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In more than one place I read that, given a moduli problem, the existence of an object with nontrivial automorphisms prevents the existence of a fine solution. I'd like to understand in which sense this is true.

Let us assume that a coarse moduli space $M$ exists for a moduli problem, where we have an object $X$ with Aut $X\neq 1_X$. Can we conclude that there is no universal family over $M$?

Here are my thoughts: at the very beginning, I imagined that, if there was such a universal family, then we would be in trouble in saying what to "attach" over $X$. But in fact now I think the answer should be no (even if I don't have a counterexample): indeed, why can't we just "identify" $X$ with all of its automorphic images? Let me explain what I mean with an example: take $M_g$. A universal family $U$ over $M_g$ would be - I guess - of the form $U=\{([C],P)\,|\,[C]\in M_g\textrm{ and }P\in C\}=M_{g,1}$. The fiber over a point $[C]\in M_g$ would be $\{([C],P)\,|\,P\in C\}$. How to "distinguish" this from $\{([C],\sigma P)\,|\,P\in C\}$, where $\sigma$ is a nontrivial automorphism of $C$?

Now let us come back to our $M$. Of course, with or without the above $X$, if there is a nontrivial isotrivial family, then $M$ cannot be fine. So the question is: are we always able to construct a nontrivial isotrivial family thanks to an object $X$ with Aut $X\neq 1_X$? In other words, how is isotriviality related to the existence of such an $X$?

Thank you!

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    @Andrew: Thank you so much! didn't check on MO...2012-11-11

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