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Stably rational algebraic tori. (English) Zbl 0946.14030

Suppose that \(T\) is a torus over a field \(k\) of characteristic zero. If \(T\) is stably rational, that is, \(T\times_k{\mathbb A}^m_k\) is rational over \(k\) for some \(m\), then it is conjectured that \(T\) is rational over \(k\). Here this conjecture is proved under the additional assumption that the splitting field of \(T\) is a cyclic extension of \(k\), by reformulating it in terms of linear representations of \(T\). The method is first used to reprove a weaker result of A. A. Klyachko [in: Arithmetic and geometry of varieties, Interuniv. Collect. Sci. Works, Kujbyshev, 73-78 (1988; Zbl 0751.14031)] and then extended to give the main result of the paper.

MSC:

14M20 Rational and unirational varieties
14G25 Global ground fields in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14L99 Algebraic groups

Citations:

Zbl 0751.14031
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References:

[1] Beauville, A., Colliot-Thélène, J.-L., Sansuc, J.-J., Swinnerton-Dyer, Sir P., Variétés stablement rationnelles non rationnelles. Ann. Math.121 (1985), 283-318. · Zbl 0589.14042
[2] Voskresenski, V., The geometry of linear algebraic groups. Proc. Steklov Inst. Math.132 (1973), 173-183. · Zbl 0309.14040
[3] Voskresenski, V., Fields of Invariants of Abelian Groups. Russian Math. Surveys28 (1973), 79-105. · Zbl 0289.14006
[4] Klyachko, A., On the rationality of tori with a cyclic splitting field. Arithmetic and Geometry of Varieties, Kuibyshev Univ., 1988, 73-78 (Russian). · Zbl 0751.14031
[5] Chistov, A., On the birational equivalence of tori with a cyclic splitting field. Zapiski Nauchnykh Seminarov LOMI64 (1976), 153-158 (Russian). · Zbl 0358.14017
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