Normal bundle of a section of a projective bundle

Let $X=\mathbb{P}^1$, $E=\mathcal{O}\oplus\mathcal{O}(1)\oplus \mathcal{O}(2)$ and let $Y=\mathbb{P}(E)$. Denote $\pi:Y\to X$ the projection map. There is a section of $\pi$ corresponding to the surjection $E\to \mathcal{O}$. Let $C$ be the image of this section in $Y$. How can we compute the normal bundle of this curve?

The case when $E$ is a rank 2 vector bundle is discussed here.Normal bundle of a section of a $\mathbb{P}^1$-bundle. However, I believe in our case, the argument cannot be directly applied.

Thanks for the help!

Solution 1:

Let us do this more generally, so that we do not have to look at many cases. Let $E$ be a vector bundle on a scheme $X$ and assume we have an exact sequence $0\to F\to E\to L\to 0$, with $L$ a line bundle. This gives a section $D$ of $\pi:P=\mathbb{P}_X(E)\to X$, the projective bundle. Let us calculate the normal bundle of $D$ in $P$.

We have a natural map $\pi^*F\to \mathcal{O}_P(1)$ and one easily checks that the cokernel is just $\mathcal{O}_D(1)$, which can easily seen to be $L$ (identifying $D$ with $X$). So, we get a surjection $\pi^*(F\otimes L^{-1})\to I_D$, the ideal sheaf of $D\subset P$. Since codimension of $D$ is just the rank of $F$, we see that $D$ is a local complete intersection, coming from a correct rank vector bundle. Thus the normal bundle of $D$ then is just $F^*\otimes L$.