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arXiv · 1410.4152

Faithful realizability of tropical curves

Man-Wai Cheung·Lorenzo Fantini·Jennifer Park·Martin Ulirsch
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Abstract

We study whether a given tropical curve $Γ$ in $\mathbb{R}^n$ can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $Γ$. 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.

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Man-Wai Cheung, Lorenzo Fantini, Jennifer Park, Martin Ulirsch. 2015-08-20. Faithful realizability of tropical curves. https://doi.org/10.1093/imrn%2Frnv269

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