{
  "id": "1612.05912",
  "version": "v1",
  "published": "2016-12-18T13:14:42.000Z",
  "updated": "2016-12-18T13:14:42.000Z",
  "title": "The Geometry of the Artin-Schreier-Mumford Curves over an Algebraically Closed Field",
  "authors": [
    "Gábor Korchmáros",
    "Maria Montanucci"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\\, c\\neq 0,$ defined over a field $\\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\\sqrt{g}(\\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\\mathcal{X})$ of all automorphisms of $\\mathcal{X}$ fixing $\\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\\mathcal{Y}$ in the three-dimensional projective space $PG(3,\\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\\mathbb{F}_{q^2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-18T13:14:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H37",
      "14H05"
    ],
    "keywords": [
      "algebraically closed field",
      "artin-schreier-mumford curve",
      "nonsingular model",
      "characteristic",
      "elementary abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}