Surfaces of general type

Question

What methods do we know about proving-disproving existence of rational points on surfaces of general type? I was recently asked. My gut answer was- 'nothing'.

Answer 1

Usually nothing, as you guessed $-$ though you might get lucky:

i) There might be a local obstruction (e.g. no rational points on the twisted Fermat sextic surfaces $x^6+y^6+z^6+t^6=0$ and $x^6+2y^6+4z^6=8t^6$).

ii) the surface, say $S$, may map to a curve with finitely many rational points.

iii) $S$ may be contained in an abelian variety, in which case Faltings II applies (the big example is symmetric squares of non-hyperelliptic curves).

[it's been noted that both (ii) and (iii) mean that $S$ has nontrivial Albanese variety.]

iv) finally $S$ may have nontrivial $\pi_1$, in which case one can lift to unramified covers and try to apply (ii) or (iii). For example, if $C$ and $C'$ are curves of genus at least $2$ with involutions $\iota,\iota'$ at least one of which has no fixed points then we can prove $(C \times C') \, / \, (\iota,\iota')$ has finitely many rational points by applying (ii) or (iii) to finitely many twists of $C \times C'$.

In each of cases (ii), (iii), (iv), some further luck may be needed to find all rational points and prove that the list is complete.