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 $\mathrm{dim} Z = k$.