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Let $k$ be an algebraically closed field. As the coarse moduli space of curves $M_g$ of genus $g$ over $k$ is not a fine moduli space, it does not have a universal family. But I am wondering if it has a family (proper and flat) such that the fiber over every point $[C]$ of $M_g$ is isomorphic to the curve $C$.

As a disclaimer: I am not that familiar with the language of stacks. As far as I understand the situation in this context, the stack $\mathcal{M}_g$ has an universal family $\mathcal{C}_g$. The corresponding coarse moduli space of $\mathcal{C}_g$ is $M_{g,1}$, so the coarse moduli space of curves with one marked point. The morphism $\pi \colon M_{g,1} \to M_g$ on the level of quasiprojective varieties is just forgetting about the marked point. This family has the property that the fiber over a point $[C]$ is isomorphic to $C$, at least if $C$ has no nontrivial automorphisms. In all other cases the fiber is isomorphic to $C/\operatorname{Aut}(C)$. Is it possible to get something better than that?

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    $\begingroup$ If you had such a family, then you'd get a section of the map $\mathcal{M}_g$ to $M_g$, and also a map from the total space of the family to $\mathcal{C}_g$. $\endgroup$
    – naf
    Sep 7 '20 at 9:57
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    $\begingroup$ There is the forgetful morphism from the coarse moduli space $M_{g,1}$ to $M_g$. However, that morphism is not everywhere flat. $\endgroup$
    – Jason Starr
    Sep 7 '20 at 16:37
  • $\begingroup$ @ulrich Is this something that should not exist? $\endgroup$
    – Fabian Ruoff
    Sep 8 '20 at 6:41
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    $\begingroup$ My previous comment would imply that there is a morphism from any family (as in the question) to $M_{g,1}$ which on fibres is the quotient by the automorphism group. This morphism would be finite birational and this is not possible since $M_{g,1}$ is normal. (I am assuming that $g \geq 3$.) $\endgroup$
    – naf
    Sep 8 '20 at 13:14
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To close up loose ends and for everyone finding this questions: Such a family does not exist in general. An argument for elliptic curves can be found in Robin Hartshorne Deformation Theory in Remark 26.3.1.

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