A relative Nash-Tognoli theorem over $\mathbb{Q}$ and application to the $\mathbb{Q}$-algebraicity problem

We prove a relative version over $\mathbb{Q}$ of Nash-Tognoli theorem, that is: Let $M$ be a compact smooth manifold with closed smooth submanifolds $M_1,\dots,M_\ell$ in general position, then there exists a nonsingular real algebraic set $M'\subset\mathbb{R}^n$ with nonsingular algebraic subsets $M_1',\dots,M_\ell'$ and a diffeomorphism $h:M\to M'$ such that $h(M_i)=M_i'$ for all $i=1,\dots,\ell$ such that $M',M_1',\dots,M_\ell'$ are described, both globally and locally, by polynomial equations with rational coefficients. In addition, if $M,M_1,\dots,M_\ell$ are nonsingular algebraic sets, then we prove the diffeomorphism $h:M\to M'$ can be chosen semialgebraic and the result can be extended to the noncompact case. In the proof we describe also the $\mathbb{Z}/2\mathbb{Z}$-homological cycles of real embedded Grassmannian manifolds by nonsingular algebraic representatives over $\mathbb{Q}$ via the Bott-Samelson resolution of Schubert varieties.