On Special Values for Pencils of Plane Curve Singularities

Let (Ft : t ∈ P) be a pencil of plane curve singularities and let μ0 be the Milnor number of the fiber Ft. We prove a formula for the jumps μ0 − inf{μ0 : t ∈ P}. As an application, we give a description of the special values of the pencil (Ft : t ∈ P). Introduction. Let (Ft : t ∈ P1), P1 = C ∪ {∞} be a pencil of plane curve singularities defined by two coprime power series f, g ∈ C{X,Y } without constant term. That is Ft = f − tg for t ∈ C and F∞ = g. Let μ0 be the Milnor number of the fiber Ft and let μ 0 = inf{μ0 : t ∈ P}. Our aim is to give a formula for the jumps μ0 − μmin 0 by means of the meromorphic fraction f/g considered on the branches of the Jacobian curve j(F ) = ∂f ∂X ∂g ∂Y − ∂f ∂Y ∂g ∂X = 0. Roughly speaking we will show that μ0−μ 0 = the number of zeros of f/g−t if t ∈ C and μ0 − μmin 0 = the number of poles of f/g on the branches of the Jacobian curve j(F ) = 0 provided that μ0 6= +∞ (resp. μ0 6= +∞). Then we prove a known result on the special values of the pencil (Ft : t ∈ P1). 2000 Mathematics Subject Classification. 14H20, 32S10.