## Kyoto Journal of Mathematics

### Affine surfaces with isomorphic $\mathbb{A}^{2}$-cylinders

#### Abstract

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

#### Article information

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

Dates
Revised: 30 January 2017
Accepted: 1 February 2017
First available in Project Euclid: 21 August 2018

https://projecteuclid.org/euclid.kjm/1534838488

Digital Object Identifier
doi:10.1215/21562261-2018-0005

Mathematical Reviews number (MathSciNet)
MR3934627

Zentralblatt MATH identifier
07081626

#### Citation

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. https://projecteuclid.org/euclid.kjm/1534838488

#### References

• [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, http://www.math.wayne.edu/~lml/ (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.