Ample vector bundle

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

<math>{s}\over{t}<math>

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

<math>s_1,\ s_2,\ ...,\ s_k<math>

where <math>k<math> is the dimension of the space of sections, it makes sense to regard

<math>[s_1:\ s_2:\ ...:\ s_k]<math>

as coordinates on <math>M<math>, in the projective space sense. Therefore this sets up a mapping

<math>M\ \rightarrow\ P^{k-1}<math>

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

There is a more general theory of ample vector bundles.

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