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Bulletin SOCIÉTÉ MA THÉMA TIQUE DE FRANCE Publié avec le concours du Centre nat

Bulletin SOCIÉTÉ MA THÉMA TIQUE DE FRANCE Publié avec le concours du Centre national de la recherche scientiﬁque de la SOCIÉTÉ MA THÉMA TIQUE DE FRANCE T om e 135 Fascicule3 2 0 0 7 pages 1- THE NUMBER OF CONJUGACY CLASSES OF THE CREMONA GROUP Jérémy Blanc Bull. Soc. math. France 135 (3), 2007, p. 419– THE NUMBER OF CONJUGACY CLASSES OF ELEMENTS OF THE CREMONA GROUP OF SOME GIVEN FINITE ORDER by Jérémy Blanc Abstract. — This note presents the study of the conjugacy classes of elements of some given ﬁnite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is inﬁnite if n is even, n = 3 or n = 5, and that it is equal to 3 (respectively 9) if n = 9 (respectively if n = 15) and to 1 for all remaining odd orders. Some precise representative elements of the classes are given. Résumé (Le nombre de classes de conjugaison du groupe de Cremona d’un ordre ﬁni donné) Cet article présente l’étude des classes de conjugaisons des éléments d’ordre ﬁni n dans le groupe de Cremona du plan. En particulier, il est montré que le nombre de classes de conjugaisons est inﬁni si n est pair, n = 3 ou n = 5, et que ce nombre est égal à 3 (respectivement 9) si n = 9 (respectivement si n = 15) et à 1 pour les nombres entiers impairs restant. Des représentants explicites des classes de conjugaisons sont donnés. 1. Introduction Let us recall that a rational transformation of P2(C) is a map of the form (x : y : z) 99K (ϕ1(x, y, z) : ϕ2(x, y, z) : ϕ3(x, y, z)) , Texte reçu le 9 janvier 2007, révisé le 27 avril 2007 Jérémy Blanc, Laboratoire J.A. Dieudonné (UMR 6621), Université de Nice Sophia An- tipolis – CNRS, Faculté des Sciences, Parc Valrose, 06108 Nice Cedex 2 (France) E-mail : blancj@math.unice.fr 2000 Mathematics Subject Classiﬁcation. — 14E07, 14E05, 20E45. Key words and phrases. — Cremona group, birational transformations, conjugacy classes, elements of ﬁnite order. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2007/419/$ 5.00 © Société Mathématique de France 420 BLANC (J.) where ϕ1, ϕ2, ϕ3 ∈C[x, y, z] are homogeneous polynomials of the same degree. If such a map has an inverse of the same type, we say that it is birational. The Cremona group is the group of birational transformations of P2(C). This group has been studied since the 19th century by many mathematicians. One of the ﬁrst natural questions that we may ask when we study some group is the following: Question 1.1. — Given some positive integer n, how many conjugacy classes of elements of order n exist in the Cremona group? First of all, it is important to note that the number of conjugacy classes is at least one, for any integer n, as the linear automorphism (x : y : z) 7− →(x : y : e2iπ/nz) is a representative element of one class. It was proved in [2] that all the linear automorphisms of the plane of the same ﬁnite order are birationally conjugate (the same is true in any dimension, see [4, Prop. 5]); to ﬁnd more conjugacy classes we have therefore to show the existence of non-linearizable birational transformations. The ﬁrst answer to Question 1.1 was given in [3] for n = 2. Inﬁnitely many involutions which are not conjugate are found. Since the proof of [3] is considered as incomplete, a precise and complete one may be found in [1]. In [9], the answer for n prime is given. It is shown that the number of conjugacy classes is inﬁnite for n = 3, 5 and is equal to 1 if n is a prime integer ≥7. For other orders, a lot of examples have been given in the ancient articles (for example in [12], [16]) and in many more recent articles, the most recent one being [8]. However, the precise answer to Question 1.1 was not given for n not prime. In this paper, we answer Question 1.1 for any integer n, proving the following theorems: Theorem 1.2. — For any even integer n, the number of conjugacy classes of elements of order n in the Cremona group is inﬁnite. This is also true for n = 3, 5. Theorem 1.3. — For any odd integer n ̸= 3, 5, the number of conjugacy classes of elements of order n in the Cremona group is ﬁnite. Furthermore this number is equal to 3 (respectively 9) if n = 9 (respectively if n = 15) and is 1 otherwise. tome 135 – 2007 – no 3 THE NUMBER OF CONJUGACY CLASSES OF THE CREMONA GROUP 421 This paper is a part of the author’s PHD Thesis (the full text is available in [5], the results without proofs have been published in [6]). The results have been a little improved and the arguments have been slightly ameliorated to reduce the length of the proofs. We thank a lot our PHD advisor Thierry Vust, and also Arnaud Beauville, Igor Dolgachev, Ivan Pan and Felice Ronga for helpful discussions. Remark 1.4. — Theorem 1.2 and a part of Theorem 1.3 may be proved using the classiﬁcation of ﬁnite Abelian subgroups of the Cremona group made in [5], but the proof is really long and intricate, whereas the results of this paper only require a much shorter and direct proof. Moreover, the precise counting of Theorem 1.3 does not follow from the classiﬁcation (in general, it is not easy to decide whether or not two elements of the same subgroup are conjugate). 2. Automorphisms of rational surfaces Let us remark the obvious but important observation: take some birational transformation ϕ of a rational surface S. Any birational map λ : S 99K P2 con- jugates ϕ to the birational transformation ϕλ = λ◦ϕ◦λ−1 of P2. Although ϕλ is not unique, all the possible ϕλ’s form an unique conjugacy class of birational transformations of P2. Conversely, taking some birational transformation of P2, we may conjugate it to a birational transformation of any rational surface. If the order of the transformation is ﬁnite, we may furthermore conjugate it to a (biregular) au- tomorphism of a rational surface. (See for example [10], Theorem 1.4.) An important family of rational surfaces are the rational surfaces with an ample anticanonical divisor, i.e. the Del Pezzo surfaces. These surfaces are P1 × P1, P2, and the blow-up of 1 ≤r ≤8 points of P2 in a general posi- tion (i.e. such that no irreducible curve of self-intersection −2 belongs to the surface). There is an extensive literature about this; some descriptions may be found for example in [13]. The degree of such a surface is the square of its canonical divisor, and is an integer between 1 and 9; it is 9 for P2, 8 for P1 ×P1 and 9 −r for the blow-up of r points in P2. Almost all of our examples of rational surfaces will be Del Pezzo surfaces. 3. Elements of order 3,5 and of any even order. The proof of Theorem 1.2 Let us give families of conjugacy classes of elements of order 2, 3 and 5 of the Cremona group. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 422 BLANC (J.) Example 3.1 (Birational transformations of order 2). — Let a1, . . . , an, b1, . . . , bn in C be all distinct. The birational map ((x1 : x2), (y1 : y2)) 99K (x1 : x2), y2 n Y i=1 (x1 −bix2) : y1 n Y i=1 (x1 −aix2) !! of P1 × P1 is an involution, which is classically called de Jonquières involution. Its ﬁxed points form a smooth curve Γ ⊂P1 × P1 of equation (y1)2 · n Y i=1 (x1 −aix2) = (y2)2 · n Y i=1 (x1 −bix2). The restriction to Γ of the projection P1 × P1 →P1 on the ﬁrst factor is a surjective morphism Γ →P1 of degree 2, ramiﬁed over the points (a1 : 1), . . . , (an : 1), (b1 : 1), . . . , (bn : 1). The curve Γ is therefore an hyperelliptic curve. These involutions are birationally equivalent to those of [1], Example 2.4 (c). Example 3.2 (Birational transformations of order 3). — Let F be a non- singular form of degree 3 in three variables and let Γ = (x : y : z) ∈P2 | F(x, y, z) = 0 be the smooth cubic plane curve associated to it. The surface S = (w : x : y : z) ∈P3 | w3 = F(x, y, z) ⊂P3 is thus a smooth cubic surface in P3, which is rational (it is a Del Pezzo surface of degree 3, see for example [13], Theorem III.3.5). The map w 7→e2iπ/3w gives rise to an automorphism of S whose set of ﬁxed points is isomorphic to the uploads/Societe et culture/ 2007-04-howmanybullsmf.pdf