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Bezout's Theorem

Definition. Suppose that and are two irreducible varieties and that their intersection has irreducible components . We say that and intersect generically transversally if, for each , and intersect transversally at a general point , i.e., are smooth at with tangent spaces spanning (the tangent space to at .)

Theorem 5 (Bezout)   Let and be subvarieties of pure dimensions and with , and suppose they intersect generically transversely. Then

degdeg   deg

In particular, if , this says that will consist of deg   deg points.

A pair of pure-dimensional varieties and intersect properly if their intersection has the expected dimension, i.e.,

dimdimdim

Theorem 6   If and intersect properly,

deg   deg   deg

where the sum is over all irreducible subvarieties of the appropriate dimension (in effect, over all irreducible components of ). Here is the intersection multiplicity of and along :
1.
for all ( otherwise.)

2.
if and intersect transversely at a general point of .

3.
is additive, i.e. for any and as long as all three numbers are defined and and have no common components.

In particular, for any subvarieties and of pure dimension in intersecting properly:

deg   deg   deg

Next: About this document ... Up: Algebraic Geometry Previous: Zariski Topology
Zvezdelina Stankova-Frenkel 2001-01-24