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Divisors

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Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 2)

Summary

If D is a Cartier divisor on a scheme X, and α is a k-cycle on X, we construct an intersection class
$$ D\cdot\alpha \, \in \,A_{k - 1} \left( {\left| D \right| \cap \left| \alpha \right|} \right) $$
where |D|1, |α| are the supports of D and α. For α = [V], V subvariety, D · [V] is defined by one of two procedures: (i) if V ⊄ |D|, D restricts to a Cartier divisor on V, and D · [V] is defined to be the associated Weil divisor of this restriction; (ii) if V ⍧ |D|, the restriction of the line bundle O X (D) to V is the line bundle of a well-defined linear equivalence class of Cartier divisors on V, and D · [V] is represented by the associated Weil divisor of any such Cartier divisor.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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