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Positivity and Excess Intersection

  • William Fulton
  • Robert Lazarsfeld
Part of the Progress in Mathematics book series (PM, volume 24)

Abstract

Consider a variety M, and a projective local complete intersection
$${\text{X }}\underline \subset {\text{ M}}$$
of pure codimension e. Then for any subvariety \({\text{Y }}\underline \subset {\text{ M}}\) of dimension k ≥ e, the intersection class
$${\text{X }} \bullet {\text{ Y }} \in {\text{ }}{{\text{A}}_{k - e}}(X)$$
is defined up to rational equivalence on X. One of the most basic facts of intersection theory is that if Y meets X, and does so properly, then X·Y is non-zero and in fact has positive degree with respect to any projective embedding of X. On the other hand, if the intersection of X and Y is improper, then X·Y may be zero or of negative degree. Our purpose here is to give some conditions on X to guarantee the non-negativity or positivity of the intersection class in the case of possibly excess intersection. These conditions take the form of hypotheses on the normal bundle NX/M to X in M, the theme being that positivity of the vector bundle NX/M forces the positivity of X·Y provided only that Y meets X. We give several simple applications and related results, including a lower bound for the multiplicity of a proper intersection, generalizing a classical result for curves on a surface.

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Copyright information

© Birkhäuser Boston, Inc. 1982

Authors and Affiliations

  • William Fulton
    • 1
    • 2
    • 3
  • Robert Lazarsfeld
    • 4
  1. 1.Institut des Hautes Études ScientifiquesFrance
  2. 2.Institute for Advanced StudyUSA
  3. 3.Brown UniversityUSA
  4. 4.Institute for Advanced StudyHarvard UniversityUSA

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