On the growth behaviour of Hironaka quotients

Abstract

We consider a finite analytic morphism φ = (f, g) : (X, p) −→ (C 2 , 0) where (X, p) is a complex analytic normal surface germ and f and g are complex analytic function germs. Let π : (Y, E Y) → (X, p) be a good resolution of φ with exceptional divisor E Y = π −1 (p). We denote G(Y) the dual graph of the resolution π. 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.