Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder

In a smooth family of projective, complex varieties, stable rationality need not be preserved under generisation. This was proved by Hassett, Pirutka and Tschinkel upon use of the specialisation method. Work of Schreieder produced many more examples and introduced a simplification of the specialisation method (no explicit desingularisation). In this text, I try to describe the combined method from scratch in one of the simplest cases. A small, further simplification consists in using specialisation of R-equivalence in place of Fulton's specialisation for the Chow group of zero-cycles.Comment: These are slightly revised version of notes written on the occasion of the conference Quadratic Forms in Chile 2018, held at IMAFI, Universitad de Talca, 8-12 January 2018. Language : Englis

Fields of cohomological dimension one versus C_1-fields

Ax gave examples of fields of cohomological dimension 1 which are not C_1-fields. Kato and Kuzumaki asked whether a weak form of the C_1-property holds for all fields of cohomological dimension 1 (existence of solutions in extensions of coprime degree rather than existence of a solution in the ground field). Using work of Merkur'ev and Suslin, and of Rost, D. Madore and I produced examples which show that the answer is in the negative. In the present note, I produce examples which require less work than the original ones. In the original paper, some of the examples were given by forms of degree 3 in 4 variables. Here, for an arbitrary prime p>3, I use forms of degree p in p+1 variables.Comment: 5 pages, in Englis

Surfaces stablement rationnelles sur un corps quasi-fini

If a smooth, geometrically rational surface over a finite field is not rational over that field, then over some finite extension of that field the Brauer group of the surface is nonzero. In particular such a surface is not stably rational. This is a special case of a general statement about geometrically rational surfaces which split over a cyclic extension of their field of definition.Comment: In French; revised version, 14th June 201

Rationalit\'e d'un fibr\'e en coniques

F. Campana had asked whether a certain threefold is rational. In arXiv:1310.3569v1 [mathAG], this variety was shown to be birational to a specific conic bundle and then to be unirational. We prove that this conic bundle is rational.Comment: in Frenc

Droites sur les hypersurfaces cubiques

Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendr\'e par les droites. Le cas lisse est un th\'eor\`eme connu. La d\'emonstration ici donn\'ee repose sur un r\'esultat sur les surfaces g\'eom\'etriquement rationnelles sur un corps quelconque (1983), obtenu via la K-th\'eorie alg\'ebrique.Comment: in Frenc

Principe local global pour les espaces lin\'eaires sur les intersections de deux quadriques

In arXiv:1410.5671, Jahnel and Loughran prove the local global principle for existence of linear spaces of dimension $r$ on smooth complete intersections of two quadrics in projective space of dimension $2r+2$. We present an alternative proof of a slightly more general result. ----- Dans arXiv:1410.5671, Jahnel et Loughran \'etablissent un principe local-global pour l'existence d'espaces lin\'eaires de dimension $r$ dans les intersections compl\`etes lisses de deux quadriques dans un espace projectif de dimension $2r+2$. On donne une d\'emonstration alternative d'un r\'esultat un peu plus g\'en\'eral.Comment: 3 pages, in Frenc

Groupe de Brauer non ramifi\'e d'espaces homog\`enes de tores

Let k be a field, X a smooth, projective k-variety. If X is geometrically rational, there is an injective map from the quotient of Brauer groups Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the geometric Picard group. In this note, where the main attention is on smooth compactifications of homogeneous spaces of algebraic k-tori, we show how under some hypotheses the map is onto, and how one may in some special case exhibit concrete generators in Br(X). This is applied to the analysis of counterexamples to the local-global principle for norms in biquadratic extensions of number fields

Surfaces de del Pezzo de degr\'e 4 sur un corps C1C_1

Sur toute surface de del Pezzo de degr\'e 4 sur un corps $C_1$ de caract\'eristique z\'ero, tous les points rationnels sont R-\'equivalents. Plus g\'en\'eralement, ceci vaut sur tout corps parfait infini de caract\'eristique diff\'erente de 2. ---- On a del Pezzo surface of degree 4 over a characteristic zero $C_1$-field, all rational points are R-equivalent.This more generally holds over any infinite perfect $C_{1}$-field of characteristic different from 2.Comment: in French, to appear in Taiwanese Journal of Mathematic

CH0CH_{0}-trivialit\'e universelle d'hypersurfaces cubiques presque diagonales

Toute hypersurface cubique lisse complexe de dimension au moins 2 dont l'\'equation est donn\'ee par l'annulation d'une somme de formes cubiques \`a variables s\'epar\'ees, chaque forme impliquant au plus trois variables, est universellement $CH_{0}$-triviale. --- If a smooth cubic hypersurface of dimension at least 2 is defined by the vanishing of a sum of cubic forms in independent variables and each of these forms involves at most 3 variables, then the cubic hypersurface is universally $CH_{0}$-trivial : there is an integral Chow decomposition of the diagonal.Comment: in French; title slightly changed, final version, 30 nov. 2016, to appear in Algebraic Geometr

Descente galoisienne sur le second groupe de Chow : mise au point et applications

Connections between the second Chow group of a smooth projective variety and its third unramified cohomology group, with coefficients the roots of unity twisted twice, feature in several recent works. In this note we revisit a 1996 paper by B. Kahn and specialize it to various types of rationally connected varieties.Comment: In French. Final version, to appear in Documenta math. Section 5 contains a discussion of the universal unramified third cohomology group for Fano hypersurfaces, and in particular for cubic hypersurface