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**Finite subgroups of the plane Cremona group.**
*(English)*
Zbl 1219.14015

Tschinkel, Yuri (ed.) et al., Algebra, arithmetic, and geometry. In honor of Yu. I. Manin on the occasion of his 70th birthday. Vol. I. Boston, MA: Birkhäuser (ISBN 978-0-8176-4744-5/hbk; 978-0-8176-4745-2/ebook). Progress in Mathematics 269, 443-548 (2009).

The Cremona group is the group of birational transformations of the complex plane or, equivalently, the group of automorphisms of the field \(\mathbb{C}(x,y)\) that are the identity on the subfield \(\mathbb{C}\).

The study of the finite subgroups of the Cremona group started a long time ago with the work of Wiman and Kantor in the 19th century. The classification was not completely achieved and the question of conjugation between the groups found was not studied.

In the article under review, the authors make the list of Kantor and Wiman more precise. They use the technique of Mori theory, as it was already done to study the cyclic groups of prime order by L. Bayle and A. Beauville [Asian J. Math. 4, No. 1, 11–17 (2000; Zbl 1055.14012)] and by T. de Fernex [Nagoya Math. J. 174, 1–28 (2004; Zbl 1062.14019)]. Any finite subgroup of the Cremona group is conjugate to a finite group of (biregular) automorphisms of a smooth projective surface. Choosing the action to be minimal, we end up with either a del Pezzo surface or a conic bundle.

The study of automorphisms of del Pezzo surfaces is made in detail in the article, together with the description of the elements in the associated Weyl groups. The case of conic bundle is more subtle. The authors find descriptions but not as precise as in the case of del Pezzo surfaces.

The description of finite subgroups of the Cremona group given in the article is the most precise given up to now. It has been completely achieved in the case of cyclic groups by the reviewer [Comment. Math. Helv. 86, No. 2, 469–497 (2011; Zbl 1213.14029)]. The case of conic bundles has also been studied in details in the PhD thesis of V. I. Tsygankov, see e.g. [“Equations of \(G\)-minimal bundles on conics of degree 4”, Vestn. Mosk. Univ., Ser. I 2010, No. 2, 39–42 (2010); translation in Mosc. Univ. Math. Bull. 65, No. 2, 72–75 (2010), doi:10.3103/S002713221002004X].

For the entire collection see [Zbl 1185.00041].

The study of the finite subgroups of the Cremona group started a long time ago with the work of Wiman and Kantor in the 19th century. The classification was not completely achieved and the question of conjugation between the groups found was not studied.

In the article under review, the authors make the list of Kantor and Wiman more precise. They use the technique of Mori theory, as it was already done to study the cyclic groups of prime order by L. Bayle and A. Beauville [Asian J. Math. 4, No. 1, 11–17 (2000; Zbl 1055.14012)] and by T. de Fernex [Nagoya Math. J. 174, 1–28 (2004; Zbl 1062.14019)]. Any finite subgroup of the Cremona group is conjugate to a finite group of (biregular) automorphisms of a smooth projective surface. Choosing the action to be minimal, we end up with either a del Pezzo surface or a conic bundle.

The study of automorphisms of del Pezzo surfaces is made in detail in the article, together with the description of the elements in the associated Weyl groups. The case of conic bundle is more subtle. The authors find descriptions but not as precise as in the case of del Pezzo surfaces.

The description of finite subgroups of the Cremona group given in the article is the most precise given up to now. It has been completely achieved in the case of cyclic groups by the reviewer [Comment. Math. Helv. 86, No. 2, 469–497 (2011; Zbl 1213.14029)]. The case of conic bundles has also been studied in details in the PhD thesis of V. I. Tsygankov, see e.g. [“Equations of \(G\)-minimal bundles on conics of degree 4”, Vestn. Mosk. Univ., Ser. I 2010, No. 2, 39–42 (2010); translation in Mosc. Univ. Math. Bull. 65, No. 2, 72–75 (2010), doi:10.3103/S002713221002004X].

For the entire collection see [Zbl 1185.00041].

Reviewer: Jérémy Blanc (Basel)