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

• If ${C, D}$ intersect transversely, then ${C.D = |C \cap D|}$.
• The intersection product is additive. That is, given curves ${C_1, C_2, D}$, we have $\displaystyle (C_1 + C_2). D = C_1.D + C_2.D,$

where ${C_1+C_2}$ 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 $\displaystyle C.D \stackrel{\mathrm{def}}{=} \chi( \mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D ) = \sum_i (-1)^i \mathbb{H}^i( \mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D) \ \ \ \ \ (1)$

would be a good definition: i.e., that the failure of $\displaystyle C.D = |C \cap D|$

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

Theorem 1 The definition of ${C.D}$ in (1) satisfies the conditions desired of an intersection product. (more…)