The conjectures on motives mentioned above indicate that varieties defined over number fields have rather special geometric properties. This raises the question of whether we can find some geometric characterisation of such varieties.
In the case of curves, such a characterisation was found by Belyi, who showed
Theorem 12.1 (Belyi). A curve X is defined over a number field if and only if it is expressed as a covering of P1 branched over at most 3 points.
We can view 3 points as a divisor in P1 which has the property that any deformation of this divisor is obtained by a global automorphism of P1. This leads to the question of what analogous divisors are in P2 (and more generally Pn).
Definition 12.2. A divisor D in P2 is said to be geometrically rigid if any equisingular deformation of D is obtained as the image of D under an automorphism of P2.
We then obtain the somewhat surprising result:
Theorem 12.3. A divisor D in P2 is defined over a number field if and only if it is a sub-divisor of a geometrically rigid divisor D′.
The following generalisation of Belyi’s theorem to surfaces follows almost immediately
Theorem 12.4. A surface X is defined over a number field if and only if can be expressed as a cover of P2 which is unramified outside a geometrically rigid divisor.
A number of open questions in higher dimensions have also been posed in the paper where the details of these results have been written up.