# The Nakai–Moishezon criterion

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

An ample line bundle has positive top self-intersection on every subvariety. The Nakai–Moishezon criterion says that this numerical condition is also sufficient: a line bundle on a projective scheme is ample as soon as \((L^{\dim V}\cdot V)>0\) for every integral subvariety \(V\) of positive dimension. The proof is an induction on dimension. The inductive hypothesis makes \(L\) ample on every divisor; cohomology sequences then show that some power of \(L\) has many sections, then that a further power is generated by its sections, and a globally generated bundle that is positive on every curve is ample because the morphism it defines has finite fibres.

We use [Intersection numbers of line bundles](intersection-numbers-of-line-bundles.md). From the courses on schemes and their cohomology we use: Serre's finiteness and vanishing theorems and the cohomological criterion for ampleness (a line bundle \(L\) on a proper scheme over a Noetherian ring is ample if and only if \(H^q(X,F\otimes L^n)=0\) for every coherent \(F\), every \(q>0\) and all \(n\gg0\)), Theorems 2.2 and 4.1 of [Coherent sheaves on projective schemes: Serre's theorems](course:AG-QC/serres-theorems-on-projective-schemes#4-vanishing-detects-ampleness); the filtration of coherent sheaves by ideals on integral subschemes, Theorem 2.2 of [Coherence of higher direct images under proper morphisms](course:AG-QC/proper-morphisms-and-coherent-direct-images#2-integral-supports-and-coherent-filtrations); that \(L\) is ample if and only if some positive power is, and that ampleness passes to closed subschemes, Proposition 1.2 of [Ample invertible sheaves](course:AG-MO/ample-invertible-sheaves#1-affine-neighbourhoods-from-sections); that a tensor product of a globally generated and a very ample bundle is very ample, Section 5 of [Very ample invertible sheaves, Segre and Veronese embeddings](course:AG-MO/very-ample-sheaves-and-embeddings#5-very-ampleness-and-tensor-products); morphisms to projective space defined by generating sections, [Projective space, relative Proj and maps to projective space](course:AG-SS/projective-space-relative-proj-and-maps-to-projective-space#2-the-universal-property-of-degree-one-proj); and that the quasi-finite locus is open and proper quasi-finite morphisms are finite, Theorems 3.1 and 5.1 of [Zariski's Main Theorem](course:AG-MO/zariskis-main-theorem#5-properness-and-normality).

Throughout, \(k\) is an algebraically closed field and \(X\) is a projective \(k\)-scheme. A *curve* in \(X\) is an integral closed subscheme of dimension one.

## 1. Three ways to recognize ampleness

**Lemma 1.1** (components). Let \(L\) be invertible on a proper \(k\)-scheme \(X\). If \(L|_V\) is ample for every irreducible component \(V\) of \(X\), with its reduced structure, then \(L\) is ample.

**Proof.** We verify the cohomological criterion. Let \(F\) be coherent on \(X\). Filter \(F\) with quotients \(i_*I\), where \(i:Z\to X\) is integral and closed and \(I\subseteq\mathcal O_Z\) is a coherent ideal. Each such \(Z\) lies in an irreducible component \(V\), and \(L|_Z\) is the restriction of the ample \(L|_V\) to a closed subscheme, hence ample. So \(H^q(X,i_*I\otimes L^n)=H^q(Z,I\otimes L|_Z^n)=0\) for \(q>0\) and \(n\gg0\). The vanishing of all higher cohomology for \(n\gg0\) is preserved under extensions, by the long exact sequence, so it holds for \(F\). \(\square\)

**Lemma 1.2** (finite pullback). Let \(f:X\to Y\) be a finite morphism of proper \(k\)-schemes and \(M\) ample on \(Y\). Then \(f^*M\) is ample.

**Proof.** For coherent \(F\) on \(X\), the sheaf \(f_*F\) is coherent and, \(f\) being affine, \(H^q(X,F\otimes f^*M^n)\) equals \(H^q(Y,f_*F\otimes M^n)\) by the projection formula. The right side vanishes for \(q>0\) and \(n\gg0\). \(\square\)

**Lemma 1.3** (globally generated and positive on curves). Let \(L\) be a globally generated invertible sheaf on a proper \(k\)-scheme \(X\) with \(\deg(L|_C)>0\) for every curve \(C\subseteq X\). Then \(L\) is ample.

**Proof.** Generating sections \(s_0,\dots,s_N\) define a morphism \(\varphi:X\to\mathbf P^N_k\) with \(\varphi^*\mathcal O(1)\cong L\). Let \(y\) be a closed point. The fibre \(\varphi^{-1}(y)\) is a closed subscheme of \(X\), proper over \(k\). If it had positive dimension, it would contain a curve \(C\), and \(L|_C\cong\varphi^*\mathcal O(1)|_C\) would be pulled back from the point \(y\), hence trivial, giving \(\deg(L|_C)=0\). So every fibre over a closed point is finite, and \(\varphi\) is quasi-finite at every closed point of \(X\) (closed points map to closed points). The locus where \(\varphi\) is not quasi-finite is closed; if it were nonempty it would contain a closed point. Hence \(\varphi\) is quasi-finite, and being proper, it is finite. Lemma 1.2 shows that \(L=\varphi^*\mathcal O(1)\) is ample. \(\square\)

## 2. The criterion

**Theorem 2.1** (Nakai–Moishezon). An invertible sheaf \(L\) on the projective \(k\)-scheme \(X\) is ample if and only if

\[
(L^{\dim V}\cdot V)>0
\]

for every integral closed subscheme \(V\subseteq X\) of positive dimension.

**Proof.** If \(L\) is ample, so is \(L|_V\), and Theorem 3.4 of the previous lesson gives positivity.

For the converse we prove, by induction on \(n\ge0\), the statement: *for every projective \(k\)-scheme \(X\) of dimension at most \(n\) and every invertible \(L\) satisfying the numerical condition, \(L\) is ample.* For \(n=0\), \(X\) is finite, hence affine, and every invertible sheaf on an affine scheme is ample. Let \(n\ge1\) and assume the statement in dimension \(n-1\).

*Step 1: reduction to an integral scheme.* The irreducible components of \(X\) are integral projective schemes satisfying the numerical condition, since their integral subschemes are integral subschemes of \(X\). By Lemma 1.1 and the inductive hypothesis for components of dimension less than \(n\), it suffices to treat an integral \(X\) of dimension \(n\).

*Step 2: ample on every proper closed subscheme.* Every closed subscheme \(D\subsetneq X\) has dimension at most \(n-1\) and satisfies the numerical condition, so \(L|_D\) is ample by the inductive hypothesis.

*Step 3: a power of \(L\) has a nonzero section.* Choose a very ample \(H\) such that \(L\otimes H\) is also very ample: for instance \(H=\mathcal O_X(m)\) with \(m\) large enough that \(L(m)\) is globally generated, using that a globally generated bundle tensored with a very ample one is very ample. Choose nonzero sections \(a\in H^0(X,L\otimes H)\) and \(b\in H^0(X,H)\), with zero schemes \(A\) and \(B\). Since \(X\) is integral, multiplication by \(a\) and by \(b\) is injective, and \(A,B\) are proper closed subschemes. For every integer \(m\) there are exact sequences

\[
\begin{aligned}
&0\to L^{m-1}\otimes H^{-1}\xrightarrow{\,a\,}L^m\\
&\qquad\to L^m|_A\to0,\\
&0\to L^{m-1}\otimes H^{-1}\xrightarrow{\,b\,}L^{m-1}\\
&\qquad\to L^{m-1}|_B\to0 .
\end{aligned}
\]

By Step 2 and Serre vanishing, \(H^i(A,L^m|_A)=H^i(B,L^m|_B)=0\) for \(i\ge1\) and \(m\ge m_1\). The two long exact sequences then give, for \(i\ge2\) and \(m>m_1\),

\[
\begin{aligned}
H^i(X,L^m)&\cong H^i(X,L^{m-1}\otimes H^{-1})\\
&\cong H^i(X,L^{m-1}).
\end{aligned}
\]

So \(h^i(L^m)=\dim H^i(X,L^m)\) is independent of \(m\ge m_1\) for each \(i\ge2\), and there is a constant \(c\) with

\[
\begin{aligned}
&\chi(L^m)=h^0(L^m)-h^1(L^m)\\
&\quad +\sum_{i\ge2}(-1)^ih^i(L^m)\le h^0(L^m)+c\\
&(m\ge m_1).
\end{aligned}
\]

The polynomial \(\chi(L^m)\) has degree at most \(n\), and its coefficient of \(m^n\) is \((L^n\cdot X)/n!\): in \(\chi(L^{n_1+\dots+n_n})\) the coefficient of \(n_1\cdots n_n\) is \(n!\) times that coefficient. By hypothesis \((L^n\cdot X)>0\), so \(\chi(L^m)\to\infty\), hence \(h^0(L^m)\to\infty\). Fix \(m_0\ge1\) and a nonzero \(s\in H^0(X,L^{m_0})\).

*Step 4: a power is globally generated.* Put \(M=L^{m_0}\). Its section \(s\) is injective on the integral \(X\); let \(D\) be its zero scheme, a proper closed subscheme, on which \(M|_D=(L|_D)^{m_0}\) is ample by Step 2. From

\[
0\to M^{m-1}\xrightarrow{\,s\,}M^m\to M^m|_D\to0
\]

and \(H^1(D,M^m|_D)=0\) for \(m\ge m_2\), we get, for these \(m\), the exact sequence

\[
\begin{aligned}
&H^0(X,M^m)\to H^0(D,M^m|_D)\\
&\quad\to H^1(X,M^{m-1})\\
&\quad\to H^1(X,M^m)\to0.
\end{aligned}
\]

So \(h^1(M^m)\le h^1(M^{m-1})\), and the nonincreasing sequence of nonnegative integers \(h^1(M^m)\) is constant for \(m\ge m_3\). For such \(m\), the surjection \(H^1(M^{m-1})\to H^1(M^m)\) between spaces of the same finite dimension is injective, and therefore \(H^0(X,M^m)\to H^0(D,M^m|_D)\) is surjective. Enlarging \(m\), we may also assume that \(M^m|_D\) is globally generated (Serre's theorem for the ample \(M|_D\)). Then the sections of \(M^m\) generate its fibres at every point of \(D\), by lifting generating sections from \(D\); and at every point outside \(D\) the section \(s^m\) does not vanish. So \(M^m\) is globally generated.

*Step 5: conclusion.* For every curve \(C\subseteq X\), \(\deg(M^m|_C)=mm_0\,(L\cdot C)>0\) by the numerical condition with \(V=C\). Lemma 1.3 shows that \(M^m=L^{mm_0}\) is ample, and so \(L\) is ample. \(\square\)

Step 3 uses only the top self-intersection \((L^n\cdot X)\), while Step 5 uses only the curves; all the intermediate dimensions enter through the induction in Step 2.

**Corollary 2.2.** Let \(L\) be invertible on \(X\). If \(L|_V\) satisfies the numerical condition for every irreducible component \(V\) of \(X\), then \(L\) is ample. On a curve, \(L\) is ample if and only if its degree on every irreducible component is positive. On an integral surface, \(L\) is ample if and only if \((L^2)>0\) and \(\deg(L|_C)>0\) for every curve \(C\).

**Proof.** These are the cases of Theorem 2.1 by dimension. \(\square\)

## 3. Exercises

**Exercise 3.1.** Show that \(\mathcal O(a)\) on \(\mathbf P^n_k\) is ample if and only if \(a>0\), using Theorem 2.1 and a line in \(\mathbf P^n\).

**Exercise 3.2.** Let \(f:X\to Y\) be a finite surjective morphism of integral projective \(k\)-schemes and \(M\) invertible on \(Y\). Using Proposition 3.1 of the previous lesson, show that \(f^*M\) satisfies the numerical condition of Theorem 2.1 if \(M\) does. Compare with Lemma 1.2.

**Exercise 3.3.** On \(X=\mathbf P^1\times\mathbf P^1\), show that \(\mathcal O(a,b)\) is ample if and only if \(a>0\) and \(b>0\), using Theorem 2.1.

**Exercise 3.4.** Show that on \(\mathbf P^2\) the bundle \(\mathcal O(-3)\) has \(h^0=0\) and \(\chi=1\). Which cohomology group is responsible, and how does Step 3 of the proof avoid the false inequality \(h^0\ge\chi\)? Show that on a curve \(h^0\ge\chi\) always holds.

**Exercise 3.5.** Let \(L\) be invertible on \(X\) and suppose that \(L^r\) is globally generated for some \(r\ge1\). Show that \(L\) is ample if and only if \(\deg(L|_C)>0\) for every curve \(C\subseteq X\).

## 4. Solutions

**3.1.** The only integral subvarieties to test are the positive-dimensional \(V\subseteq\mathbf P^n\). If \(a>0\), \(\mathcal O(a)\) is very ample and Theorem 3.4 of the previous lesson applies. If \(a\le0\), a line \(\ell\) has \((\mathcal O(a)\cdot\ell)=a\le0\).

**3.2.** For an integral \(W\subseteq X\) of dimension \(d\ge1\), \(f(W)\) is integral of dimension \(d\), since a finite morphism has finite fibres, and \((f^*M^d\cdot W)\) equals \([k(W):k(f(W))]\,(M^d\cdot f(W))>0\).

**3.3.** For an integral curve \(C\subseteq X\) put \(c=(\mathcal O(1,0)\cdot C)\) and \(d=(\mathcal O(0,1)\cdot C)\). Both bundles are globally generated, so some section does not vanish identically on \(C\), and Corollary 2.5 of the previous lesson gives \(c,d\ge0\); and \(c+d=(\mathcal O(1,1)\cdot C)>0\) because \(\mathcal O(1,1)\) is very ample. By multilinearity \((\mathcal O(a,b)\cdot C)=ac+bd\), and \((\mathcal O(a,b)^2)=2ab\) (Exercise 5.1 of the previous lesson). If \(a,b>0\), all these numbers are positive and Theorem 2.1 applies. If \(a\le0\), the curve \(C=\mathbf P^1\times\{q\}\) has \(c=1\), \(d=0\) and degree \(a\le0\); similarly if \(b\le0\).

**3.4.** \(\chi(\mathcal O(m))=(m+1)(m+2)/2\) for all \(m\), which is \(1\) at \(m=-3\), while \(H^0(\mathcal O(-3))=H^1(\mathcal O(-3))=0\) and \(H^2(\mathbf P^2,\mathcal O(-3))\) is one-dimensional, by the cohomology of projective space. So \(h^2\) is responsible. Step 3 only uses that \(h^i(L^m)\) is bounded for \(i\ge2\), which suffices because \(\chi(L^m)\to\infty\). On a curve \(h^i=0\) for \(i\ge2\), so \(\chi=h^0-h^1\le h^0\).

**3.5.** If \(L\) is ample, its degree on curves is positive. Conversely, \(L^r\) is globally generated and positive on curves, so it is ample by Lemma 1.3, and hence \(L\) is ample.

## References

- [Vakil] R. Vakil, The Rising Sea: Foundations of Algebraic Geometry, public draft of 21 October 2025. https://math.stanford.edu/~vakil/216blog/
- [Stacks] The Stacks project authors, The Stacks project, chapter Varieties, section Numerical intersections. https://stacks.math.columbia.edu/tag/0BEL