Fibrations associées à un pinceau de courbes planes

Meromorphic functions, bifurcation sets and fibred links

“…In particular Theorem 1.2 gives an alternative proof to (1) ⇒ (5) which was first observed by F. Michel and H. Maugendre in [10].…”

“…The equivalence (1) ⇔ (2) is proved by A. Pichon in [14] when f and g have 0 as an isolated critical point and generalized by A. Pichon and J. Seade in [15]. In particular Theorem 1.2 gives an alternative proof to (1) ⇒ (5) which was first observed by F. Michel and H. Maugendre in [10].…”

“…Then, by Remark 3 and Theorem 1 of [10], any c ∈ {0, ∞} is a generic value of the pencil generated by f and g. (1) The meromorphic function f /g is semitame at the origin; (2) Each c ∈ {0, ∞} is a generic value of the pencil of curves generated by f and g; (3) The multilink L f /g is fibred. Moreover, if these conditions hold, then the Milnor map…”

“…There is an alternative proof using the results of [9] and [10] about another bifurcation set B defined in terms of the Jacobian curve of the morphism…”

Pencils and critical locus on normal surfaces

“…The set Λ := {w 2 f − w 1 g, w 1 , w 2 ∈ C} is the pencil defined by f and g. We denote φ w the element of the pencil Λ equal to w 2 f − w 1 g. Its (non reduced) zero locus, denoted by Φ w , is called the fibre defined by φ w . Such linear families of curves have been studied independently and through different approach for (Z, z) equal to (C 2 , 0) in [11], [7] and [16]. In the general case (it means (Z, z) a germ of normal complex analytic surface which is not smooth anymore), Lê Dũng Tràng and R. Bondil give in [3] a definition of general elements of the pencil which are characterized by the minimality of their Milnor number.…”