Family over the coarse moduli space of curves

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?

• 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 $\mathcal{C}_g$.
– naf
Sep 7 '20 at 9:57
• There is the forgetful morphism from the coarse moduli space $M_{g,1}$ to $M_g$. However, that morphism is not everywhere flat. Sep 7 '20 at 16:37
• @ulrich Is this something that should not exist? Sep 8 '20 at 6:41
• 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 $g \geq 3$.)
– naf
Sep 8 '20 at 13:14