Arithmetic and birational properties of linear spaces on intersections of two quadrics

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.