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 …