4.5.2 Some jargon about linear systems
Suppose \(D\) is a divisor on a \(S\). The linear system \(|D|\) is the projectivization \(\mathbb {P}(\operatorname{\mathrm{H}}^0(S, \mathcal{O}_X(D)))\) of the space of global sections of the line bundle associated to \(D\) or, equivalently, \(|D| = \{ D'\sim D: D'\geq 0\} \). A linear subspace \(P\) of \(|D|\) is called a linear system on \(S\).
The base locus of a linear system \(P\) is \(\operatorname{\mathrm{bs}}P := \cap _{D'\in P} D'\). A point in the base locus is called a base point. A prime divisor \(C\) contained in the \(\operatorname{\mathrm{bs}}P\) is called a fixed component of \(P\), and the fixed part of \(P\) is the largest divisor \(F\) contained in \(\operatorname{\mathrm{bs}}P\).
There is a bijection between the following sets:
\(\{ \)rational maps \(\phi : S\dashrightarrow \mathbb {P}^n\) such that \(\phi (S)\) is contained in no hyperplane\(\} \);
\(\{ n\)-dimensional linear systems on \(S\) without fixed part\(\} \).
Suppose \(\phi \) is as a. Then we get a morphism \(\phi _U:U\to \mathbb {P}^n\), where \(U=\operatorname{\mathrm{dom}}\phi \). Moreover, \(U = S\setminus \Sigma \) for some finite set \(\Sigma \). Set \(\mathcal{L} = \phi _U^*\mathcal{O}_{\mathbb {P}^n}(1)\). Then, as \(\Sigma \) is finite, \(\mathcal{L}\) extends uniquely to a line bundle \(\overline{\mathcal{L}}\) on \(X\). We get a composition
Since \(\phi (S)\) is contained in no hyperplane, the compostion is injective. So the image is an \(n+1\)-dimensional subspace \(V\) of \(\operatorname{\mathrm{H}}^0(X,\overline{\mathcal{L}})\). Let \(P\subseteq |\overline{\mathcal{L}}| = \mathbb {P}(V)\). Then \(\dim P = n\). Moreover, \(P\) has no fixed part, because \(\operatorname{\mathrm{bs}}P\subseteq \Sigma \).
On the other hand, suppose we’re given the data in b. The identify \(\mathbb {P}^n\) with the dual projective space of \(P\). Then we get a rational map \(\phi :S\dashrightarrow \mathbb {P}^n\) by sending \(x\in S\) to the set \(H_x\) of hyperplanes in \(P\) containing \(x\). This set \(H_x\) is necessarily a codimension \(1\) linear subspace of \(P\) as long as \(x\not\in \operatorname{\mathrm{bs}}P\). So we get a rational map defined outside of the finitely many points in \(\operatorname{\mathrm{bs}}P\). Since there is no divisor \(D\in P\) containing all points \(x\in X\), \(\phi (S)\) is not contained in a hyperplane.
Suppose \(P\) is a nonemtpy linear system with empty fixed part. Then \(P\) has at most \(D^2\) fixed points where \(D\) is an element of \(P\).
The base locus \(\operatorname{\mathrm{bs}}P\) is clearly a closed algebraic subset of \(S\). So, since \(P\) has no fixed part, \(b := |\operatorname{\mathrm{bs}}P| {\lt} \infty \).
I claim that we can find divisors \(D_1,D_2\in P\) such that \(|D_1\cap D_2| {\lt} \infty \). To see this, let \(D\in P\) be a divisor with irreducible components \(C_1,\ldots , C_k\). Then, for each \(i\), we can find a divisor \(D_i\) with \(D_i\cap C_i\) finite. Let \(\mathcal{L}\) be the line bundle associated to \(D\) and, let \(D_i = \operatorname{\mathrm{div}}s_i\) for some (nonzero) section \(s_i\in \operatorname{\mathrm{H}}^0(S, \mathcal{L})\). Since \(k\) is algebraically closed and, thus, infinite, we can find \(\alpha _1,\ldots , \alpha _k\) such that \(\operatorname{\mathrm{div}}\sum \alpha _i s_i\) does not contain any of the \(C_i\) as components. And this proves the claim.
Now, with the claim proved, take \(D_1, D_2\in P\) with \(D_1\cap D_2\) finite. We have \(b = D^2 = \deg D_1\cdot D_2\). So \(|D_1\cap D_2| \leq b\).