Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}\_{g,n}(K)$ and $\overline{M}\_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing Deligne-Mumford stable curves over $K$ and $\overline{K}$ respectively. 


Is it true that $\overline{M}\_{g,n}(K)\times\_{Spec(K)}Spec(\overline{K})\cong \overline{M}\_{g,n}(\overline{K})$ ?


My question is related to
"[What is $M\_g$ over a finite field, really?](https://mathoverflow.net/questions/72903/what-is-m-g-over-a-finite-field-really?rq=1)" regarding a similar issue in positive characteristic.



Moduli of curves in characteristic zero
I will just develop a bit the comment by Jason Starr. In these notes
<http://math.stanford.edu/~conrad/papers/coarsespace.pdf>
you can find the following theorem by Keel and Mori:


Let $S$ be a scheme and let $\mathcal{X}$ be an Artin stack that is locally of finite presentation over $S$ and has ﬁnite inertia stack $I\_S(\mathcal{X})$. There exists a coarse moduli space $\pi:\mathcal{X}\rightarrow X$, such that:


* The structure map $X\rightarrow S$ is separated if $\mathcal{X}\rightarrow S$ is separated, and it is locally of finite type if $S$ is locally noetherian.
* The map $\pi$ is proper and quasi-ﬁnite.


Moreover, if $X^{'}\rightarrow X$ is a ﬂat map of algebraic spaces then $\pi:\mathcal{X}^{'}:=\mathcal{X}\times\_{X}X^{'}\rightarrow X^{'}$ is a coarse moduli space.


In you case the map $Spec(\overline{K})\rightarrow Spec(K)$ is flat. Therefore $\overline{M}\_{g,n}^{K}\times\_{Spec(K)}Spec(\overline{K})$ is a coarse moduli space for the stack $\overline{\mathcal{M}}\_{g,n}^{\overline{K}}$. Since the coarse moduli space of $\overline{\mathcal{M}}\_{g,n}^{\overline{K}}$ is unique up to a unique isomorphism you get $\overline{M}\_{g,n}^{K}\times\_{Spec(K)}Spec(\overline{K})\cong \overline{M}\_{g,n}^{\overline{K}}$.


