Moduli of curves in characteristic zero 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?" regarding a similar issue in positive characteristic.
 A: 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 $\overline{M}_{g,n}^{K}\times_{Spec(K)}Spec(\overline{K})\cong \overline{M}_{g,n}^{\overline{K}}$.
