Suppose is a projective variety, with
an inclusion (a ``closed immersion''). Projective space
has a natural line bundle
, and the pullback
is said to be a very ample line bundle on .
That is, a line bundle is very ample if it can be obtained by pulling
via a closed immersion into projective space.
Equivalently, a line bundle is very ample if its global sections
determine a closed immersion into projective space
. The tensor product of
two very ample line bundles is again very ample.
A line bundle on a projective variety is ample if some tensor
power of it is very ample. The ample cone is the convex cone in
an ample line
bundle on .
The ampleness of a line bundle is determined only by its first
Chern class. More precisely, a line bundle is ample if and only
if, for every subvariety ,
Jeffrey Herschel Giansiracusa