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Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers.

Does this imply that $X$ itself can be defined over the field of algebraic numbers?

Basically, I'm asking if the property "can be defined over the field of algebraic numbers" is "birationally invariant".

Is this something that holds for general schemes?

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It doesn't quite make sense to speak of "birational equivalence" of arbitrary Riemann surfaces. For instance, is the upper half plane "birationally equivalent" to $\mathbb{P}^1$?

Even if you restrict to nonsingular, connected algebraic curves over $\mathbb{C}$, the answer is no. Consider for instance the affine curve $C$ obtained as $\mathbb{P}^1$ with the points $0,1,\infty,\pi$ removed. The moduli space of curves of genus zero with four points removed is isomorphic to the affine line -- one can view this in terms of the cross ratio, or in terms of building an elliptic curve ramified over precisely those four points on $\mathbb{P}^1$. As soon as you have a positive dimensional moduli space, you have "generic points" which cannot be defined over $\overline{\mathbb{Q}}$. The curve $C$ has transcendental cross-ratio / $j$-invariant, so is such an example.

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    Ah, it seems I mistakenly took "four points removed" for "four points labelled". Thanks for the additional explanation.2013-08-21