Criterion for triviality of projective bundle

Question

Let $X$ be a smooth, projective variety and $\pi:W \to X$ be a $\mathbb{P}^1$-bundle of $X$. Suppose that there exists a section of $\pi$ in the sense that there exists a morphism $s:X \to W$ such that $\pi \circ s$ is identity on $X$. Does this imply that $W$ is a trivial $\mathbb{P}^1$-bundle over $X$? If not true in general, is it true if $X$ is a smooth rational curve?

Answer 1

No.

It's almost like you are asking for the case of Hirzebruch surface. By construction, write $$H_n = \mathbb{P} (O \oplus O(n)),$$ where $O, O(n)$ are line bundles on $\mathbb P^1$. $H_n$ obviously has a structure of a $\mathbb P^1$ bundle over $\mathbb P^1$ and it has a section given by $O$.

However, almost all of them are not trivial bundles.