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Jordan property for groups of birational selfmaps

Part of: Birational geometry

Published online by Cambridge University Press:  17 September 2014

Yuri Prokhorov  and
Constantin Shramov
Yuri Prokhorov
Affiliation:
Steklov Institute of Mathematics, 8 Gubkina Street, Moscow 119991, Russia Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Street, Moscow 117312, Russia email costya.shramov@gmail.com Faculty of Mathematics, Moscow State University, Moscow 117234, Russia email prokhoro@gmail.com
Constantin Shramov
Affiliation:
Steklov Institute of Mathematics, 8 Gubkina Street, Moscow 119991, Russia Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Street, Moscow 117312, Russia email costya.shramov@gmail.com
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Abstract

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Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field $\Bbbk$ of characteristic zero.

Keywords

Jordan groupbirational automorphismsminimal model programdivisor class group

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 12 , December 2014 , pp. 2054 - 2072
Copyright
© The Author(s) 2014 

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