Blocking Sets in the complement of hyperplane arrangements in projective space

TL;DR

This paper explores the existence of blocking sets in the complements of hyperplane arrangements in projective spaces, generalizing known results from affine spaces and analyzing specific cases like the braid arrangement.

Contribution

It introduces the study of blocking sets in the complement of hyperplane arrangements in projective space, extending previous affine space results and addressing the braid arrangement case.

Findings

Blocking sets exist in the complement of certain hyperplane arrangements.

The paper provides solutions for the braid arrangement case.

Poses open questions for further research in this area.

Abstract

It is well know that the theory of minimal blocking sets is studied by several author. Another theory which is also studied by a large number of researchers is the theory of hyperplane arrangements. We can remark that the affine space $AG(n,q)$ is the complement of the line at infinity in $PG(n,q)$. Then $AG(n,q)$ can be regarded as the complement of an hyperplane arrangement in $PG(n,q)$! Therefore the study of blocking sets in the affine space $AG(n,q)$ is simply the study of blocking sets in the complement of a finite arrangement in $PG(n,q)$. In this paper the author generalizes this remark starting to study the problem of existence of blocking sets in the complement of a given hyperplane arrangement in $PG(n,q)$. As an example she solves the problem for the case of braid arrangement. Moreover she poses significant questions on this new and interesting problem.