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

$\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}$

on ${S}$ 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 ${D}$ be a very ample divisor on ${S}$. Then we have:

• ${D.C > 0}$ for all curves (i.e., strictly effective divisors) on ${S}$. In fact, if ${D}$ defines an imbedding ${S \hookrightarrow \mathbb{P}^M}$, then the degree of ${C}$ under this imbedding is ${D.C}$.
• As a special case of this, ${D.D > 0}$. In fact, ${D}$ 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 ${S}$ be a smooth projective surface as above. If ${D}$ is a divisor on ${S}$ (not necessarily effective!) satisfying ${D.D>0}$ and ${D.C > 0}$ for all curves on ${S}$, then ${D}$ is ample. In particular, ampleness depends only on the numerical equivalence class of ${D}$.

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…)