Rationality of complete intersections of two quadrics over nonclosed fields

Rationality of real conic bundles with quartic discriminant curve

“…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].…”

“…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.…”

“…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.…”

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

“…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].…”

“…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.…”