Ample line bundle

Definitions

Ample invertible sheaves on quasi-compact schemes

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

Let X be a scheme, and let ${\displaystyle \mathcal{L}}$ be an invertible sheaf on X. For each ${\displaystyle x\in X}$, let ${\displaystyle {\mathfrak{m}}_x}$ denote the ideal sheaf of the reduced subscheme supported only at x. 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 at x (considered as a skyscraper sheaf supported at x), 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 of s in the tensor product.

Fix ${\displaystyle s\in \mathrm{\Gamma }(X,\mathcal{L})}$. For every s, the restriction ${\displaystyle \mathcal{L}{|}_{X_s}}$ is a free ${\displaystyle {\mathcal{O}}_X}$-module trivialized by the restriction of s, 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.

Assume X is quasi-compact. Then ${\displaystyle \mathcal{L}}$ is ample 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. For example, the trivial line bundle ${\displaystyle {\mathcal{O}}_X}$ is ample if and only if X is quasi-affine.

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 point O, and if ${\displaystyle \mathcal{L}}$ is the restriction of ${\displaystyle {\mathcal{O}}_{{\mathbf{P}}^2}(1)}$ to X, 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 ${\displaystyle {\mathcal{O}}_{{\mathbf{P}}^2}(1)}$ contains O.

It is necessary to allow powers of ${\displaystyle \mathcal{L}}$ in the definition. In fact, for every N, 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, suppose Z 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 degree N. By taking Z to be a sufficiently large set of points in general position, we may ensure that no plane curve of degree N (and hence any lower degree) contains all the points of Z. 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 ring S. The identity endomorphism of S corresponds to a homomorphism ${\displaystyle \epsilon }$. Applying the ${\displaystyle \mathrm{Proj}}$ functor produces a morphism from an open subscheme of X, denoted ${\displaystyle G(\epsilon )}$, to ${\displaystyle \mathrm{Proj}S}$.

The basic characterization of ample invertible sheaves states that if X is a quasi-compact quasi-separated scheme and ${\displaystyle \mathcal{L}}$ is an invertible sheaf on X, then the following assertions are equivalent:

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 of X.
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 of X.
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 of X with its image.
6. For every quasi-coherent sheaf ${\displaystyle \mathcal{F}}$ on X, 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 ${\displaystyle \mathcal{J}}$ on X, 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 ${\displaystyle \mathcal{F}}$ of finite type on X, 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.
9. For every quasi-coherent sheaf ${\displaystyle \mathcal{F}}$ of finite type on X, there exists integers ${\displaystyle n>0}$ and ${\displaystyle k>0}$ such that ${\displaystyle \mathcal{F}}$ is isomorphic to a quotient of ${\displaystyle {\mathcal{L}}^{\otimes (-n)}\otimes {\mathcal{O}}_X^k}$.
10. For every quasi-coherent sheaf of ideals ${\displaystyle \mathcal{J}}$ of finite type on X, there exists integers ${\displaystyle n>0}$ and ${\displaystyle k>0}$ such that ${\displaystyle \mathcal{J}}$ is isomorphic to a quotient of ${\displaystyle {\mathcal{L}}^{\otimes (-n)}\otimes {\mathcal{O}}_X^k}$.

On proper schemes

When X 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 integer r such that the tensor power ${\displaystyle {\mathcal{L}}^{\otimes r}}$ is very ample. In particular, a proper scheme over R has an ample line bundle if and only if it is projective over R. 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 scheme X over a field has positive degree on every curve in X, by the corresponding statement for very ample line bundles.

A Cartier divisor D on a proper scheme X over a field k is said to be ample if the corresponding line bundle O(D) is ample. (For example, if X is smooth over k, then a Cartier divisor can be identified with a finite linear combination of closed codimension-1 subvarieties of X 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 bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that the sheaf ${\displaystyle F\otimes L^{\otimes r}}$ is globally generated for all ${\displaystyle r\ge s}$. Here s may depend on F.

Another characterization of ampleness, known as the Cartan–Serre–Grothendieck theorem, is in terms of coherent sheaf cohomology. Namely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that

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

for all ${\displaystyle i>0}$ and all ${\displaystyle r\ge s}$. In particular, high powers of an ample line bundle kill cohomology in positive degrees. This implication is called the Serre vanishing theorem, proved by Jean-Pierre Serre in his 1955 paper Faisceaux algébriques cohérents.

Examples/Non-examples

• The trivial line bundle ${\displaystyle O_X}$ on a projective variety X of positive dimension is basepoint-free but not ample. More generally, for any morphism f from a projective variety X 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, whereas L is ample if and only if the morphism f is finite (that is, all fibers of f have dimension 0 or are empty).
• For an integer d, the space of sections of the line bundle O(d) over ${\displaystyle {\mathbb{P}}_{\mathbb{C}}^1}$ is the complex vector space of homogeneous polynomials of degree d in variables x,y. In particular, this space is zero for d < 0. For ${\displaystyle d\ge 0}$, the morphism to projective space given by O(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 a rational normal curve of degree d 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 that O(d) is ample if and only if ${\displaystyle d\ge 1}$.

• For an example where "ample" and "very ample" are different, let X be a smooth projective curve of genus 1 (an elliptic curve) over C, and let p be a complex point of X. Let O(p) be the associated line bundle of degree 1 on X. Then the complex vector space of global sections of O(p) has dimension 1, spanned by a section that vanishes at p. So the base locus of O(p) is equal to p. On the other hand, O(2p) is basepoint-free, and O(dp) is very ample for ${\displaystyle d\ge 3}$ (giving an embedding of X as an elliptic curve of degree d 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 a ramified double cover ${\displaystyle X\to {\mathbb{P}}^1}$.
• On curves of higher genus, there are ample line bundles L for which every global section is zero. (But high multiples of L have many sections, by definition.) For example, let X be a smooth plane quartic curve (of degree 4 in ${\displaystyle {\mathbb{P}}^2}$) over C, and let p and q be distinct complex points of X. Then the line bundle ${\displaystyle L=O(2p-q)}$ is ample but has ${\displaystyle H^0(X,L)=0}$.

Criteria for ampleness of line bundles

Intersection theory

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

On a smooth projective curve X over an algebraically closed field k, a line bundle L is very ample if and only if ${\displaystyle h^0(X,L\otimes O(-x-y))=h^0(X,L)-2}$ for all k-rational points x,y in X. Let g be the genus of X. By the Riemann–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.

For example, the canonical bundle ${\displaystyle K_X}$ of a curve X 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 negative curvature. The canonical bundle is very ample if and only if ${\displaystyle g\ge 2}$ and the curve is not hyperelliptic.

The Nakai–Moishezon criterion (named for Yoshikazu Nakai (1963) and Boris Moishezon (1964)) states that a line bundle L on a proper scheme X over a field is ample if and only if ${\displaystyle {\int }_Y{c}_1(L{)}^{\text{dim}(Y)}>0}$ for every (irreducible) closed subvariety Y of X (Y is not allowed to be a point). In terms of divisors, a Cartier divisor D is ample if and only if ${\displaystyle D^{\text{dim}(Y)}\cdot Y>0}$ for every (nonzero-dimensional) subvariety Y of X. For X a curve, this says that a divisor is ample if and only if it has positive degree. For X a surface, the criterion says that a divisor D is ample if and only if its self-intersection number ${\displaystyle D^2}$ is positive and every curve C on X has ${\displaystyle D\cdot C>0}$.

Kleiman's criterion

To state Kleiman's criterion (1966), let X be a projective scheme over a field. Let ${\displaystyle N_1(X)}$ be the real vector space of 1-cycles (real linear combinations of curves in X) modulo numerical equivalence, meaning that two 1-cycles A and B are equal in ${\displaystyle N_1(X)}$ if and only if every line bundle has the same degree on A and on B. By the Néron–Severi theorem, the real vector space ${\displaystyle N_1(X)}$ has finite dimension. Kleiman's criterion states that a line bundle L on X is ample if and only if L has positive degree on every nonzero element C of the closure of the cone of curves NE(X) in ${\displaystyle N_1(X)}$. (This is slightly stronger than saying that L has positive degree on every curve.) Equivalently, a line bundle is ample if and only if its class in the dual vector space ${\displaystyle N^1(X)}$ is in the interior of the nef cone.

Kleiman's criterion fails in general for proper (rather than projective) schemes X over a field, although it holds if X is smooth or more generally Q-factorial.

A line bundle on a projective variety is called strictly nef if it has positive degree on every curve. Nagata (1959) and David 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. 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 bundle L 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) curves C in X, where m(C) is the maximum of the multiplicities at the points of C.

Several characterizations of ampleness hold more generally for line bundles on a proper algebraic space over a field k. In particular, the Nakai-Moishezon criterion is valid in that generality. The Cartan-Serre-Grothendieck criterion holds even more generally, for a proper algebraic space over a Noetherian ring R. (If a proper algebraic space over R has an ample line bundle, then it is in fact a projective scheme over R.) Kleiman's criterion fails for proper algebraic spaces X over a field, even if X is smooth.

Openness of ampleness

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

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 let L be a line bundle on X. Then the set of points y in Y such that L is ample on the fiber ${\displaystyle X_y}$ is open (in the Zariski topology). More strongly, if L is ample on one fiber ${\displaystyle X_y}$, then there is an affine open neighborhood U of y such that L is ample on ${\displaystyle f^{-1}(U)}$ over U.

Kleiman's other characterizations of ampleness

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 bundle L on a proper scheme X over a field, the following are equivalent:

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

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

Generalizations

Ample vector bundles

Robin Hartshorne defined a vector bundle F on a projective scheme X over a field to be ample if the line bundle ${\displaystyle \mathcal{O}(1)}$ on the space ${\displaystyle \mathbb{P}(F)}$ of hyperplanes in F is ample.

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

Big line bundles

A useful weakening of ampleness, notably in birational geometry, is the notion of a big line bundle. A line bundle L on a projective variety X of dimension n over a field is said to be big if there is a positive real number a 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 of L, in the sense that for every line bundle L on X there is a positive number b with ${\displaystyle h^0(X,L^{\otimes j})\le b{j}^n}$ for all j > 0.

There are several other characterizations of big line bundles. First, a line bundle is big if and only if there is a positive integer r such that the rational map from X to ${\displaystyle \mathbb{P}(H^0(X,L^{\otimes r}))}$ given by the sections of ${\displaystyle L^{\otimes r}}$ is birational onto its image. Also, a line bundle L is big if and only if it has a positive tensor power which is the tensor product of an ample line bundle A and an effective line bundle B (meaning that ${\displaystyle H^0(X,B)\ne 0}$). 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.

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 on Y is big on X. (At first sight, the pullback is only a line bundle on the open subset of X where f is a morphism, but this extends uniquely to a line bundle on all of X.) For ample line bundles, one can only say that the pullback of an ample line bundle by a finite morphism is ample.

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

${\displaystyle (H+E)\cdot E=E^2=-1<0.}$

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

Relative ampleness

Given a quasi-compact morphism of schemes ${\displaystyle f:X\to S}$, an invertible sheaf L on X is said to be ample relative to f or f-ample if the following equivalent conditions are met:

1. For each open affine subset ${\displaystyle U\subset S}$, the restriction of L to ${\displaystyle f^{-1}(U)}$ is ample (in the usual sense).
2. f is quasi-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 the adjunction 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) that X can be openly compactified to a projective scheme with ${\displaystyle \mathcal{O}(1)=L}$ (not just to a proper scheme).