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Revision as of 02:26, 24 July 2004
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
- s/t
makes sense as a well-defined numerical function on M, one can take a basis for all global sections of L on M and try to use them as a set of homogeneous coordinates on M. If the basis is written out as
- s1, s2, ..., sk,
where k is the dimension of the space of sections, it makes sense to regard
as coordinates on M, in the projective space sense. Therefore this sets up a mapping
- M → P
which is required to be an embedding. (In a more invariant treatment, the RHS here is described as the projective space undelying the space of all global sections.)
An ample line bundle L is one which becomes very ample after it is raiswed 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.
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