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 ${\pi }^{\ast }\mathcal{O}(1)$ 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 $s_0,s_1,\dots ,s_n$ determine a closed immersion into projective space $[s_0,s_1,\dots ,s_n]:X\hookrightarrow {\mathbb{P}}^n$. 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 $\{c_1(L):L\text{ an ample line bundle on }X\}$.

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 $Z$, where $\dim Z=k$, we have $c_1(L{)}^k\cap [Z]>0$.