Mathematische Annalen

, Volume 373, Issue 3–4, pp 1135–1149 | Cite as

Deformations of \(\mathbb {A}^1\)-cylindrical varieties

  • Adrien DuboulozEmail author
  • Takashi Kishimoto


An algebraic variety is called \(\mathbb {A}^{1}\)-cylindrical if it contains an \(\mathbb {A}^{1}\)-cylinder, i.e. a Zariski open subset of the form \(Z\times \mathbb {A}^{1}\) for some algebraic variety Z. We show that the generic fiber of a family \(f:X\rightarrow S\) of normal \(\mathbb {A}^{1}\)-cylindrical varieties becomes \(\mathbb {A}^{1}\)-cylindrical after a finite extension of the base. This generalizes the main result of Dubouloz and Kishimoto (Nagoya Math J 223:1–20, 2016) which established this property for families of smooth \(\mathbb {A}^{1}\)-cylindrical affine surfaces. Our second result is a criterion for existence of an \(\mathbb {A}^{1}\)-cylinder in X which we derive from a careful inspection of a relative Minimal Model Program run from a suitable smooth relative projective model of X over S.

Mathematics Subject Classification

14R25 14D06 14M20 14E30 


  1. 1.
    Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc 23, 405–468 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cheltsov, I., Park, J., Won, J.: Affine cones over smooth cubic surfaces. J. Eur. Math. Soc. 18, 1537–1564 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cheltsov, I., Park, J., Won, J.: Cylinders in singular del Pezzo surfaces. Compos. Math. 152, 1198–1224 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheltsov, I., Park, J., Won, J.: Cylinders in del Pezzo surfaces. Int. Math. Res. Not. 2017(4, 1), 1179–1230 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dubouloz, A., Kishimoto, T.: Log-uniruled affine varieties without cylinder-like open subsets. Bull. de la SMF 143, 383–401 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dubouloz, A., Kishimoto, T.: Families of affine ruled surfaces: existence of cylinders. Nagoya Math. J. 223, 1–20 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dubouloz, A., Kishimoto, T.: Explicit biregular/birational geometry of affine threefolds: completions of ${\mathbb{A}}^3$ into del Pezzo fibrations and Mori conic bundles, Adv. Stud. Pure Math. 75, Mathematical Society of Japan, Tokyo, 2017, pp. 49–71Google Scholar
  8. 8.
    Dubouloz, A., Kishimoto, T.: Cylinders in del Pezzo fibrations. Israël J. Math. 225(2), 797–815 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grothendieck, A.: Éléments de Géométrie Algébrique, II, Publ. Math. IHES, 8, (1961)Google Scholar
  10. 10.
    Gurjar, R., Masuda, K., Miyanishi, M.: Deformations of $\mathbb{A}^1$-fibrations. In: Automorphisms in Birational and Affine Geometry Springer, Proceedings in Mathematics and Statistics (Vol. 79, pp. 327–361) (2014)Google Scholar
  11. 11.
    Hacon, C., McKernan, J.: On Shokurov’s rational connectedness conjecture. Duke Math. J., 138, 119–136 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math. 79 , I:109-203, II:205-326, (1964)Google Scholar
  13. 13.
    Iitaka, S., Fujita, T.: Cancellation theorem for algebraic varieties. J. Faculty Sci. Univ. Tokyo Sect. 1A Math. 24(1), 123–127 (1977)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kambayashi, T., Miyanishi, M.: On flat fibrations by the affine line. Illinois J. Math. 22(4), 662–671 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kawamata, Y.: Higher Dimensional Algebraic Varieties, Iwanami Studies in Advanced Mathematics, Iwanami Shoten, 2014. (Japanese)Google Scholar
  16. 16.
    Keel, S., McKernan, J.: Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999)Google Scholar
  17. 17.
    Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: $\mathbb{G}_a$-actions on affine cones. Transf. Groups 18, 1137–153 (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: Unipotent group actions on del Pezzo cones. Algebraic Geom. 1, 46–56 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: Affine cones over Fano threefolds and additive group actions. Osaka J. Math. 51, 1093–1112 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kollár, J.: Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32. Springer, Berlin (1996)Google Scholar
  21. 21.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Geometry, Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge (1998)Google Scholar
  22. 22.
    Miyanishi, M., Sugie, T.: Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ. 20(1), 11–42 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nagata, M.: Imbedding of an abstract variety in a complete variety. J. Math. Kyoto 2, 1–10 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Prokhorov, Y., Zaidenberg, M.: Examples of cylindrical Fano fourfolds. Eur. J. Math. 2, 262–282 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Prokhorov, Y., Zaidenberg, M.: New examples of cylindrical Fano fourfolds, Adv. Stud. Pure Math. 75, Mathematical Society of Japan, Tokyo, pp. 443–463 (2017)Google Scholar
  26. 26.
    Zhang, Q.: Rational connectedness of log $\mathbb{Q}$-Fano varieties. J. Reine Angew. Math. 590, 131–142 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMB UMR5584, CNRS, Univ. Bourgogne Franche-ComtéDijonFrance
  2. 2.Department of Mathematics, Faculty of ScienceSaitama UniversitySaitamaJapan

Personalised recommendations