Construction of Fully Faithful Tropicalizations for Curves in Ambient Dimension 3

TL;DR

This paper constructs fully faithful tropicalizations for algebraic curves in three-dimensional ambient space, ensuring the tropicalization accurately reflects the curve's structure and can be extended to smooth, faithful tropicalizations in higher-dimensional toric varieties.

Contribution

It provides a method to produce fully faithful tropicalizations for Mumford curves in dimension three and extends these to smooth, faithful tropicalizations in larger ambient spaces.

Findings

Constructed fully faithful tropicalizations for any Mumford curve in dimension three.

Extended the tropicalization to a smooth, fully faithful form in product spaces.

Ensured the tropicalization is isometric to a subgraph of the Berkovich space.

Abstract

In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the tropicalization has nice properties. We construct, for any Mumford curve $X$, a map to a three-dimensional torus, such that the tropicalization is isometric to a subgraph of the Berkovich space $X^{\rm an}$, called the extended skeleton. In this case, we say the tropicalization is "fully faithful." Additionally, given a map $X$ to a toric variety $Y$, which induces a fully faithful tropicalization, we show that we can extend the map to $X \to Y \times (\mathbf{P}^1)^n$ such that the new tropicalization is smooth and fully faithful.