Abstract
We consider threefold del Pezzo fibrations over a curve germ whose central fiber is nonrational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a one-to-one correspondence between such fibrations and certain nonsingular del Pezzo fibrations equipped with a cyclic group action.
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References
V.A. Iskovskikh and Yu. G. Prokhorov, Algebraic Geometry. V. Fano Varieties, in Encyclopaedia Math. Sci. (Springer-Verlag, Berlin, 1999), Vol. 47.
F. Hidaka and K. Watanabe, “Normal Gorenstein surfaces with ample anti-canonical divisor,” Tokyo J. Math. 04 (2), 319–330 (1981).
S. Mori and Yu. G. Prokhorov, “Multiple fibers of del Pezzo fibrations,” in Multidimensional Algebraic Geometry, Tr. Mat. Inst. Steklova (MAIK “Nauka/Interperiodika”, Moscow, 2009), Vol. 264, pp. 137–151 [Proc. Steklov Inst. Math. 264, 131–145 (2009)].
M. Kontsevich and Yu. Tschinkel, Specialization of Birational Types, arXiv: 1708.05699(2017).
B. Totaro, “Rationality does not specialise among terminal varieties,” Math. Proc. Cambridge Philos. Soc. 161 (1), 13–15 (2016).
A. Perry, “Rationality does not specialize among terminal fourfolds,” Algebra Number Theory 11 (9), 2193–2196 (2017).
T. Fujisawa, “On non-rational numerical del Pezzo surfaces,” Osaka J. Math. 32 (3), 613–636 (1995).
K. Matsuki, Introduction to the Mori Program (Springer, New York, 2002).
Y Kawamata, K. Matsuda and K. Matsuki, “Introduction to the minimal model problem,” in Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. (North-Holland, Amsterdam, 1987), Vol. 10, pp.283–360.
Y Kawamata, “Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces,” Ann. of Math. (2) 127 (1), 93–163 (1988).
M. Reid, “Nonnormal del Pezzo surfaces,” Publ. Res. Inst. Math. Sci. 30 (5), 695–727 (1994).
M. Abe and M. Furushima, “On non-normal del Pezzo surfaces,” Math. Nachr. 260, 3–13 (2003).
J. Kollár and Sh. Mori, Birational Geometry of Algebraic Varieties, in Cambridge Tracts in Math. (Cambridge Univ. Press, Cambridge, 1998), Vol. 134.
J. Kollár, Singularities of the Minimal Model Program, in Cambridge Tracts in Math. (Cambridge Univ. Press, Cambridge, 2013), Vol. 200.
I. V. Dolgachev and V. A. Iskovskikh, “Finite subgroups of the plane Cremona group,” in Algebra, Arithmetic, and Geometry, Progr. Math. (Birkhäuser Boston, Boston, MA, 2009), Vol. 269, pp. 443–548.
Vik. S. Kulikov, “Degenerations of K3 surfaces and Enriques surfaces,” Izv Akad. Nauk SSSR Ser. Mat. 41(5), 1008–1042 (1977)[Math. USSR-Izv. 11 (5), 957–989 (1977)].
Acknowledgments
The author is grateful to Yurii Prokhorov for numerous useful discussions, to Alexander Kuznetsov, Dmitrii Mineev, and Constantin Shramov for their valuable suggestions, to Jérémy Blanc for posing Question 2.6, and to the referee for comments on Theorem 4.3.
Funding
This work was supported in part supported by the Russian Academic Excellence Project “5-100”, by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”, and by the Simons Foundation.
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 6, pp. 881–893.
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Loginov, K.V. On Nonrational Fibers of del Pezzo Fibrations over Curves. Math Notes 106, 930–939 (2019). https://doi.org/10.1134/S0001434619110294
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DOI: https://doi.org/10.1134/S0001434619110294