Faithful realizability of tropical curves

TL;DR

This paper proves that a broad class of tropical curves, including all trivalent ones, can be faithfully realized as tropicalizations of algebraic curves, using toric schemes and deformation theory.

Contribution

It establishes faithful realizability of many tropical curves, including higher valence cases, and connects metric graphs with algebraic curves via tropicalization.

Findings

All trivalent tropical curves are faithfully realizable.

Many higher valence tropical curves are also faithfully realizable.

Every rational metric graph corresponds to a faithful tropicalization of an algebraic curve.

Abstract

We study whether a given tropical curve $\Gamma$ in $\mathbb{R}^n$ can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $\Gamma$. We give an affirmative answer to this question for a large class of tropical curves that includes all trivalent tropical curves, but also many tropical curves of higher valence. We then deduce that for every metric graph $G$ with rational edge lengths there exists a smooth algebraic curve in a toric variety whose analytification has skeleton $G$, and the corresponding tropicalization is faithful. Our approach is based on a combination of the theory of toric schemes over discrete valuation rings and logarithmically smooth deformation theory, expanding on a framework introduced by Nishinou and Siebert.