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Let $C\subset\mathbb{C}^2$ be an irreducible algebraic curve defined over a number field $F.$ Suppose that for any $(z, w)\in C, z\in \mathbb{\overline{Q}}, w\in\mathbb{\overline{Q}},$ either $z\in F(w)$ or $w\in F(z)$. My question is whether such a curve $C$ must necessarily be rational? If $z\in F(w)$ for all but finitely many $(z, w),$ then $C$ is easily seen to be rational by Faltings' theorem.

Any suggestion would be greatly appreciated.

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    $\begingroup$ Welcome new contributor. Much more is true: by Hilbert's Irreducibility Theorem, for each projection, the projection from $C$ to the affine line is birational. $\endgroup$
    – Jason Starr
    Commented May 4, 2022 at 23:05
  • $\begingroup$ Would you mind giving more details? Thanks! $\endgroup$
    – John Z.
    Commented May 4, 2022 at 23:15
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    $\begingroup$ I meant to write that at least one of the two projections is birational (not necessarily both). $\endgroup$
    – Jason Starr
    Commented May 5, 2022 at 1:28
  • $\begingroup$ I still don't see how to arrive at that conclusion by Hilbert's irreducibility. $\endgroup$
    – John Z.
    Commented May 5, 2022 at 1:32
  • $\begingroup$ Maybe I misunderstood your question. $\endgroup$
    – Jason Starr
    Commented May 5, 2022 at 10:52

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