Let be a smooth, projective surface over the algebraically closed field . Previous posts have set up an intersection theory

on with very convenient formal properties. We also described a historically important use of this machinery: the Weil bound on points on a smooth curve over a finite field. The purpose of this post is to prove an entirely numerical criterion for ampleness of a line bundle on a surface, due to Nakai and Moishezon.

Let be a very ample divisor on . Then we have:

- for all curves (i.e., strictly effective divisors) on . In fact, if defines an imbedding , then the degree of under this imbedding is .
- As a special case of this, . In fact, must be effective.

Since a power of an ample divisor is very ample, the same is true for an ample divisor.

The purpose of this post is to prove the converse:

Theorem 1 (Nakai-Moishezon)Let be a smooth projective surface as above. If is a divisor on (not necessarily effective!) satisfying and for all curves on , then is ample. In particular, ampleness depends only on the numerical equivalence class of .

Once again, the source for this material is Hartshorne’s *Algebraic geometry. *The goal is to get to some computations and examples as soon as possible. (more…)