Adic curves: stable reduction, skeletons and metric structure

TL;DR

This paper investigates the structure of adic curves over arbitrary rank affinoid fields, establishing classifications, semistable reduction, and metric skeletons, with new technical tools like ranger compactification.

Contribution

It introduces a novel approach to classify points, prove semistable reduction, and define metric skeletons for adic curves using ranger compactification.

Findings

Classification of points on adic curves

Semistable reduction theorem for adic curves

Construction of metric skeletons and deformational retractions

Abstract

We study the structure of adic curves over an affinoid field of arbitrary rank. In particular, quite analogously to Berkovich geometry we classify points on curves, prove a semistable reduction theorem in the version of Ducros' triangulations, define associated curve skeletons and prove that they are deformational retracts in a suitable sense. An important new technical tool is an appropriate compactification of ordered groups that we call the ranger compactification. Intervals of rangers are then used to define metric structures and construct deformational retractions.