DI_manin - Finite Subgroups of the Plane Cremona Group Igor...

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Finite Subgroups of the Plane Cremona Group Igor V. Dolgachev 1 and Vasily A. 1 Department of Mathematics, University of Michigan, 525 E. University Ave., Ann Arbor, MI, 49109 idolga@umich.edu To Yuri I. Manin Summary. This paper completes the classic and modern results on classiFcation of conjugacy classes of Fnite subgroups of the group of birational automorphisms of the complex projective plane. Key words: Cremona group, Del Pezzo surfaces, conic bundles. 2000 Mathematics Subject Classifcations : 14E07 (Primary); 14J26, 14J50, 20B25 (Secondary) 1 Introduction The Cremona group Cr k ( n ) over a Feld k is the group of birational au- tomorphisms of the projective space P n k , or equivalently, the group of k -automorphisms of the Feld k ( x 1 ,x 2 ,...,x n ) of rational functions in n independent variables. The group Cr k (1) is the group of automorphisms of the projective line, and hence it is isomorphic to the projective linear group PGL k (2) . Already in the case n =2 the group Cr k (2) is not well understood in spite of extensive classical literature (e.g., [ 21 ], [ 35 ]) on the subject as well as some modern research and expositions of classical results (e.g., [ 2 ]). Very little is known about the Cremona groups in higher-dimensional spaces. In this paper we consider the plane Cremona group over the Feld of com- plex numbers, denoted by Cr (2) . We return to the classical problem of classi- Fcation of Fnite subgroups of Cr (2) . The classiFcation of Fnite subgroups of PGL (2) is well known and goes back to ±. Klein. It consists of cyclic, dihedral, 1 The author was supported in part by NS± grant 0245203. 2 The author was supported in part by R±BR 05-01-00353-a R±BR 08-01-00395-a, grant CRD± RUMI 2692-MO-05 and grant of NSh 1987-2008.1. Y. Tschinkel and Y. Zarhin (eds.), Algebra, Arithmetic, and Geometry , 443 Progress in Mathematics 269, DOI 10.1007/978-0-8176-4745-2_11, c ± Springer Science+Business Media, LLC 2009 Iskovskikh 2
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444 Igor V. Dolgachev and Vasily A. Iskovskikh tetrahedral, octahedral, and icosahedral groups. Groups of the same type and order constitute a unique conjugacy class in PGL (2) . Our goal is to Fnd a similar classiFcation in the two-dimensional case. The history of this problem begins with the work of E. Bertini [ 10 ], who classiFed conjugacy classes of subgroups of order 2 in Cr (2) .A l r eady in this case the answer is drastically di±erent. The set of conjugacy classes is parametrized by a disconnected algebraic variety whose connected compo- nents are respectively isomorphic to either the moduli spaces of hyperelliptic curves of genus g (de Jonquières involutions), or the moduli space of canonical curves of genus 3 (Geiser involutions), or the moduli space of canonical curves of genus 4 with vanishing theta characteristic (Bertini involutions). Bertini’s proof was considered to be incomplete even according to the standards of rigor of nineteenth-century algebraic geometry. A complete and short proof was published only a few years ago by L. Bayle and A. Beauville [ 5 ].
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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DI_manin - Finite Subgroups of the Plane Cremona Group Igor...

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