Rationality of complete intersections of two quadrics over nonclosed fields

Hassett, Brendan; Tschinkel, Yuri; Colliot-Thélène, Jean-Louis

doi:10.4171/lem/1001

Cited by 12 publications

(8 citation statements)

References 23 publications

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“…The IJT obstruction has been used to great effect to study rationality of geometrically rational threefolds over non-closed fields: Hassett-Tschinkel (over R) [HT21b] and Benoist-Wittenberg (over arbitrary fields) [BW] showed this obstruction characterizes rationality for smooth complete intersections of two quadrics in P 5 , and Kuznetsov-Prokhorov used it for several cases of their classification of rationality for prime Fano threefolds [KP]. However, Benoist-Wittenberg also showed that the IJT obstruction is not sufficient to characterize rationality by constructing an example of a (non geometrically standard) real conic bundle threefold X → S whose intermediate Jacobian is trivial but such that S(R) is disconnected; hence, the IJT obstruction vanishes but X has a Brauer obstruction to (stable) rationality over R [BW20, Theorem 5.7].…”

Section: 3mentioning

confidence: 99%

“…In the one oval example [FJS + , Theorem 1.3(2)] Y (R) = ∅ is connected, and irrationality is shown using the intermediate Jacobian torsor (IJT) obstruction. This obstruction to rationality is a refinement over non-closed fields of the intermediate Jacobian obstruction of Clemens-Griffiths [CG72], and was recently introduced by Hassett-Tschinkel [HT21a,HT21b] and Benoist-Wittenberg [BW] (see Section 2.3). However, the two non-nested ovals example [FJS + , Theorem 1.3(1)] has no IJT obstruction to rationality but Y (R) is disconnected; hence Y is irrational.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 9. If X is a smooth complete intersection of two quadrics in P 5 that contains a conic C defined over R, then projection from the conic realizes the blow up of X along C as a conic bundle with quartic discriminant curve [HT21b,Remark 13]. Krasnov's topological classification of intersections of quadrics [Kra18,Theorem 5.4] shows that a conic bundle arising in such a way can have at most two real connected components, so in particular the conic bundle of Example 4.6 is not birational over R to an intersection of two quadrics.…”

mentioning

confidence: 99%

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Rationality of real conic bundles with quartic discriminant curve

Ji¹,

Ji²

2022

Preprint

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We study real double covers of P 1 ×P 2 branched over a (2, 2)-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct C-rational examples that have obstructions to rationality over R, and for the sixth class, we show that the models we consider are all rational.Moreover, for three of the five classes with irrational members, we give characterizations of rationality using the topology of the real locus and the intermediate Jacobian torsor obstruction of Hassett-Tschinkel and Benoist-Wittenberg. The double cover models we consider were introduced and previously studied by S.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Rationality of real conic bundles with quartic discriminant curve

Ji¹,

Ji²

2022

Preprint

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“…Via la proposition 8.1, ceci résulte des rappels sur les formes d'Albert donnés ci-dessus : l'indice de α ∈ Br(k(C)) divise 2 si et seulement si la forme f + tg est isotrope sur le corps k(C). Pour (i), voir aussi la démonstration géométrique [HT,§4,Cor. 7].…”

Section: Intersections Complètes Lisses Dans Punclassified

“…Elles impliquent que X contient une conique. Une démonstration du théorème suivant (avec la restriction X(k) = ∅) est esquissée dans [HT,Theorem 9]. Nous proposons une démonstration alternative.…”

Section: Intersections Complètes Lisses Dans Punclassified

Retour sur l'arithmétique des intersections de deux quadriques, avec un appendice par A. Kuznestov

Colliot-Thélène¹

2022

Preprint

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JEAN-LOUIS COLLIOT-TH ÉL ÈNERésumé. Soit k un corps p-adique. On montre que toute intersection de deux quadriques dans l'espace projectif P 4 k contient un point sur une extension quadratique, ce qui généralise un résultat de Creutz et Viray pour le cas lisse. La preuve utilise un théorème de Lichtenbaum sur les courbes de genre 1 sur un corps p-adique. On déduit de ce résultat que toute intersection lisse de deux quadriques X ⊂ P 5 k sur un corps de nombres k possède un point sur une extension quadratique. On déduit aussi de ce résultat une démonstration relativement courte d'un théorème de Heath-Brown : le principe de Hasse vaut pour les intersections complètes lisses X ⊂ P 7 k de deux quadriques sur un corps de nombres. On donne aussi une démonstration alternative d'un principe de Hasse pour certaines intersections de deux quadriques dans P 5 k , dû à Iyer et Parimala.Lichtenbaum proved that index and period coincide for a curve of genus one over a p-adic field. Salberger proved that the Hasse principle holds for a smooth complete intersection of two quadrics X ⊂ P n over a number field, if n ≥ 5 and X contains a conic. Building upon these two results, we extend recent results of Creutz and Viray (2021) on the existence of a quadratic point on intersections of two quadrics over p-adic fields and over number fields. We then recover Heath-Brown's theorem (2018) that the Hasse principle holds for smooth complete intersections of two quadrics in P 7 . We also give an alternate proof of a theorem of Iyer and Parimala (2022) on the local-global principle in the case n = 5.

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