Ultrametrics and surface singularities

TL;DR

This paper explores how ultrametrics can be constructed from intersection numbers of branches on surface singularities, linking these metrics to dual graphs of resolutions and extending classical results.

Contribution

It generalizes the construction of ultrametrics from smooth to normal surface singularities using intersection theory and interprets these in terms of resolution dual graphs.

Findings

Ultrametrics can be defined on branches of surface singularities.

Rooted trees associated with ultrametrics relate to dual graphs of resolutions.

The approach extends classical results to normal surface singularities.

Abstract

The present lecture notes give an introduction to works of Garc\'{i}a Barroso, Gonz\'alez P\'erez, Ruggiero and the author. The starting point of those works is a theorem of P\l oski, stating that one defines an ultrametric on the set of branches drawn on a smooth surface singularity by associating to any pair of distinct branches the quotient of the product of their multiplicities by their intersection number. We show how to construct ultrametrics on certain sets of branches drawn on any normal surface singularity from their mutual intersection numbers and how to interpret the associated rooted trees in terms of the dual graphs of adapted embedded resolutions. The text begins by recalling basic properties of intersection numbers and multiplicities on smooth surface singularities and the relation between ultrametrics on finite sets and rooted trees. On arbitrary normal surface singularities one has to use Mumford's definition of intersection numbers of curve singularities drawn on them, which is also recalled.