## Abstract

group G is called Jordan if there is a positive integer J = J_{G} such that every finite subgroup B of G contains a commutative subgroup A ⊂ B such that A is normal in B and the index [B: A] is at most J (V. L. Popov). In this paper, we deal with Jordan 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 and 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 P1 and a positive-dimensional abelian variety. We prove that Bir(X) is Jordan if (the uniruled variety) X is a conic bundle over a non-uniruled variety Y but is not birational to Y × ^{1}. (Such a conic bundle exists if and only if dim(Y) ≥ 2.) When Y is an abelian surface, this Jordan property result gives an answer to a question of Prokhorov and Shramov.

Original language | English |
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Pages (from-to) | 229-246 |

Number of pages | 18 |

Journal | Algebraic Geometry |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - 1 Mar 2017 |

### Bibliographical note

Publisher Copyright:© Foundation Compositio Mathematica 2017.

## Keywords

- Birational automorphisms
- Conic bundles
- Jordan groups
- Uniruled varieties