Intersection index (in algebraic geometry)
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)
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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.
Comments
References
[a1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Chapt. IV (Translated from Russian) MR0366917 Zbl 0284.14001 |
Intersection index (in algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_index_(in_algebraic_geometry)&oldid=23866