{
  "id": "1410.7535",
  "version": "v2",
  "published": "2014-10-28T07:17:53.000Z",
  "updated": "2015-04-13T10:41:56.000Z",
  "title": "Finite groups of automorphisms of Enriques surfaces and the Mathieu group $M_{12}$",
  "authors": [
    "Shigeru Mukai",
    "Hisanori Ohashi"
  ],
  "comment": "33 pages, 2 Figures ver 2: An Enriques surface with $\\mathfrak{A}_6$ action is constructed geometrically without using Torelli (in this new version). Section 7 is added to classify the tame Mathieu actions in positive characteristic",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "An action of a group $G$ on an Enriques surface $S$ is called Mathieu if it acts on $H^0(2K_S)$ trivially and every element of order 2, 4 has Lefschetz number 4. A finite group $G$ has a Mathieu action on some Enriques surface if and only if it is isomorphic to a subgroup of the symmetric group $\\mathfrak{S}_6$ of degree 6 and the order $|G|$ is not divisible by $2^4$. Explicit Mathieu actions of the three groups $\\mathfrak S_5, N_{72}$ and $\\mathfrak A_6$, together with non-Mathieu one of $H_{192}$, on polarized Enriques surfaces of degree 30, 18, 10 and 6, respectively, are constructed without Torelli type theorem to prove the if part.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-28T07:17:53.000Z",
      "abstract": "An action of a group $G$ on an Enriques surface $S$ is called Mathieu if it acts on $H^0(2K_S)$ trivially and every element of order 2, 4 has Lefschetz number 4. A finite group $G$ has a Mathieu action on some Enriques surface if and only if it is isomorphic to a subgroup of the symmetric group $\\mathfrak S_6$ of degree 6 and the order $|G|$ is not divisible by $2^4$. The 'if' part is proved constructing explicit actions of three groups $\\mathfrak S_5, N_{72}$ and $H_{192}$ on polarized Enriques surfaces of degree 30, 18 and 6, respectively and using the result by Keum--Oguiso--Zhang for the alternating group $\\mathfrak A_6$.",
      "comment": "26 pages, 2 Figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-13T10:41:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J28",
      "20D08"
    ],
    "keywords": [
      "finite group",
      "mathieu group",
      "automorphisms",
      "constructing explicit actions",
      "lefschetz number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.7535M"
    }
  }
}