Variations on the Theme of Zariski’s Cancellation Problem

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 319)


This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.


Zariski Cancellation Problem Flattenable variety Algebraic group Action 



I am grateful to the referee for thoughtful reading and suggestions.


  1. 1.
    Akbulut, S.: Lectures on algebraic spaces. In: 1992 Proceedings of KAIST Mathematics Workshop on Algebra and Topology, pp.1–15. KAIST, Daejeon, Republic of Korea (1993)Google Scholar
  2. 2.
    Białynicki-Birula, A.: Remarks on the action of an algebraic torus on \(k^n\). Bull. Acad. Polon. Sci, Sér. sci. math., astr., phys. XIV(4), 177–181 (1966)Google Scholar
  3. 3.
    Bodnár, G., Hauser, H., Schicho, J., Villamayor U.O.: Plain varieties. Bull. Lond. Math. Soc. 40(6), 965–971 (2008)Google Scholar
  4. 4.
    Bogomolov, F., Böhning, C.: On uniformly rational varieties. In: Topology, Geometry, Integrable Systems, and Mathematical Physics. American Mathematical Society Translations: Series 2, vol. 234, pp. 33–48. Advances in the Mathematical Sciences 67. American Mathematical Society, Providence, RI (2014)Google Scholar
  5. 5.
    Borel, A.: Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes. Tôhoku Math. J. 13, 216–240 (1961)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borel, A.: On affine algebraic homogeneous spaces. Arch. Math. 45, 74–78 (1985)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borel, A.: Linear Algebraic Groups. Graduate Text in Mathematics, vol. 126, 2nd edn. Springer, New York (1991)CrossRefGoogle Scholar
  8. 8.
    Bourbaki, N.: Groupes et Algèbres de Lie. Chapter IV, V, VI. Hermann (1968)Google Scholar
  9. 9.
    Burde, D., Globke, W., Minchenko, A.: Étale representations for reductive algebraic groups with factors \({\rm Sp}_n\) or \({\rm SO}_n\). Transform. Groups 24, 769–780 (2019).
  10. 10.
    Chevalley, C.: On algebraic group varieties. J. Math. Soc. Japan 6(3–4), 303–324 (1954)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chevalley, C.: Les classes d’équivalence rationnelle, I. In: Anneaux de Chow et Applications. Séminaire Claude Chevalley, Exp. no. 2, pp. 1–14 (1958)Google Scholar
  12. 12.
    Chin, C., Zhang, D.-Q.: Rationality of homogeneous varieties. Trans. Am. Math. Soc. 369, 2651–2673 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cohen, A.M., Seitz, G.M.: The \(r\)-rank of the groups of exceptional Lie type. Indag. Math. 49, 251–259 (1987)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851–897 (1989)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Grothendieck, A.: Torsion homologique et sections rationnelle. In: Anneaux de Chow et Applications, Séminaire Claude Chevalley, vol. 3, Exp. no. 5, pp. 1–29. Secrétariat math., Paris (1958)Google Scholar
  16. 16.
    Gupta, N.: A survey on Zariski cancellation problem. Indian J. Pure Appl. Math. 46(6), 865–877 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hu, S.-T.: Homotopy Theory. Academic Press, New York (1959)zbMATHGoogle Scholar
  18. 18.
    Kraft, H.: Challenging problems in affine \(n\)-space. Astérisque 47(802), 295–317 (1996)zbMATHGoogle Scholar
  19. 19.
    Luna, D.: Slices étales. Mém. SMF 33, 81–105 (1973)zbMATHGoogle Scholar
  20. 20.
    Manin, Y.: Cubic Forms. North-Holland, Amsterdam (1974)zbMATHGoogle Scholar
  21. 21.
    Petitjean, C.: Equivariantly uniformly rational varieties. Michugan Math. J. 66(2), 245–268 (2017)Google Scholar
  22. 22.
    Popov, V.L.: Sections in invariant theory. In: Proceedings of the Sophus Lie Memorial Conference, Oslo 1992, pp. 315–362. Scandinavian University Press, Oslo (1994)Google Scholar
  23. 23.
    Popov, V.L.: 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, vol. 54, pp. 289–311. American Mathematical Society (2011)Google Scholar
  24. 24.
    Popov, V.L.: Some subgroups of the Cremona groups. In: Affine Algebraic Geometry, Proceedings (Osaka, Japan, 3–6 March 2011), pp. 213–242. World Scientific, Singapore (2013)Google Scholar
  25. 25.
    Popov, V.L.: Rationality and the FML invariant. J. Ramanujan Math. Soc. 28A, 409–415 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic Geometry, IV, Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123–284. Springer, Berlin (1994)Google Scholar
  27. 27.
    Quillen, D.: Projective modules over polynomial rings. Invent. Math. 36(1), 167–171 (1976)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rosenlicht, M.: Toroidal algebraic groups. Proc. Am. Math. Soc. 12, 984–988 (1961)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Serre, J.-P.: Espaces fibrés algébriques. In: Anneaux de Chow et Applications, Séminaire Claude Chevalley, vol. 3, Exp. no. 1, pp. 1–37. Secrétariat mathématique, Paris (1958)Google Scholar
  30. 30.
    Serre, J.-P.: Sous-groupes finis des groupes de Lie. Séminaire Bourbaki, Exp. no. 864 (1998–99). In: Serre, J.-P. (ed.) Exposés de Séminaires 1950–1999, pp. 233–248. Documents Mathématiques, Soc. Math. de France, Paris (2001)Google Scholar
  31. 31.
    Suslin, A.A.: Projective modules over polynomial rings are free. Soviet Math. 17(4), 1160–1164 (1976)zbMATHGoogle Scholar
  32. 32.
    Timashev, D.A.: Homogeneous Spaces and Equivariant Embeddings. Encyclopaedia of Mathematikcal Sciences, vol. 138. Subseries Invariant Theory and Algebraic Transformation Groups, vol. VIII. Springer, Berlin (2011)Google Scholar

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Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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