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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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    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 https://arxiv.org/abs/1702.04404). – dhy Jun 02 '19 at 13:49
  • Is it known whether such fields exist at all? – Avi Steiner Jun 02 '19 at 18:05
  • @AviSteiner not known to me, that's for sure –  Jun 03 '19 at 09:51

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