In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz
numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.
Let $Gamma$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence
class of divisors of degree $g$ on $Gamma$. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an integral version of this result which is of independent interest. As an application, we provide a geometric proof of (a dual version of) Kirchhoffs celebrated Matrix-Tree Theorem. Indeed, we show that each weighted graph model $G$ for $Gamma$ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus ${rm Pic}^g(Gamma)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${rm Pic}^g(Gamma)$ is the sum of the volumes of the cells in the decomposition.
Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton i
s a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger skeletonized
