In Proposition (2) in the paper [1], in below, it is proved that:
Assume that $K$ is a number field and that $C$ has infinitely many points of degree $\leq d$ over $K$. (Here $d\geq 2$ is an integer) Then there is a $K$-rational covering $\pi: C \rightarrow {\mathbb P}^1_K$ of degree $\leq 2d$.
Now, I am interested to know:
1) Is this result true over any perfect field of characteristic zero?
2) Is there any generalization of the above proposition for surfaces or higher dimensional varieties in the literature?
[1] G. Frey, Curves with infinitely many points of fixed degree, Israel Journal of Mathematics, 85 (1994), 79-83