# Ample vector bundle

In mathematics, in algebraic geometry or the theory of complex manifolds, a very ample line bundle [itex]L[itex] is one with enough sections to set up an embedding of its base variety or manifold [itex]M[itex] into projective space. That is, considering that for any two sections [itex]s[itex] and [itex]t[itex], the ratio

[itex]{s}\over{t}[itex]

makes sense as a well-defined numerical function on [itex]M[itex], one can take a basis for all global sections of [itex]L[itex] on [itex]M[itex] and try to use them as a set of homogeneous coordinates on [itex]M[itex]. If the basis is written out as

[itex]s_1,\ s_2,\ ...,\ s_k[itex]

where [itex]k[itex] is the dimension of the space of sections, it makes sense to regard

[itex][s_1:\ s_2:\ ...:\ s_k][itex]

as coordinates on [itex]M[itex], in the projective space sense. Therefore this sets up a mapping

[itex]M\ \rightarrow\ P^{k-1}[itex]

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 [itex]L[itex] is one which becomes very ample after it is raised to some tensor power, i.e. the tensor product of [itex]L[itex] with itself enough times has enough sections. These definitions make sense for the underlying divisors (Cartier divisors) [itex]D[itex]; an ample [itex]D[itex] is one for which [itex]nD[itex] 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 [itex]D[itex] for a very ample [itex]L[itex] will correspond to the hyperplane sections (intersection with some hyperplane) of the embedded [itex]M[itex].

There is a more general theory of ample vector bundles.

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