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



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> 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}
> 
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> 


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](https://doi.org/10.1017/CBO9780511623936) (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.



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


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



Noether–Enriques using Tsen's lemma
I will assume that $X$ is proper. Then the generic fiber is a smooth projective curve of genus $0$ over the function field $K = k(Z)$ but any such curve can be embedded as a conic in $\mathbb{P}^2\_K$ using the anticanonical linear series. By Tsen's theorem, the conic has a $K$-point which we can spread out to a $U$-point of $X$ where $U$ is some open subset of $Z$. Finally by properness, this $U$-point extends to a section over $Z$.


