Hypersurfaces quartiques de dimension 3 : non-rationalité stable

Abstract

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.

On stable rationality of Fano threefolds and del Pezzo fibrations

“…• (1, 2, 52): double cover of P 3 ramified in a surface of degree 6, unirationality is unknown, very general V are not stably rational [Bea14] • (1, 4, 30): quartic in P 4 , unirationality is unknown, very general V are not stably rational [CTP14] • (1, 6, 20): intersection of a quadric and a cubic, unirational • (1, 8, 14): intersection of three quadrics in P 6 , unirational • (1, 10, 10): section of Gr(2, 5) by a subspace of codimension 2 and a quadric, general such V are non-rational, all are unirational • (1, 14, 5): section of Gr(2, 5) by a subspace of codimension 5, unirational • (2, 8, 21): V 1 , unirationality is unknown • (2, 8 · 2, 10): V 2 , double cover of P 3 , ramified in a smooth quartic, unirational, very general V 2 are not stably rational [Voi15,Bea15] • (2, 8 · 3, 5): V 3 , cubic in P 4 , unirational…”

“…An application of the results of [CTP14, §1] implies that a very general V ⊂ P(1, 1, 1, 2, 3) also fails to admit an integral decomposition of the diagonal, and thus is not stably rational.…”

“…Recent breakthroughs of Voisin [Voi15], developed and amplified by Colliot-Thélène-Pirutka [CTP14,CTP15], Beauville [Bea14], and Totaro [Tot15], have reshaped the classical study of rationality questions for higher-dimensional varieties. Failure of stable rationality is now known for large classes of rationally-connected threefolds.…”

Degenerations of Gushel–Mukai fourfolds, with a view towards irrationality proofs

“…Also, GM fourfolds have recently been studied from various perspectives, geometric, Hodge theoretic and derived categorical [6,7,18]. One may ask whether the very general GM fourfold is even not stably rational, and then one can seek to apply the degeneration method of Voisin, Colliot-Thélène-Pirutka, Totaro et al [5,20,21] that has led to such a multitude of applications recently. In fact, GM fourfolds are birational to a certain class of conic bundles over P 3 with sextic discriminant surfaces, see Proposition 2.2.…”

The Integral Cohomology of the Hilbert Scheme of Two Points

“…Configurations of two points come up naturally in geometry, but one especially relevant use of the Hilbert scheme X [2] is in Voisin's paper on the universal C H 0 group of cubic hypersurfaces [22]. The background is that major recent advances have been made in determining which algebraic varieties are stably rational, that is, become birational to projective space after multiplying by projective space of some dimension [5,19,21]. These papers are based on the observation that if a smooth projective variety is stably rational, then its Chow group of 0-cycles is universally trivial, meaning that C H 0 does not increase when the base field is increased.…”