Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective.
If $X$ has Kodaira dimension 2 and we assume that the Bombieri-Lang conjecture holds, then the rational points of $X$ are not dense and thus by Hilbert irreducibility we can conclude that there exists a section $C\to X$.
Question 1: is the Bombieri-Lang conjecture really necessary? I have this gut feeling that the finiteness of rational points of an hyperbolic curve should be enough, and that Kodaira dimension 2 is not necessary, but I cannot conclude.
Question 2: as a consequence of Hilbert irreducibility, if $C_1\to C_2$ is a map of curves such that $C_1(k')\to C_2(k')$ is surjective for every finite extension $k'/k$, then there exists a section $C_2\to C_1$. In the setting above, if we remove the assumption that $C$ is rational and instead ask that $X(k')\to C(k')$ is surjective for every finite extension $k'$, can we still conclude that a section exists?
