Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem

TL;DR

This paper proves the uniqueness of canonical effective divisors on tropical curves using combinatorial methods and offers a geometric proof of Kirchhoff's Matrix-Tree Theorem via polyhedral decompositions of the Jacobian.

Contribution

It provides a new combinatorial proof of the uniqueness of break divisors and introduces an integral version, linking tropical geometry with classical graph theory results.

Findings

Unique break divisors exist in each linear equivalence class.

Polyhedral decomposition of the Jacobian relates to spanning trees.

Volume of the Jacobian equals sum of cell volumes in the decomposition.

Abstract

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) Kirchhoff's 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.