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In ], in ] or the theory of ]s, a '''very ample ]''' ''L'' is one with enough ]s to set up an ] of its base variety or manifold ''M'' into ]. That is, considering that for any two sections ''s'' and ''t'', the ratio In ], in ] or the theory of ]s, a '''very ample ]''' ''L'' is one with enough ]s to set up an ] of its base variety or manifold ''M'' into ]. That is, considering that for any two sections ''s'' and ''t'', the ratio


:<math>{s}\over{t}</math>
:''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 ] on ''M''. If the basis is written out as 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 ] on ''M''. If the basis is written out as


:''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>k</sub>, :<math>s_1, s_2, ..., s_k</math>


where ''k'' is the dimension of the space of sections, it makes sense to regard 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 as coordinates on ''M'', in the projective space sense. Therefore this sets up a mapping


:<math>M \rightarrow P^{k-1}</math>
:''M'' &rarr; ''P''<sup>''k''&minus;1</sup>


which is required to be an embedding. (In a more invariant treatment, the ] here is described as the projective space undelying the space of all global sections.) which is required to be an embedding. (In a more invariant treatment, the ] here is described as the projective space undelying the space of all global sections.)

Revision as of 17:25, 31 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 {\displaystyle {s} \over {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

s 1 , s 2 , . . . , s k {\displaystyle s_{1},s_{2},...,s_{k}}

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

[ s 1 , s 2 , . . . , s k {\displaystyle s_{1},s_{2},...,s_{k}} ]

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

M → P k − 1 {\displaystyle M\rightarrow P^{k-1}}

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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