Cylinders in rational surfaces

Research Article
  • 39 Downloads

Abstract

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.

Keywords

Polarized cylinder Rational surface del Pezzo 

Mathematics Subject Classification

14J26 

References

  1. 1.
    Artin, M.: On isolated rational singularities of surfaces. Amer. J. Math. 88(1), 129–136 (1966)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cheltsov, I., Park, J., Won, J.: Cylinders in singular del Pezzo surfaces. Compositio Math. 152(6), 1198–1224 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cheltsov, I., Park, J., Won, J.: Cylinders in del Pezzo surfaces. Int. Math. Res. Not. IMRN 2017(4), 1179–1230 (2017)MathSciNetMATHGoogle Scholar
  8. 8.
    Flenner, H., Zaidenberg, M.: Rational curves and rational singularities. Math. Z. 244(3), 549–575 (2003)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  14. 14.
    Kishimoto, T., Prokhorov, Yu., Zaidenberg, M.: Unipotent group actions on del Pezzo cones. Algebraic Geom. 1(1), 46–56 (2014)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Reid, M.: Surfaces of small degree. Math. Ann. 275(1), 71–80 (1986)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sakai, F.: On polarized normal surfaces. Manuscripta Math. 59(1), 109–127 (1987)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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

Personalised recommendations