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Unformatted text preview: TAME AND WILD FINITE SUBGROUPS OF THE PLANE CREMONA GROUP IGOR V. DOLGACHEV To the memory of Vasily Iskovskikh Abstract. We survey some old and new results about finite subgroups of the Cremona group Cr n ( k ) of birational automorphisms of the pro- jective n-space over a field k . Contents 1. Introduction 1 2. General facts 2 2.1. G-varieties 2 2.2. Lift to characteristic 0 3 3. The case k = C 4 3.1. Conic bundles 4 3.2. De Jonqui` eres transformations 6 3.3. Automorphism groups of Del Pezzo surfaces 8 3.4. Elementary links 23 3.5. Final classification 25 4. Cyclic tame subgroups of Cr 2 ( k ), where k is a perfect field 29 4.1. Elements of finite order in reductive algebraic groups 29 4.2. Elements of order 7 30 5. Wild cyclic groups 32 5.1. Conic bundles 32 5.2. Del Pezzo surfaces 33 6. Wild simple groups 36 6.1. Projective linear groups 36 6.2. Conic bundles 38 6.3. Del Pezzo surfaces 39 References 43 1. Introduction The Cremona group Cr n ( k ) of degree n over a field k is the group of birational automorphisms of P n k . In algebraic terms Cr n ( k ) = Aut k ( k ( t 1 , . . . , t n )) . 1 2 IGOR V. DOLGACHEV In this article I will survey some old and new results on classification of of conjugacy classes of finite subgroups of Cr 2 ( k ). Recall that in the case n = 1, we have Cr 1 ( k ) = Aut( P 1 k ) = PGL 2 ( k ) . The classification of finite subgroups of PGL 2 ( k ) is well-known. If k is algebraically closed of characteristic zero, then each such group is isomorphic to either a cyclic groups C n , or a dihedral groups D n of order 2 n , or the tetrahedron group T , or the octahedron group O , or the icosahedron group I . There is only one conjugacy class for each group in Cr 1 ( k ). If char( k ) = p > 0, then G is isomorphic to a subgroup of PGL 2 ( F q ) for some q = p s . In this survey we will be concerned with the case n = 2. We will consider three essentially different cases: k is the field of complex numbers C ; k is an arbitrary field of characteristic prime to the order | G | of G ; k is algebraically closed of characteristic p dividing the order of G . Although in the first case the classification is almost complete, in the re- maining cases it is very far from being complete. This work arises from collaboration with my old, now deceased, friend and colleague Vasya Iskovskikh. His help and guidance is hard to overestimate. 2. General facts 2.1. G-varieties. Let G be a finite subgroup of Cr n ( k ). We say that a rational variety X regularizes G if there exists a birational isomorphism : X- P n such that - 1 G is a subgroup of automorphisms of X . Lemma 2.1. Each finite subgroup of Cr n ( k ) can be regularized....
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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.
- Fall '04