Definitions
The notion of ample line bundles L is slightly weaker than very ample line bundles: L is called ample if some tensor power L⊗n is very ample. This is equivalent to the following definition: L is ample if for any coherent sheaf F on X, there exists an integer n(F), such that F ⊗ L⊗n is generated by its global sections.
An equivalent, maybe more intuitive, definition of the ampleness of the line bundle is its having a positive tensorial power that is very ample. In other words, for there exists a projective embedding such that, that is the zero divisors of global sections of are hyperplane sections.
This definition makes sense for the underlying divisors (Cartier divisors) ; an ample is one where moves in a large enough linear system. Such divisors form a cone in all divisors of those that are, in some sense, positive enough. The relationship with projective space is that the for a very ample corresponds to the hyperplane sections (intersection with some hyperplane) of the embedded .
The equivalence between the two definitions is credited to Jean-Pierre Serre in Faisceaux algébriques cohérents.
Read more about this topic: Ample Line Bundle
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