## 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

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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