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.

16 pages, introduction improved and other minor modifications, to appear in IMRN, comments very welcome!

References in corpus (5)

• Enumerative tropical algebraic geometry in R2
• On the structure of nonarchimedean analytic curves
• Tropical varieties for non-archimedean analytic spaces
• Embeddings and immersions of tropical curves
• Harmonic tropical morphisms and approximation

Cited by in corpus (8)

• Moduli of stable maps in genus one and logarithmic geometry II
• Embeddings and immersions of tropical curves
• Tropicalization of the moduli space of stable maps
• Enumerative geometry of elliptic curves on toric surfaces
• Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
• Tropical Graph Curves
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Constructions of superabundant tropical curves in higher genus