Ample line bundle

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In algebraic geometry, 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. 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.

[edit] 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).

[edit] 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.

[edit] 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 $M$ .

[edit] Criteria for ampleness

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 complete algebraic 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 which 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.

[edit] 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 Hartshorne (1966).