# Intersection index (in algebraic geometry)

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The number of points in the intersection of $n$ divisors (cf. Divisor) in an $n$- dimensional algebraic variety with allowance for the multiplicities of these points. More precisely, let $X$ be an $n$- dimensional non-singular algebraic variety over a field $k$, and let $D _ {1} \dots D _ {n}$ be effective divisors in $X$ that intersect in a finite number of points. The local index (or multiplicity) of intersection of these divisors at a point $x \in X$ is the integer

$$( D _ {1} \dots D _ {n} ) _ {x} = \mathop{\rm dim} _ {k} A / ( u _ {1} \dots u _ {n} ) ,$$

where $u _ {i}$ is the local equation for the divisor $D _ {i}$ in the local ring $A = {\mathcal O} _ {X,x}$. In the complex case, the local index coincides with the residue of the form $( du _ {1} /u _ {1} ) \wedge \dots \wedge ( du _ {n} /u _ {n} )$, and also with the degree of the germ of the mapping (cf. Degree of a mapping)

$$( u _ {1} \dots u _ {n} ): ( X, x) \rightarrow ( \mathbf C ^ {n} , 0).$$

The global intersection index $( D _ {1} \dots D _ {n} )$ is the sum of the local indices over all points of the intersection $D _ {1} \cap {} \dots \cap D _ {n}$. If this intersection is not empty, then $( D _ {1} \dots D _ {n} ) > 0$.

See also Intersection theory.

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How to Cite This Entry:
Intersection index (in algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_index_(in_algebraic_geometry)&oldid=47398
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article