{
  "id": "1512.01744",
  "version": "v1",
  "published": "2015-12-06T05:52:38.000Z",
  "updated": "2015-12-06T05:52:38.000Z",
  "title": "Jordan groups, conic bundles and abelian varieties",
  "authors": [
    "Tatiana Bandman",
    "Yuri G. Zarhin"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $\\mathcal{B}$ of $G$ contains a commutative subgroup $\\mathcal{A}\\subset \\mathcal{B}$ such that $\\mathcal{A}$ is normal in $\\mathcal{B}$ and the index $[\\mathcal{B}:\\mathcal{A}] \\le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the other hand, the second named author proved that $Bir(X)$ is not Jordan if $X$ is birational to a product of the projective line and a positive-dimensional abelian variety. We prove that $Bir(X)$ is Jordan if (uniruled) $X$ is a conic bundle over a non-uniruled variety $Y$ but is not birational to a product of $Y$ and the projective line. (Such a conic bundle exists only if $\\dim(Y)\\ge 2$.) When $Y$ is an abelian surface, this Jordaness property result gives an answer to a question of Prokhorov and Shramov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-06T05:52:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E07",
      "14J50",
      "14L30",
      "14K05",
      "14H45",
      "14H37",
      "20G15"
    ],
    "keywords": [
      "conic bundle",
      "jordan groups",
      "jordaness property result",
      "projective line",
      "positive-dimensional abelian variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}