- published: 11 Sep 2015
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In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold Failed to parse (Missing texvc executable; please see math/README to configure.): M
Given a morphism Failed to parse (Missing texvc executable; please see math/README to configure.): f\ :\ X \to Y , any vector bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal F
The notions described in this article are related to this construction in the case of morphisms to projective spaces
, the line bundle corresponding to the hyperplane divisor, whose sections are the 1-homogeneous regular functions. See Algebraic geometry of projective spaces#Divisors and twisting sheaves.
Let X be a scheme or a complex manifold and F a sheaf on X. One says that F is generated by (finitely many) global sections Failed to parse (Missing texvc executable; please see math/README to configure.): a_i \in F(X) , if every stalk of F is generated as a module over the stalk of the structure sheaf by the germs of the ai. For example, if F happens to be a line bundle, i.e. locally free of rank 1, this amounts to having finitely many global sections, such that for any point x in X, there is at least one section not vanishing at this point. In this case a choice of such global generators a0, ..., an gives a morphism
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