### Notation The term "field of moduli" comes in up in different scenarios, but let's consider the following: Let X->ℙ<sup>1</sup> be a G-Galois cover, where everything is over the algebraic closure of some field L. Assume that X->ℙ<sup>1</sup> descends (without group action -- as a cover) to X<sub>L</sub>->ℙ<sub>L</sub><sup>1</sup>. Then I define the field of moduli to be the intersection of all finite extensions of L for which base change of X<sub>L</sub>->ℙ<sub>L</sub><sup>1</sup> 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<sup>1</sup> (for a D-model of X take the integral closure of ℙ<sup>1</sup> in the function field of X). Then do something like look at the moduli space of all covers of ℙ<sup>1</sup> 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<sub>D</sub>->ℙ<sub>D</sub><sup>1</sup> (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<sub>D</sub><sup>1</sup> (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.