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Wikipedia / Ample line bundle

In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective 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 global sections into projective spaces. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor.

In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bundle is very ample if it has enough sections to give a closed immersion into a projective space. A line bundle is ample if some positive power is very ample.

has positive degree on every curve. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness.

Introduction

Pullback of a line bundle and hyperplane divisors

of schemes, a vector bundle (or more generally a coherent sheaf) has a pullback is the projection on the first coordinate (see Sheaf of modules#Operations.)

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

the line bundle on projective space whose global sections are the homogeneous polynomials can also be described as the line bundle associated to a hyperplane).

Basepoint-free line bundles

be a scheme over a field. (A line bundle may also be called an invertible sheaf-vector space of global sections is not zero. Then these sections define a morphism

is a 1-dimensional vector space over the residue field 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.

The base locus is called basepoint-free is proper.. Without making a choice, this can be described as the morphism

by some morphism to a projective space.

Nef, globally generated, semi-ample

The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C has nonnegative degree (because sections of L over C, as opposed to rational sections, have no poles). In particular, every basepoint-free line bundle on a curve has nonnegative degree. As a result, a basepoint-free line bundle L on any proper scheme X over a field is nef, meaning that L has nonnegative degree on every (irreducible) curve in X.

More generally, a sheaf F-modules on a scheme X is said to be globally generated if there is a set I such that the corresponding morphism

of sheaves is surjective. A line bundle is globally generated if and only if it is basepoint-free.

For example, every quasi-coherent sheaf on an affine scheme is globally generated. Analogously, in complex geometry, Cartan's theorem A says that every coherent sheaf on a Stein manifold is globally generated.

A line bundle L on a proper scheme over a field is semi-ample if there is a positive integer r such that the tensor power is basepoint-free. A semi-ample line bundle is nef (by the corresponding fact for basepoint-free line bundles).

Very ample line bundles

is said to be very ample if it is basepoint-free and the associated morphism

. The latter definition is used to define very ampleness for a line bundle on a proper scheme over any commutative ring.

The name "very ample" was introduced by Alexander Grothendieck in 1961. Various names had been used earlier in the context of linear systems of divisors.

is the degree (because every subvariety of projective space has positive degree).

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 an invertible sheaf on X denote the ideal sheaf of the reduced subscheme supported only at x denotes the residue field at x (considered as a skyscraper sheaf supported at x is the image of s in the tensor product.

. For every s-module trivialized by the restriction of s is open in itself and affine over itself but generally not affine.

Assume X is ample is an affine scheme. is ample if and only if X is quasi-affine.

for some point O to X contains O.

in the definition. In fact, for every N. Indeed, suppose Z is a finite set 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.

and endomorphisms of the graded ring S. The identity endomorphism of S functor produces a morphism from an open subscheme of X.

The basic characterization of ample invertible sheaves states that if X is an invertible sheaf on X, then the following assertions are equivalent:

is ample.
, form a basis for the topology of X.
, form a basis for the topology of X.
is a dominant open immersion.
is a homeomorphism of the underlying topological space of X with its image.
on X is surjective.
on X is surjective.
of finite type on X is generated by its global sections.
of finite type on X.
of finite type on X.

On proper schemes

When X is ample if and only if there exists a positive integer 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. 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

. 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

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 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 is the complex vector space of homogeneous polynomials of degree d in variables x,y. In particular, this space is zero for d, the morphism to projective space given by O(d) is
by
, with image a rational normal curve of degree d. Therefore, O(d. It follows that O(d.
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 (giving an embedding of X as an elliptic curve of degree d). 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.
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) over C, and let p and q be distinct complex points of X.

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 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 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 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 of a curve X has degree 2g. 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 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 for every (irreducible) closed subvariety Y of X (Y is not allowed to be a point). In terms of divisors, a Cartier divisor D 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 is positive and every curve C on X.

Kleiman's criterion

To state Kleiman's criterion (1966), let 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 if and only if every line bundle has the same degree on A and on B. By the Néron–Severi theorem 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. (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 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. and David Mumford 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..

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, 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 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.

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 is open (in the Zariski topology). More strongly, if L, then there is an affine open neighborhood U of y such that L 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.
of positive dimension, there is a positive integer r which is not identically zero but vanishes at some point of Y.
of positive dimension, the holomorphic Euler characteristics of powers of L on Y go to infinity:
1. .

Generalizations

Ample vector bundles

Robin Hartshorne defined a vector bundle F on a projective scheme X over a field to be ample 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. Also, the Chern class of an ample vector bundle has positive degree on every r-dimensional subvariety of X.

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. 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 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 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). is in the interior of the cone of effective divisors.

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 at a point over the complex numbers. Let H be the pullback to X, and let E. Then the divisor H + E is big but not ample (or even nef) on X, because

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, an invertible sheaf L on X is said to be ample relative to f or f-ample if the following equivalent conditions are met:

, the restriction of L is ample (in the usual sense).
f is quasi-separated induced by the adjunction map:
1. .
The condition 2. without "open".

The condition 2 says (roughly) that X can be openly compactified to a projective scheme (not just to a proper scheme).

Notes

Sources

External links

The Stacks Project