I'm reading the article "Belyi's theorem for complex surfaces - Gabino Gonzalez Diez" and there are few lines of a certain proof that I don't understand (the author claims that all is trivial):
Notations:
$S$ is a projective surface $S$ embedded in some $\mathbb P^n$ and we say that it is defined over a number field if $S\cong \{g_1=0,\ldots,g_m=0\}$ for polynomials $g_i$ with coefficients in a finite extension of $\mathbb Q$.
Here a Lefschetz pencil is referred as the rational function $f$ from $S$ to $\mathbb P^1$ induced by the hyperplane sections $\{S_\lambda=H_\lambda\cap S\}$ ($\{H_\lambda\}$ is a pencil of hyperplanes in $\mathbb P^n$)
The critical points of a Lefschetz pencil are the finite singular points (nodes) of the hyperplane sections and the critical values are the images of these points through $f$.
I know the Bertini's theorem and moreover I know how to construct a Lefschetz pencil. The rational map $f$ sends a point $x\in S$ in $f(x)=\lambda$ iff $x\in S_\lambda$, but I don't understand why if $S$ is defined over a number field then $f(x)\in\mathbb P^1\left(\overline{\mathbb Q}\right)$ for any critical point $x$.
I hope the question is clear; I've tried to explain in minor space as possible the framework and the notations, but if you want more explainations I'll edit the question.
Thanks.
If $S$ can be defined over $\overline{\mathbb Q}$, then there exists a model $S_0$ for $S$ over $\overline{\mathbb Q}$ (with respect to some embedding of $\overline{\mathbb Q} \to \mathbb C$).
Choose your Lefschetz pencil on $S_0$ and note that the critical points of this Lefschetz pencil lie in $\mathbb P^1(\overline{\mathbb Q})$; see the comment of Dori Bejleri above.
The Lefschetz pencil on $S$ is the Lefschetz pencil on $S_0$ base-changed to $\mathbb C$. The critical points of this Lefschetz pencil are those of the Lefschetz pencil on $S_0$, so they still lie in $\mathbb P^1(\overline{\mathbb Q})$.