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 $R\circ L\circ R^{-1}=T$. 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 $T=R\circ L\circ R^{-1}$ being the usual conjugation map.