A constructive approach to a conjecture by Voskresenskii (Q2412566)

This paper investigates the rationality problem of a stably rational torus over an infinite field \(k\). A variety \(X\) over \(k\) is called rational if it is birational to a projective space \(\mathbb P_k^n\). A strictly weaker notion is that of stable rationality: A variety \(X\) over \(k\) is stably rational if \(X\times_k\mathbb P_k^m\) is rational for some \(m\geq 0\). Let \(T\) be a linear, i.e., affine algebraic group over \(k\). Then \(T\) is said to be an algebraic torus if, over an algebraic closure of \(k\), it becomes isomorphic to a product of \(\mathbb G_m\)'s. A conjecture of \textit{V. E. Voskresenskiĭ} [Algebraic groups and their birational invariants. Transl. from the original Russsian manuscript by Boris Kunyavskii. Rev. version of `Algebraic tori', Nauka 1977. Providence, RI: American Mathematical Society (1998; Zbl 0974.14034)] states that a stably rational torus over \(k\) must be rational. This conjecture is still an open problem. \textit{A. A. Klyachko} [in: Arithmetic and geometry of varieties, Interuniv. Collect. Sci. Works, Kujbyshev, 73--78 (1988; Zbl 0751.14031)] proved this assertion for a special type of stably rational tori. The authors of the present paper re-prove this result by constructiong explicit birational isomorphisms. In Section 4 are given some applications of the brational maps to torus-based cryptography.