### Notation

The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ^{1} be a G-Galois cover, where everything is over the algebraic closure of some field L. Assume that X->ℙ^{1} descends (without group action -- as a cover) to X_{L}->ℙ_{L}^{1}. 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_{L}^{1} 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 ℙ^{1} (for a D-model of X take the integral closure of ℙ^{1} in the function field of X). Then do something like look at the moduli space of all covers of ℙ^{1} 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_{D}^{1} (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_{D}^{1} (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.