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# When is a coarse moduli space also a fine moduli space?

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# CoarseWhenisacoarse moduli space becoming a fine moduli space?

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Given a moduli problem, it appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space(is this strictly true?).

In any case, assuming the above, what additional condition on a moduli problem in algebraic geometry will make sure that a coarse moduli space is in fact a fine moduli space?

In the n-lab page on Deligne-Mumford, the following appears.

Deligne-Mumford stacks correspond to moduli problems in which the objects being parametrized have finite automorphism groups.

Also, a few problematic examples I have heard of, mentioned a had infinite automorphism groupgroups.

Therefore, is it true that for a moduli problem in which the stack is Deligne-Mumford, and where there are no automorphisms, existence of a coarse moduli space would imply the existence of a fine moduli space?

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