[Propriétés arithmétiques et birationnelles des espaces linéaires sur les intersections de deux quadriques]
We study rationality questions for Fano schemes of linear spaces on a smooth complete intersection $X$ of two quadrics, especially over a non-closed field. Our approach is to study hyperbolic reductions of the pencil of quadrics associated to $X$. We prove that the Fano schemes $F_r(X)$ of $r$-planes are birational to symmetric powers of hyperbolic reductions, generalizing results of Reid and Colliot-Thélène–Sansuc–Swinnerton-Dyer, and we give several applications to rationality properties of $F_r(X)$.
For instance, we show that if $X$ contains an $(r+1)$-plane over a field $k$, then $F_r(X)$ is rational over $k$. When $X$ has odd dimension, we show a partial converse for rationality of the Fano schemes of second maximal linear spaces, generalizing results of Hassett–Tschinkel and Benoist–Wittenberg. When $X$ has even dimension, the analogous result does not hold, and we further investigate this situation over the real numbers. In particular, we prove a rationality criterion for the Fano schemes of second maximal linear spaces on these even-dimensional complete intersections over $\mathbb{R}$; this may be viewed as extending work of Hassett–Kollár–Tschinkel.
Nous étudions des questions de rationalité pour les schémas de Fano d’espaces linéaires sur une intersection complète et lisse $X$ de deux quadriques, en particulier sur un corps non clos. Notre approche consiste à étudier les réductions hyperboliques du pinceau de quadriques associé à $X$. Nous prouvons que les schémas de Fano $F_r(X)$ des $r$-plans sont birationnels aux puissances symétriques des réductions hyperboliques, généralisant des résultats de Reid et de Colliot-Thélène, Sansuc et Swinnerton-Dyer, et nous donnons plusieurs applications aux propriétés de rationalité de $F_r(X)$.
Par exemple, nous montrons que si $X$ contient un $(r+1)$-plan sur un corps $k$, alors $F_r(X)$ est rationnel sur $k$. Lorsque $X$ est de dimension impaire, nous montrons une réciproque partielle pour la rationalité des schémas de Fano des espaces linéaires sous-maximaux, généralisant des résultats de Hassett et Tschinkel et de Benoist et Wittenberg. Lorsque $X$ est de dimension paire, le résultat analogue n’est plus valable, et nous étudions cette situation sur les nombres réels. En particulier, nous prouvons un critère de rationalité pour les schémas de Fano des espaces linéaires sous-maximaux sur ces intersections complètes de dimension paire sur $\mathbb{R}$ ; ceci peut être considéré comme une extension des travaux de Hassett, Kollár et Tschinkel.
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Keywords: Rationality, intermediate Jacobians, quadric fibrations
Mots-clés : Rationalité, jacobiennes intermédiaires, fibrations en quadriques
Lena Ji 1 ; Fumiaki Suzuki 2
CC-BY 4.0
@article{JEP_2025__12__1161_0,
author = {Lena Ji and Fumiaki Suzuki},
title = {Arithmetic and birational properties of linear spaces on intersections of two~quadrics},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1161--1196},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.308},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.308/}
}
TY - JOUR AU - Lena Ji AU - Fumiaki Suzuki TI - Arithmetic and birational properties of linear spaces on intersections of two quadrics JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 1161 EP - 1196 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.308/ DO - 10.5802/jep.308 LA - en ID - JEP_2025__12__1161_0 ER -
%0 Journal Article %A Lena Ji %A Fumiaki Suzuki %T Arithmetic and birational properties of linear spaces on intersections of two quadrics %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 1161-1196 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.308/ %R 10.5802/jep.308 %G en %F JEP_2025__12__1161_0
Lena Ji; Fumiaki Suzuki. Arithmetic and birational properties of linear spaces on intersections of two quadrics. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1161-1196. doi : 10.5802/jep.308. https://jep.centre-mersenne.org/articles/10.5802/jep.308/
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