Résumé. -Inspirés par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses de dimension 3 sur les complexes qui ne sont pas stablement rationnelles, plus précisément dont le groupe de Chow de degré zéro n'est pas universellementégalà Z. La méthode de spécialisation adoptée ici permet de construire des exemples définis sur un corps de nombres.Abstract. -There are (many) smooth quartic threefolds over the complex field which are not stably rational. More precisely, their degree zero Chow group is not universally equal to Z. The proof uses a variation of a method due to C. Voisin. The specialisation argument we use yields examples defined over a number field.

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers. GeneralitiesWe recall implications of the "integral decomposition of the diagonal and specialization" method, following [CTP14], [Voi15b], and [Pir16].A projective variety X over a field k is universally CH 0 -trivial if for all field extensions k ′ /k the natural degree homomorphism from the Chow group of zero-cycles CH 0 (X k ′ ) → Z is an isomorphism. Examples include smooth k-rational varieties. More complicated examples arise as follows:

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Using the methods developed by Kollár, Voisin, ourselves, Totaro, we prove that a cyclic cover of P n C , n 3 of prime degree p, ramified along a very general hypersurface of degree mp is not stably rational if n + 1  mp. In small dimensions, we recover double covers of P 3 C , ramified along a quartic (Voisin), and double covers of P 3 C ramified along a sextic (Beauville), and we also find double covers of P 4 C ramified along a sextic. This method also allows one to produce examples over a number field.

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension 2 over a rational surface.

We prove that a very general double cover of the projective four-space, ramified in a quartic threefold, is not stably rational.

We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P 3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen-Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P 2 . We also prove the existence of universally CH0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.

We study rationality properties of smooth complete intersections of three quadrics in P 7 . We exhibit a smooth family of such intersections with both rational and non-rational fibers. Contents 1. Introduction 259 2. Strategy 260 3. Computations 263 4. Differentiating quadric bundles 271 References 273