Algebraic models defined over \(\mathbb{Q}\) of differential manifolds (Q1189642)

Let \(V\) be a compact differential manifold of dimension \(q\) in \(\mathbb{R}^ n\). The classical Nash-Tognoli theorem says that \(V\) can be approximated in \(\mathbb{R}^ n\) (if \(n\geq 2q+1)\) by a regular affine algebraic variety \(M\) --- here it is proved that \(M\) can be taken such that it is defined over \(\mathbb{Q}\). The same is possible for a rational model of \(V\) if \(\dim V=2\), i.e. \(V\) has an algebraic model which is a surface birationally equivalent to \(\mathbb{P}_ 2\), moreover defined over \(\mathbb{Q}\).