TL;DR
This paper demonstrates that any balanced 1-dimensional polyhedral complex can be realized as the tropicalization of a smooth algebraic curve over a non-Archimedean field, mapping to a toric Artin fan, thus linking tropical and algebraic geometry.
Contribution
It establishes a universal realization result connecting tropical curves with algebraic curves mapping to toric Artin fans, expanding the understanding of tropicalization.
Findings
Any balanced 1-dimensional polyhedral complex is realizable as a tropicalization of a smooth curve.
The construction involves non-Archimedean fields and toric Artin fans.
This bridges tropical geometry with algebraic geometry via toric Artin fans.
Abstract
We prove that every balanced 1-dimensional polyhedral complex arises as the tropicalization of a smooth curve over a non-Archimedean field mapping to a toric Artin fan, namely the quotient of a toric variety by its dense torus.
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