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…)