Let be a smooth, projective surface over an algebraically closed field and let be curves (subschemes pure of codimension one) on . In the previous post, we discussed what a good theory of intersections would look like. We wanted to be able to define the intersection in such a manner that:

- If intersect transversely, then .
- The intersection product is
**additive**. That is, given curves , we havewhere is treated as an effective Cartier divisor.

- The intersection product is invariant under linear equivalence and descends to a pairing on the Picard group.

**1. Definition of the intersection product**

In the previous post, we saw that any intersection theory as above was necessarily unique, and suggested that the Euler characteristic formula

would be a good definition: i.e., that the failure of

in general was due to two factors: the existence of nilpotents in the (scheme-theoretic as opposed to set-theoretic) intersection and higher homotopy groups in the (derived as opposed to scheme-theoretic) intersection . The main goal of this post is to prove that (1)does give a good theory. That is, we would like to prove:

Theorem 1The definition of in (1) satisfies the conditions desired of an intersection product. (more…)