{
  "id": "math/0311508",
  "version": "v3",
  "published": "2003-11-27T19:13:48.000Z",
  "updated": "2005-03-14T20:29:41.000Z",
  "title": "Surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2",
  "authors": [
    "Francesco Polizzi"
  ],
  "comment": "36 pages. To appear in Transactions of the American Mathematical Society",
  "journal": "Trans. Amer. Math. Soc. 358 (2006), no. 2, 759--798",
  "doi": "10.1090/S0002-9947-05-03673-1",
  "categories": [
    "math.AG"
  ],
  "abstract": "We classify the minimal algebraic surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that if $S$ is such a surface then there exist two smooth curves $C, F$ and a finite group $G$ acting freely on $C \\times F$ such that $S = (C \\times F)/G$. We describe the $C, F$ and $G$ that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map $\\phi$ of $S$ is composed with the involution $\\sigma$ induced on $S$ by $\\tau \\times id: C \\times F \\longrightarrow C \\times F$, where $\\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1, K^2=8$ which yield the first known examples of surfaces with these invariants. We compute their dimension, and we show that they are three smooth and irreducible components of the moduli space $\\mathcal{M}$ of surfaces with $p_g=q=1, K^2=8$. For each of these families, an alternative description as a double cover of the plane is also given, and the index of the paracanonical system is computed.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-03-14T20:29:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J29",
      "14J10",
      "14H37"
    ],
    "keywords": [
      "bicanonical map",
      "general type",
      "minimal algebraic surfaces",
      "moduli space",
      "smooth curves"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....11508P"
    }
  }
}