# Ample cone

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 back 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 generated by 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 , , where .

Jeffrey Herschel Giansiracusa 2005-06-27