Intersection multiplicity

Summary

Two subvarieties meet along the components of their intersection. The intersection multiplicity along such a component is a positive integer that says how many times it should be counted. For curves in the plane it is the dimension of a local ring, and in general it is given by Serre's Tor formula. With these multiplicities, counts of intersection points become stable under deformation, which is what makes Bézout's theorem and intersection theory work.