{
  "id": "math/0107100",
  "version": "v1",
  "published": "2001-07-13T08:06:55.000Z",
  "updated": "2001-07-13T08:06:55.000Z",
  "title": "The classification of double planes of general type with $K^2=8$ and $p_g=0$",
  "authors": [
    "Rita Pardini"
  ],
  "comment": "LaTeX2e, 23 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study minimal {\\em double planes} of general type with $K^2=8$ and $p_g=0$, namely pairs $(S,\\sigma)$, where $S$ is a minimal complex algebraic surface of general type with $K^2=8$ and $p_g=0$ and $\\sigma$ is an automorphism of $S$ of order 2 such that the quotient $S/\\sigma$ is a rational surface. We prove that $S$ is a free quotient $(F\\times C)/G$, where $C$ is a curve, $F$ is an hyperelliptic curve, $G$ is a finite group that acts faithfully on $F$ and $C$, and $\\sigma$ is induced by the automorphism $\\tau\\times Id$ of $F\\times C$, $\\tau$ being the hyperelliptic involution of $F$. We describe all the $F$, $C$ and $G$ that occur: in this way we obtain 5 families of surfaces with $p_g=0$ and $K^2=8$, of which we believe only one was previously known. Using our classification we are able to give an alternative description of these surfaces as double covers of the plane, thus recovering a construction proposed by Du Val. In addition we study the geometry of the subset of the moduli space of surfaces of general type with $p_g=0$ and $K^2=8$ that admit a double plane structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-07-13T08:06:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J29"
    ],
    "keywords": [
      "general type",
      "double plane",
      "classification",
      "minimal complex algebraic surface",
      "study minimal"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......7100P"
    }
  }
}