotients are exactly the polar quotients of f and we affirm the results of D. T. Le, F. Michel and C. Weber.

Soient f et g deux germes de fonctions analytiques de (C 2 , 0) dans (C, 0) sans branche commune, c'est-a-dire que si Formula math. sont des decompositions de f et g en leurs facteurs irreductibles distincts, alors fi ¬= g j pour tout i et tout j modulo une unite. Dans [14], nous avons defini les quotients jacobiens de (g, f) et demontre que ce sont des invariants du type topologique de (g, f). Nous les avons calcules en fonction de la topologie de (g, f). Nous les etudions ici a travers la resolution minimale de f. g. Ceci donne un autre moyen de calcul explicite des quotients jacobiens de (g, f) a l'aide de la resolution minimale de f, g (sect. 3). On decrit ensuite le comportement de croissance de ces quotients dans la resolution minimale de f. g (sect. 4). Dans le cas ou g est une forme lineaire transverse a f, on retrouve les resultats de D. T. Le, F. Michel et C. Weber dans [9]. Nous terminons enfin en donnant un exemple d'application de ces resultats, lequel constitue la reponse a une question de D. T. Le et C. Weber (voir [11]) relative a leur etude de la conjecture jacobienne (sect. 5).

Etant donne f un germe de fonction analytique a l'origine de C2, nous comparons la complexite topologique de la courbe discriminante de f a celle de sa courbe polaire. Let f be an analytic function germ at 0 in C2. We compare the topological complexity of the discriminant curve of f to the one of its polar curve.

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.