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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 <math>M</math> into ]. That is, considering that for any two sections <math>s</math> and <math>t</math>, the ratio


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


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 <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 ] on <math>M</math>. If the basis is written out as


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


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


:<math></math> :<math></math>


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


:<math>M\ \rightarrow\ P^{k-1}</math> :<math>M\ \rightarrow\ P^{k-1}</math>
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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.)


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 large 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''. An '''ample line bundle''' <math>L</math> 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) <math>D</math>; an ample <math>D</math> is one for which <math>nD</math> ''moves in a large 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 <math>D</math> for a very ample <math>L</math> will correspond to the ]s (intersection with some ]) of the embedded <math>M</math>.


There is a more general theory of ample ]s. There is a more general theory of ample ]s.

Revision as of 18:29, 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 {\displaystyle M} into projective space. That is, considering that for any two sections s {\displaystyle s} and t {\displaystyle t} , the ratio

s t {\displaystyle {s} \over {t}}

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

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

where k {\displaystyle k} is the dimension of the space of sections, it makes sense to regard

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

as coordinates on M {\displaystyle 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 {\displaystyle 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 {\displaystyle D} ; an ample D {\displaystyle D} is one for which n D {\displaystyle 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 {\displaystyle D} for a very ample L {\displaystyle L} will correspond to the hyperplane sections (intersection with some hyperplane) of the embedded M {\displaystyle M} .

There is a more general theory of ample vector bundles.

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