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Faithful realizability of tropical curves. (English) Zbl 1404.14071
Summary: 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 T. Nishinou and B. Siebert [Duke Math. J. 135, No. 1, 1–51 (2006; Zbl 1105.14073)].

##### MSC:
 14T05 Tropical geometry (MSC2010) 14H99 Curves in algebraic geometry
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