2.2 Ample line bundles

In this section, \(X\) is a \(n\)-dimensional projective variety over a field \(k\). We write \(\operatorname{\mathrm{Pic}}X = \operatorname{\mathrm{H}}^1(X,\mathcal{O}_X^{\times })\) for the Picard group of \(X\), i.e., the group of isomorphism classes of line bundles on \(X\).

Recall that a line bundle \(L\in \operatorname{\mathrm{Pic}}X\) is very ample if there exists a closed immersion \(f:X\to \mathbb {P}^N\) for some \(N\) such that \(L\cong f^*\mathcal{O}_{\mathbb {P}^N}(1)\). A big part of algebraic geometry consists of looking for ways to find very ample line bundles. One reason for this is that very ample line bundles are not only a good way to get morphisms to projective space, but, as any morphism \(g:X\to Y\) to a projective variety \(Y\xrightarrow {i} \mathbb {P}^N\) gives a morphism \(ig:X\to \mathbb {P}^N\), ample line bundles can be used to get morphisms to arbitrary projective varieties.

A line bundle \(L\in \operatorname{\mathrm{Pic}}X\) is ample if it satisfies the following equivalent conditions:

1. \(L^k\) is very ample for \(k\gg 0\).

2. For every coherent sheaf \(\mathcal{F}\), \(\mathcal{F}\otimes \mathcal{L}^k\) is generated by its global sections for \(k\gg 0\).

3. For every coherent sheaf \(\mathcal{F}\) and every positive integer \(i\), \(\operatorname{\mathrm{H}}^0(X,\mathcal{F}\otimes \mathcal{L}^k)=0\) for \(k\gg 0\).

Let’s write \(\operatorname{\mathrm{Div}}X\) for the group of Cartier divisors on \(X\). Then, for every \(D\in \operatorname{\mathrm{Div}}X\), we get a line bundle \(\mathcal{O}_X(D)\) on \(X\).