Self intersection of zero section of decomposable ruled surface

Question

Let $L_1, L_2$ be two line bundles over a curve $C$. Then we can consider the ruled surface $S=\Bbb P (L_1\oplus L_2)$ over $C$. Let $s_0$ and $s_\infty$ denote the zero section and the infinity section, respectively. In this case is there a way to express the self intersections of $s_0$ and $s_\infty$ by the degrees of $L_1$ and $L_2$?

Answer 1

The normal bundle for a section is isomorphic to the restriction of the relative tangent bundle. The relative tangent bundle $\mathcal{T}$ fits into the relative Euler exact sequence $$ 0 \to \mathcal{O} \to \pi^*(L_1 \oplus L_2) \otimes \mathcal{O}(1) \to \mathcal{T} \to 0, $$ where $\pi$ is the projection and $\mathcal{O}(1)$ is the Grothendieck bundle. If you restrict this to the section, say, $C_1$ and use the fact that the pullback of $\mathcal(1)$ is, say, $L_1^\vee$, you obtain the exact sequence $$ 0 \to \mathcal{O} \to (L_1 \oplus L_2) \otimes L_1^\vee \to \mathcal{T}\vert_{C_1} \to 0, $$ and it follows that $$ \mathcal{N}_{C_1/S} \cong \mathcal{T}\vert_{C_1} \cong L_2 \otimes L_1^\vee. $$