Definable sets of Berkovich curves

TL;DR

This paper develops a framework to associate definable sets and maps to $k$-analytic curves, enabling a logical perspective on Berkovich geometry and extending prior results on radiality and iso-definability.

Contribution

It introduces a functorial association of definable sets to $k$-analytic curves, extending Hrushovski-Loeser's theorem and applying to a broad class of curves and morphisms.

Findings

Definable sets correspond bijectively to radial subsets of curves.

The approach recovers and extends Temkin's results on radiality.

Applicable to strictly $k$-affinoid curves and arbitrary morphisms.

Abstract

In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.