Skip to main content
SpringerLink
Log in

A Lefschetz hyperplane theorem for Mori dream spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let X be a smooth Mori dream space of dimension ≥ 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Néron–Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking “Mori dream hypersurfaces” of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Batyrev, V.V., Popov, O.N.: The Cox ring of a del Pezzo surface. In: Arithmetic of Higher-dimensional Algebraic Varieties. Progress in Mathematics, vol. 226, pp. 85–103. Birkhäuser, Boston (2004)

  2. Berchtold F., Hausen J.: Homogeneous coordinates for algebraic varieties. J. Algebra 266, 636–670 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. Preprint, arXiv:math/0610203

  4. Borcea, C.: Homogeneous vector bundles and families of Calabi-Yau threefolds, II, Several complex variables and complex geometry, Part 2. In: Proceedings of the Symposium on Pure Mathematics, vol. 52, pp. 83–91. American Mathematical Society, Providence (1991)

  5. Castravet, A.-M.: The Cox ring of \({\bar{M}_{0,6}}\). Trans. Am. Math. Soc. (to appear). arXiv:0705.0070

  6. Castravet A.-M., Tevelev J.: Hilbert’s 14-th problem and Cox rings. Compos. Math. 142, 1479–1498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cox D.A.: The homogeneous coordinate ring of a toric variety. J. Algebra Geom. 4, 17–50 (1995)

    MATH  Google Scholar 

  8. Debarre O.: Higher-dimensional Algebraic Geometry, Universitext. Springer, New York (2001)

    Google Scholar 

  9. Dolgachev, I.: Lectures on invariant theory. In: London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003)

  10. Drezet J.-M., Narasimhan M.S.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97, 53–94 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Elizondo E.J., Kurano K., Watanabe K.: The total coordinate ring of a normal projective variety. J. Algebra 276(2), 625–637 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fulton W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

    Google Scholar 

  13. Gretenkort J., Kleinschmidt P., Sturmfels B.: On the existence of certain smooth toric varieties. Discret. Comput. Geom. 5, 255–262 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hartshorne R.: Algebraic Geometry Graduate. Texts in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  15. Hassett B., Lin H.-W., Wang C.-L.: The weak Lefschetz principle is false for ample cones. Asian J. Math. 6(1), 95–100 (2002)

    MathSciNet  MATH  Google Scholar 

  16. Hu, Y.: Geometric invariant theory and birational geometry. arXiv:math.AG/0502462

  17. Hu Y., Keel S.: Mori dream spaces and GIT. Mich. Math. J. 48, 331–348 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kleinschmidt, P., Schwartz, N., Sturmfels, B.: Unimodular fans, linear codes, and toric manifolds. In: Discrete and Computational Geometry: Papers from the DIMACS Special Year, pp. 179–186. American Mathematical Society, Providence (1991)

  19. Lazarsfeld R.: Positivity in Algebraic Geometry I–II. Ergeb. Math. Grenzgeb., vols. 48–49. Springer, Berlin (2004)

    Google Scholar 

  20. Miller E., Sturmfels B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)

    Google Scholar 

  21. Munkres J.R.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984)

    MATH  Google Scholar 

  22. Serganova V., Skorobogatov A.: Del Pezzo surfaces and representation theory. Algebra Number Theory 1, 393–419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stillman M., Testa D., Velasco M.: Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces. J. Algebra 316(2), 777–801 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wiśniewski J.A.: On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417, 141–157 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

    Shin-Yao Jow

Authors
  1. Shin-Yao Jow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shin-Yao Jow.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jow, SY. A Lefschetz hyperplane theorem for Mori dream spaces. Math. Z. 268, 197–209 (2011). https://doi.org/10.1007/s00209-010-0666-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-010-0666-9

Keywords

Search

Navigation