TITLE: generalization of Bezout's Theorem?
QUESTION [12 upvotes]: I learned Bezout's Theorem in class, stated for plane curves (if irreducible, sum of intersection multiplicities equals product of degrees). What is the proper general statement, for projective varieties of degree n?
I think it is something like: If finite, the sum of multiplicities equals the product of degrees.. else the (dimension? degree? sums over irreducible components?) of the intersection is less than or equal to the difference in degrees.
Help is appreciated!

REPLY [10 votes]: In that wonderful book of Fulton, the more general result is at the bottom of page 223, where it says that for r pure dimensional schemes in P^n, whose codimensions add to at most n, the product of their degrees is at least as great as the sum of the degrees of the irreducible components of their intersection.  Thus for instance, if three quadric hypersurfaces in P^4 have a common curve of degree 8, their intersection has no further components.