Year: 2025
Type: article
Abstract: 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 ℝ ; this may be viewed as extending work of Hassett–Kollár–Tschinkel.
DOI: 10.5802/jep.308
Authors: Lena Ji, Fumiaki Suzuki
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4413282123