Obstructions to lifting tropical curves in surfaces in 3-space

Details Obstructions to lifting tropical curves in surfaces in 3-space Obstructions to lifting tropical curves in surfaces in 3-space

2010-12-31

arxiv.org/abs/1101.0297v2

Mathematics

Algebraic Geometry

19 pages, 2 figures. Revised and reorganized, with a clearer focus on the nature of the combinatorial obstructions

Scientific paper

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.

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