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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)].

14T05 Tropical geometry (MSC2010)
14H99 Curves in algebraic geometry
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