Advertisement

JORDAN PROPERTIES OF AUTOMORPHISM GROUPS OF CERTAIN OPEN ALGEBRAIC VARIETIES

  • TATIANA BANDMAN
  • YURI G. ZARHIN
Article

Abstract

Let W be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that W is birational to a product of a smooth projective variety A and the projective line. We prove that if A contains no rational curves then the automorphism group G := Aut (W) of W is Jordan. That means that there is a positive integer J = J (W) such that every finite subgroup ℬ of G contains a commutative subgroup \( \mathcal{A} \) such that \( \mathcal{A} \) is normal in ℬ and the index \( \left[\mathrm{\mathcal{B}}:\mathcal{A}\right]\le J \).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W. Barth, K. Hulek, C. Peters, A. van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin, 2004.CrossRefMATHGoogle Scholar
  2. 2.
    T. Bandman, Yu. G. Zarhin, Jordan groups and algebraic surfaces, Transform. Groups 20 (2015), no. 2, 327–334.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    T. Bandman, Yu. G. Zarhin, Jordan groups, conic bundles and abelian varieties, Algebraic Geometry 4 (2017), no. 2, 229–246.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    C. Birkar, Singularities of linear systems and boundedness of Fano varieties, arXiv:1609.05543 (2016).Google Scholar
  5. 5.
    E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p, II, in: Complex Analysis and Algebraic Geometry (W. L. Baily Jr., T. Shioda, eds.), Cambridge Univ. Press, Cambridge, 1977, pp. 23–43.CrossRefGoogle Scholar
  6. 6.
    B. Conrad, A modern proof of Chevalley’ s theorem on algebraic groups, J. Ramunajam Math. Soc. 17 (2002), no. 1, 1–18.MathSciNetMATHGoogle Scholar
  7. 7.
    F. Cossec, I. Dolgachev, Enriques Surfaces I, Progress in Mathematics, Vol. 76, Birkhäuser, Berlin, 1989.Google Scholar
  8. 8.
    C. W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley, New York, 1962.MATHGoogle Scholar
  9. 9.
    O. Debarre, Higher-Dimensional Algebraic Geometry, Springer-Verlag, New York, 2001.CrossRefMATHGoogle Scholar
  10. 10.
    A. Grothendieck, Technique de construction et théorḿes d’existence en géométrie algébrique IV: les schémas de Hilbert, Séminaire N. Bourbaki, 1960–1961, exp. 221, 249–276.Google Scholar
  11. 11.
    A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de J. Dieudonné) : IV, Étude locale des schémas et des morphismes de schémas, Troisiéme partie, Publ. Math. IHES 28 (1966), 5–255.Google Scholar
  12. 12.
    Sh. Iitaka, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 76, Springer-Verlag, Berlin, 1982.Google Scholar
  13. 13.
    T. Kambayashi, D. Wright, Flat families of affine lines are affine-line bundles, Illinois J. Math. 29 (1985), no. 4, 672–681.MathSciNetMATHGoogle Scholar
  14. 14.
    J. Kollar, Rational Curves on Algebraic Varieties, Ergebnisse der Math. 3 Folge, Vol. 32, Springer-Verlag, Berlin, 1996.Google Scholar
  15. 15.
    S. Lang, Algebra, 2nd edition, Addison-Wesley, Reading, MA, 1993.MATHGoogle Scholar
  16. 16.
    Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, Vol. 6, Oxford Univ. Press, New York, 2002.Google Scholar
  17. 17.
    T. Matsusaka, Polarized varieries, fields of moduli and generalized Kummer varieties of polarized varieties, American J. Math. 80, no. 1, 45–82.Google Scholar
  18. 18.
    D. Mumford, T. Oda, Algebraic Geometry II, Texts and Reading in Mathematics, Vol.73, Hindustan Book Agency, Mumbai, 2015.Google Scholar
  19. 19.
    Sh. Meng, D.-Q. Zhang, Jordan property for non-linear algebraic groups and projective varieties, American J. Math. 140 (2018), no. 4, 1133–1145.CrossRefGoogle Scholar
  20. 20.
    D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. Vol. 1358, Springer-Verlag, Berlin, 1999.Google Scholar
  21. 21.
    D. Mumford, Abelian Varieties, 3rd edition, Hindustan Book Agency, India, Mumbai, 2008.MATHGoogle Scholar
  22. 22.
    I. Mundet i Riera, A. Turull, Boosting an analogue of Jordan’s theorem for finite groups, Adv. Math. 272 (2015), 820–836.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    V. L. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in: Affine Algebraic Geometry: the Russell Festschrift, CRM Proceedings and Lecture Notes, Vol. 54, Amer. Math. Soc., Providence, 2011, pp. 289–311Google Scholar
  24. 24.
    V. L. Popov, Jordan groups and automorphism groups of algebraic varieties, in: Automorphisms in Birational and Affine Geometry, Springer Proceedings in Mathematics and Statistics, Vol. 79, Springer, Cham, 2014, pp. 185–213.Google Scholar
  25. 25.
    Yu. Prokhorov, C. Shramov, Jordan property for Cremona groups, Amer. J. Math. 138 (2016), no. 2, 403–418.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yu. Prokhorov, C. Shramov, Jordan Property for groups of birational Selfmaps, Compositio Math. 150 (2014), 2054–2072.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Yu. Prokhorov, C. Shramov, Finite groups of birational selfmaps of threefolds, arXiv:1611.00789 (2016).Google Scholar
  28. 28.
    M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil Ci. 35 (1963), 487–489.MathSciNetMATHGoogle Scholar
  29. 29.
    F. Sakai, Kodaira dimension of complements of divisors, in: Complex Analysis and Algebraic Geometry (W.L.Baily Jr., T. Shioda, eds.), Cambridge University Press, Cambridge, 1977, pp. 239–259.Google Scholar
  30. 30.
    F. Serrano, Divisors of Bielliptic surfaces and Embedding in4, Math. Z. 203 (1990), 527–533.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    J.-P. Serre, Finite Groups: an Introduction, International Press, Somerville, MA, 2016.MATHGoogle Scholar
  32. 32.
    И. Р. Шафаревич (ред.), Алгебраические поверхности, Труды мат. инст. им. В. А. Стеклова, т. LXXV, Наука, М., 1965. Engl. transl.: I. R. Shafarevich et al., Algebraic Surfaces, American Mathematical Society, Providence, RI, 1967.Google Scholar
  33. 33.
    H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    H. Sumihiro, Equivariant completion, II, J. Math. Kyoto Univ. 15 (1975), no. 3, 573–605.MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Yu. G. Zarhin, Theta groups and products of abelian and rational varieties, Proc. Edinburgh Math. Soc. 57 (2014), no. 1, 299–304.MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Yu. G. Zarhin, Jordan groups and elliptic ruled surfaces, Transform. Groups 20 (2015), no. 2, 557–572.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations