Homotopy theory of schemes and $R$-equivalence

Abstract

We prove that, for any smooth and projective scheme $X$ over a field $k$ of char. $0$, the set of maps from Spec $k$ to $X$ in the $\mathbf{A}^1$-homotopy category of schemes $\mathcal{H}_{\mathbf{A}^1}(k)$ is in bijection with the quotient of $X(k)$ by $R$-equivalence, and is a birational invariant of $X$. This is achieved by establishing a precise relation between the localization of the category of smooth $k$-schemes by birational maps and the category $\mathcal{H}_{\mathbf{A}^1}(k)$, and by applying results of the second named author and R. Sujatha on birational invariants. This gives a new proof of results obtained by A. Asok and F. Morel.