On the growth behaviour of Hironaka quotients

TL;DR

This paper investigates how Hironaka quotients associated with a finite analytic morphism on a complex surface germ behave along the resolution graph, revealing increasing patterns along certain arcs and constancy elsewhere.

Contribution

It establishes the existence of maximal oriented arcs in the resolution graph where Hironaka quotients strictly increase, and describes their behavior on the graph.

Findings

Hironaka quotients increase along maximal oriented arcs.

Quotients are constant on connected components outside these arcs.

The structure of the resolution graph influences quotient behavior.

Abstract

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.