This is related to this 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?
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4$\begingroup$ Already in "Geometric Invariant Theory" (which was Mumford's thesis), Mumford constructed the coarse moduli scheme $M_g$ over $\text{Spec}\ \mathbb{Z}$. You can read about what this means in Mumford's book. The great irony of GIT is that, although GIT was largely motivated by this problem, in fact Mumford constructs $M_g$ over $\text{Spec}\ \mathbb{Z}$ without using GIT (precisely because neither had Haboush yet extended GIT to char $p$, nor had Seshadri extended GIT to mixed characteristic). $\endgroup$– Jason StarrCommented Jul 23, 2019 at 9:03
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$\begingroup$ @JasonStarr The $K$-points of moduli space constructed via GIT gives exactly the isomorphic classes of curves over $K$? For any field (especially not algabraically closed)? $\endgroup$– UpcCommented Jul 23, 2019 at 12:19
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1$\begingroup$ @JasonStarr: could you post this as an answer? You seem to be better aware of the history than myself, and I think it would be a better answer than mine. $\endgroup$– R. van Dobben de BruynCommented Jul 23, 2019 at 13:47
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2 Answers
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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:
If $C$ is a stable curve of genus $g \geq 2$, then $\omega_C^{\otimes 3}$ is very ample.
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).
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 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 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.
References.
[DM69] P. Deligne and D. Mumford, The irreducibility of the space of curves of a given genus. Inst. Hautes Études Sci. Publ. Math. 36, p. 75-109 (1969). DOI: 10.1007/BF02684599.
answered Jul 22, 2019 at 22:47
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As requested, I am posting my comment as an answer. Mumford discusses this in the preface to the first edition of "Geometric Invariant Theory".
Already in "Geometric Invariant Theory" (which was Mumford's thesis), Mumford constructed the coarse moduli scheme $M_g$ over $\text{Spec}\ \mathbb{Z}.$ You can read about what this means in Mumford's book. The great irony of GIT is that, although GIT was largely motivated by this problem, in fact Mumford constructs $M_g$ over $\text{Spec}\ \mathbb{Z}$ without using GIT (precisely because neither had Haboush yet extended GIT to char $p$, nor had Seshadri extended GIT to mixed characteristic).