Ample vector bundle

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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. That is, considering that for any two sections s and t, the ratio

makes sense as a well-defined which is required to be an embedding. (In a more invariant treatment, the RHS here is described as the projective space underlying the space of all global sections.)

An ample line bundle L is one which becomes very ample after it is raised to some tensor power, i.e. the tensor product of L with itself enough times has enough sections. These definitions make sense for the underlying divisors (Cartier divisors) D; an ample D is one for which nD 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.

There is a more general theory of ample vector bundles.

[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 example. for a smooth algebraic surface S, the Nakai-Moishezon criterion states that D is ample if 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. There are other criteria such as the Kleiman condition and Seshadri condition, to characterise the ample cone.