# An explicit descent of real algebraic varieties

Hidalgo, Riben (2018) An explicit descent of real algebraic varieties. In: Algebraic curves and their applications. Contemporary Math., 724 . American Mathematical Society, USA, pp. 246-257. (In Press)

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Let $X$ be an smooth complex affine 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 $\overline{\mathbb Q}$ so that $Z=R(X)$ is defined over ${\mathbb R} \cap \overline{\mathbb Q}$, $R:X \to Z$ is an isomorphisms and $T=R \circ L \circ R^{-1}$ being the usual conjugation map.