%0 Journal Article
%T Faithful realizability of tropical curves
%A Man-Wai Cheung
%A Lorenzo Fantini
%A Jennifer Park
%A Martin Ulirsch
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1410.4152v3