Blocking subspaces with points and hyperplanes

TL;DR

This paper characterizes minimal sets of points and hyperplanes in projective space that intersect all k-dimensional subspaces, revealing different minimal configurations depending on the relation between k and n.

Contribution

It provides a complete characterization of the smallest blocking sets with points and hyperplanes in projective geometry, including new mixed configurations for the critical case.

Findings

For k > (n-1)/2, smallest sets are points only.

For k < (n-1)/2, smallest sets are hyperplanes only.

For k = (n-1)/2, mixed sets of points and hyperplanes are smaller than pure sets.

Abstract

In this paper, we characterise the smallest sets $B$ consisting of points and hyperplanes in $\text{PG}(n,q)$, such that each $k$-space is incident with at least one element of $B$. If $k > \frac {n-1} 2$, then the smallest construction consists only of points. Dually, if $k < \frac{n-1}2$, the smallest example consists only of hyperplanes. However, if $k = \frac{n-1}2$, then there exist sets containing both points and hyperplanes, which are smaller than any blocking set containing only points or only hyperplanes.