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". 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. Understanding the ample line bundles on a given variety amounts to understanding the different ways of mapping into projective spaces. In view of the correspondence between line bundles and divisors, 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 on a complete variety is very ample if it has enough sections to give a closed immersion of into a projective space. A line bundle is ample if some positive power is very ample.
An ample line bundle on a projective variety has positive degree on every curve in. 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

Given a morphism of schemes, a vector bundle has a pullback to, where the projection is the projection on the first coordinate. 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.
The notions described in this article are related to this construction in the case of a morphism to projective space
with the line bundle on projective space whose global sections are the homogeneous polynomials of degree 1 in variables. The line bundle can also be described as the line bundle associated to a hyperplane in . If is a closed immersion, for example, it follows that the pullback is the line bundle on associated to a hyperplane section.

Basepoint-free line bundles

Let be a scheme over a field with a line bundle. Let be elements of the -vector space of global sections of. The zero set of each section is a closed subset of ; let be the open subset of points at which at least one of is not zero. Then these sections define a morphism
In more detail: for each point of, the fiber of over is a 1-dimensional vector space over the residue field. Choosing a basis for this fiber makes into a sequence of 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 to is isomorphic to the pullback.
The base locus of a line bundle on a scheme is the intersection of the zero sets of all global sections of. A line bundle is called basepoint-free if its base locus is empty. That is, for every point of there is a global section of which is nonzero at. If is proper over a field, then the vector space of global sections has finite dimension; the dimension is called. So a basepoint-free line bundle determines a morphism over, where, given by choosing a basis for. Without making a choice, this can be described as the morphism
from to the space of hyperplanes in, canonically associated to the basepoint-free line bundle. This morphism has the property that is the pullback.
Conversely, for any morphism from a scheme to a projective space over, the pullback line bundle is basepoint-free. Indeed, is basepoint-free on, because for every point in there is a hyperplane not containing. Therefore, for every point in, there is a section of over that is not zero at, and the pullback of is a global section of that is not zero at. In short, basepoint-free line bundles are exactly those that can be expressed as the pullback of 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 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 such that has nonnegative degree. 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 curve in X.
More generally, a sheaf F of -modules on a scheme X is said to be globally generated if there is a set I of global sections 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.

Very ample line bundles

A line bundle on a proper scheme over a field is said to be very ample if it is basepoint-free and the associated morphism
is an immersion. Here. Equivalently, is very ample if can be embedded into a projective space of some dimension over in such a way that is the restriction of the line bundle to. 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.
For a very ample line bundle on a proper scheme over a field with associated morphism, the degree of on a curve in is the degree of as a curve in. So has positive degree on every curve in .

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 be an invertible sheaf on X. For each, let denote the ideal sheaf of the reduced subscheme supported only at x. For, define
Equivalently, if denotes the residue field at x, then
where is the image of s in the tensor product.
Fix. For every s, the restriction is a free -module trivialized by the restriction of s, meaning the multiplication-by-s morphism is an isomorphism. The set is always open, and the inclusion morphism is an affine morphism. Despite this, need not be an affine scheme. For example, if, then is open in itself and affine over itself but generally not affine.
Assume X is quasi-compact. Then is ample if, for every, there exists an and an such that and is an affine scheme. For example, the trivial line bundle is ample if and only if X is quasi-affine.
In general, it is not true that every is affine. For example, if for some point O, and if is the restriction of to X, then and have the same global sections, and the non-vanishing locus of a section of is affine if and only if the corresponding section of contains O.
It is necessary to allow powers of in the definition. In fact, for every N, it is possible that is non-affine for every with. Indeed, suppose Z is a finite set of points in,, and. The vanishing loci of the sections of 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 contains all the points of Z. In particular their non-vanishing loci are all non-affine.
Define. Let denote the structural morphism. There is a natural isomorphism between -algebra homomorphisms and endomorphisms of the graded ring S. The identity endomorphism of S corresponds to a homomorphism. Applying the functor produces a morphism from an open subscheme of X, denoted, to.
The basic characterization of ample invertible sheaves states that if X is a quasi-compact quasi-separated scheme and is an invertible sheaf on X, then the following assertions are equivalent:
  1. is ample.
  2. The open sets, where and, form a basis for the topology of X.
  3. The open sets with the property of being affine, where and, form a basis for the topology of X.
  4. and the morphism is a dominant open immersion.
  5. and the morphism is a homeomorphism of the underlying topological space of X with its image.
  6. For every quasi-coherent sheaf on X, the canonical map is surjective.
  7. For every quasi-coherent sheaf of ideals on X, the canonical map is surjective.
  8. For every quasi-coherent sheaf of ideals on X, the canonical map is surjective.
  9. For every quasi-coherent sheaf of finite type on X, there exists an integer such that for, is generated by its global sections.
  10. For every quasi-coherent sheaf of finite type on X, there exists integers and such that is isomorphic to a quotient of.
  11. For every quasi-coherent sheaf of ideals of finite type on X, there exists integers and such that is isomorphic to a quotient of.

    On proper schemes

When X is separated and finite type over an affine scheme, an invertible sheaf is ample if and only if there exists a positive integer r such that the tensor power 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 is ample.
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 is ample if and only if for every coherent sheaf F on X, there is an integer s such that the sheaf is globally generated for all. 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 is ample if and only if for every coherent sheaf F on X, there is an integer s such that
for all and all. 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.