# Pencilled regular parallelisms

## TL;DR

This paper characterizes pencilled hyperflock determining line sets in projective spaces over any field, providing a construction method, classification, and algebraic conditions for their existence, and relating them to regular parallelisms.

## Contribution

It introduces a construction and classification of pencilled hfd line sets, linking them to regular parallelisms and providing algebraic existence conditions.

## Key findings

- All pencilled hfd line sets are determined by the presented construction.
- Regular parallelisms correspond to pencilled hfd line sets.
- Necessary and sufficient algebraic conditions for existence are established.

## Abstract

Over any field $\mathbb K$, there is a bijection between regular spreads of the projective space ${\rm PG}(3,{\mathbb K})$ and $0$-secant lines of the Klein quadric in ${\rm PG}(5,{\mathbb K})$. Under this bijection, regular parallelisms of ${\rm PG}(3,{\mathbb K})$ correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be \emph{pencilled} if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being \emph{pencilled}. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.