Field of definition for general type surfaces

Question

In the survey paper

https://arxiv.org/abs/1004.2583

of Bauer-Catanese-Pignatelli, they mention a question of Mumford:

Can a computer classify all surfaces of general type with $p_g=0$?

I've been playing a bit with the Craighero-Gattazzo (CG) surface (a particular surface of this kind) using various computer algebra systems, and my life has been made difficult by the fact that the standard equations for this surface are defined over a cubic extension of $\mathbb{Q}$ rather than over $\mathbb{Q}$ itself (it seems to make computations run for longer and various algorithms are not implemented).

This made me worry about Mumford's question: because $\bar{\mathbb{Q}}$ is only countable, a generic complex surface in this moduli space will only be defined over some transcendental extension of $\mathbb{Q}$, which presumably makes Groebner basis calculations even less tractable. So my question is:

In the moduli space of general type surfaces, is anything known about the existence or density of surfaces defined over $\mathbb{Q}$ or $\bar{\mathbb{Q}}$? Should I be able to perturb the pluricanonical ring of the CG surface and find a "nearby" surface defined over $\mathbb{Q}$? Should every component of moduli space contain a surface defined over $\bar{\mathbb{Q}}$?

If this question is too general, I would be happy to know the answer to the following more concrete question.

The CG surface has an explicit birational model as a quintic in $\mathbb{P}^3$ with four simple elliptic singularities. The standard model is defined over $\mathbb{Q}[r]/(r^3+r^2-1)$. Is it known that it is necessary to work over this cubic extension, or could there be a similar quintic defined over $\mathbb{Q}$ with the requisite properties, i.e. whose minimal resolution is biholomorphic to (or at least deformation equivalent to) the CG surface?

Answer 1

Firstly, since you are interested in the field of definition of the surfaces, you should work with the moduli stack, rather than the coarse moduli space. A $k$-rational point on the moduli stack corresponds to a surface defined over $k$, but the same is not true for the coarse moduli space (this is the issue that ``Field of definition $\neq$ Field of moduli'' in general for coarse moduli spaces; though this issue only occurs for surfaces with non-trivial automorphism group).

So let $\mathcal{X}$ be moduli stack of interest. The functor of points is as follows: for a $\mathbb{Q}$-scheme $S$, we have that $\mathcal{X}(S)$ is the set of all smooth proper morphisms $Y \to S$ of relative dimensional two with connected fibres such that the relative canonical bundle $\omega_{Y/S}$ is relatively ample.

You seem to be also interested in singular surfaces, so you should adapt the definition as required; but this will correspond to some compactification of this stack. As you are interested in density of rational points, one can just work with the open subset corresponding to smooth surfaces. Moreover, if you want a stack of finite type, you should restrict the Hodge numbers or Hilbert polynomial, as you have done in the question.

Now note that the definition makes sense for any $\mathbb{Q}$-scheme $S$, so this means that $\mathcal{X}$ is defined over $\mathbb{Q}$. In fact, you can easily modify the functor of points definition for an arbitrary scheme $S$, which means that one can define a moduli stack over $\mathbb{Z}$, just as Deligne-Mumford defined the moduli stack over curves of given genus over $\mathbb{Z}$.

Next, in general, unless there is a good reason otherwise, one expects that moduli spaces are of general type. Therefore conjectures of Lang-Vojta predict that the $\mathbb{Q}$-rational points are not Zariski dense. Hence in general there is no reason to expect to be able to approximate a surface over $\mathbb{C}$ by a surface over $\mathbb{Q}$.

As for $\bar{\mathbb{Q}}$-points, as explained in the comments, for a non-empty finite type $\bar{\mathbb{Q}}$ scheme, the $\bar{\mathbb{Q}}$-points are Zariski dense.