Abstract
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.
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References
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)
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)
Bodnár, G., Hauser, H., Schicho, J., Villamayor U.O.: Plain varieties. Bull. Lond. Math. Soc. 40(6), 965–971 (2008)
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)
Borel, A.: Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes. Tôhoku Math. J. 13, 216–240 (1961)
Borel, A.: On affine algebraic homogeneous spaces. Arch. Math. 45, 74–78 (1985)
Borel, A.: Linear Algebraic Groups. Graduate Text in Mathematics, vol. 126, 2nd edn. Springer, New York (1991)
Bourbaki, N.: Groupes et Algèbres de Lie. Chapter IV, V, VI. Hermann (1968)
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). https://doi.org/10.1007/s00031-018-9483-8
Chevalley, C.: On algebraic group varieties. J. Math. Soc. Japan 6(3–4), 303–324 (1954)
Chevalley, C.: Les classes d’équivalence rationnelle, I. In: Anneaux de Chow et Applications. Séminaire Claude Chevalley, Exp. no. 2, pp. 1–14 (1958)
Chin, C., Zhang, D.-Q.: Rationality of homogeneous varieties. Trans. Am. Math. Soc. 369, 2651–2673 (2017)
Cohen, A.M., Seitz, G.M.: The \(r\)-rank of the groups of exceptional Lie type. Indag. Math. 49, 251–259 (1987)
Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851–897 (1989)
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)
Gupta, N.: A survey on Zariski cancellation problem. Indian J. Pure Appl. Math. 46(6), 865–877 (2015)
Hu, S.-T.: Homotopy Theory. Academic Press, New York (1959)
Kraft, H.: Challenging problems in affine \(n\)-space. Astérisque 47(802), 295–317 (1996)
Luna, D.: Slices étales. Mém. SMF 33, 81–105 (1973)
Manin, Y.: Cubic Forms. North-Holland, Amsterdam (1974)
Petitjean, C.: Equivariantly uniformly rational varieties. Michugan Math. J. 66(2), 245–268 (2017)
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)
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)
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)
Popov, V.L.: Rationality and the FML invariant. J. Ramanujan Math. Soc. 28A, 409–415 (2013)
Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic Geometry, IV, Encyclopaedia of Mathematical Sciences, vol. 55, pp. 123–284. Springer, Berlin (1994)
Quillen, D.: Projective modules over polynomial rings. Invent. Math. 36(1), 167–171 (1976)
Rosenlicht, M.: Toroidal algebraic groups. Proc. Am. Math. Soc. 12, 984–988 (1961)
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)
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)
Suslin, A.A.: Projective modules over polynomial rings are free. Soviet Math. 17(4), 1160–1164 (1976)
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)
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Popov, V.L. (2020). Variations on the Theme of Zariski’s Cancellation Problem. In: Kuroda, S., Onoda, N., Freudenburg, G. (eds) Polynomial Rings and Affine Algebraic Geometry. PRAAG 2018. Springer Proceedings in Mathematics & Statistics, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-030-42136-6_11
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