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In short, the answer to the question is yes. I'm aware of the existence of moduli spaces for canonically polarized varieties with fixed Hilbert polynomial over $\mathbf C$. I think they require the minimal model program, but that should be fine.

My real question is whether these moduli spaces extend to stacks defined over $\mathbf Z$.

What about the 2-dimensional case?

Is there a moduli stack over $\mathbf Z$ of canonically polarized surfaces of general type with fixed Hilbert polynomial?

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Do you want something more than "moduli stack"? Surely the abstract stack exists because you just define it in the same way. The hard part would be proving it is DM or Artin or some extra "nice" adjective. – Matt Jan 13 '13 at 18:48
When I say moduli stack I mean a Deligne-Mumford stack solving the moduli problem. Sorry for not saying that explicitly. – Harry Jan 13 '13 at 19:55

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