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We prove that for any prime p there exists an algebraic action of the two-dimensional Witt group W2 (p) on an algebraic variety X such that the closure in X of the W2(p)-orbit of some point x ∈ X contains infinitely many W2(p)-orbits. This is related to the problem of extending, from the case of characteristic zero to the case of characteristic p, the classification of connected affine algebraic groups G such that every algebraic G-variety with a dense open G-orbit contains only finitely many G-orbits.

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I am grateful to J.-P. Serre for comments.

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Correspondence to Vladimir L. Popov.

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In memory of I. R. Shafarevich

This article was submitted by the author simultaneously in Russian and English

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 307, pp. 212–216.

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Popov, V.L. Orbit Closures of the Witt Group Actions. Proc. Steklov Inst. Math. 307, 193–197 (2019).

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