Let \(X\) be the ruled surface \(\PP(\mathcal{E})\) over a curve \(C\). Show that \(\mathcal{E}\) is decomposable if and only if there exist two sections \(C^{\prime}, C^{\prime \prime}\) of \(X\) such that \(C^{\prime} \cap C^{\prime \prime}=\varnothing\).
Solution
For a section \[ \sigma:C\longrightarrow \PP(\mathcal E), \] write \[ q_\sigma:\mathcal E\twoheadrightarrow\mathcal L_\sigma \] for the corresponding quotient line bundle from Proposition V.2.6, and put \[ \mathcal N_\sigma=\ker q_\sigma. \] Proposition V.2.6 also shows that \(\mathcal N_\sigma\) is an invertible sheaf.
1. For two sections \(\sigma',\sigma''\), their images are disjoint if andonly if\[ (\mathcal N_{\sigma'})_c\ne(\mathcal N_{\sigma''})_c \quad\text{inside }\mathcal E_c \]for every \(c\in C\).
Both sections lie over the identity of \(C\). Hence their images meet if and only if there is a point \(c\in C\) for which \[ \sigma'(c)=\sigma''(c) \] in the fibre \(\PP(\mathcal E_c)\).
Under Hartshorne’s quotient convention for \(\PP(\mathcal E)\), a point of \(\PP(\mathcal E_c)\) is a one-dimensional quotient of the two-dimensional vector space \(\mathcal E_c\). Two such quotient maps determine the same point exactly when they have the same one-dimensional kernel. Since the quotient sequences of Proposition V.2.6 remain exact on fibres, those kernels are \((\mathcal N_{\sigma'})_c\) and \((\mathcal N_{\sigma''})_c\). This proves the claim.
2. If \(\mathcal E\) is decomposable, then \(\PP(\mathcal E)\) has twodisjoint sections.
Write \[ \mathcal E\cong\mathcal L'\oplus\mathcal L'' \] with \(\mathcal L'\) and \(\mathcal L''\) invertible. The two projections \[ \mathcal E\twoheadrightarrow\mathcal L', \qquad \mathcal E\twoheadrightarrow\mathcal L'' \] give sections \(\sigma'\) and \(\sigma''\) by Proposition V.2.6. Their kernels are respectively \(\mathcal L''\) and \(\mathcal L'\). In every fibre these are the two distinct direct-summand lines of \[ \mathcal E_c=\mathcal L'_c\oplus\mathcal L''_c. \] Step 1 therefore shows that the two sections are disjoint.
3. If \(\PP(\mathcal E)\) has two disjoint sections, then\[ \boxed{ \mathcal E\cong \mathcal N_{\sigma'}\oplus\mathcal N_{\sigma''}. } \]
Let \(\sigma',\sigma''\) be disjoint sections and consider the natural map of rank-two locally free sheaves \[ \theta: \mathcal N_{\sigma'}\oplus\mathcal N_{\sigma''} \longrightarrow \mathcal E \] induced by the two kernel inclusions.
For each \(c\in C\), step 1 says that the one-dimensional subspaces \[ (\mathcal N_{\sigma'})_c, \qquad (\mathcal N_{\sigma''})_c \] of the two-dimensional vector space \(\mathcal E_c\) are distinct. They therefore span \(\mathcal E_c\), so \[ \theta_c: (\mathcal N_{\sigma'})_c\oplus(\mathcal N_{\sigma''})_c \longrightarrow\mathcal E_c \] is an isomorphism for every \(c\).
The determinant \(\det\theta\) is consequently a nowhere-vanishing morphism between line bundles. Hence \(\det\theta\) is an isomorphism, and therefore \(\theta\) itself is an isomorphism. Thus \(\mathcal E\) is a direct sum of two invertible sheaves.
4. Q.E.D.
Step 2 proves the forward implication, and step 3 proves the converse.