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.

MSC:

 1.4e+08 Birational automorphisms, Cremona group and generalizations 1.4e+06 Rational and birational maps 1.4e+31 Minimal model program (Mori theory, extremal rays) 2e+46 Conjugacy classes for groups
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