{
  "id": "1601.06698",
  "version": "v1",
  "published": "2016-01-25T18:01:27.000Z",
  "updated": "2016-01-25T18:01:27.000Z",
  "title": "A lower bound for $K^2_S$",
  "authors": [
    "Vincenzo Di Gennaro",
    "Davide Franco"
  ],
  "comment": "12 pages, Dedicated to Philippe Ellia on his sixtieth birthday",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $(S,\\mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\\mathcal L$ of degree $d > 35$. In this paper we prove that $K^2_S\\geq -d(d-6)$. The bound is sharp, and $K^2_S=-d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,\\mathcal L)|$ embeds $S$ in a smooth rational normal scroll $T\\subset \\mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $\\frac{d}{2}Q$, where $Q$ is a quadric on $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-25T18:01:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J99",
      "14M20",
      "14N15",
      "51N35"
    ],
    "keywords": [
      "lower bound",
      "smooth rational normal scroll",
      "ample line bundle",
      "complex surface",
      "linear system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}