# 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 $p$.

• 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. Dec 29, 2009 at 23:54