Three plots about Cremona groups

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Abstract

The first group of results of the paper concerns the compressibility offinite subgroups of the Cremona groups. The second concerns the embeddability ofother groups in the Cremona groups and, conversely, of the Cremona groups in othergroups. The third concerns the connectedness of the Cremona groups.

About the authors

Vladimir Leonidovich Popov

Steklov Mathematical Institute of Russian Academy of Sciences

Email: popovvl@mi-ras.ru
Doctor of physico-mathematical sciences, Professor

References

  1. J.-P. Serre, “Le groupe de Cremona et ses sous-groupes finis”, Seminaire Bourbaki, Exposes 997–1011, v. 2008/2009, Asterisque, 332, Soc. Math. France, Paris, 2010, vii, 75–100, Exp. No. 1000
  2. J. Blanc, J.-P. Furter, “Topologies and structures of the Cremona groups”, Ann. of Math. (2), 178:3 (2013), 1173–1198
  3. V. L. Popov, “Some subgroups of the Cremona groups”, Affine algebraic geometry (Osaka, 2011), World Sci. Publ., Hackensack, NJ, 2013, 213–242
  4. В. Л. Попов, “Торы в группах Кремоны”, Изв. РАН. Сер. матем., 77:4 (2013), 103–134
  5. V. L. Popov, “On infinite dimensional algebraic transformation groups”, Transform. Groups, 19:2 (2014), 549–568
  6. В. Л. Попов, “Борелевские подгруппы групп Кремоны”, Матем. заметки, 102:1 (2017), 72–80
  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, Inc., Boston, MA, 2009, 443–548
  8. V. L. Popov, “Jordan groups and automorphism groups of algebraic varieties”, Automorphisms in birational and affine geometry (Levico Terme, 2012), Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 185–213
  9. V. L. Popov, Cremona conference–Basel 2016, Question session, 2016
  10. 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
  11. Yu. Prokhorov, C. Shramov, “Jordan property for Cremona groups”, Amer. J. Math., 138:2 (2016), 403–418
  12. C. Birkar, Birational geometry of algebraic varieties, 2018
  13. S. Cantat, “Morphisms between Cremona groups, and characterization of rational varieties”, Compositio Math., 150:7 (2014), 1107–1124
  14. J. Blanc, S. Zimmermann, “Topological simplicity of the Cremona groups”, Amer. J. Math., 140:5 (2018), 1297–1309
  15. J. Blanc, “Groupes de Cremona, connexite et simplicite”, Ann. Sci. Ec. Norm. Super. (4), 43:2 (2010), 357–364
  16. J. W. Alexander, “On the deformation of an $n$-cell”, Proc. Nat. Acad. Sci. USA, 9:12 (1923), 406–407
  17. И. Р. Шафаревич, “О некоторых бесконечномерных группах. II”, Изв. АН СССР. Сер. матем., 45:1 (1981), 214–226
  18. M. Rosenlicht, “Some basic theorems on algebraic groups”, Amer. J. Math., 78:2 (1956), 401–443
  19. E. Bierstone, P. D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math., 128:2 (1997), 207–302
  20. Э. Б. Винберг, В. Л. Попов, “Теория инвариантов”, Алгебраическая геометрия – 4, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 55, ВИНИТИ, М., 1989, 137–309
  21. Z. Reichstein, “On the notion of essential dimension for algebraic groups”, Transform. Groups, 5:3 (2000), 265–304
  22. Z. Reichstein, “Compressions of group actions”, Invariant theory in all characteristics, CRM Proc. Lecture Notes, 35, Amer. Math. Soc., Providence, RI, 2004, 199–202
  23. M. Garcia-Armas, “Strongly incompressible curves”, Canad. J. Math., 68:3 (2016), 541–570
  24. И. Р. Шафаревич, Основы алгебраической геометрии, 3-е изд., МЦНМО, М., 2007, 590 с.
  25. Б. Л. ван дер Варден, Алгебра, 2-е изд., Наука, М., 1979, 624 с.
  26. T. A. Springer, “Poincare series of binary polyhedral groups and McKay's correspondence”, Math. Ann., 278:1-4 (1987), 99–116
  27. N. Lemire, V. L. Popov, Z. Reichstein, “Cayley groups”, J. Amer. Math. Soc., 19:4 (2006), 921–967
  28. I. Dolgachev, A. Duncan, “Fixed points of a finite subgroup of the plane Cremona group”, Algebr. Geom., 3:4 (2016), 441–460
  29. K. A. Nguyen, M. van der Put, J. Top, “Algebraic subgroups of $operatorname{GL}_2(mathbb C)$”, Indag. Math. (N.S.), 19:2 (2008), 287–297
  30. J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, Ph.D. thèse No. 3777, Univ. Genève, Genève, 2006, 186 pp.
  31. С. Ленг, Алгебра, Мир, М., 1968, 564 с.
  32. J. Buhler, Z. Reichstein, “On the essential dimension of a finite group”, Compositio Math., 106:2 (1997), 159–179
  33. Z. Reichstein, B. Youssin, “Essential dimensions of algebraic groups and a resolution theorem for $G$-varieties”, Canad. J. Math., 52:5 (2000), 1018–1056
  34. Э. Б. Винберг, Линейные представления групп, Наука, М., 1985, 144 с.
  35. V. L. Popov, “On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties”, Affine algebraic geometry, The Russell festschrift, CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 289–311
  36. Yu. Prokhorov, C. Shramov, “Jordan constant for Cremona group of rank $3$”, Mosc. Math. J., 17:3 (2017), 457–509
  37. Ch. Urech, Letter of October 11, 2018 to V. L. Popov, 2018
  38. J. Blanc, S. Lamy, S. Zimmermann, Quotients of higher dimensional Cremona groups, 2019, 84 pp.
  39. J. Deserti, “Le groupe de Cremona est hopfien”, C. R. Math. Acad. Sci. Paris, 344:3 (2007), 153–156
  40. L. N. Mann, J. C. Su, “Actions of elementary $p$-groups on manifolds”, Trans. Amer. Math. Soc., 106 (1963), 115–126
  41. A. Borisov, Letters of May $1$ and $2$, 2017 to V. L. Popov, 2017

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