The Intersection Algorithm and Applications

Abstract

The central topic in this chapter is the Refined Bezout Theorem, which applies to possibly improper intersections. To treat this, we need to refine our considerations of the process of reduction to the diagonal. This is done in general in Section 2.1 in a scheme-theoretic version of the Intersection Algorithm; a more concrete version for treating intersections of closed subschemes in projective space is given in Section 2.2. For instance, in the latter case, roughly speaking, the diagonal is viewed as the intersection of hyperplanes in general position and, starting with the join variety, we intersect with a general hyperplane, gather any components that already fall into the diagonal into the so-called Vogel cycle, intersect the residual scheme with the next general hyperplane, and so on; the process certainly stops once we have used all the hyperplanes, since these intersect in the diagonal: we note that the varieties appearing in the Vogel cycle are counted with a new measure of multiplicity. Here again a (more general) associativity formula results and the former Samuel multiplicities are now attached just to the irreducible components of the intersection. In the concrete case, when one of the schemes is already linear, we see that we can immediately apply our Intersection Algorithm in the ambient space, without going to the join variety in the product space. In a theme that we will return to, we look at the range in which the Vogel cycle vanishes and examine the behaviour of this cycle under coning.