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For which fields $k$ does there exist a proper morphism $S\rightarrow \mathrm{Spec}\:k$ of relative dimension $\leq 2$ such that for every geometrically connected smooth proper $C \rightarrow \mathrm{Spec}\:k$ of relative dimension $1$ there exists a $k$-closed immersion $C\rightarrow S$?

Complex numbers are not such a field (in fact, a stronger result holds), some "massaging" might extend this to all subfields of $\mathbb{C}$. No idea what is going on in $\mathrm{char}\:p$.

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    $\begingroup$ The proof you linked to works for all uncountable algebraically closed fields of characteristic $0$. I think you need a new idea to extend it to say, $\overline{\mathbb{Q}}.$ For (uncountable algebraically closed) fields of characteristic $p$, this seems closely related to asking if $M_g$ is unirational for large $g$, and to my knowledge this is still open (see the final remarks in arxiv.org/abs/1702.04404). $\endgroup$
    – dhy
    Commented Jun 2, 2019 at 13:49
  • $\begingroup$ Is it known whether such fields exist at all? $\endgroup$
    – Avi Steiner
    Commented Jun 2, 2019 at 18:05
  • $\begingroup$ @AviSteiner not known to me, that's for sure $\endgroup$
    – user140765
    Commented Jun 3, 2019 at 9:51

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