Intersection Indices

Abstract

The theorems on the dimension of an intersection of varieties, which we have proved in Ch. I, often allow us to assert that certain systems of equations have solutions. However, they say nothing about the number of solutions, assuming it to be finite. The difference is like that between the theorem on the existence of roots of a polynomial and the theorem that the total number of roots of a polynomial is equal to its degree. The latter theorem is true only if we count each root with its multiplicity. Similarly, to formulate general theorems on the number of points of intersection of subvarieties, we ought to assign certain multiplicities to these points. This will be done in the present subsection.