Faithful realizability of tropical curves
Abstract
We study whether a given tropical curve $\Gamma$ in $\mathbb{R}^n$ can be realized as the tropicalization of an algebraic curve whose nonarchimedean 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 DOI:
 10.48550/arXiv.1410.4152
 arXiv:
 arXiv:1410.4152
 Bibcode:
 2014arXiv1410.4152C
 Keywords:

 Mathematics  Algebraic Geometry;
 14T05
 EPrint:
 16 pages, introduction improved and other minor modifications, to appear in IMRN, comments very welcome!