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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.

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2 Answers 2

up vote 11 down vote accepted

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.

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1  
Thank you very much. –  Ariyan Javanpeykar May 11 at 17:00
    
LateX wasn't rendering so good anymore (on my pc), so I changed some of the "$\overline{\mathbb{Q}}$" into "Qbar". –  Ariyan Javanpeykar May 11 at 23:32
    
It seems to me that hyperelliptic is overkill and you could do exactly the same thing in the elliptic case, ramified at $0,1,\infty,\pi$. –  Ben Wieland May 12 at 0:07
    
@Ari: What you added contains a mistake. "Bonus: we can find $\mathcal{X}$ such that $Y$ and $X$ are defined over $\mathbb{Q}$, but $\mathcal{X}$ isn't. (Just take $Y = X = \mathbb{P}^1$)". That is impossible. This is the reason that I chose the example I chose, where the morphism $\pi$ is "characteristic". –  Jason Starr May 12 at 12:58
    
@JasonStarr I see. Thank you. –  Ariyan Javanpeykar May 12 at 15:28

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.

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I think I'm missing the argument. Assume that the coarse moduli space $X$ has a model $X_0$ over $\overline{\mathbb {Q}}$. Where exactly do you show that $\mathcal X$ has a model over $\overline{\mathbb {Q}}$? –  Ariyan Javanpeykar May 10 at 23:08
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@CamSar your definition of "$X$ is defined over $\overline{\mathbb Q}$" is not correct. –  John Pardon May 10 at 23:48
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Indeed, every $X$ over $\mathbb C$ has a morphism to $\operatorname{Spec}\overline{\mathbb Q}$ since there is a morphism $\operatorname{Spec}\mathbb C\to\operatorname{Spec}\overline{\mathbb Q}$. –  John Pardon May 10 at 23:49
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If $X$ is a scheme over $\mathbb C$, then "$X$ is defined over $\overline{\mathbb Q}$" means there exists a scheme $Y$ over $\overline{\mathbb Q}$ and an isomorphism $X\cong Y\times_{\overline{\mathbb Q}}\mathbb C$ (and similarly for stacks, etc.). –  John Pardon May 10 at 23:51

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