Normal bundle of a section of a $\mathbb{P}^1$-bundle

Solution 1:

Here is how such normal bundles are calculated. Let $\pi:P\to X$ and $\mathcal{O}_P(1)$, the relative ample bundle. We have $\pi^*(\mathcal{O}_X\oplus\mathcal{L})\to\mathcal{O}_P(1)$, the canonical surjective map. The surjection $\mathcal{O}_X\oplus\mathcal{L}\to \mathcal{O}_X$ has kernel $\mathcal{L}$ and thus we get a map $\pi^*\mathcal{L}\to\mathcal{O}_P(1)$ and then the cokernel is supported on $D$ (this is how $D$ is defined) and we get an exact sequence, $0\to \pi^*\mathcal{L}(-1)\to\mathcal{O}_P\to\mathcal{O}_D\to 0$. The normal bundle is $\pi^*\mathcal{L}^{-1}(1)\otimes \mathcal{O}_D$, which is just $\mathcal{L}^{-1}$ (identifying $D$ with $X$), snce $\mathcal{O}_D(1)$ is just $\mathcal{O}_D$.