Belyi's theorem and ramification over $\mathbb{P}^n$ (after K. H. Paranjape)

Belyi's theorem states that a smooth complex projective curve can be defined over $\overline{\mathbb{Q}}$ if and only if there exists a finite map $f \colon X \rightarrow \mathbb{P}^1$ branched over no more than three points. Last year we have discussed higher-dimensional analogues of Belyi's theorem (after G. González-Diez and A. Javanpeykar), in these results the finite map $f$ is replaced by a Lefschetz pencil on $X$.
But there exists a different approach — we can realise $X$ as a cover of $ \mathbb{P}^n$, and characterise when $X$ can be defined over $ \overline{\mathbb{Q}}$ in terms of the branching divisor on $\mathbb{P}^n$. Following K. H. Paranjape, we will discuss such a Belyi-type result for projective surfaces.