# Noether–Enriques using Tsen's lemma

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.

• @LaurentMoret-Bailly Opps, it is a silly mistake.
– user485190
Jul 3 at 14:12

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 $$Z$$.
• How can we make the $U$-point $U\rightarrow X_U$ to a section $Z\rightarrow X$?