  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.
