Linear subspaces of minimal codimension in hypersurfaces

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 $r\le 2$.

15 pages, v2 substantially rewritten: added Conjecture B and a result on hypersurfaces of rank 2; the result on Schmidt rank is moved to another paper; v3: modified Conjecture B and added examples in the introduction

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