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I am trying to understand why the (coarse) moduli space $\overline{M}_g$ of stable curves of genus $g$ does not admit a universal family. I am following the proof in p.267 of this book.

A key step is the fact that every stable curve admits a Kuranishi family. So, suppose we have a universal family $\pi:\mathcal{C}\rightarrow \overline{M}_g$ and let $[C]\in \overline{M}_g$. Let $\xi:\mathcal{X}\rightarrow (X,x_0)$ be a Kuranishi family of $C$.

On the one hand there exists a neighborhood $V$ of $[C]$ in $\overline{M}_g$ and a map $f:V\rightarrow X$ such that $\pi|_V$ is the pullback of $\xi$ via $f$.

On the other hand, there exists a map $h:X\rightarrow \overline{M}_g$ such that $\xi$ is the pullback of $\pi$ via $h$.

By considering the composition $f\circ h$ (and restricting to a neighborhood of $x_0$) we are supposed to get a family differing from a Kuranishi family by at most an automorphism of the central fiber. I do not understand why this is true.

Anyway, this implies that $f\circ h$ is an automorphism and therefore $h$ is injective. Nevertheless, if $\gamma\in Aut(C)\setminus\{id\}$ then $\gamma$ acts non-trivially on $V$ and $h$ sends any orbit of $Aut(C)$ to the same point of $\overline{M}_g$. We reach a contradiction.

In conclusion, why $f\circ h$ gives a family differing from a Kuranishi family by at most an automorphism of the central fiber?

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  • $\begingroup$ I'm not familiar with this specific argument, but if you simply want to know why there does not exist a fine moduli space of stable curves it suffices to show that there exists non-trivial but isotrivial families (i.e. all fibres are isomorphic, but the total space is not isomorphic to the trivial family). $\endgroup$ – loch Sep 21 '18 at 12:58

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