| Revision as of 17:48, 17 June 2004 editCharles Matthews (talk | contribs)Autopatrolled, Administrators360,392 editsm odd format bug?← Previous edit | Revision as of 17:48, 17 June 2004 edit undoCharles Matthews (talk | contribs)Autopatrolled, Administrators360,392 editsmNo edit summaryNext edit → | ||
| Line 17: | Line 17: | ||
| 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 |
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''. | ||
| There is a more general theory of ample ]s. | There is a more general theory of ample ]s. | ||
Revision as of 17:48, 17 June 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.