In mathematics, a distinctive feature ofalgebraic geometry is that someline bundles on aprojective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many globalsections. Understanding the ample line bundles on a given variety${\displaystyle X}$ amounts to understanding the different ways of mapping${\displaystyle X}$ intoprojective spaces. In view of the correspondence between line bundles anddivisors (built fromcodimension-1 subvarieties), there is an equivalent notion of anample divisor.

In more detail, a line bundle is calledbasepoint-free if it has enough sections to give amorphism to projective space. A line bundle issemi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, aline bundle on a complete variety${\displaystyle X}$ isvery ample if it has enough sections to give aclosed immersion (or "embedding") of${\displaystyle X}$ into a projective space. A line bundle isample if some positive power is very ample.

An ample line bundle on a projective variety${\displaystyle X}$ has positive degree on everycurve in${\displaystyle X}$. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness.

Given a morphism${\displaystyle f:X\to Y}$ ofschemes, avector bundle${\displaystyle p:E\to Y}$ (or more generally acoherent sheaf on${\displaystyle Y}$) has apullback to${\displaystyle X}$,${\displaystyle f^{\ast }E=\{(x,e)\in X\times E,f(x)=p(e)\}}$ where the projection${\displaystyle p^{\prime }:f^{\ast }E\to X}$ is the projection on the first coordinate (seeSheaf of modules#Operations). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber of${\displaystyle f^{\ast }E}$ at a point${\displaystyle x\in X}$ is the fiber of${\displaystyle E}$ at${\displaystyle f(x)\in Y}$.)

The notions described in this article are related to this construction in the case of a morphism to projective space

${\displaystyle f:X\to {\mathbb{P}}^n,}$

with${\displaystyle E=\mathcal{O}(1)}$ theline bundle on projective space whose global sections are thehomogeneous polynomials of degree 1 (that is, linear functions) in variables${\displaystyle x_0,\dots ,x_n}$. The line bundle${\displaystyle \mathcal{O}(1)}$ can also be described as the line bundle associated to ahyperplane in${\displaystyle {\mathbb{P}}^n}$ (because the zero set of a section of${\displaystyle \mathcal{O}(1)}$ is a hyperplane). If${\displaystyle f}$ is a closed immersion, for example, it follows that the pullback${\displaystyle f^{\ast }O(1)}$ is the line bundle on${\displaystyle X}$ associated to a hyperplane section (the intersection of${\displaystyle X}$ with a hyperplane in${\displaystyle {\mathbb{P}}^n}$).

Let${\displaystyle X}$ be a scheme over afield${\displaystyle k}$ (for example, an algebraic variety) with a line bundle${\displaystyle L}$. (A line bundle may also be called aninvertible sheaf.) Let${\displaystyle a_0,...,a_n}$ be elements of the${\displaystyle k}$-vector space${\displaystyle H^0(X,L)}$ ofglobal sections of${\displaystyle L}$. The zero set of each section is a closed subset of${\displaystyle X}$; let${\displaystyle U}$ be the open subset of points at which at least one of${\displaystyle a_0,\dots ,a_n}$ is not zero. Then these sections define a morphism

${\displaystyle f:U\to {\mathbb{P}}_k^n,x\mapsto [a_0(x),\dots ,a_n(x)].}$

In more detail: for each point${\displaystyle x}$ of${\displaystyle U}$, the fiber of${\displaystyle L}$ over${\displaystyle x}$ is a 1-dimensional vector space over theresidue field${\displaystyle k(x)}$. Choosing a basis for this fiber makes${\displaystyle a_0(x),\dots ,a_n(x)}$ into a sequence of${\displaystyle n+1}$ numbers, not all zero, and hence a point in projective space. Changing the choice of basis scales all the numbers by the same nonzero constant, and so the point in projective space is independent of the choice.

Moreover, this morphism has the property that the restriction of${\displaystyle L}$ to${\displaystyle U}$ is isomorphic to the pullback${\displaystyle f^{\ast }\mathcal{O}(1)}$.^[1]

Thebase locus of a line bundle${\displaystyle L}$ on a scheme${\displaystyle X}$ is the intersection of the zero sets of all global sections of${\displaystyle L}$. A line bundle${\displaystyle L}$ is calledbasepoint-free if its base locus is empty. That is, for every point${\displaystyle x}$ of${\displaystyle X}$ there is a global section of${\displaystyle L}$ which is nonzero at${\displaystyle x}$. If${\displaystyle X}$ isproper over a field${\displaystyle k}$, then the vector space${\displaystyle H^0(X,L)}$ of global sections has finite dimension; the dimension is called${\displaystyle h^0(X,L)}$.^[2] So a basepoint-free line bundle${\displaystyle L}$ determines a morphism${\displaystyle f:X\to {\mathbb{P}}^n}$ over${\displaystyle k}$, where${\displaystyle n=h^0(X,L)-1}$, given by choosing a basis for${\displaystyle H^0(X,L)}$. Without making a choice, this can be described as the morphism

${\displaystyle f:X\to \mathbb{P}(H^0(X,L))}$

from${\displaystyle X}$ to the space of hyperplanes in${\displaystyle H^0(X,L)}$, canonically associated to the basepoint-free line bundle${\displaystyle L}$. This morphism has the property that${\displaystyle L}$ is the pullback${\displaystyle f^{\ast }\mathcal{O}(1)}$.

Conversely, for any morphism${\displaystyle f}$ from a scheme${\displaystyle X}$ to a projective space${\displaystyle {\mathbb{P}}^n}$ over${\displaystyle k}$, the pullback line bundle${\displaystyle f^{\ast }\mathcal{O}(1)}$ is basepoint-free. Indeed,${\displaystyle \mathcal{O}(1)}$ is basepoint-free on${\displaystyle {\mathbb{P}}^n}$, because for every point${\displaystyle y}$ in${\displaystyle {\mathbb{P}}^n}$ there is a hyperplane not containing${\displaystyle y}$. Therefore, for every point${\displaystyle x}$ in${\displaystyle X}$, there is a section${\displaystyle s}$ of${\displaystyle \mathcal{O}(1)}$ over${\displaystyle {\mathbb{P}}^n}$ that is not zero at${\displaystyle f(x)}$, and the pullback of${\displaystyle s}$ is a global section of${\displaystyle f^{\ast }\mathcal{O}(1)}$ that is not zero at${\displaystyle x}$. In short, basepoint-free line bundles are exactly those that can be expressed as the pullback of${\displaystyle \mathcal{O}(1)}$ by some morphism to a projective space.

Thedegree of a line bundleL on a proper curveC overk is defined as the degree of the divisor (s) of any nonzero rational sections ofL. The coefficients of this divisor are positive at points wheres vanishes and negative wheres has a pole. Therefore, any line bundleL on a curveC such that${\displaystyle H^0(C,L)\ne 0}$ has nonnegative degree (because sections ofL overC, as opposed to rational sections, have no poles).^[3] In particular, every basepoint-free line bundle on a curve has nonnegative degree. As a result, a basepoint-free line bundleL on any proper schemeX over a field isnef, meaning thatL has nonnegative degree on every (irreducible) curve inX.^[4]

More generally, a sheafF of${\displaystyle O_X}$-modules on a schemeX is said to beglobally generated if there is a setI of global sections${\displaystyle s_i\in H^0(X,F)}$ such that the corresponding morphism

${\displaystyle \underset{i\in I}{⨁}{O}_X\to F}$

of sheaves is surjective.^[5] A line bundle is globally generatedif and only if it is basepoint-free.

For example, everyquasi-coherent sheaf on anaffine scheme is globally generated.^[6] Analogously, incomplex geometry,Cartan's theorem A says that every coherent sheaf on aStein manifold is globally generated.

A line bundleL on a proper scheme over a field issemi-ample if there is a positive integerr such that thetensor power${\displaystyle L^{\otimes r}}$ is basepoint-free. A semi-ample line bundle is nef (by the corresponding fact for basepoint-free line bundles).^[7]

A line bundle${\displaystyle L}$ on a proper scheme${\displaystyle X}$ over a field${\displaystyle k}$ is said to bevery ample if it is basepoint-free and the associated morphism

${\displaystyle f:X\to {\mathbb{P}}_k^n}$

is an immersion. Here${\displaystyle n=h^0(X,L)-1}$. Equivalently,${\displaystyle L}$ is very ample if${\displaystyle X}$ can be embedded into a projective space of some dimension over${\displaystyle k}$ in such a way that${\displaystyle L}$ is the restriction of the line bundle${\displaystyle \mathcal{O}(1)}$ to${\displaystyle X}$.^[8] The latter definition is used to define very ampleness for a line bundle on a proper scheme over anycommutative ring.^[9]

The name "very ample" was introduced byAlexander Grothendieck in 1961.^[10] Various names had been used earlier in the context oflinear systems of divisors.

For a very ample line bundle${\displaystyle L}$ on a proper scheme${\displaystyle X}$ over a field with associated morphism${\displaystyle f}$, the degree of${\displaystyle L}$ on a curve${\displaystyle C}$ in${\displaystyle X}$ is thedegree of${\displaystyle f(C)}$ as a curve in${\displaystyle {\mathbb{P}}^n}$. So${\displaystyle L}$ has positive degree on every curve in${\displaystyle X}$ (because every subvariety of projective space has positive degree).^[11]

Ample line bundles are used most often on proper schemes, but they can be defined in much wider generality.

LetX be a scheme, and let${\displaystyle \mathcal{L}}$ be an invertible sheaf onX. For each${\displaystyle x\in X}$, let${\displaystyle {\mathfrak{m}}_x}$ denote theideal sheaf of the reduced subscheme supported only atx. For${\displaystyle s\in \mathrm{\Gamma }(X,\mathcal{L})}$, define$${\displaystyle X_s=\{x\in X:s_x\notin {\mathfrak{m}}_x{\mathcal{L}}_x\}.}$$Equivalently, if${\displaystyle \kappa (x)}$ denotes the residue field atx (considered as a skyscraper sheaf supported atx), then$${\displaystyle X_s=\{x\in X:{\overline{s}}_x\ne 0\in \kappa (x)\otimes {\mathcal{L}}_x\},}$$where${\displaystyle {\overline{s}}_x}$ is the image ofs in the tensor product.

Fix${\displaystyle s\in \mathrm{\Gamma }(X,\mathcal{L})}$. For everys, the restriction${\displaystyle \mathcal{L}{|}_{X_s}}$ is a free${\displaystyle {\mathcal{O}}_X}$-module trivialized by the restriction ofs, meaning the multiplication-by-s morphism${\displaystyle {\mathcal{O}}_{X_s}\to \mathcal{L}{|}_{X_s}}$ is an isomorphism. The set${\displaystyle X_s}$ is always open, and the inclusion morphism${\displaystyle X_s\to X}$ is an affine morphism. Despite this,${\displaystyle X_s}$ need not be an affine scheme. For example, if${\displaystyle s=1\in \mathrm{\Gamma }(X,{\mathcal{O}}_X)}$, then${\displaystyle X_s=X}$ is open in itself and affine over itself but generally not affine.

AssumeX is quasi-compact. Then${\displaystyle \mathcal{L}}$ isample if, for every${\displaystyle x\in X}$, there exists an${\displaystyle n\ge 1}$ and an${\displaystyle s\in \mathrm{\Gamma }(X,{\mathcal{L}}^{\otimes n})}$ such that${\displaystyle x\in X_s}$ and${\displaystyle X_s}$ is an affine scheme.^[12] For example, the trivial line bundle${\displaystyle {\mathcal{O}}_X}$ is ample if and only ifX isquasi-affine.^[13]

In general, it is not true that every${\displaystyle X_s}$ is affine. For example, if${\displaystyle X={\mathbf{P}}^2\setminus \{O\}}$ for some pointO, and if${\displaystyle \mathcal{L}}$ is the restriction of${\displaystyle {\mathcal{O}}_{{\mathbf{P}}^2}(1)}$ toX, then${\displaystyle \mathcal{L}}$ and${\displaystyle {\mathcal{O}}_{{\mathbf{P}}^2}(1)}$ have the same global sections, and the non-vanishing locus of a section of${\displaystyle \mathcal{L}}$ is affine if and only if the corresponding section of${\displaystyle {\mathcal{O}}_{{\mathbf{P}}^2}(1)}$ containsO.

It is necessary to allow powers of${\displaystyle \mathcal{L}}$ in the definition. In fact, for everyN, it is possible that${\displaystyle X_s}$ is non-affine for every${\displaystyle s\in \mathrm{\Gamma }(X,{\mathcal{L}}^{\otimes n})}$ with${\displaystyle n\le N}$. Indeed, supposeZ is a finite set of points in${\displaystyle {\mathbf{P}}^2}$,${\displaystyle X={\mathbf{P}}^2\setminus Z}$, and${\displaystyle \mathcal{L}={\mathcal{O}}_{{\mathbf{P}}^2}(1){|}_X}$. The vanishing loci of the sections of${\displaystyle {\mathcal{L}}^{\otimes N}}$ are plane curves of degreeN. By takingZ to be a sufficiently large set of points ingeneral position, we may ensure that no plane curve of degreeN (and hence any lower degree) contains all the points ofZ. In particular their non-vanishing loci are all non-affine.

Define${\displaystyle S=\underset{n\ge 0}{⨁}\mathrm{\Gamma }(X,{\mathcal{L}}^{\otimes n})}$. Let${\displaystyle p:X\to \mathrm{Spec}\mathbf{Z}}$ denote the structural morphism. There is a natural isomorphism between${\displaystyle {\mathcal{O}}_X}$-algebra homomorphisms${\displaystyle p^{\ast }(\tilde{S})\to \underset{n\ge 0}{⨁}{\mathcal{L}}^{\otimes n}}$ and endomorphisms of the graded ringS. The identity endomorphism ofS corresponds to a homomorphism${\displaystyle \epsilon }$. Applying the${\displaystyle \mathrm{Proj}}$ functor produces a morphism from an open subscheme ofX, denoted${\displaystyle G(\epsilon )}$, to${\displaystyle \mathrm{Proj}S}$.

The basic characterization of ample invertible sheaves states that ifX is a quasi-compact quasi-separated scheme and${\displaystyle \mathcal{L}}$ is an invertible sheaf onX, then the following assertions are equivalent:^[14]

1. ${\displaystyle \mathcal{L}}$ is ample.
2. The open sets${\displaystyle X_s}$, where${\displaystyle s\in \mathrm{\Gamma }(X,{\mathcal{L}}^{\otimes n})}$ and${\displaystyle n\ge 0}$, form a basis for the topology ofX.
3. The open sets${\displaystyle X_s}$ with the property of being affine, where${\displaystyle s\in \mathrm{\Gamma }(X,{\mathcal{L}}^{\otimes n})}$ and${\displaystyle n\ge 0}$, form a basis for the topology ofX.
4. ${\displaystyle G(\epsilon )=X}$ and the morphism${\displaystyle G(\epsilon )\to \mathrm{Proj}S}$ is a dominant open immersion.
5. ${\displaystyle G(\epsilon )=X}$ and the morphism${\displaystyle G(\epsilon )\to \mathrm{Proj}S}$ is a homeomorphism of the underlying topological space ofX with its image.
6. For every quasi-coherent sheaf${\displaystyle \mathcal{F}}$ onX, the canonical map${\displaystyle \underset{n\ge 0}{⨁}\mathrm{\Gamma }(X,\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\otimes n}){\otimes }_{\mathbf{Z}}{\mathcal{L}}^{\otimes -n}\to \mathcal{F}}$ is surjective.
7. For every quasi-coherent sheaf of ideals${\displaystyle \mathcal{J}}$ onX, the canonical map${\displaystyle \underset{n\ge 0}{⨁}\mathrm{\Gamma }(X,\mathcal{J}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\otimes n}){\otimes }_{\mathbf{Z}}{\mathcal{L}}^{\otimes -n}\to \mathcal{J}}$ is surjective.
8. For every quasi-coherent sheaf of ideals${\displaystyle \mathcal{J}}$ onX, the canonical map${\displaystyle \underset{n\ge 0}{⨁}\mathrm{\Gamma }(X,\mathcal{J}{\otimes }_{{\mathcal{O}}_X}{\mathcal{L}}^{\otimes n}){\otimes }_{\mathbf{Z}}{\mathcal{L}}^{\otimes -n}\to \mathcal{J}}$ is surjective.
9. For every quasi-coherent sheaf${\displaystyle \mathcal{F}}$ of finite type onX, there exists an integer${\displaystyle n_0}$ such that for${\displaystyle n\ge n_0}$,${\displaystyle \mathcal{F}\otimes {\mathcal{L}}^{\otimes n}}$ is generated by its global sections.
10. For every quasi-coherent sheaf${\displaystyle \mathcal{F}}$ of finite type onX, there exists integers${\displaystyle n>0}$ and${\displaystyle k>0}$ such that${\displaystyle \mathcal{F}}$ is isomorphic to a quotient of${\displaystyle {\mathcal{L}}^{\otimes (-n)}\otimes {\mathcal{O}}_X^k}$.
11. For every quasi-coherent sheaf of ideals${\displaystyle \mathcal{J}}$ of finite type onX, there exists integers${\displaystyle n>0}$ and${\displaystyle k>0}$ such that${\displaystyle \mathcal{J}}$ is isomorphic to a quotient of${\displaystyle {\mathcal{L}}^{\otimes (-n)}\otimes {\mathcal{O}}_X^k}$.

WhenX is separated and finite type over an affine scheme, an invertible sheaf${\displaystyle \mathcal{L}}$ is ample if and only if there exists a positive integerr such that the tensor power${\displaystyle {\mathcal{L}}^{\otimes r}}$ is very ample.^[15][16] In particular, a proper scheme overR has an ample line bundle if and only if it is projective overR. Often, this characterization is taken as the definition of ampleness.

The rest of this article will concentrate on ampleness on proper schemes over a field, as this is the most important case. An ample line bundle on a proper schemeX over a field has positive degree on every curve inX, by the corresponding statement for very ample line bundles.

ACartier divisorD on a proper schemeX over a fieldk is said to be ample if the corresponding line bundleO(D) is ample. (For example, ifX is smooth overk, then a Cartier divisor can be identified with a finitelinear combination of closed codimension-1 subvarieties ofX with integer coefficients.)

Weakening the notion of "very ample" to "ample" gives a flexible concept with a wide variety of different characterizations. A first point is that tensoring high powers of an ample line bundle with any coherent sheaf whatsoever gives a sheaf with many global sections. More precisely, a line bundleL on a proper schemeX over a field (or more generally over aNoetherian ring) is ample if and only if for every coherent sheafF onX, there is an integers such that the sheaf${\displaystyle F\otimes L^{\otimes r}}$ is globally generated for all${\displaystyle r\ge s}$. Heres may depend onF.^[17][18]

Another characterization of ampleness, known as theCartan–Serre–Grothendieck theorem, is in terms ofcoherent sheaf cohomology. Namely, a line bundleL on a proper schemeX over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheafF onX, there is an integers such that

${\displaystyle H^i(X,F\otimes L^{\otimes r})=0}$

for all${\displaystyle i>0}$ and all${\displaystyle r\ge s}$.^[19][18] In particular, high powers of an ample line bundle kill cohomology in positive degrees. This implication is called theSerre vanishing theorem, proved byJean-Pierre Serre in his 1955 paperFaisceaux algébriques cohérents.

• The trivial line bundle${\displaystyle O_X}$ on a projective varietyX of positive dimension is basepoint-free but not ample. More generally, for any morphismf from a projective varietyX to some projective space${\displaystyle {\mathbb{P}}^n}$ over a field, the pullback line bundle${\displaystyle L=f^{\ast }O(1)}$ is always basepoint-free, whereasL is ample if and only if the morphismf isfinite (that is, all fibers off have dimension 0 or are empty).^[20]
• For an integerd, the space of sections of the line bundleO(d) over${\displaystyle {\mathbb{P}}_{\mathbb{C}}^1}$ is thecomplex vector space of homogeneous polynomials of degreed in variablesx,y. In particular, this space is zero ford < 0. For${\displaystyle d\ge 0}$, the morphism to projective space given byO(d) is

${\displaystyle {\mathbb{P}}^1\to {\mathbb{P}}^d}$

by

${\displaystyle [x,y]\mapsto [x^d,x^{d-1}y,\dots ,y^d].}$

This is a closed immersion for${\displaystyle d\ge 1}$, with image arational normal curve of degreed in${\displaystyle {\mathbb{P}}^d}$. Therefore,O(d) is basepoint-free if and only if${\displaystyle d\ge 0}$, and very ample if and only if${\displaystyle d\ge 1}$. It follows thatO(d) is ample if and only if${\displaystyle d\ge 1}$.

• For an example where "ample" and "very ample" are different, letX be a smooth projective curve ofgenus 1 (anelliptic curve) overC, and letp be a complex point ofX. LetO(p) be the associated line bundle of degree 1 onX. Then the complex vector space of global sections ofO(p) has dimension 1, spanned by a section that vanishes atp.^[21] So the base locus ofO(p) is equal top. On the other hand,O(2p) is basepoint-free, andO(dp) is very ample for${\displaystyle d\ge 3}$ (giving an embedding ofX as an elliptic curve of degreed in${\displaystyle {\mathbb{P}}^{d-1}}$). Therefore,O(p) is ample but not very ample. Also,O(2p) is ample and basepoint-free but not very ample; the associated morphism to projective space is aramified double cover${\displaystyle X\to {\mathbb{P}}^1}$.
• On curves of higher genus, there are ample line bundlesL for which every global section is zero. (But high multiples ofL have many sections, by definition.) For example, letX be a smooth plane quartic curve (of degree 4 in${\displaystyle {\mathbb{P}}^2}$) overC, and letp andq be distinct complex points ofX. Then the line bundle${\displaystyle L=O(2p-q)}$ is ample but has${\displaystyle H^0(X,L)=0}$.^[22]

To determine whether a given line bundle on a projective varietyX is ample, the followingnumerical criteria (in terms of intersection numbers) are often the most useful. It is equivalent to ask when a Cartier divisorD onX is ample, meaning that the associated line bundleO(D) is ample. The intersection number${\displaystyle D\cdot C}$ can be defined as the degree of the line bundleO(D) restricted toC. In the other direction, for a line bundleL on a projective variety, thefirst Chern class${\displaystyle c_1(L)}$ means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section ofL.

On asmooth projective curveX over analgebraically closed fieldk, a line bundleL is very ample if and only if${\displaystyle h^0(X,L\otimes O(-x-y))=h^0(X,L)-2}$ for allk-rational pointsx,y inX.^[23] Letg be the genus ofX. By theRiemann–Roch theorem, every line bundle of degree at least 2g + 1 satisfies this condition and hence is very ample. As a result, a line bundle on a curve is ample if and only if it has positive degree.^[24]

For example, thecanonical bundle${\displaystyle K_X}$ of a curveX has degree 2g − 2, and so it is ample if and only if${\displaystyle g\ge 2}$. The curves with ample canonical bundle form an important class; for example, over the complex numbers, these are the curves with a metric of negativecurvature. The canonical bundle is very ample if and only if${\displaystyle g\ge 2}$ and the curve is nothyperelliptic.^[25]

TheNakai–Moishezon criterion (named for Yoshikazu Nakai (1963) andBoris Moishezon (1964)) states that a line bundleL on a proper schemeX over a field is ample if and only if${\displaystyle {\int }_Y{c}_1(L{)}^{\text{dim}(Y)}>0}$ for every (irreducible) closed subvarietyY ofX (Y is not allowed to be a point).^[26] In terms of divisors, a Cartier divisorD is ample if and only if${\displaystyle D^{\text{dim}(Y)}\cdot Y>0}$ for every (nonzero-dimensional) subvarietyY ofX. ForX a curve, this says that a divisor is ample if and only if it has positive degree. ForX a surface, the criterion says that a divisorD is ample if and only if itsself-intersection number${\displaystyle D^2}$ is positive and every curveC onX has${\displaystyle D\cdot C>0}$.

To stateKleiman's criterion (1966), letX be a projective scheme over a field. Let${\displaystyle N_1(X)}$ be thereal vector space of 1-cycles (real linear combinations of curves inX) modulo numerical equivalence, meaning that two 1-cyclesA andB are equal in${\displaystyle N_1(X)}$ if and only if every line bundle has the same degree onA and onB. By theNéron–Severi theorem, the real vector space${\displaystyle N_1(X)}$ has finite dimension. Kleiman's criterion states that a line bundleL onX is ample if and only ifL has positive degree on every nonzero elementC of theclosure of thecone of curves NE(X) in${\displaystyle N_1(X)}$. (This is slightly stronger than saying thatL has positive degree on every curve.) Equivalently, a line bundle is ample if and only if its class in thedual vector space${\displaystyle N^1(X)}$ is in the interior of thenef cone.^[27]

Kleiman's criterion fails in general for proper (rather than projective) schemesX over a field, although it holds ifX is smooth or more generallyQ-factorial.^[28]

A line bundle on a projective variety is calledstrictly nef if it has positive degree on every curve.Nagata (1959) andDavid Mumford constructed line bundles on smooth projective surfaces that are strictly nef but not ample. This shows that the condition${\displaystyle c_1(L{)}^2>0}$ cannot be omitted in the Nakai–Moishezon criterion, and it is necessary to use the closure of NE(X) rather than NE(X) in Kleiman's criterion.^[29] Every nef line bundle on a surface has${\displaystyle c_1(L{)}^2\ge 0}$, and Nagata and Mumford's examples have${\displaystyle c_1(L{)}^2=0}$.

C. S. Seshadri showed that a line bundleL on a proper scheme over an algebraically closed field is ample if and only if there is a positive real number ε such that deg(L|_C) ≥ εm(C) for all (irreducible) curvesC inX, wherem(C) is the maximum of the multiplicities at the points ofC.^[30]

Several characterizations of ampleness hold more generally for line bundles on a properalgebraic space over a fieldk. In particular, the Nakai-Moishezon criterion is valid in that generality.^[31] The Cartan-Serre-Grothendieck criterion holds even more generally, for a proper algebraic space over a Noetherian ringR.^[32] (If a proper algebraic space overR has an ample line bundle, then it is in fact a projective scheme overR.) Kleiman's criterion fails for proper algebraic spacesX over a field, even ifX is smooth.^[33]

On a projective schemeX over a field, Kleiman's criterion implies that ampleness is an open condition on the class of anR-divisor (anR-linear combination of Cartier divisors) in${\displaystyle N^1(X)}$, with its topology based on the topology of the real numbers. (AnR-divisor is defined to be ample if it can be written as a positive linear combination of ample Cartier divisors.^[34]) An elementary special case is: for an ample divisorH and any divisorE, there is a positive real numberb such that${\displaystyle H+aE}$ is ample for all real numbersa of absolute value less thanb. In terms of divisors with integer coefficients (or line bundles), this means thatnH +E is ample for all sufficiently large positive integersn.

Ampleness is also an open condition in a quite different sense, when the variety or line bundle is varied in an algebraic family. Namely, let${\displaystyle f:X\to Y}$ be a proper morphism of schemes, and letL be a line bundle onX. Then the set of pointsy inY such thatL is ample on thefiber${\displaystyle X_y}$ is open (in theZariski topology). More strongly, ifL is ample on one fiber${\displaystyle X_y}$, then there is an affine open neighborhoodU ofy such thatL is ample on${\displaystyle f^{-1}(U)}$ overU.^[35]

Kleiman also proved the following characterizations of ampleness, which can be viewed as intermediate steps between the definition of ampleness and numerical criteria. Namely, for a line bundleL on a proper schemeX over a field, the following are equivalent:^[36]

• L is ample.
• For every (irreducible) subvariety${\displaystyle Y\subset X}$ of positive dimension, there is a positive integerr and a section${\displaystyle s\in H^0(Y,{\mathcal{L}}^{\otimes r})}$ which is not identically zero but vanishes at some point ofY.
• For every (irreducible) subvariety${\displaystyle Y\subset X}$ of positive dimension, theholomorphic Euler characteristics of powers ofL onY go to infinity:

${\displaystyle \chi (Y,{\mathcal{L}}^{\otimes r})\to \mathrm{\infty }}$ as${\displaystyle r\to \mathrm{\infty }}$.

Robin Hartshorne defined avector bundleF on a projective schemeX over a field to beample if the line bundle${\displaystyle \mathcal{O}(1)}$ on the space${\displaystyle \mathbb{P}(F)}$ of hyperplanes inF is ample.^[37]

Several properties of ample line bundles extend to ample vector bundles. For example, a vector bundleF is ample if and only if high symmetric powers ofF kill the cohomology${\displaystyle H^i}$ of coherent sheaves for all${\displaystyle i>0}$.^[38] Also, theChern class${\displaystyle c_r(F)}$ of an ample vector bundle has positive degree on everyr-dimensional subvariety ofX, for${\displaystyle 1\le r\le \text{rank}(F)}$.^[39]

A useful weakening of ampleness, notably inbirational geometry, is the notion of abig line bundle. A line bundleL on a projective varietyX of dimensionn over a field is said to be big if there is a positive real numbera and a positive integer${\displaystyle j_0}$ such that${\displaystyle h^0(X,L^{\otimes j})\ge a{j}^n}$ for all${\displaystyle j\ge j_0}$. This is the maximum possible growth rate for the spaces of sections of powers ofL, in the sense that for every line bundleL onX there is a positive numberb with${\displaystyle h^0(X,L^{\otimes j})\le b{j}^n}$ for allj > 0.^[40]

There are several other characterizations of big line bundles. First, a line bundle is big if and only if there is a positive integerr such that therational map fromX to${\displaystyle \mathbb{P}(H^0(X,L^{\otimes r}))}$ given by the sections of${\displaystyle L^{\otimes r}}$ isbirational onto its image.^[41] Also, a line bundleL is big if and only if it has a positive tensor power which is the tensor product of an ample line bundleA and an effective line bundleB (meaning that${\displaystyle H^0(X,B)\ne 0}$).^[42] Finally, a line bundle is big if and only if its class in${\displaystyle N^1(X)}$ is in the interior of the cone of effective divisors.^[43]

Bigness can be viewed as a birationally invariant analog of ampleness. For example, if${\displaystyle f:X\to Y}$ is a dominant rational map between smooth projective varieties of the same dimension, then the pullback of a big line bundle onY is big onX. (At first sight, the pullback is only a line bundle on the open subset ofX wheref is a morphism, but this extends uniquely to a line bundle on all ofX.) For ample line bundles, one can only say that the pullback of an ample line bundle by a finite morphism is ample.^[20]

Example: LetX be theblow-up of the projective plane${\displaystyle {\mathbb{P}}^2}$ at a point over the complex numbers. LetH be the pullback toX of a line on${\displaystyle {\mathbb{P}}^2}$, and letE be the exceptional curve of the blow-up${\displaystyle \pi :X\to {\mathbb{P}}^2}$. Then the divisorH +E is big but not ample (or even nef) onX, because

This negativity also implies that the base locus ofH +E (or of any positive multiple) contains the curveE. In fact, this base locus is equal toE.

Given aquasi-compact morphism of schemes${\displaystyle f:X\to S}$, an invertible sheafL onX is said to beample relative tof orf-ample if the following equivalent conditions are met:^[44][45]

1. For each open affine subset${\displaystyle U\subset S}$, the restriction ofL to${\displaystyle f^{-1}(U)}$ isample (in the usual sense).
2. f isquasi-separated and there is an open immersion${\displaystyle X\hookrightarrow {\mathrm{Proj}}_S(\mathcal{R}),\mathcal{R}:=f_{\ast }\left(\underset{0}{\overset{\mathrm{\infty }}{⨁}}{L}^{\otimes n}\right)}$ induced by theadjunction map:

   ${\displaystyle f^{\ast }\mathcal{R}\to \underset{0}{\overset{\mathrm{\infty }}{⨁}}{L}^{\otimes n}}$.

3. The condition 2. without "open".

The condition 2 says (roughly) thatX can be openly compactified to aprojective scheme with${\displaystyle \mathcal{O}(1)=L}$ (not just to a proper scheme).

• Algebraic geometry of projective spaces
• Fano variety: a variety whose canonical bundle is anti-ample
• Matsusaka's big theorem
• Divisorial scheme: a scheme admitting an ample family of line bundles

• Holomorphic vector bundle
• Kodaira embedding theorem: on a compact complex manifold, ampleness and positivity coincide.
• Kodaira vanishing theorem
• Lefschetz hyperplane theorem: an ample divisor in a complex projective varietyX is topologically similar toX.

• The Stacks Project

• Algebraic geometry
• Geometry of divisors
• Vector bundles

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