This is related to [this](https://mathoverflow.net/questions/151292/number-of-curves-over-a-finite-field) question. I learnt about moduli problem mainly with the book Harris and Morrison. Therefore, I have only seen the construction of moduli spaces $M\_{g}$ over $\mathbb{C}$. But now I want to change the base field. It does not really make sense to me to take $\mathbb{F}\_q$-points as described in the comments of that post, because $M\_g$ is a $\text{Spec}(\mathbb{C})$-scheme not a $\text{Spec}(\mathbb{F}\_q)$-scheme. So how should I understand $M\_{g}$ as a coarse moduli space properly?



Moduli of curves over finite field
The constructions of the Deligne–Mumford stack $\mathscr M\_g$ and its coarse moduli space $M\_g$ are very similar, and Deligne–Mumford's original article [DM69] is surprisingly readable. Note that Deligne and Mumford write $\mathscr M\_g$ (resp. $M\_g$) for what is now commonly known as the moduli of *stable* (rather than smooth) curves $\overline{\mathscr M\_g}$ (resp. $\overline{M\_g}$); feel free to stick to the smooth case if you prefer.


The construction goes as follows:


1. If $C$ is a stable curve of genus $g \geq 2$, then $\omega\_C^{\otimes 3}$ is very ample.
2. Construct a space $H\_g \subseteq \mathbf{Hilb}\_{\mathbf P^{5g-6}}$ of tricanonically embedded stable curves (using Hilbert scheme methods; *the* number one tool for representing moduli problems).
3. Construct $\mathscr M\_g$ (resp. $M\_g$) as the stacky (resp. GIT) quotient of $H\_g$ by the automorphism action $\mathbf{PGL}(5g-6) \circlearrowright \mathbf{Hilb}\_{\mathbf P^{5g-6}}$.


I [just learned](https://mathoverflow.net/q/72903/82179) that there is a bit of an issue whether you take the GIT quotient first and then specialise to characteristic $p$ or vice versa. Perhaps Alper's good moduli spaces can say something about this question. Or you can use the stacky definition.


To some extent the main goal of the [stacks project](https://stacks.math.columbia.edu/) is to provide all details needed to define $\overline{\mathscr M\_g}$, so this is another reference. But this is thousands of pages, so not really useful as an introduction.




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**References.**


[DM69] P. Deligne and D. Mumford, [*The irreducibility of the space of curves of a given genus*](http://www.numdam.org/item/PMIHES_1969__36__75_0/). Inst. Hautes Études Sci. Publ. Math. **36**, p. 75-109 (1969). DOI: [10.1007/BF02684599](http://dx.doi.org/10.1007/BF02684599).


