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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

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    $\begingroup$ 1) is obviously false for $\mathbb{C}$ and $d=2$ as not all curves are quartic covers of $\mathbb{P}^1$ (gonality is not bounded). $\endgroup$ Mar 28, 2019 at 4:28
  • $\begingroup$ Thank you Felipe for your response. But, I am interested in the answers for perfect fields that are not algebraically closed, say ${\mathbb C}(t)$ or $k(t)$ with $k$ a number field! $\endgroup$
    – user131222
    Mar 29, 2019 at 1:39

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