Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}\_{g,n}$ represents the functor $$\overline{\mathcal{M}}\_{g,n}(S)$$
assigning to a scheme $S$ the groupoid of proper stable nodal curves $C\to S$ over $S$ with genus $g$ and $n$ marked sections, taken up to isomorphism. If we are careful about Galois actions, it becomes clear that in fact $\overline{\mathcal{M}\_{g,n}}$ is defined over $\mathbb{Q}$, and classifies families of curves together with certain descent data to $\mathbb{Q}$. Using GIT, Mumford defined certain natural models of $\mathcal{\overline{M}}\_{g,n}$ over $\mathbb{Z}$. I would like to know whether these models represent any meaningful moduli functor, or if not, whether there is still a notion of stable nodal curve with marked points over an integral base that has some of the nice properties of the Deligne-Mumford functor. 



Moduli problem of stable nodal curves over the integers
Already Deligne and Mumford's original paper constructs $\overline{\mathcal M}\_g$ over $\operatorname{Spec} \mathbb Z$ [DM, §5]. They do not do the pointed version, so they restrict themselves to $g \geq 2$ (otherwise it will only exist as an Artin stack). Note also that they write $\mathcal M\_g$ for what is now commonly known as $\overline{\mathcal M}\_g$. Another reference is of course the Stacks project; in particular [Stacks, Tag [0E99](https://stacks.math.columbia.edu/tag/0E99)].


The same should true for the pointed version $\overline{\mathcal M}\_{g,n}$, and I think it should not be so hard to modify the proof from the unpointed to the pointed case (especially if one also studies the versions for $g \in \{0,1\}$). I suspect that this has been carried out in the literature somewhere, but I do not know a reference off hand.




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**References.**


[DM] Deligne, Pierre; Mumford, D., [*The irreducibility of the space of curves of a given genus*](http://www.numdam.org/item?id=PMIHES_1969__36__75_0). Publ. Math. Inst. Hautes Étud. Sci. **36** (1969), p. 75-109. [ZBL0181.48803](https://zbmath.org/?q=an:0181.48803).


[Stacks] A.J de Jong et al, [*The stacks project*](http://stacks.math.columbia.edu).


