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Finite 3-subgroups in the Cremona group of rank 3. (English. Russian original) Zbl 1469.14028

Math. Notes 108, No. 5, 697-715 (2020); translation from Mat. Zametki 108, No. 5, 725-749 (2020).
The paper under review studies \(p\)-subgroups with \(p=3\) of the birational automorphism group \(\mathrm{Bir}(X)\) for \(X\) a complex projective rationally connected 3-fold, using the language of \(G\)-varieties. The author gives (sharp) bounds on the number of generators of such 3-subgroups \(G\). In addition, the author finds even tighter bounds in the case where \(G\) acts by regular automorphisms on a minimal terminal \(G\)-Mori fibre space, with the exception of a few singular Fano varieties.
The approach presented is different from the previous work of [Y. Prokhorov and C. Shramov, Math. Nachr. 291, No. 8–9, 1374–1389 (2018; Zbl 1423.14099)], which relied on finding fixed points of the action of \(G\) on \(X\): in fact, this does not always happen when \(p=3\). An explanatory example of this fact is given in Section 2. The author carries out an in-depth study in the case where \(X\) is either a del Pezzo 3-fold or a Fano 3-fold.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations

Citations:

Zbl 1423.14099
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References:

[1] Dolgachev, I. V.; Iskovskikh, V. A., Finite subgroups of the plane Cremona group, Algebra, Arithmetic, and Geometry, I, 443-548 (2009) · Zbl 1219.14015 · doi:10.1007/978-0-8176-4745-2_11
[2] Beauville, A., \(p\)-elementary subgroups of the Cremona group, J. Algebra, 314, 2, 553-564 (2007) · Zbl 1126.14017 · doi:10.1016/j.jalgebra.2005.07.040
[3] Prokhorov, Yu., 2-elementary subgroups of the space Cremona group, Automorphisms in Birational and Affine Geometry, 79, 215-229 (2014) · Zbl 1327.14070
[4] Prokhorov, Yu., \(p\)-elementary subgroups of the Cremona group of rank 3, Classification of Algebraic Varieties, 0, 327-338 (2011) · Zbl 1221.14015 · doi:10.4171/007-1/16
[5] Prokhorov, Yu.; Shramov, C., \(p\)-subgroups in the space Cremona group, Math. Nachr., 291, 8-9, 1374-1389 (2018) · Zbl 1423.14099 · doi:10.1002/mana.201700030
[6] Prokhorov, Yu., \(G\)-Fano threefolds. I, Adv. Geom., 13, 3, 389-418 (2013) · Zbl 1291.14024
[7] Prokhorov, Y., On the number of singular points of terminal factorial Fano threefolds, Math. Notes, 101, 6, 1068-1073 (2017) · Zbl 1391.14082 · doi:10.1134/S0001434617050364
[8] Kuznetsov, A. G.; Prokhorov, Yu. G.; Shramov, C. A., Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math., 13, 1, 109-185 (2018) · Zbl 1406.14031 · doi:10.1007/s11537-017-1714-6
[9] Popov, Vl., Jordan groups and automorphism groups of algebraic varieties, Automorphisms in Birational and Affine Geometry, 79, 185-213 (2014) · Zbl 1325.14024
[10] Prokhorov, Yu.; Shramov, C., Jordan property for groups of birational selfmaps, Compos. Math., 150, 12, 2054-2072 (2014) · Zbl 1314.14022 · doi:10.1112/S0010437X14007581
[11] Chow, W.-L., On the geometry of algebraic homogeneous spaces, Ann. of Math. (2), 50, 1, 32-67 (1949) · Zbl 0040.22901 · doi:10.2307/1969351
[12] Hall Jr., M., The Theory of Groups (1976), New York: Chelsea Publ., New York · Zbl 0354.20001
[13] Schweizer, A., On the exponent of the automorphism group of a compact Riemann surface, Arch. Math. (Basel), 107, 4, 329-340 (2016) · Zbl 1352.14020 · doi:10.1007/s00013-016-0933-z
[14] Iskovskikh, V. A.; Prokhorov, Y., Fano varieties, Algebraic Geometry, V, 47, 0 (1999) · Zbl 0912.14013
[15] Reid, M., Young person’s guide to canonical singularities, Algebraic Geometry, Bowdoin, 1985, 46, 345-414 (1987) · Zbl 0634.14003 · doi:10.1090/pspum/046.1/927963
[16] Namikawa, Yo., Smoothing Fano 3-folds, J. Algebraic Geom., 6, 2, 307-324 (1997) · Zbl 0906.14019
[17] Kawamata, Y., Boundedness of \(\mathbb{Q} \)-Fano threefolds, Proceedings of the International Conference on Algebra, 131, 439-445 (1992) · Zbl 0785.14024
[18] Prokhorov, Yu., \(G\)-Fano threefolds. II, Adv. Geom., 13, 3, 419-434 (2013) · Zbl 1291.14025
[19] Shin, K., \(3\)-dimensional Fano varieties with canonical singularities, Tokyo J. Math., 12, 2, 375-385 (1989) · Zbl 0708.14025 · doi:10.3836/tjm/1270133187
[20] Mukai, Sh., Curves, K3 surfaces and Fano 3-folds of genus \(\le 10\), Algebraic Geometry and Commutative Algebra, I, 357-377 (1988) · Zbl 0701.14044
[21] Prokhorov, Yu., Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I (2019) · Zbl 1471.14032 · doi:10.1134/S0081543819060130
[22] Iskovskikh, V., Fano 3-folds. I, Math. USSR-Izv., 11, 3, 485-527 (1977) · Zbl 0382.14013 · doi:10.1070/IM1977v011n03ABEH001733
[23] Debarre, O.; Kuznetsov, A. G., Gushel-Mukai varieties: classification and birationalities, Algebr. Geom., 5, 1, 15-76 (2018) · Zbl 1408.14053
[24] Feit, W., The current situation in the theory of finite simple groups, Actes du Congrès International des Mathématiciens, 0, 55-93 (1971) · Zbl 0344.20008
[25] Borel, A., Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tohoku Math. J. (2), 13, 216-240 (1961) · Zbl 0109.26101 · doi:10.2748/tmj/1178244298
[26] Borel, A.; Serre, J.-P., Sur certains sous-groupes des groupes de Lie compacts, Comment. Math. Helv., 27, 128-139 (1953) · Zbl 0051.01902 · doi:10.1007/BF02564557
[27] Tahara, K., On the finite subgroups of \(\operatorname{GL}(3,\mathbb{Z})\), Nagoya Math. J., 41, 169-209 (1971) · Zbl 0194.33603
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