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**The number of conjugacy classes of elements of the Cremona group of some given finite order.**
*(English)*
Zbl 1158.14017

One of the classical problems involving the plane Cremona group \(Cr_2(\mathbb{C})\) of birational automorphisms of \(\mathbb{P}^2_{\mathbb{C}}\), is the classification of the conjugacy classes of its elements. The history of the problem starts with the work of E. Bertini who classified the conjugacy classes of order 2 subgroups of \(Cr_2(\mathbb{C})\). However, his classification was incomplete and the first complete proof was given by L. Bayle and A. Beauville [Asian J. Math. 4, 11–17 (2000; Zbl 1055.14012)] who showed that any proper birational automorphism of \(\mathbb{P}^2_{\mathbb{C}}\) of order 2 is conjugate to either a De Jonquiéres involution, a Geiser involution or a Bertini involution. In fact there are infinitely many of each kind. The conjugacy classes of elements of prime order were classified by T. de Fernex [Nagoya Math. J. 174, 1–28 (2004; Zbl 1062.14019)].

The paper under review studies the problem of whether there are finitely many or infinitely many conjugacy classes or elements of a given order \(n\). If \(n\) is even, then it is shown by explicit examples that the number of conjugacy classes of elements of order \(n\) is infinite. The same holds for \(n=3,5\). If \(n\) is odd and \(n \not= 3,5\), then the number of conjugacy classes of elements or order \(n\) is finite. In fact, this number is equal to 3 or 9 if \(n=9\) or \(15\), respectively, and 1 otherwise. The idea of the proof of the second result is the following. According to the ideas of Y. Manin and V. Iskovskikh [Russ. Math. Surv. 51, No. 4, 585–652 (1996; Zbl 0914.14005)] there is a one to one correspondence between conjugacy classes of elements of order \(n\) and birational equivalence classes of pairs \((S, \theta)\), where \(S\) is a rational surface and \(\theta \in \mathrm{Aut}(S)\) of order \(n\). Then by using the two dimensional minimal model program, one can assume that the pair \((S,\theta)\) is minimal and in fact \(S\) is either a Del Pezzo surface or a conic bundle. A case by case study gives the result.

The paper under review studies the problem of whether there are finitely many or infinitely many conjugacy classes or elements of a given order \(n\). If \(n\) is even, then it is shown by explicit examples that the number of conjugacy classes of elements of order \(n\) is infinite. The same holds for \(n=3,5\). If \(n\) is odd and \(n \not= 3,5\), then the number of conjugacy classes of elements or order \(n\) is finite. In fact, this number is equal to 3 or 9 if \(n=9\) or \(15\), respectively, and 1 otherwise. The idea of the proof of the second result is the following. According to the ideas of Y. Manin and V. Iskovskikh [Russ. Math. Surv. 51, No. 4, 585–652 (1996; Zbl 0914.14005)] there is a one to one correspondence between conjugacy classes of elements of order \(n\) and birational equivalence classes of pairs \((S, \theta)\), where \(S\) is a rational surface and \(\theta \in \mathrm{Aut}(S)\) of order \(n\). Then by using the two dimensional minimal model program, one can assume that the pair \((S,\theta)\) is minimal and in fact \(S\) is either a Del Pezzo surface or a conic bundle. A case by case study gives the result.

Reviewer: Nikolaos Tziolas (Nicosia)

### MSC:

14E07 | Birational automorphisms, Cremona group and generalizations |

14E05 | Rational and birational maps |

14E30 | Minimal model program (Mori theory, extremal rays) |

20E45 | Conjugacy classes for groups |