On k-Jet Ampleness

  • Mauro C. Beltrametti
  • Andrew J. Sommese
Part of the The University Series in Mathematics book series (USMA)


Let X be an n-dimensional projective manifold mapped into a projective space Ψ:X → ℙ. Let L be the pullback, Ψ*O(1), of the hyperplane section bundle. If Ψ is an embedding, L is said to be very ample. This is an intensively studied and well-understood concept. In this chapter we study a particular notion of higher-order embedding. We say that L is k-jet ample for a nonnegative integer k if, given any r integers k 1 , ., k r , such that \( k + 1 = \sum\nolimits_{i = 1}^r {{k_i}} \) and any r distinct points {x 1 ,. . ., x r }X, the evaluation map
$$ X \times \Gamma (L) \to L/L \otimes m_{{x_1}}^{{k_1}} \otimes ... \otimes m_{xr}^{{k_r}} \to 0 $$
is surjective, where m xi . denotes the maximal ideal at x t . Note that L is spanned (respectively, very ample) if and only if L is 0-jet ample (respectively, 1-jet ample).


Line Bundle Distinct Point Exceptional Divisor Hilbert Scheme Ample Line Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Ballico and M. Beltrametti, On 2-spannedness for the adjunction mapping, Mannscripta Math. 61, 447–458 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    G. Barthel, F. Hirzebruch, and T. Höfer, Geradenkonfigurationen und Algebraische Flächen, in Aspects of Mathematics, D4, Vieweg and Sohn, Wiesbaden (1987).Google Scholar
  3. 3.
    M. Beltrametti, P. Francia, and A. J. Sommese, On Reider’s method and higher order embeddings, Duke Math. J. 58, 425–439 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    M. Beltrametti and A. J. Sommese, On k-spannedness for projective surfaces, in 1988 L’Aquila Proceedings: Hyperplane sections, Lecture Notes in Math., Vol. 1417, pp. 24–51, Springer-Verlag, Berlin and New York (1990).Google Scholar
  5. 5.
    M. Beltrametti and A. J. Sommese, Zero cycles and k-th order embeddings of smooth projective surfaces, in 1988 Cortona Proceedings: Projective Surfaces and Their Classification, Symposia Mathematica, INDAM, Vol. 32, pp. 33–48, Academic Press, New York (1991).Google Scholar
  6. 6.
    E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Etudes Sci. Publ. Math. 42, 171–219 (1973).MathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Catanese and L. Göttsche, d-very ample line bundles and embeddings of Hilbert schemes of 0-cycles, Manuscripta Math. 68, 337–341 (1988).CrossRefGoogle Scholar
  8. 8.
    J.-P. Demailly, A numerical criterion for very ample line bundles, preprint (1990).Google Scholar
  9. 9.
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., Vol. 52, Springer-Verlag, Berlin and New York (1977).zbMATHCrossRefGoogle Scholar
  10. 10.
    R. Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43, 73–89 (1971).MathSciNetGoogle Scholar
  11. 11.
    F. Hirzebruch, Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, pp. 113–140, Birkhauser (1983).CrossRefGoogle Scholar
  12. 12.
    A. Lanteri, M. Palleschi, and A. J. Sommese, Very ampleness of K x£ dimX for ample and spanned line bundles £, Osaka J. Math. 26, 647–664 (1989).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann. 261, 43–46 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. Kumpera and D. Spencer, Lie Equations, Vol. I, General theory, Ann. of Math. Stud. No. 73, Princeton University Press, Princeton (1972).zbMATHGoogle Scholar
  15. 15.
    C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. 36, 41–51 (1972).MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. J. Sommese, Compact complex manifolds possessing a line bundle with a trivial jet bundle, Abh. Math. Sem. Univ. Hamburg 55, 151–170 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    A. J. Sommese, On the density of ratios of Chern numbers of algebraic surfaces, Math. Ann. 268, 207–221 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A. J. Sommese and A. Van de Ven, Homotopy groups of pullbacks of varieties, Nagoya Math. J. 102, 79–90 (1986).MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Van de Ven, On the 2-connectedness of very ample divisors on a surface, Duke Math. J. 46, 403–407 (1979).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Mauro C. Beltrametti
    • 1
  • Andrew J. Sommese
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di GenovaGenovaItaly
  2. 2.Department of MathematicsUniversity of Notre DameSouth BendIndiana

Personalised recommendations