{
  "id": "math/0608360",
  "version": "v3",
  "published": "2006-08-14T17:25:07.000Z",
  "updated": "2007-07-09T18:06:35.000Z",
  "title": "Riemann-Roch and Abel-Jacobi theory on a finite graph",
  "authors": [
    "Matthew Baker",
    "Serguei Norine"
  ],
  "comment": "35 pages. v3: Several minor changes made, mostly fixing typographical errors. This is the final version, to appear in Adv. Math",
  "categories": [
    "math.CO",
    "math.AG",
    "math.NT"
  ],
  "abstract": "It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-07-09T18:06:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite graph",
      "abel-jacobi theory",
      "riemann surface",
      "linear equivalence",
      "discrete analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8360B"
    }
  }
}