Ample line bundle

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 $M$ 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 $f\colon X\to Y$ , any vector bundle ${\mathcal {F}}$ on Y, or more generally any sheaf in ${\mathcal {O}}_{Y}$ 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

f\colon X\to {\mathbb P}^{N},

and

{\mathcal F}={\mathcal O}(1)\in {\mathrm {Pic}}({\mathbb P}^{N})

,

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 $a_{i}\in F(X)$ , 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

f\colon X\rightarrow {\mathbb {P}}^{{n}},\ x\mapsto [a_{0}(x):\dotsb :a_{n}(x)],

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.

A=\Gamma (F,X)

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

\rho =\rho _{{X,U}}

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:

i^{{*}}({\mathcal {O}}(1))\cong 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 $H^{i}(X,F)$ 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 ${\mathcal L}$ is its having a positive tensorial power that is very ample. In other words, for $n\gg 0$ there exists a projective embedding $j:X\to {\mathbb P}^{N}$ such that ${\mathcal L}^{{\otimes n}}=j^{*}({\mathcal O}(1))$ , that is the zero divisors of global sections of ${\mathcal L}^{{\otimes n}}$ are hyperplane sections.

This definition makes sense for the underlying divisors (Cartier divisors) $D$ ; an ample $D$ is one where $nD$ 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 $D$ for a very ample $L$ corresponds to the hyperplane sections (intersection with some hyperplane) of the embedded $M$ .

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

Examples/Non-examples

The line bundle ${\mathcal {O}}_{\mathbb {P} ^{1}}(2)$ defines an embedding into $\mathbb {P} ^{2}$ by the map $[s:t]\mapsto [s^{2}:st:t^{2}]$ since $\Gamma (\mathbb {P} ^{1},{\mathcal {O}}_{\mathbb {P} ^{1}}(2))=\mathbb {k} \cdot s^{2}\oplus \mathbb {k} \cdot st\oplus \mathbb {k} \cdot t^{2}$ . This gives the closed subscheme of $\mathbb {P} ^{2}$ by ${\text{Proj}}(\mathbb {k} [x,y,z]/(xz-y^{2}))$ .
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 $C\hookrightarrow \mathbb {P} ^{2}$ , the restriction of ${\mathcal {O}}_{\mathbb {P} ^{2}}(2)$ to $C$ gives a smooth curve in $\mathbb {P} ^{5}$ which is not a complete intersection curve.
The line bundle ${\mathcal {O}}_{\mathbb {P} _{x,y}^{1}\times \mathbb {P} _{s,t}^{1}}(0,2)$ is not ample.

Criteria for ampleness of line bundles

Intersection theory

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 (Nakai 1963, 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.

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.

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.

Seshadri (1972, 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 ${\mathcal L}$ on a variety $X$ , the following conditions are equivalent:

${\mathcal L}$ is ample
for m big enough, ${\mathcal L}^{{\otimes m}}$ is very ample
for any coherent sheaf ${\mathcal {F}}$ on X, the sheaf ${\mathcal F}\otimes {\mathcal L}^{{\otimes m}}$ is generated by global sections, for m big enough

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

for any coherent sheaf ${\mathcal {F}}$ on X, the higher cohomology groups $H^{i}(X,{\mathcal F}\otimes {\mathcal L}^{{\otimes m}}),\ i\geq 1$ vanish for m big enough.

Generalizations

Vector bundles of higher rank

A locally free sheaf (vector bundle) $F$ on a variety is called ample if the invertible sheaf ${\mathcal {O}}(1)$ on ${\mathbb {P}}(F)$ is ample 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 ${\mathcal L}$ on X is said to be big if the equivalent following conditions are satisfied:

${\mathcal L}$ is the tensor product of an ample line bundle and an effective line bundle
the Hilbert polynomial of the finitely generated graded ring $\bigoplus _{{k=0}}^{\infty }\Gamma (X,{\mathcal L}^{{\otimes k}})$ has degree the dimension of X
the rational mapping of the total system of divisors $X\to {\mathbb P}\Gamma (X,{\mathcal L}^{{\otimes k}})$ is birational on its image for $k\gg 0$ .

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

References

Study references

Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Lazarsfeld, Robert (2004), Positivity in Algebraic Geometry, Berlin: Springer-Verlag
The slides on ampleness in Vladimir Lazić's Lectures on algebraic geometry

Research texts

Hartshorne, Robin (1966), "Ample vector bundles", Publications Mathématiques de l'IHÉS (29): 63–94, ISSN 1618-1913, MR 0193092
Kleiman, Steven L. (1966), "Toward a numerical theory of ampleness", Annals of Mathematics, Second Series, Annals of Mathematics, 84 (3): 293–344, doi:10.2307/1970447, ISSN 0003-486X, JSTOR 1970447, MR 0206009
Moishezon, B. G. (1964), "A projectivity criterion of complete algebraic abstract varieties", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 28: 179–224, ISSN 0373-2436, MR 0160782
Nagata, Masayoshi (1959), "On the 14th problem of Hilbert", American Journal of Mathematics, The Johns Hopkins University Press, 81 (3): 766–772, doi:10.2307/2372927, JSTOR 2372927, MR 0154867
Nakai, Yoshikazu (1963), "A criterion of an ample sheaf on a projective scheme", American Journal of Mathematics, The Johns Hopkins University Press, 85 (1): 14–26, doi:10.2307/2373180, ISSN 0002-9327, JSTOR 2373180, MR 0151461
Seshadri, C. S. (1972), "Quotient spaces modulo reductive algebraic groups", Annals of Mathematics, Second Series, Annals of Mathematics, 95 (3): 511–556, doi:10.2307/1970870, ISSN 0003-486X, JSTOR 1970870, MR 0309940

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