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 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 Nishinou and Siebert.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- 10.48550/arXiv.1410.4152
- arXiv:
- arXiv:1410.4152
- Bibcode:
- 2014arXiv1410.4152C
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14T05
- E-Print:
- 16 pages, introduction improved and other minor modifications, to appear in IMRN, comments very welcome!