Rational linear subspaces of hypersurfaces over finite fields

TL;DR

This paper investigates conditions under which smooth hypersurfaces over finite fields contain rational linear subspaces, establishing existence results based on inequalities involving the dimensions and degree.

Contribution

It provides new criteria guaranteeing the presence of rational linear subspaces in hypersurfaces over finite fields, extending previous results to broader cases.

Findings

Existence of rational r-planes under certain inequalities

Results hold for smooth hypersurfaces with large characteristic p

Stronger conditions yield similar results without smoothness or large p

Abstract

Fix positive integers $n,r,d$. We show that if $n,r,d$ satisfy a suitable inequality, then any smooth hypersurface $X\subset \mathbb{P}^n$ defined over a finite field of characteristic $p$ sufficiently large contains a rational $r$-plane. Under more restrictive hypotheses on $n,r,d$ we show the same result without the assumption that $X$ is smooth or that $p$ is sufficiently large.