Fix a natural number $g$, a prime $p$, and a $p$-group $P$.

Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ acts without stabilizers on points lying over $\mathbb A^1 \subset \mathbb P^1$. Then I will call $C$ a $(P,\infty)$-curve. In other words $(P,\infty)$-curves are finite Galois covers of $\mathbb P^1$ with Galois group $P$, etale away from $\infty$.

I think by general nonsense a moduli space for $(P,\infty)$-curves can be constructed as a Deligne-Mumford stack from the moduli space of smooth curves of genus $g$. There is some subtlety in that the condition that $P$ acts simply away from $\infty$ defines a closed subset and not a closed subscheme/substack, but one can give it a closed subscheme/substack structure by writing down a polynomial that vanishes on the fixed points (the discriminant of the cover) and setting it equal to a polynomial vanishing only at $\infty$. Because $(P,\infty)$-curves can only exist in characteristic $p$, it is most natural to view this as a space over $\mathbb F_p$.

Is this moduli space smooth?

My motivation is that $(P,\infty)$-curves over a perfect field $k$ are in bijection with surjective homomorphisms $\pi_1(\mathbb A^1_k) \to P$, and hence in bijection with surjective homomorphisms $\operatorname{Gal} ( k((x))) \to P$, because the maximal pro-$p$ quotients of those groups are isomorphic. I want to understand quotients of that group, so I want to understand this space.

nota smooth morphism. This is why the Oort conjecture is nontrivial. $\endgroup$ – Jason Starr Mar 5 '15 at 20:23