{
  "id": "1311.7271",
  "version": "v4",
  "published": "2013-11-28T10:54:55.000Z",
  "updated": "2015-02-11T09:18:23.000Z",
  "title": "On the slope of hyperelliptic fibrations with positive relative irregularity",
  "authors": [
    "Xin Lu",
    "Kang Zuo"
  ],
  "comment": "final version, accepted by Trans. Amer. Math. Soc",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $f:\\, S \\to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\\lambda_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\\lambda_f<4$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-26T13:45:03.000Z",
      "comment": "22 pages, comments are welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-02-11T09:18:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D06",
      "14J99",
      "14H10",
      "14D99",
      "14J29"
    ],
    "keywords": [
      "positive relative irregularity",
      "hyperelliptic fibrations",
      "sharp lower bound",
      "locally non-trivial relatively minimal fibration",
      "hyperelliptic curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7271L"
    }
  }
}