On the growth behaviour of Hironaka quotients

Journal of Singularities
volume 20 (2020), 31-53

Received: 4 June 2019. Received in revised form: 19 January 2020

DOI: 10.5427/jsing.2020.20b

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Abstract:

We consider a finite analytic morphism φ =(f,g): (X,p)-->(C ^2, 0) where (X,p) is a complex analytic irreducible 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.

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