European Journal of Mathematics

, Volume 4, Issue 3, pp 1161–1196 | Cite as

Cylinders in rational surfaces

  • Lisa Marquand
  • Joonyeong Won
Research Article


Let S be a smooth rational surface with \(K^2_S\geqslant 3\). We show that there exist A-polar cylinders for a polarized pair (SA) except when S is a smooth cubic surface and A is an anticanonical divisor.


Polarized cylinder Rational surface del Pezzo 

Mathematics Subject Classification



  1. 1.
    Artin, M.: On isolated rational singularities of surfaces. Amer. J. Math. 88(1), 129–136 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bass, H.: A nontriangular action of \(\mathbb{G}_a\) on \(\mathbb{A}^3\). J. Pure Appl. Algebra 33(1), 1–5 (1984)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brieskorn, E.: Rationale Singularitäten komplexer Flächen. Invent. Math. 4, 336–358 (1967/1968)Google Scholar
  4. 4.
    Cheltsov, I.: Cylinders in rational surfaces (2016). arXiv:1611.05514
  5. 5.
    Cheltsov, I., Park, J., Won, J.: Affine cones over smooth cubic surfaces. J. Eur. Math. Soc. (JEMS) 18(7), 1537–1564 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheltsov, I., Park, J., Won, J.: Cylinders in singular del Pezzo surfaces. Compositio Math. 152(6), 1198–1224 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheltsov, I., Park, J., Won, J.: Cylinders in del Pezzo surfaces. Int. Math. Res. Not. IMRN 2017(4), 1179–1230 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Flenner, H., Zaidenberg, M.: Rational curves and rational singularities. Math. Z. 244(3), 549–575 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Halphen, G.-H.: Sur la réduction des équations différentielles linéaires aux formes intégrales. In: Oeuvres de G.-H. Halphen, vol. III. Gauthier-Villars, Paris (1921)Google Scholar
  10. 10.
    Hidaka, F., Watanabe, K.: Normal Gorenstein surfaces with ample anti-canonical divisor. Tokyo J. Math. 4(2), 319–330 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kishimoto, T., Prokhorov, Yu., Zaidenberg, M.: Group actions on affine cones. In: Daigle, D., Ganong, R., Koras, M. (eds.) Affine Algebraic Geometry CRM Proceedings and Lecture Notes, vol. 54, pp. 123–163. American Mathematical Society, Providence, (2011)Google Scholar
  12. 12.
    Kishimoto, T., Prokhorov, Yu., Zaidenberg, M.: \(\mathbb{G}_a\)-actions on affine cones. Transform. Groups 18(4), 1137–1153 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kishimoto, T., Prokhorov, Yu., Zaidenberg, M.: Affine cones over Fano threefolds and additive group actions. Osaka J. Math. 51(4), 1093–1112 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kishimoto, T., Prokhorov, Yu., Zaidenberg, M.: Unipotent group actions on del Pezzo cones. Algebraic Geom. 1(1), 46–56 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaliman, S.: Free \(\mathbf{C}_{+}\)-actions on \(\mathbf{C}^3\) are translations. Invent. Math. 156(1), 163–173 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Laufer, H.B.: On rational singularities. Amer. J. Math. 94(2), 597–608 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Masuda, K., Miyanishi, M.: The additive group actions on \(\mathbb{Q}\)-homology planes. Ann. Inst. Fourier (Grenoble) 53(2), 429–464 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Perepechko, A.Yu.: Flexibility of affine cones over del Pezzo surfaces of degree \(4\) and \(5\). Funct. Anal. Appl. 47(4), 284–289 (2013)Google Scholar
  19. 19.
    Park, J., Won, J.: Flexible affine cones over del Pezzo surfaces of degree 4. Eur. J. Math. 2(1), 304–318 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Reid, M.: Surfaces of small degree. Math. Ann. 275(1), 71–80 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sakai, F.: On polarized normal surfaces. Manuscripta Math. 59(1), 109–127 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schwartz, H.A.: Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische function ihres vierten Elementes darstellt. J. Reine Angew. Math. 75, 292–335 (1873)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Snow, D.M.: Unipotent actions on affine space. In: Kraft, H. (ed.) Topological Methods in Algebraic Transformation Groups. Progress in Mathematics, vol. 80, pp. 165–176. Birkhäuser, Boston (1989)CrossRefGoogle Scholar
  24. 24.
    Winkelmann, J.: On free holomorphic \(\mathbf{C}\)-actions on \(\mathbf{C}^n\) and homogeneous Stein manifolds. Math. Ann. 286(1–3), 593–612 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Center for Geometry and PhysicsInstitute for Basic Science (IBS)PohangRepublic of Korea

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