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| In ], in ] or the theory of ]s, a '''very ample line bundle''' ''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 | |||
| :''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 | |||
| :''s''<sub>1</sub>, ''s''<sub>2</sub>, ..., ''s''<sub>k</sub>, | |||
| 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''<sup>''k''−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.) | |||
| An '''ample line bundle''' ''L'' is one which becomes very ample after it is raiswed to some tensor power, i.e. the ] of ''L'' with itself enough times has enough sections. These definitions make sense for the underlying ''divisors'' (]s) ''D''; an ample ''D'' is one for which ''nD'' ''moves in a llarge enough ]''. Such divisors form a ] 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 ]s (intersection with some ]) of the embedded ''M''. | |||
| There is a more general theory of ample ]s. | |||
Latest revision as of 20:29, 7 April 2007
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