Lifting Tropical Curves in Space and Linear Systems on Graphs

TL;DR

This paper investigates which graphs can be realized as tropicalizations of algebraic curves, providing necessary conditions using linear systems and revealing new combinatorial structures in tropical geometry.

Contribution

It introduces a necessary condition for graphs to be tropicalizations of curves, generalizing Speyer's well-spacedness condition and proposing new combinatorial insights.

Findings

Reproduces a generalization of Speyer's well-spacedness condition

Provides new necessary conditions for tropicalizations of algebraic curves

Suggests a novel combinatorial structure on tropicalizations

Abstract

Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition reproduces a generalization of Speyer's well-spacedness condition and also gives new conditions. In addition, it suggests a new combinatorial structure on tropicalizations of algebraic curves.