{
  "id": "math/0608094",
  "version": "v1",
  "published": "2006-08-03T14:58:48.000Z",
  "updated": "2006-08-03T14:58:48.000Z",
  "title": "The automorphism group of an affine quadric",
  "authors": [
    "Burt Totaro"
  ],
  "comment": "9 pages, to appear in Math. Proc. Camb. Phil. Soc",
  "doi": "10.1017/S0305004107000357",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We determine the automorphism group for a large class of affine quadric hypersurfaces over a field, viewed as affine algebraic varieties. In particular, we find that the group of real polynomial automorphisms of the n-sphere is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. The proof uses Karpenko's theorem that certain projective quadrics over a field are not ruled. That is, they are not birational over the given field to the product of any variety with the projective line. We also formulate a general result on automorphisms of affine varieties. We conclude by conjecturing a converse to Karpenko's theorem, predicting exactly which projective quadrics are ruled.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-03T14:58:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E04",
      "14R20"
    ],
    "keywords": [
      "automorphism group",
      "karpenkos theorem",
      "affine quadric hypersurfaces",
      "real polynomial automorphisms",
      "projective quadrics"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}