Faithful Realizability of Tropical Curves
Man-Wai Cheung, Lorenzo Fantini, Jennifer M. Park, Martin Ulirsch · International Mathematics Research Notices · 2015
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.