{
  "id": "math/0406239",
  "version": "v2",
  "published": "2004-06-11T12:12:45.000Z",
  "updated": "2004-06-14T18:29:02.000Z",
  "title": "On the uniqueness of ${\\bf C}^*$-actions on affine surfaces",
  "authors": [
    "Hubert Flenner",
    "Mikhail Zaidenberg"
  ],
  "comment": "11/06/2004 2 version 14/06/2004",
  "categories": [
    "math.AG"
  ],
  "abstract": "We prove that a normal affine surface $V$ over $\\bf C$ admits an effective action of a maximal torus ${\\bf T}={\\bf C}^{*n}$ ($n\\le 2$) such that any other effective ${\\bf C}^*$-action is conjugate to a subtorus of $\\bf T$ in Aut $(V)$, in the following particular cases: (a) the Makar-Limanov invariant ML$(V)$ is nontrivial, (b) $V$ is a toric surface, (c) $V={\\bf P}^1\\times {\\bf P}^1\\backslash \\Delta$, where $\\Delta$ is the diagonal, and (d) $V={\\bf P}^2\\backslash Q$, where $Q$ is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth) and (d) is a result of Danilov-Gizatullin and Doebeli.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-06-14T18:29:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14R05",
      "14R20",
      "14J50"
    ],
    "keywords": [
      "uniqueness",
      "normal affine surface",
      "makar-limanov invariant ml",
      "affine plane",
      "maximal torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6239F"
    }
  }
}