A higgledy-piggledy set of planes based on the ABB-representation of linear sets

TL;DR

This paper explores the geometric structures of linear sets in projective spaces using the André/Bruck-Bose representation, linking them to quadrics and designs, and constructs optimal sets of planes in PG(5,q).

Contribution

It establishes new geometric correspondences for linear sets and introduces a novel construction of higgledy-piggledy plane sets in PG(5,q).

Findings

Linear sets of rank 3 correspond to hyperbolic quadrics.

Linear clubs relate to subspaces of a 2-design based on rational curves.

Constructed optimal higgledy-piggledy sets of planes in PG(5,q).

Abstract

In this paper, we investigate the Andr\'e/Bruck-Bose representation of certain $\mathbb{F}_q$-linear sets contained in a line of $\text{PG}(2,q^t)$. We show that scattered $\mathbb{F}_q$-linear sets of rank $3$ in $\text{PG}(1,q^3)$ correspond to particular hyperbolic quadrics and that $\mathbb{F}_q$-linear clubs in $\text{PG}(1,q^t)$ are linked to subspaces of a certain $2$-design based on normal rational curves; this design extends the notion of a circumscribed bundle of conics. Finally, we use these results to construct optimal higgledy-piggledy sets of planes in $\text{PG}(5,q)$.