An effective descent of arithmetical real algebraic varieties

TL;DR

This paper provides an explicit rational map to descend complex algebraic varieties with antiholomorphic symmetries to real algebraic varieties, making the process constructive rather than existential.

Contribution

It introduces a method to explicitly compute the descent of algebraic varieties with antiholomorphic automorphisms over the rationals.

Findings

Constructs an explicit rational map over ${f Q}$ for descent.

Ensures the descended variety is defined over ${f R} igcap ar{f Q}$.

Provides a concrete procedure for explicit equations in descent.

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.