# Coarse moduli spaces over Z and F_p

I would like to know to what extent it is possible to compare fibers over $\mathbb{F}_p$ of coarse moduli spaces over $\mathbb{Z}$, and coarse moduli spaces over $\mathbb{F}_p$. I ask a more precise question below.

Let $\mathcal{M}_g^{\mathbb{Z}}$ be the moduli stack of smooth genus $g$ curves over $\mathbb{Z}$. Let $M_g^{\mathbb{Z}}$ be its coarse moduli space, and $(M_g^{\mathbb{Z}})_p$ the fiber of this coarse moduli space over $\mathbb{F}_p$. Let $\mathcal{M}_g^{\mathbb{F}_p}$ be the moduli stack of smooth genus $g$ curves over $\mathbb{F}_p$ and $M_g^{\mathbb{F}_p}$ its coarse moduli space.

The universal property gives a map $\phi:M_g^{\mathbb{F}_p}\rightarrow(M_g^{\mathbb{Z}})_p$. My question is : is $\phi$ an isomorphism ?

In fact, since $\phi$ is a bijection between geometric points, and $M_g^{\mathbb{F}_p}$ is normal, the question can be reformulated as : is $(M_g^{\mathbb{Z}})_p$ normal ? This shows that when $g$ is fixed, the answer is "yes" except for a finite number of primes $p$.

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## 1 Answer

So you're asking if formation of coarse spaces commutes with (certain types of) base change. In general the answer is no; one needs the notion of a tame moduli space. A good starting point for this is Jarod Alper's paper "Good Moduli Spaces for Artin Stacks", available on his web page; he explains the notion and cites the relevant papers for tame moduli spaces. This should help you to work out your particular example (I don't know the answer off the top of my head).

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Thanks for the reference ! However, I don't think it applies in this situation. Indeed, being "tame" is a condition on the automorphism groups of the geometric points. In the situation here, these groups are reduced, and I think the copndition is exactly "being of order prime to the characteristic". And there exist curves in characteristic $p$ with $p$-groups of automorphisms. Such arguments apply, however, for primes $p$ bigger than the known upper bounds for the order of the automorphism group of a smooth genus $g$ curve. – Olivier Benoist Dec 29 '09 at 23:54