La descente sur les variétés rationnelles, II

On Albanese torsors and the elementary obstruction

“…In the remark which follows their proof, they note that the converse is known to hold by a theorem of Colliot-Thélène and Sansuc if Pic(X ⊗ k k) is a free abelian group (as is the case for instance if X is geometrically unirational); see [5,Prop. 3.3.2].…”

“…Thus, if we define the generic period P gen (X) as the supremum of P (U ) when U ranges over all dense open subsets of X, then for a 0-cycle of degree 1 to exist on X, it is necessary that P gen (X) = 1. Colliot-Thélène and Sansuc defined and studied in [5] another general obstruction to the existence of 0-cycles of degree 1, the so-called elementary obstruction. Namely, denoting by k a separable closure of k and by k(X) the function field of X ⊗ k k, the elementary obstruction is said to vanish if and only if the inclusion k ⋆ ⊂ k(X) ⋆ admits a Galois-equivariant retraction.…”

“…The elementary obstruction to the existence of 0-cycles of degree 1 on X is known to vanish if and only if the Yoneda equivalence class of a certain 2-extension e(X) of Pic(X ⊗ k k) by k ⋆ in the category of discrete Galois modules is trivial (see [5,Proposition 2.2.4]). One is naturally led to consider an analogous 2-extension E(X) of the relative Picard functor Pic X/k by the multiplicative group G m ; it is a straightforward generalisation of the elementary obstruction.…”

Del Pezzo surfaces and representation theory

“…Universal torsors were introduced by Colliot-Thélène and Sansuc in the 1970's in a seemingly unrelated line of research; see [Colliot-Thélène and Sansuc 1987] or [Skorobogatov 2001]. If X is a smooth projective variety over a field k, then an X -torsor under a torus T is a pair (Y, f ), where Y is a variety over k with a free action of T , and f is an affine morphism Y → X whose fibres are the orbits of T .…”

The Brauer group of torsors and its arithmetic applications

“…Let 1~ be a field of characteristic 0, and let X be a variety over k. Denote by 1~ an algebraic closure of k, and set Let Br X = H6t (X, Gm) be the cohomological Brauer-Grothendieck group of X. In their fundamental paper [CS87a] Colliot-Thelene and Sansuc defined universal torsors and studied their various properties. For a universal torsor f : Y -X over a variety X such that all invertible functions on X are constants, and Pic X has no torsion, they prove that the kernel of the natural map Br Y -Br Y is naturally isomorphic to Br k. In this paper we show (Theorem 1.7) that the map Br X -Br Y/Br k defined by f is canonically identified with the map Br X -(Br X )r (provided k is such that H3 (r,1~* ) = 0).…”