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On the Berkovich double residue fields and birational models

Keita Goto

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.

On the Berkovich double residue fields and birational models

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 of the Berkovich analytification of an algebraic variety and identify the Berkovich double residue field at with the union of the residue fields at the center of in birational models. Besides, we concretely compute the Berkovich double residue field for any quasi monomial valuation.

Paper Structure

This paper contains 9 sections, 23 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Berkovich Spectra
  4. Berkovich analytifications
  5. Centers
  6. Some basic properties of $\widetilde{\mathscr{H} (x)}$
  7. $\widetilde{\mathscr{H} (x)}$ for quasi monomial valuations
  8. $\widetilde{\mathscr{H} (x)}$ for finite group action
  9. $\widetilde{\mathscr{H} (x)}$ over CDVF

Key Result

Theorem 1.2

Let $X$ be a variety over a non-Archimedean field $k$. For any $x\in X^{\rm val}$, we define the directed set $B(X,x)$ as above. If $B(X,x)\neq \emptyset$, then it follows that where $\bigcup_{\mathcal{X}\in B(X,x)} \kappa(c_{\mathcal{X}}(x))$ means the union of all $\kappa(c_\mathcal{X}(x))$'s as subfields of $\widetilde{\mathscr{H}(x)}$ under the canonical injections $\kappa(c_\mathcal{X}(x)) \

Theorems & Definitions (77)

  • Definition 1.1: $=$ Definition \ref{['model']}
  • Theorem 1.2: $=$ Theorem \ref{['union']}
  • Theorem 1.3: $=$ Theorem \ref{['smon']}
  • Theorem 1.4: $=$ Theorem \ref{['quasi']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 67 more