Intersection index (in algebraic geometry)

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The number of points in the intersection of divisors (cf. Divisor) in an -dimensional algebraic variety with allowance for the multiplicities of these points. More precisely, let be an -dimensional non-singular algebraic variety over a field , and let be effective divisors in that intersect in a finite number of points. The local index (or multiplicity) of intersection of these divisors at a point is the integer

where is the local equation for the divisor in the local ring . In the complex case, the local index coincides with the residue of the form , and also with the degree of the germ of the mapping (cf. Degree of a mapping)

The global intersection index is the sum of the local indices over all points of the intersection . If this intersection is not empty, then .

See also Intersection theory.



[a1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Chapt. IV (Translated from Russian)
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Intersection index (in algebraic geometry). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article