Ample line bundle.html

In mathematics, in algebraic geometry or the theory of complex manifolds, a very ample line bundle $L$ is one with enough sections to set up an embedding of its base variety or manifold $M$ into projective space. The notions of ample line bundles and globally generated sheaves are precursors of very ample line bundles.

1 Sheaves generated by their global sections
2 Very ample line bundles
3 Ample line bundles
4 Criteria for ampleness
5 Ample vector bundles of higher rank
6 Cohomology of a very ample line bundle
7 References

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 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: X → Pⁿ, x ↦ a₀(x): ... : a_n(x)],

such that the pullback f*(O(1)) is F. The converse statement is also true: given such a morphism f, the pullback of O(1) is generated by its global sections (on X).

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 immersion 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^*(O(1)) ≅ 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 immersion.

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

Hⁱ(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.

Ample line bundles

The notion of ample line bundles L is slightly weaker than very ample line bundles: L is called ample if some tensor power L^⊗n is very ample. This is equivalent to the following definition: 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.

These definitions make sense for the underlying divisors (Cartier divisors) $D$ ; an ample $D$ is one for which $n D$ moves in a large enough linear system. Such divisors form a cone in all divisors, of those which are in some sense positive enough. The relationship with projective space is that the $D$ for a very ample $L$ will correspond to the hyperplane sections (intersection with some hyperplane) of the embedded $M$ .

Criteria for ampleness

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

For example. for a smooth algebraic surface S, the Nakai-Moishezon criterion states that D is ample iff its self-intersection number is strictly positive, and for any irreducible curve C on S we have

D.C > 0

in the sense of intersection theory.

Another useful criterion is the Kleiman condition. This states that for any complete algebraic scheme X, a divisor D on X is ample iff D.x > 0 for any nonzero element x in the closure of NE(X), the cone of curves of X. (Note that taking the closure is necessary here; it is possible (Nagata 1959) to construct divisors on surfaces which have positive intersection with every effective divisor, but are not ample.)

Other criteria such as the Seshadri condition give further characterisations of the ample cone.

Ample 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.

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

Cohomology of a very ample line bundle

All sheaf cohomology groups of a very ample line bundle vanish, except for the zeroeth group. In particular this implies that the zeroeth sheaf cohomology group, which is the vector space of global sections, has a rank equal to the Euler characteristic and so is given by the Hirzebruch-Riemann-Roch theorem.

References

Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, MR 0463157, ISBN 978-0-387-90244-9
Lazarsfeld, Robert (2004), Positivity in Algebraic Geometry, Berlin: Springer-Verlag
Nagata, Masayoshi (1959), "On the 14th problem of Hilbert", American Journal of Mathematics 81: 766–772, MR 0154867, ISSN 0002-9327

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