Geodesic
Birational automorphisms of Severi-Brauer surfaces
Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 466-480 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that a finite group acting by birational automorphisms of a nontrivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most $3$. Also, we find an explicit bound for the orders of such finite groups in the case when the base field contains all roots of $1$. Bibliography: 25 titles.
Keywords: Severi-Brauer surface, group of birational automorphisms. 
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С. A. Shramov. Birational automorphisms of Severi-Brauer surfaces. Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 466-480. http://geodesic.mathdoc.fr/item/SM_2020_211_3_a5/

[1] M. Artin, “Brauer–Severi varieties”, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., 917, Springer, Berlin–New York, 1982, 194–210 | DOI | MR | Zbl

[2] H. F. Blichfeldt, Finite collineation groups, Univ. Chicago Press, Chicago, IL, 1917, xi+194 pp.

[3] A. Borel, “Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes”, Tôhoku Math. J. (2), 13:2 (1961), 216–240 | DOI | MR | Zbl

[4] R. W. Carter, “Conjugacy classes in the Weyl group”, Compositio Math., 25 (1972), 1–59 | MR | Zbl

[5] F. Châtelet, “Variations sur un thème de H. Poincaré”, Ann. Sci. École Norm. Sup. (3), 61 (1944), 249–300 | DOI | MR | Zbl

[6] P. Corn, “Del Pezzo surfaces of degree $6$”, Math. Res. Lett., 12:1 (2005), 75–84 | DOI | MR | Zbl

[7] I. V. Dolgachev, V. A. Iskovskikh, “Finite subgroups of the plane Cremona group”, Algebra, arithmetic, and geometry. In honor of Yu. I. Manin, v. I, Progr. Math., 269, Birkhäuser Boston, Boston, MA, 2009, 443–548 | DOI | MR | Zbl

[8] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge Univ. Press, Cambridge, 2012, xii+639 pp. | DOI | MR | Zbl

[9] M. Garcia-Armas, “Finite group actions on curves of genus zero”, J. Algebra, 394 (2013), 173–181 | DOI | MR | Zbl

[10] M. H. Gizatullin, “Defining relations for the Cremona group of the plane”, Math. USSR-Izv., 21:2 (1983), 211–268 | DOI | MR | Zbl

[11] S. Gorchinskiy, C. Shramov, Unramified Brauer group and its applications, Transl. Math. Monogr., 246, Amer. Math. Soc., Providence, RI, 2018, xvii+179 pp. | DOI | MR | Zbl

[12] V. A. Iskovskih, “Minimal models of rational surfaces over arbitrary fields”, Math. USSR-Izv., 14:1 (1980), 17–39 | DOI | MR | Zbl

[13] V. A. Iskovskikh, “A simple proof of a theorem of Gizatullin”, Proc. Steklov Inst. Math., 183 (1991), 127–133 | MR | Zbl

[14] V. A. Iskovskikh, “Factorization of birational maps of rational surfaces from the viewpoint of Mori theory”, Russian Math. Surveys, 51:4 (1996), 585–652 | DOI | DOI | MR | Zbl

[15] V. A. Iskovskikh, F. K. Kabdykairov, S. L. Tregub, “Relations in the two-dimensional Cremona group over a perfect field”, Russian Acad. Sci. Izv. Math., 42:3 (1994), 427–478 | DOI | MR | Zbl

[16] V. A. Iskovskikh, S. L. Tregub, “On birational automorphisms of rational surfaces”, Math. USSR-Izv., 38:2 (1992), 251–275 | DOI | MR | Zbl

[17] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, viii+320 pp. | DOI | MR | Zbl

[18] J. Kollár, Severi–Brauer varieties; a geometric treatment, arXiv: 1606.04368

[19] Yu. I. Manin, M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, North-Holland Math. Library, 4, North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., New York, 1974, vii+292 pp. | MR | MR | Zbl | Zbl

[20] J.-P. Serre, “A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field”, Mosc. Math. J., 9:1 (2009), 183–198 | MR | Zbl

[21] C. Shramov, V. Vologodsky, Automorphisms of pointless surfaces, arXiv: 1807.06477

[22] A. Trepalin, “Quotients of cubic surfaces”, Eur. J. Math., 2:1 (2016), 333–359 | DOI | MR | Zbl

[23] F. W. Weinstein, On birational automorphisms of Severi–Brauer surfaces, Prepr. Rep. Math. Univ. Stockholm No. 1, 1989, 14 pp.

[24] F. W. Weinstein, On birational automorphisms of Severi–Brauer surfaces, arXiv: 1907.08115

[25] E. Yasinsky, “The Jordan constant for Cremona group of rank 2”, Bull. Korean Math. Soc., 54:5 (2017), 1859–1871 | DOI | MR | Zbl