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.

10 pages. Major changes in Section 3: Proof of Theorem 3.1 was shortened and a new theorem about linear tropicalizations of smooth curves was added. More details were given for the proof of Theorem 4.2