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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

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

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

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

where k is the dimension of the space of sections, it makes sense to regard

${\displaystyle }$

as coordinates on M, in the projective space sense. Therefore this sets up a mapping

${\displaystyle M\ \rightarrow \ P^{k-1}}$

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 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.

• Bundles (mathematics)