As in the previous two posts, let be a smooth, projective surface over an algebraically closed field . In the previous posts, we set up an intersection theory for divisors, which was a symmetric bilinear form

that gave the “natural” answer for the intersection of two transversely intersecting curves. Specifically, we had

the bilinearity of this map had to do with the fact that the Euler characteristic was a **quadratic** function on the Picard group. The purpose of this post is to prove a few more general and classical facts about this intersection pairing. As usual, Hartshorne’s *Algebraic geometry *and Mumford’s *Lectures on curves on an algebraic surface *are very helpful sources for this material; I also found Abhinav Kumar’s lecture notes useful.

**1. The Riemann-Roch theorem**

The Euler characteristic of a line bundle on is a “topological” invariant: it is unchanged under deformations. Given an algebraic family of line bundles on — in other words, a scheme and a line bundle on which restricts on the fibers to — the Euler characteristics are constant. This is one of the parts of the semicontinuity theorem on the cohomology of a flat family of sheaves. Over the complex numbers, one can see this by observing that the Euler characteristic of a line bundle is the index of an elliptic operator — more specifically, the index of the Dolbeault complex associated to — and can therefore be computed in purely topological terms via the Hirzebruch-Riemann-Roch formula.

In algebraic geometry, the fact that the Euler characteristic is a topological invariant is reflected in the following result, which computes it solely in terms of intersection numbers:

Theorem 1Let be a line bundle on . Then

where is the canonical divisor on . (more…)