Editing Intersection number (section)

== Definition for algebraic varieties ==

The usual constructive definition in the case of algebraic varieties proceeds in steps. The definition given below is for the intersection number of [[Divisor (algebraic geometry)|divisor]]s on a nonsingular variety ''X''.

1. The only intersection number that can be calculated directly from the definition is the intersection of hypersurfaces (subvarieties of ''X'' of codimension one) that are in general position at ''x''. Specifically, assume we have a nonsingular variety ''X'', and ''n'' hypersurfaces ''Z''<sub>''1''</sub>, ..., ''Z''<sub>''n''</sub> which have local equations ''f''<sub>''1''</sub>, ..., ''f''<sub>''n''</sub> near ''x'' for polynomials ''f''<sub>''i''</sub>(''t''<sub>''1''</sub>, ..., ''t''<sub>''n''</sub>), such that the following hold:

* <math>n = \dim_k X</math>.
* <math>f_i(x) = 0</math> for all ''i''. (i.e., ''x'' is in the intersection of the hypersurfaces.)
* <math>\dim_x \cap_{i=1}^n Z_i = 0</math> (i.e., the divisors are in general position.)
* The <math>f_i</math> are nonsingular at ''x''.

Then the intersection number at the point ''x'' (called the '''intersection multiplicity''' at ''x'') is

:<math>(Z_1 \cdots Z_n)_x := \dim_k \mathcal{O}_{X, x} / (f_1, \dots, f_n)</math>,

where <math>\mathcal{O}_{X, x}</math> is the local ring of ''X'' at ''x'', and the dimension is dimension as a ''k''-vector space. It can be calculated as the [[Localization of a ring|localization]] <math>k[U]_{\mathfrak{m}_x}</math>, where <math>\mathfrak{m}_x</math> is the maximal ideal of polynomials vanishing at ''x'', and ''U'' is an open affine set containing ''x'' and containing none of the singularities of the ''f''<sub>''i''</sub>.

2. The intersection number of hypersurfaces in general position is then defined as the sum of the intersection numbers at each point of intersection.

:<math>(Z_1 \cdots Z_n) = \sum_{x \in \cap_i Z_i} (Z_1 \cdots Z_n)_x</math>

3. Extend the definition to ''effective'' divisors by linearity, i.e.,

:<math>(n Z_1 \cdots Z_n) = n(Z_1 \cdots Z_n)</math> and <math>((Y_1 + Z_1) Z_2 \cdots Z_n) = (Y_1 Z_2 \cdots Z_n) + (Z_1 Z_2 \cdots Z_n)</math>.

4. Extend the definition to arbitrary divisors in general position by noticing every divisor has a unique expression as ''D'' = ''P'' – ''N'' for some effective divisors ''P'' and ''N''. So let ''D''<sub>''i''</sub> = ''P''<sub>''i''</sub> – ''N''<sub>i</sub>, and use rules of the form

:<math>((P_1 - N_1) P_2 \cdots P_n) = (P_1 P_2 \cdots P_n) - (N_1 P_2 \cdots P_n)</math>

to transform the intersection.

5. The intersection number of arbitrary divisors is then defined using a "[[Chow's moving lemma]]" that guarantees we can find linearly equivalent divisors that are in general position, which we can then intersect.

Note that the definition of the intersection number does not depend on the order in which the divisors appear in the computation of this number.