{
  "id": "math/0610368",
  "version": "v1",
  "published": "2006-10-11T13:33:54.000Z",
  "updated": "2006-10-11T13:33:54.000Z",
  "title": "Finite abelian subgroups of the Cremona group of the plane",
  "authors": [
    "Jérémy Blanc"
  ],
  "comment": "PHD Thesis, 189 pages, 34 figures, original text may be found at http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.html",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on birational transformations, as for example the existence of infinitely many conjugacy classes of elements of order n, for any even number n, a result false in the odd case. We prove also that a root of some linear transformation of finite order is itself conjugate to a linear transformation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-11T13:33:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E07",
      "14J26",
      "14J50",
      "14L30"
    ],
    "keywords": [
      "finite abelian subgroups",
      "cremona group",
      "conjugacy classes",
      "linear transformation",
      "del pezzo surfaces"
    ],
    "tags": [
      "dissertation"
    ],
    "publication": {
      "journal": "Ph.D. Thesis",
      "year": 2006,
      "month": "Oct"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 189,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006PhDT.......289B"
    }
  }
}