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On k-Jet Ampleness

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

Abstract

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).

Keywords

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.

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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

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