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The Nakai–Moishezon criterion says that a Cartier divisor $D$ on a projective scheme is ample exactly whenIn particular the test includes each positive-dimensional irreducible component. Here and below, “proper” means proper over the ground field, not necessarily a strict subset of $X$. Interpreting it as a strict subset would make the later criteria false even for an integral projective curve, whose strict closed subvarieties have dimension zero.

The intersection product of Cartier divisors with cycles depends only on their numerical equivalence of divisors classes. One way to see this is to intersect all but one factor first, obtaining a one-cycle; replacing the remaining factor by a numerically equivalent divisor does not change its pairing with that cycle. Multilinearity then handles replacement of every factor. Consequently $D\equiv D^{\prime }$ makes all the displayed numbers equal. ThusThe criterion applies to the integral reduced subvarieties of a possibly nonreduced scheme; ampleness on reduced components explains why the nilpotent structure does not change this condition.

Choose an ample Cartier divisor $H$. Put initially $A=\delta H$, $A^{\prime }=tH$ for small positive real $\delta ,t$. Since $D$ is nef, the nef-plus-ample ampleness lemma makes $D+tH$ ample. The polynomialhas $P(0,0)=D^n\cdot [X]>0$. Thus the desired strict inequality holds when $\delta ,t$ are sufficiently small and positive.

We must also arrange rationality of the two specified classes; $D$ itself need not be rational. Choose rational ample classes $B$ and $C$ sufficiently near $D+tH$ and $(\delta +t)H$, and defineThen $A^{\prime }$ is close to $tH$ and $A$ is close to $\delta H$, so both are ample real divisors by openness of the ample cone. Also $D+A^{\prime }=B$ and $A+A^{\prime }=C$ are rational and ample. Continuity preserves the strict inequality, giving

Here is the needed algebraic Morse inequality for ample divisors, with its section-count proof. Choose rational Cartier divisor representatives of $B,C$ and a common positive integer $q$ making $qB,qC$ very ample integral Cartier divisors. For the section-count argument rename these scaled representatives $B,C$; undoing this scaling restricts section indices to sufficiently divisible multiples and leaves bigness unchanged. Choose an effective Cartier divisor $G\in |C|$ by taking a defining section that avoids the associated points of ${\mathcal{O}}_X$. Repeated divisor restriction exact sequences giveBecause $C$ is very ample, for each $j$ a section of $jC$ avoiding the finitely many associated points of ${\mathcal{O}}_G$ gives an injection into ${\mathcal{O}}_G(mB)$. Thus every summand is at most $h^0(G,{\mathcal{O}}_G(mB))$. By Serre vanishing and asymptotic Riemann–Roch for the ample $B$,For $n=1$, the restriction term is the constant length of $G$, giving the same formula directly. The positive coefficient proves that $B-C$, hence $D-A$, is big. Scaling back preserves bigness, so

For a general projective scheme, enforce the same strict inequality separately on each positive-dimensional reduced irreducible component $V_j$, using its own dimension $n_j$. At $(\delta ,t)=(0,0)$ every such expression equals the positive number $D^{n_j}\cdot [V_j]$. Finitely many conditions are preserved by one sufficiently small choice and one sufficiently close rational approximation on $N^1(X{)}_{\mathbb{R}}$. The top-dimensional inequalities imply the printed inequality for $[X]$ with its positive generic multiplicities; the section proof on each component makes $D-A$ componentwise big. This avoids inferring bigness on every component from just a positive sum. The displayed inequality is used for $n\ge 1$. For $n=0$, its literal $n-1$ intersection power is undefined; handle this vacuous positivity case separately. Every line bundle is ample, all numerical classes are zero, and the componentwise bigness convention makes the conclusions automatic.

Choose $0<ϵ\ll 1$ as in (c). Thenis the sum of a nef divisor and an ample real divisor. The nef-plus-ample ampleness lemma givesFor clarity, this last lemma follows from Kleiman's criterion and the convex cone property: if $x$ is an interior point of the nef cone and $y$ lies in that cone, translating a small neighbourhood of $x$ by $y$ stays in the cone. Thus $x+y$ remains in its interior, which is the ample cone on a projective scheme.

The complete argument proves the real Nakai–Moishezon criterion rather than assuming it: curve positivity gives nefness, rational approximation and a proved section-count inequality give bigness, induction and the finite-support argument give a uniform ample subtraction, and the nef-plus-ample lemma concludes ampleness. The zero-dimensional case is automatic, and ampleness on reduced components handles reducibility and nilpotents.

First work on an integral component. Write the big real divisor as $D{\sim }_{\mathbb{R}}B+E$ with $B$ ample real divisor and $E$ effective real Cartier, using Kodaira's lemma. Let $E_1,\dots ,E_k$ be the finitely many integral components of its support. For an integral projective curve $C$ not contained in this support, restriction of each effective Cartier summand to $C$ is effective, soThereforeThis proves negative curves of a big real divisor lie in finitely many divisors.

Now let $A$ be the given ample divisor. Openness of the ample cone gives a $\delta >0$ such that $B-ϵA$ is ample for $0<ϵ\le \delta$. For each of the finitely many $E_i$, the assumption that $D{|}_{E_i}$ is ample similarly gives a ${\delta }_i>0$ such that $(D-ϵA){|}_{E_i}$ is ample for $0<ϵ\le {\delta }_i$. Choose a single positive $ϵ$ smaller than all these bounds.

If $C$ is contained in some $E_i$, its intersection with $D-ϵA$ is positive by that restriction. OtherwiseIn particular $D-ϵA$ is nef:Only the finitely many exceptional support components are needed for the restriction test; no uniform bound over all subvarieties was assumed.

For a reducible projective scheme, use componentwise bigness on a projective scheme and repeat this argument on each reduced irreducible component. Collect their exceptional supports and take the minimum of all the finitely many positive bounds. Codimension one here is measured in the relevant irreducible component. Every integral curve lies in a component, so the same conclusion holds on $X$. Nilpotent structure does not affect these curve intersection numbers.

An ample real divisor is a finite positive real combination of ample Cartier divisors:Equivalently its numerical class lies in the ample cone. This is a numerical condition even when the coefficients are irrational; it does not mean that some integer multiple of $D$ must be an integral divisor.

On an integral projective variety, a big real divisor is a finite positive real combination of big Cartier divisors. Equivalently, by the real form of Kodaira's lemma,For an integral Cartier divisor, bigness means maximal section-growth order $h^0(X,mD)\ge c{m}^{\dim X}$ along sufficiently divisible positive $m$, or Iitaka dimension $\dim X$. The real linear equivalence of divisors formulation permits finite positive combinations of effective Cartier divisors, whose supports are codimension one. The definitions and the ample-plus-effective formulation on integral varieties are discussed in Fujino's notes on big real divisors.

For the paper's assertions on a general projective scheme, use componentwise bigness on a projective scheme: require bigness on every reduced irreducible component. All arguments below can then be carried out on those finitely many integral components; ampleness is also detected there. On a reducible scheme, merely asking for maximal total section growth on one component is insufficient. For example $X={\mathbb{P}}^2⨿{\mathbb{P}}^2$, with $D$ restricting to $\mathcal{O}(1)$ on the first component and $\mathcal{O}(-1)$ on the second, has quadratic total section growth, but negative intersection with every line in the second component. No finite collection of codimension-one subvarieties can contain all those lines. Thus that weaker meaning would make part (ii) false. In dimension zero the positivity statements are vacuous and every line bundle is ample; the compatible bigness convention also regards it as big.

The Nakai–Moishezon criterion says that a Cartier divisor $D$ on a projective scheme is ample exactly whenfor every positive-dimensional integral closed subvariety $Y$. Kleiman's criterion says that the ample cone is the interior of the nef cone; equivalently, the numerical class of $D$ is ample exactly when it is strictly positive on every nonzero element of the closed cone of curves $\overline{\mathrm{NE}}(X)$. Positivity merely on individual curves is insufficient: the closure of the cone is essential. A nef divisor has nonnegative intersection number with every integral curve, and its restriction to every closed subscheme is nef.

For a projective scheme $X$ of dimension of a scheme $n$ and a Cartier divisor $D$, asymptotic Riemann–Roch givesHere Euler characteristic of a coherent sheaf means $\chi ={\sum }_{i=0}^n(-1{)}^i{h}^i$, and $D^n$ is the degree of the top intersection product with the fundamental cycle of $X$, including its component multiplicities. No ampleness assumption on $D$ is needed.

First suppose $X$ is integral. Induct on its dimension of a scheme. The previous two parts handle all $j>1$ and also handle $j=1$ when $h^0(X,mD)$ eventually vanishes. Otherwise choose $r>0$ with a nonzero global section of $rD$. On an integral variety it defines an effective Cartier divisor $E$, possibly empty, and multiplication by the section givesThus $h^1(X,mD)\le h^1(X,(m-r)D)+h^1(E,mD)$. For $n\ge 2$, the induction hypothesis bounds the last term by $O(m^{n-2})$; summing on each residue class modulo $r$ gives $O(m^{n-1})$. For $n=1$, $E$ is zero-dimensional and its positive-degree sheaf cohomology vanishes, so the same recurrence is bounded.

For a general projective scheme, a nonzero section can be a zero divisor, so that argument requires an additional step. The general cohomology growth for nef twists supplies it: for every coherent sheaf $\mathcal{F}$ with support dimension $d$,Its proof uses Fujita vanishing, an ample section avoiding the associated points of $\mathcal{F}$, and induction on support dimension. Taking $\mathcal{F}={\mathcal{O}}_X$ gives the required $O(m^{n-1})$ estimate for every $j\ge 1$, including nonreduced and reducible schemes; degrees above $n$ vanish.

For a coherent sheaf $\mathcal{F}$ of support dimension $d$ on a projective scheme and a fixed Cartier divisor $D$, $h^0(X,\mathcal{F}(mD))=O(m^d)$. Choose a sufficiently high ample divisor $A$ whose section avoids the associated points of $\mathcal{F}$ and for which $D+A$ is ample. Multiplication by the $m$th power of that section injects $\mathcal{F}(mD)$ into $\mathcal{F}(m(D+A))$. The ample Hilbert polynomial, together with Serre vanishing, bounds the latter section space by $O(m^d)$.

A real Cartier class on a projective scheme is ample if and only if its top self-intersection on every positive-dimensional integral subvariety is positive. For the converse curve tests give nefness. Small ample perturbations and rational approximation give ample rational $B,C$ with $D-A=B-C$ satisfying the algebraic Morse inequality for ample divisors, hence bigness. Induction gives ample restrictions on codimension-one subvarieties; uniform ample subtraction from a big divisor with ample exceptional restrictions makes $D-ϵA$ nef. The nef-plus-ample ampleness lemma concludes. For reducible schemes perform the finite component tests simultaneously.

If $E$ is an effective Cartier divisor on a projective scheme and ${\mathcal{O}}_E(E)$ is ample, then $E$ is semiample. The divisor restriction exact sequence and Serre vanishing make $H^1(X,(m-1)E)\to H^1(X,mE)$ surjective for large $m$. Their finite dimensions stabilize, so restriction on global sections is eventually surjective. Lift generators on $E$; off $E$, the canonical section of $mE$ generates. Together these generate ${\mathcal{O}}_X(mE)$, including on nonreduced $X$.

For an ample line bundle $A$ on a projective scheme and a coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F}\otimes A^m)=0$ for every $i>0$ and all sufficiently large $m$.

For ample line bundles $L,B$ and a coherent sheaf $\mathcal{F}$ on a projective scheme, one $M$ gives $H^i(X,\mathcal{F}\otimes L^m\otimes B^t)=0$ for every $i>0$, $m\ge M$ and $t\ge 0$. If $B$ is very ample, Serre vanishing for the finitely many fixed twists $\mathcal{F}(-iB)$ makes $\mathcal{F}\otimes L^m$ zero-regular. Persistence of Castelnuovo–Mumford regularity gives all nonnegative $B$-twists. For general ample $B$, use a very ample power and treat finitely many residues of $t$.