Intersection index (in algebraic geometry)

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.

References

[a1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1974) pp. Chapt. IV (Translated from Russian) MR0366917 Zbl 0284.14001