Ultrametric spaces of branches on arborescent singularities

TL;DR

This paper proves that a specific intersection-based function defines an ultrametric on branches of arborescent surface singularities, revealing topological and resolution structure insights, generalizing previous smooth-case results.

Contribution

It extends the ultrametric property of the intersection ratio to arborescent surface singularities and links it to resolution topology, generalizing prior smooth germ results.

Findings

U_L is an ultrametric on branches of arborescent singularities.

Maximum of U_L relates to topological invariants of the singularity.

U_L encodes resolution structure of branches.

Abstract

Let $S$ be a normal complex analytic surface singularity. We say that $S$ is arborescent if the dual graph of any resolution of it is a tree. Whenever $A,B$ are distinct branches on $S$, we denote by $A \cdot B$ their intersection number in the sense of Mumford. If $L$ is a fixed branch, we define $U_L(A,B)= (L \cdot A)(L \cdot B)(A \cdot B)^{-1}$ when $A \neq B$ and $U_L(A,A) =0$ otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of surfaces, by proving that whenever $S$ is arborescent, then $U_L$ is an ultrametric on the set of branches of $S$ different from $L$. We compute the maximum of $U_L$, which gives an analog of a theorem of Teissier. We show that $U_L$ encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both $S$ and $L$ are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of $S$. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.