Linear Tropicalizations

TL;DR

This paper establishes a connection between Berkovich spaces of algebraic varieties over non-Archimedean fields and linear tropicalizations, providing new insights into their topological structure and applications to line arrangements.

Contribution

It introduces a surjective continuous map from Berkovich spaces to inverse limits of linear tropicalizations, and characterizes when this map is a homeomorphism, with applications to realizability of line arrangements.

Findings

The map is a homeomorphism for non-singular algebraic curves.

Existence of tropical line arrangements realizable over complex numbers but not over reals.

Provides a new topological perspective on algebraic varieties via linear tropicalizations.

Abstract

Let $X$ be a closed algebraic subset of $\mathbb{A}^{n}(K)$ where $K$ is an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. We show that there is a surjective continuous map from the Berkovich space of $X$ to an inverse limit of a certain family of embeddings of $X$ called linear tropicalizations of $X$. This map is injective on the subset of the Berkovich space $X^{an}$ which contains all seminorms arising from closed points of $X$. We show that the map is a homeomorphism if $X$ is a non-singular algebraic curve. Some applications of these results to transversal intersections are given. In particular we prove that there exists a tropical line arrangement which is realizable by a complex line arrangement but not realizable by any real line arrangement.