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In mathematics, in algebraic geometry or the theory of complex manifolds, a very ample line bundle ${\displaystyle L}$ is one with enough sections to set up an embedding of its base variety or manifold ${\displaystyle M}$ into projective space. That is, considering that for any two sections ${\displaystyle s}$ and ${\displaystyle t}$, the ratio

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

makes sense as a well-defined numerical function on ${\displaystyle M}$, one can take a basis for all global sections of ${\displaystyle L}$ on ${\displaystyle M}$ and try to use them as a set of homogeneous coordinates on ${\displaystyle M}$. If the basis is written out as

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

where ${\displaystyle k}$ is the dimension of the space of sections, it makes sense to regard

${\displaystyle }$

as coordinates on ${\displaystyle M}$, 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 underlying the space of all global sections.)

An ample line bundle ${\displaystyle L}$ is one which becomes very ample after it is raised to some tensor power, i.e. the tensor product of ${\displaystyle L}$ with itself enough times has enough sections. These definitions make sense for the underlying divisors (Cartier divisors) ${\displaystyle D}$; an ample ${\displaystyle D}$ is one for which ${\displaystyle nD}$ 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 ${\displaystyle D}$ for a very ample ${\displaystyle L}$ will correspond to the hyperplane sections (intersection with some hyperplane) of the embedded ${\displaystyle M}$.

There is a more general theory of ample vector bundles.

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 algebraic surface S, the Nakai-Moishezon criterion states that D is ample if its self-intersection number is strictly positive, and for any irreducible curve C on S we have

D.C > 0

in the sense of intersection theory. There are other criteria such as the Kleiman condition and Seshadri condition, to characterise the ample cone.

• Vector bundles
• Algebraic geometry