Abstract:
Two points $x$ and $y$ of an algebraic variety $X$ are called $R$-equivalent, if there exists a collection of points $x=x_1,x_2,\ldots,x_n=y$ in $X$ such that every two consecutive points are in the image of a morphism $f\colon U\to X$, where $U$ is an open subset of the affine line. This relation was first considered by Yury Manin in his “Cubic forms” book. It turned out that the corresponding set $X(k)/R$ of $R$-equivalence classes of all points of $X$ is a very interesting invariant of varieties that allows to solve many different problems. For example, with the help of this notion one can prove that the general linear group $GL_n(H)$ of invertible matrices with quaternion entries is generated (similarly to the usual general linear group) by real scalars and elementary transvections.
One of the popular problems on $R$-equivalence is the specialization problem: what is the relation between the $R$-equivalence class sets of varieties parametrized by a smooth curve at the generic point of this curve and at a special point? It appears that if one considers infinitesimally small curves, then the corresponding $R$-equivalence class sets are in bijection. Janos Kollar (2004) proved this fact for smooth projective varieties, and Philippe Gille together with the speaker proved it for simple algebraic groups.
