Kyoto Journal of Mathematics

Affine surfaces with isomorphic A2-cylinders

Adrien Dubouloz

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We show that all complements of cuspidal hyperplane sections of smooth projective cubic surfaces have isomorphic A2-cylinders. As a consequence, we derive that the A2-cancellation problem fails in every dimension greater than or equal to 2.

Article information

Kyoto J. Math., Volume 59, Number 1 (2019), 181-193.

Received: 5 December 2016
Revised: 30 January 2017
Accepted: 1 February 2017
First available in Project Euclid: 21 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R05: Classification of affine varieties
Secondary: 14R25: Affine fibrations [See also 14D06] 14E07: Birational automorphisms, Cremona group and generalizations

affine surfaces cylinders cancellation problem


Dubouloz, Adrien. Affine surfaces with isomorphic $\mathbb{A}^{2}$ -cylinders. Kyoto J. Math. 59 (2019), no. 1, 181--193. doi:10.1215/21562261-2018-0005.

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  • [1] S. S. Abhyankar, W. Heinzer, and P. Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310–342.
  • [2] W. Danielewski, On the cancellation problem and automorphism groups of affine algebraic varieties, preprint, 1989.
  • [3] R. Dryło, Non-uniruledness and the cancellation problem, II, Ann. Polon. Math. 92 (2007), 41–48.
  • [4] A. Dubouloz, Additive group actions on Danielewski varieties and the cancellation problem, Math. Z. 255 (2007), 77–93.
  • [5] A. Dubouloz, Flexible bundles over rigid affine surfaces, Comment. Math. Helv. 90 (2015), 121–137.
  • [6] A. Dubouloz and T. Kishimoto, Log-uniruled affine varieties without cylinder-like open subsets, Bull. Soc. Math. France 143 (2015), 383–401.
  • [7] K.-H. Fieseler, On complex affine surfaces with $\mathbb{C}^{+}$-action, Comment. Math. Helv. 69 (1994), 5–27.
  • [8] G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sci. 136, Springer, Berlin, 2006.
  • [9] T. Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 106–110.
  • [10] J. Giraud, Cohomologie non abélienne, Grundlehren Math. Wiss. 179, Springer, New York, 1971.
  • [11] A. Grothendieck, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III, Inst. Hautes Études Sci. Publ. Math. 28, Presses Univ. France, Paris, 1966.
  • [12] A. Grothendieck, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie 1960-61 (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
  • [13] N. Gupta, On the cancellation problem for the affine space $\mathbb{A}^{3}$ in characteristic $p$, Invent. Math. 195 (2014), 279–288.
  • [14] M. Hochster, Nonuniqueness of coefficient rings in a polynomial ring, Proc. Amer. Math. Soc. 34 (1972), 81–82.
  • [15] S. Iitaka and T. Fujita, Cancellation theorem for algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 123–127.
  • [16] Z. Jelonek, On the cancellation problem, Math. Ann. 344 (2009), 769–778.
  • [17] D. Knutson, Algebraic Spaces, Lecture Notes in Math. 203, Springer, Berlin, 1971.
  • [18] L. Makar-Limanov, Locally nilpotent derivations, a new ring invariant and applications, lecture notes, Bar-Ilan University, Ramat Gan, 1998, (accessed 31 July 2018).
  • [19] Y. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland, Amsterdam, 1974.
  • [20] M. Miyanishi and T. Sugie, Affine surfaces containing cylinder-like open sets, J. Math. Kyoto Univ. 20 (1980), 11–42.
  • [21] M. P. Murthy, Vector bundles over affine surfaces birationally equivalent to a ruled surface, Ann. of Math. (2) 89 (1969), 242–253.
  • [22] P. Russell, “Cancellation,” in Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat. 79, Springer, Cham, 2014, 495–518.
  • [23] B. Segre, The Non-Singular Cubic Surfaces, Oxford Univ. Press, Oxford, 1942.