Mathematical Research Letters

Volume 30 (2023)

Number 1

Linear subspaces of minimal codimension in hypersurfaces

Pages: 143 – 166



David Kazhdan (Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel)

Alexander Polishchuk (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.; and National Research University Higher School of Economics, Moscow, Russia)


Let $\mathbf{k}$ be a perfect field and let $X \subset \mathbb{P}^N$ be a hypersurface of degree $d$ defined over $\mathbf{k}$ and containing a linear subspace $L$ defined over $\overline{\mathbf{k}}$ with $\operatorname{codim}_{\mathbb{P}^N} L = r$. We show that $X$ contains a linear subspace $L_0$ defined over $\mathbf{k}$ with $\operatorname{codim}_{\mathbb{P}^N} L \leq dr$. We conjecture that the intersection of all linear subspaces (over $\mathbf{k}$) of minimal codimension $r$ contained in $X$, has codimension bounded above only in terms of $r$ and $d$. We prove this when either $d \leq 3$ or $r \leq 2$.

A.P. is partially supported by the NSF grant DMS-2001224, and within the framework of the HSE University Basic Research Program and by the Russian Academic Excellence Project ‘5-100’.

Received 22 October 2021

Received revised 6 June 2022

Accepted 17 July 2022

Published 21 June 2023