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

authored by: Lena Ji, Fumiaki Suzuki
Abstract: We study rationality questions for Fano schemes of linear spaces on smooth complete intersections of two quadrics, especially over non-closed fields. 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-Th\'el\`ene--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--Koll\'ar--Tschinkel.
Organisation(s): Institute of Algebraic Geometry
Type: Preprint
Publication date: 29.02.2024
Publication status: E-pub ahead of print

BibTeX

@techreport{ad8e94b6618e499c88a357c9eb5b02b2,
title = "Arithmetic and birational properties of linear spaces on intersections of two quadrics",
abstract = " We study rationality questions for Fano schemes of linear spaces on smooth complete intersections of two quadrics, especially over non-closed fields. 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-Th\'el\`ene--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--Koll\'ar--Tschinkel. ",
keywords = "math.AG, Primary: 14E08, Secondary: 14G20, 14C25, 14D10",
author = "Lena Ji and Fumiaki Suzuki",
note = "26 pages, comments are welcome",
year = "2024",
month = feb,
day = "29",
language = "English",
type = "WorkingPaper",
}