Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Question

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.

If $\mathcal X$ can be defined over $\overline{\mathbb Q}$, then $X$ can be defined over $\overline{\mathbb Q}$. This is because "the base-change of the coarse moduli space is the coarse moduli space of the base-change".

Does the converse hold?

What if we drop the properness condition?

My motivation is mere curiosity.

Answer 1

No. Even if $X$ is "defined over $\overline{\mathbb{Q}}$" in the sense that $X$ is isomorphic to $X_0\otimes_{\overline{\mathbb{Q}}} \mathbb{C}$ for some variety $X_0$ over $\overline{\mathbb{Q}}$, nonetheless the stack $\mathcal{X}$ may not be defined over $\overline{\mathbb{Q}}$.

For instance, let $X_0$ be $\mathbb{P}^1_{\overline{\mathbb{Q}}}$ so that $X$ is $\mathbb{P}^1_{\mathbb{C}}$. Now let $t_0,t_1,t_2\in \mathbb{P}^1_{\mathbb{C}}(\mathbb{C})\setminus\{0,1,\infty\}$ be three distinct points such that at least one is transcendental, e.g., $2$, $3$, $\pi$. Denote by $$f:Y\to X,$$ the genus $2$, hyperelliptic curve branched over $$\{0,1,\infty,t_0,t_1,t_2\}.$$ There is an action of $\mathbb{Z}/2\mathbb{Z}$ on $Y$ via the hyperelliptic involution. Let $\mathcal{X}$ be the quotient Deligne-Mumford stack, $$\mathcal{X} = [Y /(\mathbb{Z}/2\mathbb{Z})].$$ The coarse moduli space is $X$, via the unique morphism $\pi: \mathcal{X}\to X$ that factors $f$. Of course $X$ is defined over $\overline{\mathbb{Q}}$. However, the branch divisor of the morphism $\pi$ is not defined over $\overline{\mathbb{Q}}$ as a subscheme of $X$. Thus, also $\mathcal{X}$ is not defined over $\overline{\mathbb{Q}}$.

Added by OP:

It might be useful to (slightly) generalize the above construction.

Let $X=\mathbb P^1_{\mathbb C}$ and let $f:Y\to X$ be a hyperelliptic curve with branch locus $D$ (and $Y$ of positive genus). Write $U=X\backslash D$. Let $\mathcal X$ be the stack $[Y/G]$. Then the unique morphism $\pi:\mathcal X \to X$ that factors $f:Y\to X$ is the coarse moduli space of $\mathcal X$. Suppose that $U$ can't be defined over Qbar. Then $Y\to X$ (and $Y$) can't be defined over $\overline{\mathbb Q}$ (as the hyperelliptic involution is unique). Therefore, $\mathcal X\to X$ and $\mathcal X$ can't be defined over Qbar.

Answer 2

Let $\mathcal{X}$ be an algebraic stack over $k$. A coarse moduli space for $\mathcal{X}$ over $k$ is a morphism $\pi:\mathcal{X}\rightarrow X$, where $X$ is a scheme such that:

• the morphism $\pi$ is universal for morphisms to schemes;
• $\pi$ induces a bijection between $|\mathcal{X}|$ and the closed points of $X$, where $|\mathcal{X}|$ denotes the set of isomorphism classes in $\mathcal{X}$.

If $\mathcal{X}$ admits a coarse moduli space $\pi:\mathcal{X}\rightarrow X$ then this is unique up to unique isomorphism. A separated algebraic stack has a coarse moduli space which is a separated algebraic space.

Let $\mathcal{X}$ be a separated stack admitting a scheme $X$ as coarse moduli space $\pi:\mathcal{X}\rightarrow X$. The map $\pi$ is universal for morphisms in schemes, that is for any morphism $f:\mathcal{X}\rightarrow Y$, with $Y$ scheme, there exists a unique morphisms of schemes $g:X\rightarrow Y$ such that $f = g\circ\pi$.

Now, if $\mathcal{X}$ is defined over $k$ we have a morphism $f:\mathcal{X}\rightarrow Spec(k)$. By universality there is a morphism of schemes $g:X\rightarrow Spec(k)$ such that $g\circ\pi = f$. In particular $X$ is defined over $k$.

Conversely, if $X$ is defined over $k$ then there is a morphism $g:X\rightarrow Spec(k)$. By composition we get a morphism $f:=g\circ\pi:\mathcal{X}\rightarrow X$. Therefore $\chi$ is defined over $k$.

For instance both the stack $\overline{\mathcal{M}}_{g,n}$ and its coarse moduli space $\overline{M}_{g,n}$ parametrizing Deligne-Mumford stable curves are defined over $\mathbb{Z}$. In particular over any commutative ring and hence over any field.