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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 [email protected] Faculty of Mathematics, Moscow State University, Moscow 117234, Russia email [email protected]
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 [email protected]
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Abstract

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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References

Birkar, C., Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 325–368.Google Scholar
Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.Google Scholar
Bierstone, E. and Milman, P. D., Functoriality in resolution of singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 609–639.Google Scholar
Borisov, A., Boundedness theorem for Fano log-threefolds, J. Algebraic Geom. 5 (1996), 119–133.Google Scholar
Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, vol. XI (Wiley-Interscience, New York, 1962).Google Scholar
Problems for the workshop ‘Subgroups of Cremona groups: classification’, 29–30 March 2010,ICMS, Edinburgh, available at http://www.mi.ras.ru/∼prokhoro/preprints/edi.pdf.Google Scholar
Fakhruddin, N., Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), 109–122.Google Scholar
Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 57–67.Google Scholar
Hanamura, M., On the birational automorphism groups of algebraic varieties, Compositio Math. 63 (1987), 123–142.Google Scholar
Hanamura, M., Structure of birational automorphism groups. I. Nonuniruled varieties, Invent. Math. 93 (1988), 383–403.CrossRefGoogle Scholar
Hacon, C., McKernan, J. and Xu, C., On the birational automorphisms of varieties of general type, Ann. of Math. (2) 177 (2013), 1077–1111.CrossRefGoogle Scholar
Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283–360.CrossRefGoogle Scholar
Kollár, J. (ed.), Flips and abundance for algebraic threefolds: A summer seminar at the University of Utah (Salt Lake City, 1991), Astérisque, vol. 211 (Société Mathématique de France, 1992).Google Scholar
Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32 (Springer, Berlin, 1996).CrossRefGoogle Scholar
Kollár, J., Miyaoka, Y., Mori, S. and Takagi, H., Boundedness of canonical Q-Fano 3-folds, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 73–77.Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998), with the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original.Google Scholar
Lang, S., Fundamentals of Diophantine geometry (Springer, New York, 1983).Google Scholar
Matsuki, K., Introduction to the Mori program, Universitext (Springer, New York, 2002).Google Scholar
Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449.Google Scholar
Miyaoka, Y. and Mori, S., A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), 65–69.CrossRefGoogle Scholar
Morton, P. and Silverman, J. H., Rational periodic points of rational functions, Int. Math. Res. Not. IMRN 2 (1994), 97–110.CrossRefGoogle Scholar
Poonen, B., Uniform boundedness of rational points and preperiodic points, Preprint (2012), arXiv:1206.7104.Google Scholar
Popov, V. L., On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in Peter Russell’s Festschrift, Proceedings of the conference on Affine Algebraic Geometry held in Professor Russell’s honour, 1–5 June 2009, McGill Univ., Montreal, CRM Proceedings and Lecture Notes, vol. 54 (Centre de Recherches Mathématiques, Montréal, 2011), 289–311.Google Scholar
Prokhorov, Y. and Shramov, C., Jordan property for Cremona groups, Amer. J. Math. (2015), to appear, arXiv:1211.3563.Google Scholar
Serre, J.-P., Bounds for the orders of the finite subgroups of G (k), in Group representation theory (EPFL Press, Lausanne, 2007), 405–450.Google Scholar
Serre, J.-P., A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Mosc. Math. J. 9 (2009), 193–208.Google Scholar
Zarhin, Y. G., Theta groups and products of abelian and rational varieties, Proc. Edinb. Math. Soc. (2) 57 (2014), 299–304.Google Scholar
Zhang, Q., Rational connectedness of log Q-Fano varieties, J. Reine Angew. Math. 590 (2006), 131–142.Google Scholar