An effective descent of arithmetical real algebraic varieties

Abstract: Let $X$ be a complex smooth algebraic variety admitting a symmetry $L$, that is, an antiholomorphic automorphism of order two. If both, $X$ and $L$ are defined over $\overline{\mathbb Q}$, then Koeck, Lau and Singerman showed the existence of a complex smooth algebraic variety $Z$ admitting a symmetry $T$, both defined over ${\mathbb R} \cap \overline{\mathbb Q}$, and of an isomorphism $R:X \to Z$ so that $R \circ L \circ R<sup>{-1}=T$. The provided proof is existential and, if explicit equations for $X$ and $L$ are given over $\overline{\mathbb Q}$, then it is not described how to get the explicit equations for $Z$ and $T$ over ${\mathbb R} \cap \overline{\mathbb Q}$. In this paper we provide an explicit rational map $R$ defined over ${\mathbb Q}$ so that $Z=R(X)$ is defined over ${\mathbb R} \cap \overline{\mathbb Q}$ and with $T=R \circ L \circ R<sup>{-1}$ being the usual conjugation map.