Abstract
We prove that an analogue of Jordan’s theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir L. Popov.
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M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 5 (1955), 407–434.
T. Bandman, Yu. G. Zarhin, Jordan groups and algebraic surfaces, Transformation Groups, to appear, arXiv:1404.1581.
S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Springer-Verlag, Berlin, 1990.
C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962. Russian transl.: Ч. Кэртис, И. Райнер, Теорuяnредсmавленuй, конечных груnn u ассоцuаmuвных алгебр, Наука, M., 1969.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Хартсхорн, Алгебраuческая геомеmрuя, Мир, М., 1981.
Sh. Iitaka, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 76, Springer-Verlag, New York, 1982.
M. Maruyama, On Classification of Ruled Surfaces, Lectures in Mathematics, Kyoto University, Kinokunia, Tokyo, 1970, available at http://www.math.kyoto-u.ac.jp/library/lmku/lmku03.pdf.
M. Maruyama, On automorphism groups of ruled surfaces. J. Math. Kyoto Univ. 11 (1971), 89–112.
D. Mumford, Abelian Varieties, 2nd ed., Oxford University Press, 1974.
D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, Vol. 1358, Springer-Verlag, Berlin, 1999.
V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in: Affine Algebraic Geometry (The Russell Festschrift), CRM Proceedings and Lecture Notes 54, American Mathematical Society, Providence, RI, 2011, pp. 289–311.
V. L. Popov, Jordan groups and automorphism groups of algebraic varieties, in: Automorphisms in Birational and Affine Geometry, Levico Terme, Italy, October 2012, Springer Proceedings in Mathematics & Statistics, Vol. 79, 2014, Springer, Heidelberg, pp. 185–213.
V. L. Popov, Finite subgroups of diffeomorphism groups, arXiv:1310.6548.
Yu. Prokhorov, C. Shramov, Jordan property for groups of birational selfmaps, arXiv:1307.1784.
I. Mundet i Riera, A. Turull, Boosting an analogue of Jordan’s theorem for finite groups, arXiv:1310.6518.
J-P. Serre, A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Moscow Math. J. 9 (2009), no. 1, 183-198.
И. Р. Шафаревич и др., Алгебраические nоверхносmи, Труды Мат. ин-та им. В. А. Стеклова LXXV (1965). Engl. transl.: I. R. Shafarevich et al., Algebraic Surfaces, American Mathematical Society, Providence, RI, 1967.
И. Р. Шафаревич, Основы алгебраической геомеmрии, 2-e изд., т. 1, Наука, M., 1988. Engl. transl.: I. R. Shafarevich, Basic Algebraic Geometry, 2nd ed., Vol. I, Springer-Verlag, Heidelberg, 1994.
Ю. Г. Зархин, Некоммуmаmивные когомологии и груnnы Мамфорда, Maт. заметки 15 (1974), 415–419. Engl. transl.: Yu. G. Zarhin, Noncommutative cohomology and Mumford groups, Mathematical Notes 15 (1974), 241–244.
Yu. G. Zarhin, Theta groups and products of abelian and rational varieties, Proc. Edinburgh Math. Soc. 57 (2014), 299–304.
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In memoriam of Emmanuil El’evich Shnol’
This work was partially supported by a grant from the Simons Foundation (# 246625 to Yuri Zarkhin).
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ZARHIN, Y.G. JORDAN GROUPS AND ELLIPTIC RULED SURFACES. Transformation Groups 20, 557–572 (2015). https://doi.org/10.1007/s00031-014-9292-7
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DOI: https://doi.org/10.1007/s00031-014-9292-7