{
  "id": "0706.2261",
  "version": "v1",
  "published": "2007-06-15T10:27:59.000Z",
  "updated": "2007-06-15T10:27:59.000Z",
  "title": "Uniqueness of $\\bf C^*$- and $\\bf C_+$-actions on Gizatullin surfaces",
  "authors": [
    "Hubert Flenner",
    "Shulim Kaliman",
    "Mikhail Zaidenberg"
  ],
  "comment": "43 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "A Gizatullin surface is a normal affine surface $V$ over $\\bf C$, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $\\bf C^*$-actions and $\\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\\bf C_+$-actions on $V$ considered up to a \"speed change\". Non-Gizatullin surfaces are known to admit at most one $\\bf A^1$-fibration $V\\to S$ up to an isomorphism of the base $S$. Moreover an effective $\\bf C^{*}$-action on them, if it does exist, is unique up to conjugation and inversion $t\\mapsto t^{-1}$ of $\\bf C^*$. Obviously uniqueness of $\\bf C^*$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of $\\bf A^1$-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $\\bf C^*$-actions and $\\bf A^1$-fibrations. In the present paper we obtain a criterion as to when $\\bf A^1$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of $V$ and the base $S$. We exhibit as well a large subclasses of Gizatullin $\\bf C^{*}$-surfaces for which a $\\bf C^*$-action is essentially unique and for which there are at most two conjugacy classes of $\\bf A^1$-fibrations over $\\bf A^1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-15T10:27:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14R20",
      "14R25"
    ],
    "keywords": [
      "uniqueness",
      "conjugacy classes",
      "smooth rational curves",
      "affine toric surfaces",
      "normal affine surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.2261F"
    }
  }
}