Unirationality and Existence of Infinitely Transitive Models

  • Fedor Bogomolov
  • Ilya Karzhemanov
  • Karine Kuyumzhiyan
Chapter

Abstract

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.

Keywords

Manifold Stratification Neron 

Notes

Acknowledgements

The first author was supported by NSF grant DMS-1001662 and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023. The third author was supported by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023, RFBR grants 12-01-31342 and 12-01-33101, by the “EADS Foundation Chair in Mathematics”, Russian-French Poncelet Laboratory (UMI 2615 of CNRS MK-983.2013.1), and Dmitry Zimin fund “Dynasty.” The first author wants to thank Sh. Kaliman and M. Zaidenberg for useful discussions. The second author would like to thank Courant Institute for hospitality. The second author has also benefited from discussions with I. Cheltsov, Yu. Prokhorov, V. Shokurov, and K. Shramov. The third author would like to thank I. Arzhantsev for fruitful discussions. The authors are grateful to the referee for valuable comments. Finally, the authors thank the organizers of the summer school and the conference in Yekaterinburg (2011), where the work on the article originated.

References

  1. 1.
    I. V. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch,   and  M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162, No 4 (2013), 60p. (to appear), arXiv:1011.5375, (2010), 41p.Google Scholar
  2. 2.
    I. V. Arzhantsev, K. Kuyumzhiyan,  and  M. Zaidenberg, Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity, Sbornik:Mathematics, russian version: 203 7, 3–30, (2012), arXiv:1003.3164.Google Scholar
  3. 3.
    C. H. Clemens  and  P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95, 281–356, (1972).Google Scholar
  4. 4.
    G. Freudenburg, Algebraic theory of locally nilpotent derivations, Encyclopaedia of Mathematical Sciences, 136, Springer, Berlin (2006).Google Scholar
  5. 5.
    M. H. Gizatullin  and  V. I. Danilov, Examples of nonhomogeneous quasihomogeneous surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 38, 42–58, (1974).Google Scholar
  6. 6.
    J. Huisman  and  F. Mangolte, The group of automorphisms of a real rational surface is n-transitive, Bull. Lond. Math. Soc. 41, no. 3, 563–568, (2009).Google Scholar
  7. 7.
    V. A. Iskovskikh  and  Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, 1–247, Encyclopaedia Math. Sci., 47 Springer, Berlin (2000).Google Scholar
  8. 8.
    Sh. Kaliman and M. Zaidenberg, Affine modifications and affine hypersurfaces with a very transitive automorphism group, Transform. Groups 4, no. 1, 53–95, (1999).Google Scholar
  9. 9.
    T. Kishimoto, Y. Prokhorov,  and  M. Zaidenberg, Group actions on affine cones, in Affine algebraic geometry, 123–163, CRM Proc. Lecture Notes, 54 Amer. Math. Soc., Providence, RI (2011).Google Scholar
  10. 10.
    J. Kollár, Nonrational hypersurfaces, J. Amer. Math. Soc. 8, no. 1, 241–249, (1995).Google Scholar
  11. 11.
    J. Kollár, Y. Miyaoka,  and  S. Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36, no. 3, 765–779, (1992).Google Scholar
  12. 12.
    K. Kuyumzhiyan  and  F. Mangolte, Infinitely transitive actions on real affine suspensions, Journal of Pure and Applied Algebra 216, 2106–2112, (2012).Google Scholar
  13. 13.
    K. Paranjape, V. Srinivas, Unirationality of the general Complete Intersection of sufficiently small multidegree, in Flips and abundance for algebraic threefolds, (ed. J. Kollár), Proc. of Summer Seminar at University of Utah, Salt Lake City, U.S.A. 1991, Astérisque 211 (1992), 241–248.Google Scholar
  14. 14.
    A. Perepechko, Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5, arXiv:1108.5841 (2011), to appear in: Func. Analysis and Its Appl.Google Scholar
  15. 15.
    V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, CRM Proceedings and Lecture Notes 54 , 289–311, (2011), arXiv:1001.1311.Google Scholar
  16. 16.
    A. N. Tjurin, The intersection of quadrics, Uspehi Mat. Nauk 30, no. 6 (186), 51–99, (1975).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Fedor Bogomolov
    • 1
    • 2
  • Ilya Karzhemanov
    • 1
  • Karine Kuyumzhiyan
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.National Research University Higher School of EconomicsMoscowRussia

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