Obstructions to lifting tropical curves in surfaces in 3-space

TL;DR

This paper investigates the conditions under which tropical curves in surfaces in 3-space can be lifted to algebraic curves, identifying combinatorial obstructions that explain why some tropical curves are not liftable.

Contribution

The paper introduces specific combinatorial obstructions to lifting tropical curves in surfaces, linking the problem to polynomial factorization in characteristic zero.

Findings

Identified combinatorial obstructions to lifting tropical curves.

Explained non-liftability of certain tropical curves by Vigeland.

Reduced lifting problem to polynomial support factorization in characteristic zero.

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.