Obstructions to lifting tropical curves in surfaces in 3-space
Abstract
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational weighted graphs in tropical surfaces. Specifically, we study the `lifting' problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a surface? We develop specific combinatorial obstructions to lifting a graph by reducing the problem to the question of whether or not one can factor a polynomial with particular support in the characteristic 0 case. This explains why some unusual tropical curves constructed by Vigeland are not liftable.
Keywords
Cite
@article{arxiv.1101.0297,
title = {Obstructions to lifting tropical curves in surfaces in 3-space},
author = {Tristram Bogart and Eric Katz},
journal= {arXiv preprint arXiv:1101.0297},
year = {2012}
}
Comments
19 pages, 2 figures. Revised and reorganized, with a clearer focus on the nature of the combinatorial obstructions