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Notes on cylinders in smooth projective surfaces

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Abstract

A Zariski open subset of an algebraic variety is called a cylinder if it is isomorphic to the direct product of the affine line and an algebraic variety. We consider the existing condition of relative cylinders with respect to a projective dominant morphism of relative dimension two. Since this consideration is essentially a determination of the existence of cylinders in the generic fiber, we study smooth projective surfaces defined over a perfect field from the point of view of cylinders. In previous work, the existing condition of cylinders in smooth minimal del Pezzo surfaces over a field of characteristic zero is known. In this article, we completely determine the existing condition of cylinders in smooth minimal geometrically rational surfaces over a perfect field. Furthermore, we show that for any birational map between smooth projective surfaces over a perfect field, one contains a cylinder if and only if so does the other.

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References

  1. Cheltsov, I., Park, J., Prokhorov, Y., Zaidenberg, M.: Cylinders in Fano varieties. EMS Surv. Math. Sci. 8, 39–105 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheltsov, I., Park, J., Won, J.: Affine cones over smooth cubic surfaces. J. Eur. Math. Soc. 18, 1537–1564 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheltsov, I., Park, J., Won, J.: Cylinders in singular del Pezzo surfaces. Compos. Math. 152, 1198–1224 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coray, D.F., Tsfasman, M.A.: Arithmetic on singular Del Pezzo surfaces. Proc. Lond. Math. Soc. 3(57), 25–87 (1988)

  5. Corti, A.: Singularities of linear systems and 3-fold birational geometry, Explic. Birational Geom. 3-folds. 281, 259–312 (2000)

  6. Dubouloz, A., Kishimoto, T.: Cylinders in del Pezzo fibrations. Israel J. Math. 225, 797–815 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iskovskikh, V.A.: Rational surfaces with a pencil of rational curves. USSR-Sb. 3, 563–587 (1967)

    Article  MATH  Google Scholar 

  8. Iskovskikh, V.A.: Minimal models of rational surfaces over arbitrary fields. Math. USSR-Izv. 14, 17–39 (1980)

    Article  MATH  Google Scholar 

  9. Iskovskikh, V.A.: Factorization of birational maps of rational surfaces from the viewpoint of Mori theory. Russ. Math. Surv. 51, 585–652 (1996)

    Article  MATH  Google Scholar 

  10. Kambayashi, T., Miyanishi, M.: On flat fibrations by the affine line. Ill. J. Math. 22, 662–671 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: Group actions on affine cones. Affine Algebraic Geom. 54, 123–163 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kishimoto, T., Prokhorov, Y., Zaidenberg, M.: Unipotent group actions on del Pezzo cones. Algebr. Geom. 1, 46–56 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kollár, J., Smith, K.E., Corti, A.: Rational and nearly rational varieties, vol. 92. Cambridge Univ. Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  14. Perepechko, A.Y.: Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5. Funct. Anal. Appl. 47, 284–289 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Poonen, B.: Rational points on varieties, Grad. Stud. Math. 186 (2017)

  16. Sawahara, M.: Cylinders in weak del Pezzo fibrations, Transform. Groups (2022). https://doi.org/10.1007/s00031-022-09730-y

  17. Sawahara, M.: Cylinders in canonical del Pezzo fibrations, Ann. Inst. Fourier (to appear). arXiv:2012.10062v2

  18. Swinnerton-Dyer, H.P.F.: Rational points on del Pezzo surfaces of degree 5, In: Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), 287–290 (1972)

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Acknowledgements

The author would like to express his sincere appreciation to the referee for the careful reading and the invaluable comments.

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  1. Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku Saitama-shi, Saitama, 338-8570, Japan

    Masatomo Sawahara

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  1. Masatomo Sawahara
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Correspondence to Masatomo Sawahara.

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Throughout this article, let \(\Bbbk \) be a perfect field of an arbitrary characteristic and let \(\overline{\Bbbk }\) be an algebraic closure of \(\Bbbk \).

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Sawahara, M. Notes on cylinders in smooth projective surfaces. Geom Dedicata 217, 6 (2023). https://doi.org/10.1007/s10711-022-00741-3

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  • DOI: https://doi.org/10.1007/s10711-022-00741-3

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