Linear subspaces of minimal codimension in hypersurfaces

TL;DR

This paper investigates the existence and properties of linear subspaces within hypersurfaces over perfect fields, establishing bounds on their codimensions and proposing a conjecture about their intersections.

Contribution

It proves bounds on the codimension of linear subspaces defined over the base field within hypersurfaces containing a given subspace, and verifies the conjecture for specific degrees and codimensions.

Findings

Existence of a linear subspace over the base field with bounded codimension

Verification of the conjecture for degrees d ≤ 3 or codimension r ≤ 2

Bound on the intersection codimension of minimal subspaces

Abstract

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