In the survey

https://arxiv.org/abs/1004.2583

from Bauer-Catanese-Pignatelli mention a question from Mumford:

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

I played a bit with the surface of Craighero-Gattazzo (CG) (a

certain surface of this type) using various computer algebra

Systems, and my life was complicated by the fact that

Standard equations for this area are defined by a cubic extension

from $ mathbb {Q} $ rather than over $ mathbb {Q} $ self

Run calculations longer and not use different algorithms

) Reacted.

That worried me about Mumford's question: Because

$ bar { mathbb {Q}} $ is only countable, a generic complex surface in

This module space is only defined by a certain transcendental

Extension of $ mathbb {Q} $, which probably makes Groebner the base

Calculations even less understandable. My question is:

Everything is known in the module space of general surfaces

the existence or density of surfaces over defined $ mathbb {Q} $ or

$ bar { mathbb {Q}} $? Should I be able to disrupt the pluricanonical

Ring the CG surface and find a "near" defined surface over

$ mathbb {Q} $? Should each component of the module space contain an area defined above it $ bar { mathbb {Q}} $?

If this question is too general, I would be glad to know the answer

to the next more specific question.

The CG surface has one

Explicit birational model as a quintet in $ mathbb {P} ^ 3 $ with four

simple elliptic singularities. The standard model is defined via

$ mathbb {Q} (r) / (r ^ 3 + r ^ 2-1) $, It is known that it is necessary to work

above this cubic extension or could there be a similar quintessence defined

over $ mathbb {Q} $ with the required properties, i. whose minimal

Resolution is biholomorphic to (or at least deformation equivalent to)

the CG surface?