Transformation Groups

, Volume 19, Issue 2, pp 549–568 | Cite as


  • Vladimir L. Popov


We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given.


Algebraic Group Natural Action Maximal Torus Dense Open Subset Connected Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ale 23]
    J. W. Alexander, On the deformation of an n-cell, Proc. Nat. Acad. Sci. USA 9 (1923), 406-407.CrossRefGoogle Scholar
  2. [AFKKZ 13]
    I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767-823.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [Bar 55]
    I. Barsotti, Structure theorems for group-varieties, Ann. Mat. Pura Appl. (4)38 (1955), no. 4, 77-119.Google Scholar
  4. [Ber 14]
    Y. Berest, Letter to V. L. Popov, February 18, 2014.Google Scholar
  5. [BEE 14]
    Y. Berest, A. Eshmatov, F. Eshmatov, Dixmier groups and Borel subgroups, arXiv:1401.7356 (2014).Google Scholar
  6. [Bor 91]
    A. Borel, Linear Algebraic Groups, 2nd Enlarged Edition, Springer, New York, 1991.Google Scholar
  7. [Bou 59]
    N. Bourbaki, Algèbre, Chapitre V, Hermann, Paris, 1959. Russian transl.: Н. Бypбаĸи, Aлгeбpa. Mнoгoчлены u noля. Уnopя∂oчeнныe гpynnы, Нayка M., 1965.Google Scholar
  8. [Jon 96]
    A. J. de Jong, Smoothness, semi-stability and alterations, Publ. math. l’IH ÉS 83 (1996), 51-93.CrossRefzbMATHGoogle Scholar
  9. [Èl’-H 74]
    M. X. Эль-Хyти, Кyбuчecкue noвepxнocmu мapкoвcкoгo muna, Мaт. cб. 93(135) (1974), no. 3, 331-346. Engl. transl.: M. H. Èl’-Huti, Cubic surfaces of Markov type, Math. USSR-Sbornik 22 (1974), no. 3, 333-348.Google Scholar
  10. [Gro 65]
    A. Grothendieck, EGA IV, 2, Publ. Math. I.H. É.S. 24 (1965), 5-231.Google Scholar
  11. [Jel 94]
    Z. Jelonek, Affine smooth varieties with finite group automorphisms, Math. Z. 216 (1994), 575-591.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [KZ 99]
    S. Kaliman, M. Zaidenberg, Affine modification and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4 (1999), 53-95.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [LBP 87]
    L. Le Bruyn, C. Procesi, Étale local structure of matrix invariants and concomitants, in: Algebraic Groups (Utrecht 1986), Lecture Notes in Mathematics, Vol. 1271, Springer-Verlag, Berlin, 1986, pp. 143-175.Google Scholar
  14. [Lun 73]
    D. Luna, Slices étales, Bull. Soc. Math. France 33 (1973), 81-105.zbMATHGoogle Scholar
  15. [Mat 63]
    H. Matsumura, On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 34 (1963), no. 8, 151-155.zbMATHMathSciNetGoogle Scholar
  16. [MM 64]
    H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964), no. 3, 347-361.Google Scholar
  17. [MO 67]
    H. Matsumura, F. Oort, Representability of group functors, and automorphisms of algebraic schemes, Invent. math. 4 (1967), 1-25.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [Pop 87]
    V. L. Popov, On actions of G a on A n, in: Algebraic Groups (Utrecht 1986), Lecture Notes in Mathematics, Vol. 1271, Springer-Verlag, Berlin, 1987, pp. 237-242.Google Scholar
  19. [Pop 05]
    V. L. Popov, Open problems, in: Affine Algebraic Geometry, Contemporary Math. 369, Amer. Math. Soc., Providence, RI, 2005, pp. 12-16.Google Scholar
  20. [Pop 07]
    V. L. Popov, Generically multiple transitive algebraic group actions, in: Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces (Mumbai, 2004), Tata Institute of Fundamental Research, Vol. 19, Narosa, Internat. distrib. by Amer. Math. Soc., New Delhi, 2007, 481-523.Google Scholar
  21. [Pop 11]
    V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in: Affine Algebraic Geometry: the Russell Festschrift, CRM Proceedings and Lecture Notes 54, Amer. Math. Soc., Providence, RI, 2011, pp. 289-311.Google Scholar
  22. [Pop 13]
    V. L. Popov, Rationality and the FML invariant, J. Ramanujan Math. Soc. 28A (2013), special issue, 409-415.Google Scholar
  23. [PV 94]
    Э. Б. Винбepг, В. Л. Пoпoв, Teopuя uнвapuaнмoв, Итoги нayки и тexники, Сoвpeмeнныe пpoблeмы мaтeмeмaтики, Фyндaмeнтaльныe нaпpaвлeния, т. 55, Aлгeбpauчecкaя гeoмeмpuя–4, BИHИTИ, M., 1989, cтp. 137-314. Engl. transl.: V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry, IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123-284.Google Scholar
  24. [Ram 64]
    C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Annalen 156 (1964), 25-33.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [Rei 93]
    Z. Reichstein, On automorphisms of matrix invariants, Trans. Amer. Math. Soc. 340 (1993), no. 1, 353-371.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [Ros 56]
    M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401-443.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [Ros 61]
    M. Rosenlicht, Toroidal algebraic groups, Proc. Amer. Math. Soc. 112 (1961), no. 6, 984-988.CrossRefMathSciNetGoogle Scholar
  28. [Ser 10]
    J.-P. Serre, Le groupe de Cremona et ses sous-groupes finis, in: Séminaire N. Bourbaki 2008/09, Exp. no. 1000, Astérisque, Vol. 332, Société Mathématique de France, 2010, pp. 75-100.Google Scholar
  29. [Sha 66]
    I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5)25 (1966), no. 1-2, 208-212.Google Scholar
  30. [Sha 82]
    И. P. Шaфapeвич, O нeкоmopыx бecкoнeчнoмepныx гpynnax, II, Изв. AH CCCP. Cepия мaтeмaтичecкaя 45 (1981), no. 1, 214-226. Engl. transl.: I. R. Shafarevich, On some infinite-dimensional groups II, Math. USSR Izv. 18 (1982), no. 1, 185-194.Google Scholar
  31. [Wil 98]
    G. Wilson, Collisions of Calogero-Moser particles and adelic Grassmannian (with an Appendix by I. G. MacDonald), Invent. math. 133 (1998), 1-41.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.National Research University, Higher School of EconomicsMoscowRussia

Personalised recommendations