Consider the following weak version of the Noether–Enriques theorem (field is $\mathbb{C}$):

Let $\varphi:X\rightarrow Z$ be a morphism from a smooth projective surface onto a smooth curve with $F_z:=\varphi^{-1}(z)\cong\mathbb{P}^1$ for every point $z$. Then there exists a Zariski open neighborhood $U$ of any point $z$ making the following diagram commute. $\require{AMScd}$ \begin{CD} \varphi^{-1}(U) @>{\sim}>> U\times\mathbb{P}^1\\ @V{\varphi}VV @VV{\text{projection}}V\\ U @>{\sim}>> U \end{CD}

The critical point is to find a divisor $D$ on $X$ such that $D.F_z=1$ or to find a section $s:Z\rightarrow X$ for $\varphi$. Beauville states in his book Complex Algebraic Surfaces (c.f. Remark III.6) that this step can be done by using Tsen's lemma, and I want to know how exactly can we do this.

In particular, I want to know how to make $X$ a conic?

Indeed, Beauville even claims that we can use Tsen's lemma to show this step for the usual Noether–Enriques theorem.