On the growth behaviour of Hironaka quotients

Abstract

We consider a finite analytic morphism $ϕ= (f,g) : (X,p)\to (\C^2,0)$ 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 $\varphi$ 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.