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, Volume 61, Issue 2, pp 145–162 | Cite as

The geometric and numerical properties of duality in projective algebraic geometry

  • Audun Holme


In this paper we investigate some fundamental geometric and numerical properties ofduality for projective varieties inP k N =P N . We take a point of view which in our opinion is somewhat moregeometric and lessalgebraic andnumerical than what has been customary in the literature, and find that this can some times yield simpler and more natural proofs, as well as yield additional insight into the situation. We first recall the standard definitions of thedual variety and theconormal scheme, introducing classical numerical invariants associated with duality. In section 2 we recall the well known duality properties these invariants have, and which was noted first byT. Urabe. In section 3 we investigate the connection between these invariants andChern classes in the singular case. In section 4 we give a treatment of the dual variety of a hyperplane section of X, and the dual procedure of taking the dual of a projection of X. This simplifies the proofs of some very interesting theorems due toR. Piene. Section 5 contains a new and simpler proof of a theorem ofA. Hefez and S. L. Kleiman. Section 6 contains some further results, geometric in nature.


Number Theory Algebraic Geometry Topological Group Additional Insight Simple Proof 
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-Verlag 1988

Authors and Affiliations

  • Audun Holme
    • 1
  1. 1.Matematisk Institutt Allégaten 55Universitetet i BergenBergenNorway

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