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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?

Giulio Bresciani
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  • 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. – Ariyan Javanpeykar Oct 03 '20 at 19:16
  • An obstruction to the existence of a section is the presence of multiple fibres. A subquestion would therefore be: Can you show that your assumption on $k$-surjectivity implies that $X\to \mathbb{P}^1_k$ has no multiple fibres (without assuming Lang's conjecture)? Same subquestion can be posed for Question 2: Given a fibration $X\to C$ which is $k'$-surjective for every number field $k'$ containing $k$, does it follow that $X\to C$ has no multiple fibres? – Ariyan Javanpeykar Oct 03 '20 at 19:23
  • Are you really sure that you get a section? It seems more like you get a geometrically integral curve $D \subset X$ which maps birationally into $C$. A priori, this need not be an isomorphism as $D$ need not be smooth. – Daniel Loughran Oct 03 '20 at 20:21
  • @DanielLoughran But $C$ is normal, so $D\to C$ will be an isomorphism by ZMT. – Ariyan Javanpeykar Oct 03 '20 at 20:22
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    Yes of course, thanks – Daniel Loughran Oct 03 '20 at 20:25
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    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) https://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q – Daniel Loughran Oct 03 '20 at 20:27
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    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. – Daniel Loughran Oct 03 '20 at 20:31
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    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 https://arxiv.org/abs/1705.10740 for details. – Daniel Loughran Oct 04 '20 at 07:39
  • @AriyanJavanpeykar Of course we want $C(k)$ non-empty, thanks, corrected. About the singular fibers, $C$ is not necessarily complete: it seems to me that is just easier to state the problem this way, then one can consider completions. – Giulio Bresciani Oct 06 '20 at 15:16
  • @DanielLoughran Thanks for the references. – Giulio Bresciani Oct 06 '20 at 15:22
  • I should add that removing the singular fibres is a bad idea: in fact it is all about the singular fibres in such problems, as in demonstrated in the arxiv preprint I mention above. – Daniel Loughran Oct 06 '20 at 16:02
  • I might also want to add that, since $X$ is of Kodaira dimension 2, there will always be singular fibres of $X\to \mathbb{P}^1$. (There are no smooth non-isotrivial families of higher genus curves over $\mathbb{P}^1$.) – Ariyan Javanpeykar Oct 06 '20 at 17:53
  • For Question 2: Have you considered what happens if $X$ is a Kodaira surface over $k$? That is, $X$ is a smooth projective geometrically connected surface over $k$ with $\omega_X$ ample which admits a smooth proper morphism $X\to C$ to a smooth projective curve of genus at least two. In this case, $X(k)$ is finite for every number field $k$ by Mordell. Assume $X\to C$ is $k$-surjective for every number field $k$. Does $X\to C$ have a section? It's a good question (and there are no singular fibres!). – Ariyan Javanpeykar Oct 06 '20 at 17:56
  • Edit: the morphism $X\to C$ should be non-isotrivial. (Otherwise $X$ will just be a product up to a finite etale cover of $X$.) – Ariyan Javanpeykar Oct 06 '20 at 18:41

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