| Revision as of 03:33, 27 October 2006 edit128.239.157.156 (talk) →Criteria for ampleness← Previous edit | Latest revision as of 20:29, 7 April 2007 edit undoJakob.scholbach (talk | contribs)Extended confirmed users, Pending changes reviewers, Rollbackers6,807 editsm ←Redirected page to Ample line bundle | ||
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| In ], in ] or the theory of ]s, a '''very ample ]''' <math>L</math> is one with enough ]s to set up an ] of its base ] 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> | |||
| makes sense as a well-defined numerical ] 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> | |||
| where <math>k</math> is the ] of the space of sections, it makes sense to regard | |||
| :<math></math> | |||
| as coordinates on <math>M</math>, in the projective space sense. Therefore this sets up a mapping | |||
| :<math>M\ \rightarrow\ P^{k-1}</math> | |||
| which is required to be an embedding. (In a more invariant treatment, the ] here is described as the projective space underlying the space of all global sections.) | |||
| An '''ample line bundle''' <math>L</math> is one which becomes very ample after it is raised to some tensor power, i.e. the ] of <math>L</math> 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. | |||
| ==Criteria for ampleness== | |||
| To decide in practice when a Cartier divisor ''D'' corresponds to an ample line bundle, there are some geometric criteria. | |||
| For example. for a smooth ] ''S'', the '''Nakai-Moishezon criterion''' states that ''D'' is ample if its ] is strictly positive, and for any irreducible curve ''C'' on ''S'' we have | |||
| :''D''.''C'' > 0 | |||
| in the sense of ]. There are other criteria such as the Kleiman condition and Seshadri condition, to characterise the '''ample cone'''. | |||
| Also, Laura is ample. | |||
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Latest revision as of 20:29, 7 April 2007
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