“Ample line bundle”的意思、由来-开放百科全书

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Ample line bundle

释义

Introduction
Inverse image of line bundle and hyperplane divisors Sheaves generated by their global sections Very ample line bundles
Definitions
Examples/Non-examples
Criteria for ampleness of line bundles
Intersection theory Sheaf cohomology Kleiman's characterizations of amplitude
Generalizations
Vector bundles of higher rank Big line bundles
See also
General algebraic geometry Ampleness in complex geometry
References
Study references Research texts

In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold into projective space. An ample line bundle is one such that some positive power is very ample. Globally generated sheaves are those with enough sections to define a morphism to projective space.

Introduction

Inverse image of line bundle and hyperplane divisors

Given a morphism , any vector bundle on Y, or more generally any sheaf of modules, e.g. a coherent sheaf, can be pulled back to X, (see Inverse image functor). This construction preserves the condition of being a line bundle, and more generally the rank.

The notions described in this article are related to this construction in the case of morphisms to projective spaces

and ,

the line bundle corresponding to the hyperplane divisor, whose sections are the 1-homogeneous regular functions. See Algebraic geometry of projective spaces#Divisors and twisting sheaves.

Sheaves generated by their global sections

Let X be a scheme or a complex manifold and F a sheaf on X. One says that F is generated by (finitely many) global sections , if every stalk of F is generated as a module over the stalk of the structure sheaf by the germs of the a_i. For example, if F happens to be a line bundle, i.e. locally free of rank 1, this amounts to having finitely many global sections, such that for any point x in X, there is at least one section not vanishing at this point. In this case a choice of such global generators a₀, ..., a_n gives a morphism

such that the pullback f*(O(1)) is F (Note that this evaluation makes sense when F is a subsheaf of the constant sheaf of rational functions on X). The converse statement is also true: given such a morphism f, the pullback of O(1) is generated by its global sections (on X).

More generally, a sheaf generated by global sections is a sheaf F on a locally ringed space X, with structure sheaf O_X that is of a rather simple type. Assume F is a sheaf of abelian groups. Then it is asserted that if A is the abelian group of global sections, i.e.

then for any open set U of X, ρ(A) spans F(U) as an O_U-module. Here

is the restriction map. In words, all sections of F are locally generated by the global sections.

An example of such a sheaf is that associated in algebraic geometry to an R-module M, R being any commutative ring, on the spectrum of a ring Spec(R).

Another example: according to Cartan's theorem A, any coherent sheaf on a Stein manifold is spanned by global sections.

Very ample line bundles

Given a scheme X over a base scheme S or a complex manifold, a line bundle (or in other words an invertible sheaf, that is, a locally free sheaf of rank one) L on X is said to be very ample, if there is an embedding i : X → Pⁿ_S, the n-dimensional projective space over S for some n, such that the pullback of the standard twisting sheaf O(1) on Pⁿ_S is isomorphic to L:

Hence this notion is a special case of the previous one, namely a line bundle is very ample if it is globally generated and the morphism given by some global generators is an embedding.

Given a very ample sheaf L on X and a coherent sheaf F, a theorem of Serre shows that (the coherent sheaf) F ⊗ L^⊗n is generated by finitely many global sections for sufficiently large n. This in turn implies that global sections and higher (Zariski) cohomology groups

are finitely generated. This is a distinctive feature of the projective situation. For example, for the affine n-space Aⁿ_k over a field k, global sections of the structure sheaf O are polynomials in n variables, thus not a finitely generated k-vector space, whereas for Pⁿ_k, global sections are just constant functions, a one-dimensional k-vector space.

Definitions

The notion of ample line bundles L is slightly weaker than very ample line bundles: a line bundle L is ample if for any coherent sheaf F on X, there exists an integer n(F), such that F ⊗ L^⊗n is generated by its global sections for n > n(F).

An equivalent, maybe more intuitive, definition of the ampleness of the line bundle is its having a positive tensorial power that is very ample. In other words, for there exists a projective embedding such that , that is the zero divisors of global sections of

are hyperplane sections.

This definition makes sense for the underlying divisors (Cartier divisors) ; an ample is one where moves in a large enough linear system. Such divisors form a cone in all divisors of those that are, in some sense, positive enough. The relationship with projective space is that the for a very ample corresponds to the hyperplane sections (intersection with some hyperplane) of the embedded .

The equivalence between the two definitions is credited to Jean-Pierre Serre in Faisceaux algébriques cohérents.

Amplitude is an open condition under small perturbations of a divisor. If H is an ample Q-divisor on X, and E an arbitrary Q-divisor. Then is ample for all sufficiently small rational numbers

Examples/Non-examples

The line bundle defines an embedding into by the map since . This gives the closed subscheme of by .
Restricting an ample line bundle to a closed subscheme is still an ample line bundle since the global sections induce an embedding into projective space. For instance, given a plane curve , the restriction of to gives a smooth curve in which is not a complete intersection curve.
The line bundle is not ample.

Criteria for ampleness of line bundles

Intersection theory

{{see|intersection theory#Intersection theory in algebraic geometry}}

To decide in practice when a Cartier divisor D corresponds to an ample line bundle, there are some geometric criteria.

For curves, a divisor D is very ample if and only if

l(D) = 2 + l(D − A − B) whenever A and B are points. By the Riemann–Roch theorem every divisor of degree

at least 2g + 1 satisfies this condition so is very ample. This implies that a divisor is ample if and only if it has positive degree. The canonical divisor of degree 2g − 2 is very ample if and only if the curve is not

a hyperelliptic curve.

The Nakai–Moishezon criterion ({{harvnb|Nakai|1963}}, {{harvnb|Moishezon|1964}}) states that a Cartier divisor D on a proper scheme X over an algebraically closed field is ample if and only if D^dim(Y).Y > 0 for every closed integral subscheme Y of X. In the special case of curves this says that a divisor is ample if and only if it has positive degree, and for a smooth projective algebraic surface S, the Nakai–Moishezon criterion states that D is ample if and only if its self-intersection number D.D is strictly positive, and for any irreducible curve C on S we have D.C > 0.

For an -divisor D Nakai criterion is slightly more subtle. In fact, if D is an ample divisor, then certainly we have Nakai inequality, but it is no longer clear that Nakai inequality characterize amplitude. In the case when X is projective, then Nakai inequality characterize amplitude.

The Kleiman condition states that for any projective scheme X, a divisor D on X is ample if and only if D.C > 0 for any nonzero element C in the closure of NE(X), the cone of curves of X. In other words, a divisor is ample if and only if it is in the interior of the real cone generated by nef divisors.

{{harvtxt|Nagata|1959}} constructed divisors on surfaces that have positive intersection with every curve, but are not ample.

This shows that the condition D.D > 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 the Kleiman condition.

{{harvtxt|Seshadri|1972|loc=Remark 7.1, p. 549}} showed that a line bundle L on a complete algebraic scheme is ample if and only if there is some positive ε such that

deg(L|_C) ≥ εm(C) for all integral curves C in X, where m(C) is the

maximum of the multiplicities at the points of C.

Sheaf cohomology

The theorem of Cartan-Serre-Grothendieck states that for a line bundle on a variety , the following conditions are equivalent:

is ample
for m big enough, is very ample
for any coherent sheaf on X, the sheaf is generated by global sections, for m big enough

If is proper over some noetherian ring, this is also equivalent to:

for any coherent sheaf on X, the higher cohomology groups vanish for m big enough. {{citation needed|date=May 2016}}

Kleiman's characterizations of amplitude

Let D be a Cartier divisor on a projective algebraic scheme X. Then D is ample if and only if it satisfies either of the following properties:

(I). For every irreducible subvariety of positive dimension, there is a positive integer m, together with a non-zero section , such that s vanishes at some point of V.

(II) For every irreducible subvariety of positive dimension,

Generalizations

Vector bundles of higher rank

A locally free sheaf (vector bundle) on a variety is called ample if the invertible sheaf on is ample {{harvtxt|Hartshorne|1966}}.

Ample vector bundles inherit many of the properties of ample line bundles.

Big line bundles

An important generalization, notably in birational geometry, is that of a big line bundle. A line bundle on X is said to be big if the equivalent following conditions are satisfied:

is the tensor product of an ample line bundle and an effective line bundle
the Hilbert polynomial of the finitely generated graded ring has degree the dimension of X
the rational mapping of the total system of divisors is birational on its image for .

The interest of this notion is its stability with respect to rational transformations.

References

Study references

{{Citation | last1=Hartshorne | first1=Robin | author1-link= Robin Hartshorne | title=Algebraic Geometry | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | year=1977}}
{{Citation | last1=Lazarsfeld | first1=Robert | author1-link= Robert Lazarsfeld | title=Positivity in Algebraic Geometry | publisher=Springer-Verlag | location=Berlin | year=2004}}
The slides on ampleness in Vladimir Lazić's [https://www.yumpu.com/en/document/fullscreen/11752434/algebraic-geometry-lecture-7-canonical-bundle-i Lectures on algebraic geometry]

Research texts

{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Ample vector bundles | url=http://www.numdam.org/item?id=PMIHES_1966__29__63_0 | mr=0193092 | year=1966 | journal=Publications Mathématiques de l'IHÉS | issn=1618-1913 | issue=29 | pages=63–94}}
{{Citation | doi=10.2307/1970447 | last1=Kleiman | first1=Steven L. | author1-link=Steven Kleiman | title=Toward a numerical theory of ampleness | jstor=1970447 | mr=0206009 | year=1966 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=84 | pages=293–344 | issue=3 | publisher=Annals of Mathematics}}
{{Citation | last1=Moishezon | first1=B. G. | authorlink1 = Boris Moishezon | title=A projectivity criterion of complete algebraic abstract varieties | mr=0160782 | year=1964 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=28 | pages=179–224}}
{{Citation | last1=Nagata | first1=Masayoshi | author1-link= Masayoshi Nagata | title=On the 14th problem of Hilbert | mr=0154867 | year=1959 | journal=American Journal of Mathematics | volume=81 | pages=766–772 | doi=10.2307/2372927 | jstor=2372927 | issue=3 | publisher=The Johns Hopkins University Press}}
{{Citation | doi=10.2307/2373180 | last1=Nakai | first1=Yoshikazu | title=A criterion of an ample sheaf on a projective scheme | jstor=2373180 | mr=0151461 | year=1963 | journal=American Journal of Mathematics | issn=0002-9327 | volume=85 | pages=14–26 | issue=1 | publisher=The Johns Hopkins University Press}}
{{Citation | doi=10.2307/1970870 | last1=Seshadri | first1=C. S. | title=Quotient spaces modulo reductive algebraic groups | jstor=1970870 | mr=0309940 | year=1972 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=95 | pages=511–556 | issue=3 | publisher=Annals of Mathematics}}

3 : Vector bundles|Algebraic geometry|Geometry of divisors

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