2014/10/31 by Man-Wai Cheung, Lorenzo Fantini, Jennifer Park +1 · 1 citation
Mathematics · #math.AG #msc:14T05
paper · pdf · doi:10.1093/imrn/rnv269
published as Int. Math. Res. Not. IMRN, no. 15, 4706-4727, 2016 · 16 pages, introduction improved and other minor modifications, to appear in IMRN, comments very welcome!
arxiv created 2015/08/20 · arxiv updated 2017/02/17
We study whether a given tropical curve Γ in ℝn 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.