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 XL->ℙL1. Then I define the field of moduli to be the intersection of all finite extensions of L for which base change of XL->ℙL1 becomes G-Galois.
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?)
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 XD->ℙD1 (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 ℙD1 (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.