Ample line bundle

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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. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. 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 on X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space. A line bundle is ample if some positive power is very ample.

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

Introduction[edit]

Pullback of a line bundle and hyperplane divisors[edit]

Given a morphism of schemes, a vector bundle E on Y (or more generally a coherent sheaf on Y) has a pullback to X, (see Sheaf 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 at a point x in X is the fiber of E at f(x).)

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

with E = O(1) the line bundle on projective space whose global sections are the homogeneous polynomials of degree 1 (that is, linear functions) in variables . The line bundle O(1) can also be described as the line bundle associated to a hyperplane in (because the zero set of a section of O(1) is a hyperplane). If f is a closed immersion, for example, it follows that the pullback is the line bundle on X associated to a hyperplane section (the intersection of X with a hyperplane in ).

Basepoint-free line bundles[edit]

Let X be a scheme over a field k (for example, an algebraic variety) with a line bundle L. (A line bundle may also be called an invertible sheaf.) Let be elements of the k-vector space of global sections of L. The zero set of each section is a closed subset of X; let U 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 x of U, the fiber of L over x is a 1-dimensional vector space over the residue field k(x). Choosing a basis for this fiber makes into a sequence of 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 L to U is isomorphic to the pullback .[1]

The base locus of a line bundle L on a scheme X is the intersection of the zero sets of all global sections of L. A line bundle L is called basepoint-free if its base locus is empty. That is, for every point x of X there is a global section of L which is nonzero at x. If X is proper over a field k, then the vector space of global sections has finite dimension; the dimension is called .[2] So a basepoint-free line bundle L determines a morphism over k, where , given by choosing a basis for . Without making a choice, this can be described as the morphism

from X to the space of hyperplanes in , canonically associated to the basepoint-free line bundle L. This morphism has the property that L is the pullback .

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

Nef, globally generated, semi-ample[edit]

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 such that has nonnegative degree (because sections of L over C, 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 bundle L on any proper scheme X over a field is nef, meaning that L has nonnegative degree on every (irreducible) curve in X.[4]

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.[5] 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.[6] 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).[7]

Very ample line bundles[edit]

A line bundle L on a proper scheme X over a field k is said to be very ample if it is basepoint-free and the associated morphism

is a closed immersion. Here . Equivalently, L is very ample if X can be embedded into projective space of some dimension over k in such a way that L is the restriction of the line bundle O(1) to X.[8] The latter definition is used to define very ampleness for a line bundle on a proper scheme over any commutative ring.[9]

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

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

Definitions[edit]

A line bundle L on a proper scheme X over a commutative ring R is said to be ample if there is a positive integer r such that the tensor power is very ample.[12] In particular, a proper scheme over R has an ample line bundle if and only if it is projective over R. 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.)

On an arbitrary scheme X, Grothendieck defined a line bundle L to be ample if X is quasi-compact and for every point x in X there is a positive integer r and a section such that s is nonzero at x and the open subscheme is affine.[13] For example, the trivial line bundle is ample if and only if X is quasi-affine.[14] The rest of this article will concentrate on ampleness on proper schemes over a field.

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 is globally generated for all . Here s may depend on F.[15][16]

Another characterization of ampleness, known as the CartanSerreGrothendieck 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

for all and all .[17][16] 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[edit]

  • The trivial line bundle 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 over a field, the pullback line bundle 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).[18]
  • For an integer d, the space of sections of the line bundle O(d) over 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 , the morphism to projective space given by O(d) is
by
This is a closed immersion for , with image a rational normal curve of degree d in . Therefore, O(d) is basepoint-free if and only if , and very ample if and only if . It follows that O(d) is ample if and only if .
  • 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.[19] 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 (giving an embedding of X as an elliptic curve of degree d in ). 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 .
  • 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 ) over C, and let p and q be distinct complex points of X. Then the line bundle is ample but has .[20]

Criteria for ampleness of line bundles[edit]

Intersection theory[edit]

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 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 is very ample if and only if for all k-rational points x,y in X.[21] 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.[22]

For example, the canonical bundle of a curve X has degree 2g − 2, and so it is ample if and only if . 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 and the curve is not hyperelliptic.[23]

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 for every (irreducible) closed subvariety Y of X (Y is not allowed to be a point).[24] In terms of divisors, a Cartier divisor D is ample if and only if 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 has .

Kleiman's criterion[edit]

To state Kleiman's criterion (1966), let X be a projective scheme over a field. Let 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 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 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 . (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.[25]

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.[26].

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 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. [27] Every nef line bundle on a surface has , and Nagata and Mumford's examples have .

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.[28]

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.[29] The Cartan-Serre-Grothendieck criterion holds even more generally, for a proper algebraic space over a Noetherian ring R.[30] (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.[31]

Openness of ampleness[edit]

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 , 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.[32]) An elementary special case is: for an ample divisor H and any divisor E, there is a positive real number b such that 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 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 is ample on one fiber , then there is an affine open neighborhood U of y such that L is ample on over U.[33]

Kleiman's other characterizations of ampleness[edit]

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:[34]

  • L is ample.
  • For every (irreducible) subvariety of positive dimension, there is a positive integer r and a section which is not identically zero but vanishes at some point of Y.
  • For every (irreducible) subvariety of positive dimension, the holomorphic Euler characteristics of powers of L on Y go to infinity:
as .

Generalizations[edit]

Ample vector bundles[edit]

Robin Hartshorne defined a vector bundle F on a projective scheme X over a field to be ample if the line bundle on the space of hyperplanes in F is ample.[35]

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 of coherent sheaves for all .[36] Also, the Chern class of an ample vector bundle has positive degree on every r-dimensional subvariety of X, for .[37]

Big line bundles[edit]

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 such that for all . 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 for all j > 0.[38]

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 given by the sections of is birational onto its image.[39] 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 ).[40] Finally, a line bundle is big if and only if its class in is in the interior of the cone of effective divisors.[41]

Bigness can be viewed as a birationally invariant analog of ampleness. For example, if 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.[18]

Example: Let X be the blow-up of the projective plane at a point over the complex numbers. Let H be the pullback to X of a line on , and let E be the exceptional curve of the blow-up . 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[edit]

Given a quasi-compact morphism of schemes , an invertible sheaf L on X is said to be ample relative to f or f-ample if the following equivalent conditions are met:[42][43]

  1. For each open affine subset , the restriction of L to is ample (in the usual sense).
  2. f is quasi-separated and there is an open immersion induced by the adjunction map:
    .
  3. The condition 2. without "open".

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

See also[edit]

General algebraic geometry[edit]

Ampleness in complex geometry[edit]

Notes[edit]

  1. ^ Hartshorne (1977), Theorem II.7.1.
  2. ^ Hartshorne (1977), Theorem III.5.2; Stacks Project, Tag 02O6.
  3. ^ Hartshorne (1977), Lemma IV.1.2.
  4. ^ Lazarsfeld (2004), Example 1.4.5.
  5. ^ Stacks Project, Tag 01AM.
  6. ^ Hartshorne (1977), Example II.5.16.2.
  7. ^ Lazarsfeld (2004), Definition 2.1.26.
  8. ^ Hartshorne (1977), section II.5.
  9. ^ Stacks Project, Tag 02NP.
  10. ^ Grothendieck, EGA II, Definition 4.2.2.
  11. ^ Hartshorne (1977), Proposition I.7.6 and Example IV.3.3.2.
  12. ^ Stacks Project, Tag 01VU.
  13. ^ Stacks Project, Tag 01PS.
  14. ^ Stacks Project, Tag 01QE.
  15. ^ Hartshorne (1977), Theorem II.7.6
  16. ^ a b Lazarsfeld (2004), Theorem 1.2.6.
  17. ^ Hartshorne (1977), Proposition III.5.3
  18. ^ a b Lazarsfeld (2004), Theorem 1.2.13.
  19. ^ Hartshorne (1977), Example II.7.6.3.
  20. ^ Hartshorne (1977), Exercise IV.3.2(b).
  21. ^ Hartshorne (1977), Proposition IV.3.1.
  22. ^ Hartshorne (1977), Corollary IV.3.3.
  23. ^ Hartshorne (1977), Proposition IV.5.2.
  24. ^ Lazarsfeld (2004), Theorem 1.2.23, Remark 1.2.29; Kleiman (1966), Theorem III.1.
  25. ^ Lazarsfeld (2004), Theorems 1.4.23 and 1.4.29; Kleiman (1966), Theorem IV.1.
  26. ^ Fujino (2005), Corollary 3.3; Lazarsfeld (2004), Remark 1.4.24.
  27. ^ Lazarsfeld (2004), Example 1.5.2.
  28. ^ Lazarsfeld (2004), Theorem 1.4.13; Hartshorne (1970), Theorem I.7.1.
  29. ^ Kollár (1990), Theorem 3.11.
  30. ^ Stacks Project, Tag 0D38.
  31. ^ Kollár (1996), Chapter VI, Appendix, Exercise 2.19.3.
  32. ^ Lazarsfeld (2004), Definition 1.3.11.
  33. ^ Lazarsfeld (2004), Theorem 1.2.17 and its proof.
  34. ^ Lazarsfeld (2004), Example 1.2.32; Kleiman (1966), Theorem III.1.
  35. ^ Lazarsfeld (2004), Definition 6.1.1.
  36. ^ Lazarsfeld (2004), Theorem 6.1.10.
  37. ^ Lazarsfeld (2004), Theorem 8.2.2.
  38. ^ Lazarsfeld (2004), Corollary 2.1.38.
  39. ^ Lazarsfeld (2004), section 2.2.A.
  40. ^ Lazarsfeld (2004), Corollary 2.2.7.
  41. ^ Lazarsfeld (2004), Theorem 2.2.26.
  42. ^ https://stacks.math.columbia.edu/tag/01VG
  43. ^ EGA, Proposition 4.6.3.

References[edit]

External links[edit]