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Compressible finite groups of birational automorphisms. (English. Russian original) Zbl 1405.14040

Dokl. Math. 98, No. 2, 413-415 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 482, No. 1, 16-18 (2018).
From the text: In recent years, there has been a noticeable increase in research activity studying the subgroups of the birational automorphism groups Bir\((X)\) of various irreducible algebraic varieties \(X\) and references therein). In particular, in a number of cases a classification of finite such subgroups up to conjugacy in Bir\((X)\) was obtained. Amging the most impressive examples is tour de force [I. V. Dolgachev and V. A. Iskovskikh, Prog. Math. 269, 443–548 (2009; Zbl 1219.14015)], where the finite subgroups in the Cremona group of rank 2 are classified: the list obtained by Dolgachev and Iskovskikh [Dolgachev, Iskovskikh, loc. cit.] contains representatives of all conjugacy classes of finite subgroups of this group (but a complete solution to the problem of finding a unique such representative has not yet been given).
In these studies, all finite subgroups in Bir\((X)\) are treated on an equal footing. However, in reality some of them should be considered as “not basic”, since they are obtained from others by some standard general “base change” construction (see Sect. 2). This leads to the problem, posed by the author [Automorphisms in birational and affine geometry. Papers based on the presentations at the conference, Levico Terme, Italy (2012). Cham: Springer. 185–213 (2014; Zbl 1325.14024), Section 3.4; 6], on finding those subgroups in the available classification lists which can be obtained by a nontrivial base change. In this note, developing this topic, we formulate several results on such subgroups. Some of them are of a general nature, while some concern finite subgroups in the Cremona group of rank 2.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14J50 Automorphisms of surfaces and higher-dimensional varieties
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References:

[1] J. Blanc, These, No. 3777 (Genève, 2006). arXiv:math.AG/0610368.
[2] I. V. Dolgachev and V. A. Iskovskikh, “Finite subgroups of the plane Cremona group,” Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin, Vol. 1: Progress in Mathematics (Birkhäuser, Boston, 2009), vol. 269, pp. 443-548. · Zbl 1219.14015
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