{
  "id": "1609.02091",
  "version": "v1",
  "published": "2016-09-07T17:50:11.000Z",
  "updated": "2016-09-07T17:50:11.000Z",
  "title": "A note on Brill--Noether existence for graphs of low genus",
  "authors": [
    "Stanislav Atanasov",
    "Dhruv Ranganathan"
  ],
  "comment": "19 pages, 36 TikZ figures. Comments are welcome!",
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "abstract": "In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence conjecture for special divisors. For $g\\leq 5$ and $\\rho(g,r,d)$ non-negative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$. Moreover, the conjecture is shown to hold in rank $1$ for a number of families of highly connected combinatorial types of graphs of arbitrarily high genus. In the relevant genera, our arguments give the first combinatorial proof of the Brill--Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-07T17:50:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14T05"
    ],
    "keywords": [
      "low genus",
      "bakers brill-noether existence conjecture",
      "first combinatorial proof",
      "brill-noether existence theorem",
      "analogous combinatorial theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}