{
  "id": "1304.1256",
  "version": "v3",
  "published": "2013-04-04T07:03:55.000Z",
  "updated": "2014-01-06T21:25:02.000Z",
  "title": "A combinatorial analysis of Severi degrees",
  "authors": [
    "Fu Liu"
  ],
  "comment": "38 pages, 1 figure, 1 table. Major revision: generalized main results in previous version. The old results only applies to classical Severi degrees. The current version also applies to Severi degrees coming from special families of toric surfaces",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "Based on results by Brugall\\'e and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees $N^{d, \\delta}$ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version of a special function associated to long-edge graphs appeared in Fomin-Mikhalkin's formula, and conjectured it to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we consider a special multivariate function associated to long-edge graphs that generalizes their function. The main result of this paper is that the multivariate function we define is always linear. A special case of our result gives an independent proof of Block-Colley-Kennedy's conjecture. The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of $Q^{d, \\delta}$ and a bound $\\delta$ for the threshold of polynomiality of $N^{d, \\delta}.$ Next, in joint work with Osserman, we apply the linearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the G\\\"ottsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series $B_1(q)$ and $B_2(q)$ appearing in the G\\\"ottsche-Yau-Zaslow formula. The proof of our linearity result is completely combinatorial. We define $\\tau$-graphs which generalize long-edge graphs, and a closely related family of combinatorial objects we call $(\\tau, n)$-words. By introducing height functions and a concept of irreducibility, we describe ways to decompose certain families of $(\\tau, n)$-words into irreducible words, which leads to the desired results.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-01-06T21:25:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "14N10"
    ],
    "keywords": [
      "long-edge graphs",
      "combinatorial analysis",
      "linearity result",
      "classical severi degrees",
      "universal polynomial results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1256L"
    }
  }
}