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.

The strategy behind the proof of the Nakai-Moisezhon criterion is a three-step process: first to show that some multiple of ${D}$ is an effective divisor, second to show that some multiple of ${D}$ is generated by global sections, and third to analyze the map to projective space that ${nD, n \gg 0}$ defines.

1. ${nD}$ is effective, ${n \gg 0}$

The first step is to show that ${nD}$ for ${n \gg 0}$ is an effective divisor. In order to do that, we’ll use the Riemann-Roch theorem, in the form for surfaces

$\displaystyle \chi(nD) = \frac{1}{2}(nD).(nD - K) + \chi(\mathcal{O}_S).$

Since ${D.D > 0}$, this expression tends to ${\infty}$ as ${n \rightarrow \infty}$. It follows by Serre duality, as we saw in the proof of the Hodge index theorem, that

$\displaystyle h^0(nD) + h^0(K - nD) \rightarrow \infty, n \rightarrow \infty$

where

$\displaystyle h^0(D') = \dim_k H^0(\mathcal{O}_{D'}).$

In particular, for ${n \gg 0}$, either ${nD}$ or ${K-nD}$ is effective; but since ${D}$ intersected with any ample divisor is positive, it follows that ${K-nD}$ can’t be effective for ${n \gg 0}$. Thus

$\displaystyle h^0(nD) \rightarrow \infty, \quad n \rightarrow \infty,$

and ${nD}$ is effective for ${n \gg 0}$. (This is the only place where we needed to use that ${D.D > 0}$.)

2. ${nD}$ is generated by global sections, ${n \gg 0}$

Replacing ${D}$ by a large multiple ${nD}$, we may assume that ${D}$ is thus effective, and is associated to a curve ${E}$ on the surface ${S}$. We want to show that some power of the associated line bundle ${\mathcal{L}}$ is generated by global sections.

We have a global section of ${\mathcal{L}}$ which is nonzero away from the curve ${E}$ (namely, “${1}$”) and an exact sequence

$\displaystyle 0 \rightarrow \mathcal{O}_S \rightarrow \mathcal{L} \rightarrow \mathcal{L}|_E \rightarrow 0.$

We will show that, after tensoring with a high power of ${\mathcal{L}}$, we get a surjection

$\displaystyle H^0(\mathcal{L}^{\otimes n}) \twoheadrightarrow H^0( \mathcal{L}^{\otimes (n)}|_E) \ \ \ \ \ (1)$

and that ${\mathcal{L}^{\otimes n}|_E}$ is generated by global sections. Together, these will prove the claim of this section.

First of all, ${\mathcal{L}|_E}$ is ample. It suffices for this to check that for each irreducible, reduced component ${E_i}$ of ${E}$, that ${\mathcal{L}|_{E_{i}}}$ is ample. It suffices to check this after pulling back to the normalization ${\widetilde{E}_i}$ of ${E_i}$. But we know that

$\displaystyle D.E_i = \deg \mathcal{L}|_{\widetilde{E}_i} > 0,$

which proves ampleness. In general, we use the formula

$\displaystyle D. C = \deg \mathcal{L}_{\widetilde{C}}$

valid for a reduced, irreducible (not necessarily smooth!) curve ${C}$ on ${S}$ with normalization ${\widetilde{C}}$; this is evident when the divisor ${D}$ doesn’t intersect the singularities of ${C}$ and follows in general by additivity.

Anyway, we conclude from this that ${\mathcal{L}|_E}$ is ample, so it suffices to show that (1) is a surjection. So let’s prove that.

Applying cohomology to the short exact sequence of sheaves

$\displaystyle 0 \rightarrow \mathcal{L}^{\otimes n} \rightarrow \mathcal{L}^{\otimes (n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)}|_E \rightarrow 0 ,$

we get an exact sequence

$\displaystyle H^1(\mathcal{L}^{\otimes n}) \rightarrow H^1(\mathcal{L}^{\otimes (n+1)}) \rightarrow H^1(\mathcal{L}^{\otimes (n+1)}|_E) .$

Since the last group is zero for ${n \gg 0}$ by ampleness, it follows that we have a sequence of surjections for ${n \gg 0}$

$\displaystyle H^1(\mathcal{L}^{\otimes n}) \twoheadrightarrow H^1(\mathcal{L}^{\otimes (n+1)}) \twoheadrightarrow H^1(\mathcal{L}^{\otimes (n+2)}) ,$

which must eventually become isomorphisms by finite-dimensionality. Using the long exact sequence again, we find that ${\mathcal{L}^{\otimes (n+1)} \rightarrow \mathcal{L}^{\otimes (n+1)}|_E}$ induces a surjection in ${H^0}$ for ${n \gg 0}$.

This proves the claim of this section.

3. Completion of the proof

In the previous sections, we considered a divisor ${D}$ on a surface ${S}$ satisfying the conditions ${D.D > 0}$, ${D.C >0}$ for all curves ${C}$. We showed that, at least after replacing ${D}$ by some multiple, that ${D}$ could be assumed effective and, even better, generated by its global sections. Thus ${D}$ defines a map

$\displaystyle f: S \rightarrow \mathbb{P}^N$

for some ${N}$, with the property that ${f^* \mathcal{O}(1)}$ is the line bundle associated to ${D}$.

Unfortunately, this map need not be a closed immersion. But we do know that ${D.C > 0}$ for any curve ${C}$ on ${S}$, or ${f^*\mathcal{O}(1)|_C}$ is ample. In particular, it is nontrivial, so no curve ${C}$ is mapped to a point under ${f}$. It follows that the fibers of ${f}$ are finite; Zariski’s main theorem implies that ${f}$ is in fact a finite morphism.

Now we use the following fact, applied to ${f}$ and ${\mathcal{O}(1)}$ on ${\mathbb{P}^N}$, to conclude.

Lemma 2 A finite morphism preserves ampleness. In other words, if ${f: X \rightarrow Y}$ is a finite morphism of finite type ${k}$-schemes, and ${\mathcal{L}}$ is a line bundle on ${Y}$ which is ample, then ${f^*\mathcal{L}}$ is ample.

Proof: Let’s check this in the case that ${X, Y}$ are proper (which is all that we need). In this case, we need to show that for any coherent sheaf ${\mathcal{F}}$ on ${X}$,

$\displaystyle H^i(X, \mathcal{F} \otimes f^* \mathcal{L}^{\otimes n}) = 0, \quad i > 0, n \gg 0.$

Since ${f}$ is finite, it is equivalent to show that

$\displaystyle H^i(Y, f_*(\mathcal{F} \otimes f^* \mathcal{L}^{\otimes n})) = 0, \quad i > 0, n \gg 0.$

But we have an isomorphism

$\displaystyle f_*(\mathcal{F} \otimes f^* \mathcal{L}^{\otimes n}) \simeq \mathcal{F} \otimes \mathcal{L}^{\otimes n}$

from the projection formula, so that

$\displaystyle H^i(Y, f_*(\mathcal{F} \otimes f^* \mathcal{L}^{\otimes n})) \simeq H^i(Y, f_*\mathcal{F} \otimes \mathcal{L}^{\otimes n})= 0, \quad i > 0, n \gg 0,$

using ampleness of ${\mathcal{L}}$ on ${Y}$. $\Box$

Advertisements