2.1 The goal

The goal in the chapter state and prove two of the main theorems about ample line bundles on projective varieties: the Nakai-Moishezon theorem and Kleiman’s criterion. In order to do that, but also for other things, we’re going to need some intersection theory. Basically we need to be able to take Cartier divisors and intersect them with subvarieties. The challenge is that setting up the intersection theory needed to make sense of this takes a considerable amount of work.

There are a few options for solutions to the challenge.

1. Just do the work. For example, we could follow Fulton’s book  [ 3 ] . Most of the work we need to set things up is done in the first 8 chapters, and we could then go to the Chapter on the Grothendieck Riemann-Roch and have a fairly complete understanding of how to intersect algebraic cycles.

2. Use the technology in Kleiman’s paper  [ 5 ] to set up a numerical version of intersection theory in an ad hoc way. This is what Beauville  [ 1 ] and Bădescu  [ 2 ] do.

3. Appeal to intuition and to cohomology and simply define intersections in terms of cup products. This is what Lazaarsfeld does in  [ 6 ] . It’s faster than either of the above solutions and it has the advantage that, for some things, like understanding numerical equivalence, we would want to compare to cohomology anyway.