# Reference request: parametrizing covers of the projective line

Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways.

For example, one could fix the number $r$ of branch points, the degree $n$ of the cover and look only at simple covers of $\mathbf{P}^1$. This is usually denoted by $H_{r,n}$. Fulton defined this space as a scheme over $\textrm{Spec} \mathbf{Z}$ and showed that $H_{r,n} \otimes \mathbf{F}_p$ is irreducible for $p$ big enough.

One could also fix a subset $B\subset \mathbf{P}^1$, the degree $n$ of the cover and look at covers of degree $n$ unramified outside $B\cup \{\lambda \}$, where $\lambda \in \mathbf{P}^1-B$ is allowed to vary. One can show that most curves arise as an irreducible component of such a space (Diaz, Donagi, Harbater).

One could also look at Galois covers with a fixed Galois group, etc.

In the end, there are many ways to parametrize covers of the projective line.

Are there any standard references that contain the basics of Hurwitz spaces?

At the moment I have at my disposal

Work of M. Romagny, J. Bertin and S. Wewers (available on Romagny's website). These are very stacky.

The article of Fulton Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves.

Notes by Brian Osserman available on his website (The representation theory, geometry, and combinatorics of branched covers.).

The article of Diaz, Donagi and Harbater: Every curve is a Hurwitz space.

Question. What are the standard references for the basics of Hurwitz spaces/schemes?

One modern way to view admissible covers is via twisted stable maps to the stack $BG$ (where $G$ is the group of the Galois closure of the cover). The objects in the stack $M_{0,n}(BS_d)$ can be regarded as degree $d$ covers of a $\mathbb{P}^1$ which is ramified over $n$ marked points. The ramification type is determine by the evaluation map $ev:M_{0,n}(BS_d)\to IBS_d$ where the components of the inertia stack are indexed by conjugacy classes of $G$. The stable map compactification $\overline{M}_{0,n}(BS_d)$ is essentially the same as the admissible covers compactification, in fact it is actually a bit better behaved (I think it is the normalization of the later).