{
  "id": "1304.4259",
  "version": "v2",
  "published": "2013-04-15T20:33:25.000Z",
  "updated": "2014-09-24T18:35:17.000Z",
  "title": "Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem",
  "authors": [
    "Yang An",
    "Matthew Baker",
    "Greg Kuperberg",
    "Farbod Shokrieh"
  ],
  "comment": "20 pages -- Final version to appear in Forum of Math, Sigma",
  "doi": "10.1017/fms.2014.25",
  "categories": [
    "math.CO",
    "math.AG",
    "math.MG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-15T20:33:25.000Z",
      "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 for each linear equivalence class of divisors of degree $g$ on $\\Gamma$. We characterize these canonical effective representatives as the set of break divisors on $\\Gamma$ and present a new combinatorial proof that there is a unique break divisor in each equivalence class. We also establish the closely related fact that for fixed $q$ in $\\Gamma$, every divisor of degree $g-1$ is linearly equivalent to a unique $q$-orientable divisor. For both of these results, we also prove discrete versions for finite unweighted graphs which do not follow from the results of Mikhalkin-Zharkov. As an application of the theory of break divisors, 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 $\\Pic^g(\\Gamma)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of $\\Pic^g(\\Gamma)$ is the sum of the volumes of the cells in the decomposition.",
      "comment": "32 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-24T18:35:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tropical curve",
      "divisor classes",
      "canonical representatives",
      "unique break divisor",
      "linear equivalence class"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.4259A"
    }
  }
}