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V. E. Voskresenskii Samara State University Samara, Russia voskres@ssu.samara.ru

Birational geometry of algebraic tori
Ї Let k be a field of zero characteristic, k its algebraic closure, and the Galois group of Ї/k . Let X be an algebraic variety over k and X = X k k . The group acts Ї Ї the extension k Ї and on ob jects defined by the scheme X . The k -variety Y is called a k -form Ї naturally on X Ї Ї Ї of X if varieties X and Y are isomorphic over the field k . If X is quasipro jective over k , 1 Ї ) describes the set of classes of all k -forms of X . Two -modules A then the set H (, AutX and B will be called similar if there exist two permutations -modules S1 and S2 such that A S1 B S2 . The similarity class of a -module A is denoted by [A]. The following = assertion is known. Let X and Y be k -birationally equivalent nonsingular pro jective varieties Ї Ї Ї over field k . Then the -modules PicX and PicY are similar and groups H 1 (, PicX ) and -1 Ї ) are birational invariants of X . H (, PicX ^ Let G be a connected linear algebraic group over k and G be the -module of rational characters of G. We consider a nonsingular pro jective variety X over k that contains G as an open subset. The variety X is called a pro jective model of the group G. The class Ї [PicX ] = p(G) is a birational invariant of G. The embedding G X induces an exact ^ ^ Ї Ї sequence of modules 0 G S PicX PicG 0. If G = T is an algebraic torus then Ї = 0 and we have an exact sequence of torsion-free modules 0 T S PicXL 0, ^ ^ PicT ^ is a permutation where L is a splitting field of T , = Gal(L/k ), (L : k ) < , and S -module. We have an unexpected result which is one of the most important results in birational classification of algebraic tori: H -1 (L/F, PicXL ) = 0, k F L. Now we can give a purely algebraic description of the birational invariants [PicXL ]. For algebraic tori there is a sufficiently simple, one can say, canonical construction of complete models suggested by M. Demazure. A complete toric variety is constructed by gluing affine toric varieties, and the scheme of gluing is determined by some set of cones called a fan. As one may expect, tori T with p(T ) = [0] have some special features. Such a torus T can be described as the following factor-group: 1 S1 S2 T 1, where S1 and S2 are quasisplit k -tori. We have just obtained the main result of this article. Theorem. Any stably rational k -torus is rational over k . References [1] V. E. Voskresenskii. Algebraic Groups and Their Birational Invariants, Translation of Mathematical Monographs, vol.179, American Mathematical Society, Providence, 1998.

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