Intersection Number - Definition For Algebraic Varieties

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 divisors 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 Z₁, ..., Z_n which have local equations f₁, ..., f_n near x for polynomials f_i(t₁, ..., t_n), such that the following hold:

• .
• for all i. (i.e., x is in the intersection of the hypersurfaces.)
• (i.e., the divisors are in general position.)
• The are nonsingular at x.

Then the intersection number at the point x is

,

where 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, where 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_i.

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

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

and .

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_i = P_i - N_i, and use rules of the form

to transform the intersection.

5. The intersection number of arbitrary divisors is then defined using a "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 of the divisors.