Abstract

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 (Q) over bar, then Koeck, Lau and Singerman showed the existence of a complex smooth algebraic variety Z admitting a symmetry T, both defined over R boolean AND (Q) over bar, and of an isomorphism R : X -> Z so that R circle L circle R-1 = T. The provided proof is existential and, if explicit equations for X and L are given over Q, then it is not described how to get the explicit equations for Z and T over R boolean AND (Q) over bar. In this paper we provide an explicit rational map R defined over Q so that Z = R(X) is defined over R boolean AND (Q) over bar, R : X -> Z is an isomorphisms and T = R circle L circle R-1 being the usual conjugation map.