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I am looking for an example of a surface $X \subset \mathbb{P}^3$ defined over $\mathbb{Q}$ with the following properties: 1) $X(\mathbb{Q}) \ne \emptyset$; and 2) we know all of the elements $X(\mathbb{Q})$ explicitly. Here explicitly means that we can write $X(\mathbb{Q})$ as the union of rational points on an explicit finite collection of curves on $X$, and possibly a finite collection of explicitly determined rational points on $X$.

Note that the assumption that $X(\mathbb{Q}) \ne \emptyset$ is critical; certainly there are many surfaces $X$ which provably have no rational points (perhaps due to having a local obstruction, say).

Here is a conjectured example: for $n \geq 5$ consider the surface $X_n$ defined by

$$\displaystyle X_n : x_0^n + x_1^n = x_2^n + x_3^n.$$

Then $X_n$ contains the rational lines defined by $(x_0, x_1) = (x_2, x_3)$ and $(x_0, x_1) = (x_3, x_2)$, and it is a folklore conjecture that $X_n$ does not contain any other rational points.

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    $\begingroup$ You mention general type in the title, but not in the text; I assume you are imposing this in the text as well. If you allow singular surfaces then this is very easy as any variety is birational to hypersurface. If you are restricting to smooth surfaces then this is almost surely completely open; there are just no techniques available for dealing with rational points on simply connected surfaces of general type. $\endgroup$
    – Daniel Loughran
    Aug 3, 2021 at 8:12
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    $\begingroup$ @Daniel Loughran. I think they are smooth surfaces which are simply connected and have ample cotangent bundle (in the sense of Hartshorne). I think Bogomolov gave an example but I do not know where anymore. In that case a result of Noguchi implies that the rational points are not dense. $\endgroup$
    – Damian Rössler
    Aug 3, 2021 at 12:34
  • $\begingroup$ Yes, but in any case, I highly doubt that this result can be made effective, which is what the OP wants. $\endgroup$
    – Daniel Loughran
    Aug 3, 2021 at 13:01
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    $\begingroup$ @DamianRössler Which result of Noguchi? $\endgroup$
    – Will Sawin
    Aug 3, 2021 at 13:55
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    $\begingroup$ Take a look at Remark F.5.2.4 in [Hindry-Silverman]. $\endgroup$
    – Pasten
    Aug 4, 2021 at 16:02

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