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.

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.