Embedded $\mathbb{Q}$-desingularization of real Schubert varieties and application to the relative $\mathbb{Q}$-algebrization of nonsingular algebraic sets

Abstract

A real algebraic set V ⊂ R n is said to be a Q-nonsingular Q-algebraic set if it is described, both globally and locally, by polynomials with rational coefficients. We prove that every nonsingular real algebraic set V ⊂ R n with nonsingular algebraic subsetsA key result in the proof is the description of Z/2Z-homological cycles of real Grassmannian manifolds by Q-nonsingular Q-algebraic representatives via an explicit desingularization of real embedded Schubert varieties.