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

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 ${\mathrm{Pic}}^g(\Gamma )$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${\mathrm{Pic}}^g(\Gamma )$ is the sum of the volumes of the cells in the decomposition.

20 pages -- Final version to appear in Forum of Math, Sigma

References in corpus (1)

• Chip-firing games, potential theory on graphs, and spanning trees

Cited by in corpus (27)

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• The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian
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• Divisors on graphs, binomial and monomial ideals, and cellular resolutions
• The Kodaira dimensions of ${\overline{\mathcal{M}}}_{22}$ and ${\overline{\mathcal{M}}}_{23}$
• Kirchhoff's theorem for Prym varieties
• Canonical measures on metric graphs and a Kazhdan's theorem
• Abelian sandpile model and Biggs-Merino polynomial for directed graphs
• Chip-Firing Games and Critical Groups
• Compactified Jacobians as Mumford models
• Counting Finite Index Subrings of ${\mathbb{Z}}^n$
• Tropicalization of the universal Jacobian
• A Polyhedral Proof of the Matrix Tree Theorem
• The distribution of Weierstrass points on a tropical curve
• Combinatorial and inductive methods for the tropical maximal rank conjecture
• A universal tropical Jacobian over $M_g^{\trop}$
• On the strong maximal rank conjecture in genus 22 and 23
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• The rotor-routing torsor and the Bernardi torsor disagree for every non-planar ribbon graph
• Matrix polynomials, generalized Jacobians, and graphical zonotopes
• A multidimensional generalization of Symanzik polynomials
• The tropical Manin-Mumford conjecture
• Stability conditions for line bundles on nodal curves
• Generalized Bijective Maps between $G$-Parking Functions, Spanning Trees, and the Tutte Polynomial
• A Torelli theorem for graphs via quasistable divisors
• Divisors on complete multigraphs and Donaldson-Thomas invariants of loop quivers
• An arithmetic variant of Raynaud's theorem