Radiality of definable sets

TL;DR

This paper extends Temkin's radiality result for ramification loci to a broader class of definable sets in Berkovich spaces and demonstrates that these results can be applied uniformly in families.

Contribution

It generalizes Temkin's radiality theorem to definable sets and shows the uniformity of these properties across families of morphisms.

Findings

Radiality holds for a broader class of definable sets.

The radiality result applies uniformly in families of morphisms.

Techniques from Hrushovski-Loeser are effectively used in this context.

Abstract

In this article we use techniques developed by Hrushovski-Loeser to study certain metric properties of the Berkovich analytification of a finite morphism of smooth connected projective curves. In recent work, M. Temkin proved a radiality statement for the topological ramification locus associated to such finite morphisms. We generalize this result in two directions. We prove a radiality statement for a more general class of sets which we call definable sets. In another direction, we show that the result of Temkin can be obtained in families.