Ample cone

Suppose $ X$ is a projective variety, with $ \pi: X \hookrightarrow
\mathbb{P}^n$ an inclusion (a ``closed immersion''). Projective space has a natural line bundle $ \mathcal{O}(1)$, and the pullback $ \pi^*
\mathcal{O}(1)$ is said to be a very ample line bundle on $ X$. That is, a line bundle is very ample if it can be obtained by pulling back $ \mathcal{O}(1)$ via a closed immersion into projective space. Equivalently, a line bundle is very ample if its global sections $ s_0,
\ldots, s_n$ determine a closed immersion into projective space $ [s_0,
\ldots, s_n]: X \hookrightarrow \mathbb{P}^n$. 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 $ H^2(X, \mathbb{Q})$ generated by $ \{ c_1(L) : L$ an ample line bundle on $ X \}$.

The ampleness of a line bundle $ L$ is determined only by its first Chern class. More precisely, a line bundle $ L$ is ample if and only if, for every subvariety $ Z$, $ c_1(L)^k \cap [Z]>0$, where $ \mathrm{dim} Z = k$.

Jeffrey Herschel Giansiracusa 2005-05-17