On the Berkovich double residue fields and birational models

TL;DR

This paper explores the concept of Berkovich double residue fields at points of Berkovich analytic spaces, linking them to residue fields in birational models and providing explicit computations for quasi-monomial valuations.

Contribution

It introduces the notion of Berkovich double residue fields, identifies them with unions of residue fields in birational models, and computes them explicitly for quasi-monomial valuations.

Findings

Berkovich double residue fields can be identified with unions of residue fields in birational models.

Explicit computation of the double residue field for quasi-monomial valuations.

Provides a new perspective on residue fields in Berkovich analytic spaces.

Abstract

Just as a residue field can be considered for a point of an algebraic variety, we can also consider a residue field for a point of a Berkovich analytic space. This residue field is a valuation field in the algebraic sense. Then we can consider its residue field as a valuation field. We call it the Berkovich double residue field at the point. In this paper, we consider a point $x$ of the Berkovich analytification of an algebraic variety and identify the Berkovich double residue field at $x$ with the union of the residue fields at the center of $x$ in birational models. Besides, we concretely compute the Berkovich double residue field for any quasi monomial valuation.