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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?

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  • $\begingroup$ Minor nitpicks: $C(k)$ could be empty in which case the claim about the existence of a section does not follow. Also, I know it is common to say "family of smooth proper hyperbolic curves" but you certainly want to allow for singular fibres, so strictly speaking it is not a family of (just) smooth proper hyperbolic curves. $\endgroup$ Oct 3, 2020 at 19:16
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    $\begingroup$ Yes of course, thanks $\endgroup$ Oct 3, 2020 at 20:25
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    $\begingroup$ In any case, I highly doubt you can avoid Bombieri-Lang in Q1. This problem is quite similar to the one asked here (despite the apparent difference in Kodaira dimension, it is related to general type surfaces in the stated paper of Poonen) mathoverflow.net/questions/21003/… $\endgroup$ Oct 3, 2020 at 20:27
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    $\begingroup$ For Q2 this is also wide open and out of reach in general. See this paper for various results and conjectures: T. Graber, J. Harris, B. Mazur, J. Starr, Jumps in Mordell-Weil rank and arithmetic surjectivity. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 141-147, Progr. Math., 226, Birkh¨auser Boston, Boston, MA, 2004. $\endgroup$ Oct 3, 2020 at 20:31
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    $\begingroup$ As $C$ satisfies weak approximation, for the map on $k$-points to be surjective the map on $k_v$-points must be surjective for every place $v$ of $k$. This condition is actually now well understood and imposes various geometric conditions on the singular fibres in the family, but it doesn't say anything at all about the smooth members of the family, e.g. high genus doesn't help. See arxiv.org/abs/1705.10740 for details. $\endgroup$ Oct 4, 2020 at 7:39

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