February 17, 2022. A. K. Stavrova (PDMI). Manin's R-equivalence relation on points of varieties and groups.

Two points x and y of an algebraic variety X are called R-equivalent, if there exists a collection of points x=x₁,x₂,...,x_n=y in X such that every two consecutive points are in the image of a morphism f:U→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.

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