5 - Intersection Multiplicities

Summary

In this chapter we discuss the definition of intersection multiplicities, a topic that we will take up again in the last chapter. The basic idea is to give a definition of the multiplicity of intersection of two subschemes of a scheme in such a way that it agrees with the intuitive notion of counting the number of intersections at a point and so that certain reasonable conditions will be satisfied. One condition is that the result should be independent of rational equivalence. A second is that the intersection of two smooth subschemes that meet transversally at a point should be one; algebraically, this condition states that if x₁, … x_d form a minimal set of generators of the maximal ideal of a regular local ring, and if the subschemes are defined by the ideals x₁, … x_k and x_k+1, …x_d, respectively, their intersection multiplicity should be one. Another condition, which we will discuss in a later chapter, is that Bézout's theorem must hold; that is, in projective space the total number of intersections of two subschemes should be the product of the degrees of the subschemes.

Another version of this problem is to attempt to define an intersection product on the Chow group to create a ring structure on the Chow group.