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
Contents
Inverse image of line bundle and hyperplane divisors
Given a morphism
The notions described in this article are related to this construction in the case of morphisms to projective spaces
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
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 OX 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 OU-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 → PnS, the n-dimensional projective space over S for some n, such that the pullback of the standard twisting sheaf O(1) on PnS 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
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
This definition makes sense for the underlying divisors (Cartier divisors)
The equivalence between the two definitions is credited to Jean-Pierre Serre in Faisceaux algébriques cohérents.
Examples/Non-examples
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 Ddim(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
If
Vector bundles of higher rank
A locally free sheaf (vector bundle)
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
The interest of this notion is its stability with respect to rational transformations.