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?


There is really no universal reference, unfortunately, and what you should look at depends on what you're interested in. Are you interested in arithmetic or are you working over the complex numbers? Are you interested in compactification? Do you care mostly about simple branching or about more general branching or even about more general Galois groups than S_n? Are you interested in these guys as subvarieties of M_g, or as covers of M_{0,n}, or both, or as abstract moduli spaces or....?

But I guess this is not an answer yet, so let me add to your already good list Harris and Morrison's book on algebraic curves, which has a nice discussion of admissible covers that should be very helpful for understanding how one kind of branched cover can degenerate to another. For the topologist's point of view, you might also look at any paper containing the words "braid group" and "Nielsen classes," or McReynolds's exposition of Thurston's proof of the congruence subgroup property for the braid group.

  • $\begingroup$ Thanks for your helpful answer. My goal is to (eventually) study the arithmetic side of Hurwitz schemes, but I think it wouldn't hurt to have a good understanding of what happens over the complex numbers. Moreover, I'm interested in just learning the general theory at the moment. So compactification, connectedness, irreducibility, abstract moduli spaces point of view, etc. Could you elaborate a bit on what you mean by viewing these guys as subvarieties of M_g or as covers of M_{o,n}? $\endgroup$ – Ariyan Javanpeykar Sep 15 '11 at 14:43
  • $\begingroup$ By the way, the only book I can find written by Harris and Morisson is "Moduli of Curves". I couldn't find anything about Hurwitz spaces in this book. Is there another book I'm unaware of? $\endgroup$ – Ariyan Javanpeykar Sep 15 '11 at 15:00
  • $\begingroup$ "Moduli of Curves" is the right book -- look up "admissible covers." A Hurwitz space parametrizes branched covers C -> P^1. If you forget the cover and just remember C, you get a point of M_g. If you forget C and just remember the locations of the r branch points, you get a point of M_{0,r}. $\endgroup$ – JSE Sep 15 '11 at 15:03
  • $\begingroup$ The degree of the map to M_{0,4} is called the Hurwitz number, and looking at (the beginning of) some of Ravi Vakil's papers about Hurwitz numbers would also be a good option. $\endgroup$ – JSE Sep 15 '11 at 15:04
  • $\begingroup$ Nice. This gives me something to do for a while. $\endgroup$ – Ariyan Javanpeykar Sep 15 '11 at 15:07

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).


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