What, precisely, is the relationship between "fields of moduli" and "moduli spaces"?

Question

Notation

The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ¹ be a G-Galois cover, where everything is over the algebraic closure of some field L. Assume that X->ℙ¹ descends (without group action -- as a cover) to X_L->ℙ_L¹. Then I define the field of moduli to be the intersection of all finite extensions of L for which base change of X_L->ℙ_L¹ becomes G-Galois.

Question

There is the saying that the field of moduli is the function field of the (coarse?) moduli space of when you let the branch points vary. What is the precise statement of that? (and why is it true?)

Thoughts

It would seem that we should fix a dedekind ring whose quotient field is L (ℤ if L is ℚ), and call it D. Then descend to a D-model of ℙ¹ (for a D-model of X take the integral closure of ℙ¹ in the function field of X). Then do something like look at the moduli space of all covers of ℙ¹ with that number of (distinct) branch points, and in it look at the subscheme of all covers that can be achieved by deforming any of the fibers of our X_D->ℙ_D¹ (pick a fiber such that there's no coalescence of branch points) by a family. But there's a lot missing here, even in terms of making this precise. For example: IS there a coarse moduli space of all covers with n branch points over ℙ_D¹ (where by n branch points, I mean n branch point on each geometric fiber)? What does it look like? Why should the function field of said subscheme be the field of moduli?

Thanks in advance.

Answer 1

A few responses to different parts of your question.

1. In my experience, the phrase "field of moduli" doesn't usually refer to the function field of a coarse moduli space. Rather: the base change of your cover to Lbar corresponds to a point of M(Lbar), where M/L is the coarse moduli space. This point has a well-defined field of definition, which is by definition the field of moduli of your cover. The phrase "field of moduli" is usually used in distinction with "field of definition" -- if your cover is actually defined over L', then the field of moduli is certainly contained in L', but it may not be equal to L'. This phenomenon isn't restricted to Hurwitz spaces; there are abelian varieties over Qbar whose field of moduli is Q (that is, they correspond to points of A_g(Q)) but which don't descend to Q. This can only happen when g is even. Off the top of my head I don't remember a reference for an example, nor for the assertion of the previous sentence; maybe somebody can help me out in comments. Certainly when g=1 you don't have this problem; given a rational number j, there is an elliptic curve E/Q with j(E) = j. But you prove this by writing it down -- it's not completely obvious "by pure thought" that it should be so.

2. The most complete description of the Hurwitz stack (the moduli stack of finite covers with fixed combinatorial invariants) its associated coarse moduli space, etc., is in the Ph.D. thesis of Stefan Wewers, which is unfortunately not available online. However, the survey paper of Romagny and Wewers should give you most of what you need.

Answer 2

What I usually see called field of moduli is the following. Suppose $C$ is some member of a collection which admits a coarse moduli space $M$ say defined over $Q$. E.g. $C$ could be a curve or a $G$ cover, as in your example, and so on. So $C$ corresponds to a point $[C]$ of $M$. The field generated by the coordinates of $[C]$ is the field of moduli of $C$. But in your post, it seems you might be taking $C$ to be a generic element of your collection and what you get as field of moduli is $Q(M)$, the function field of $M$, but that sounds tautological to me.

If $C$, as above, is defined over a field $K$, then $K$ contains the field of moduli of $C$ but it's not always true that $C$ can be defined over the field of moduli.

Moduli spaces of branched covers are usually called Hurwitz schemes. I don't know the precise conditions for their existence offhand.