A higgledy-piggledy set of planes based on the ABB-representation of linear sets
Lins Denaux, Jozefien D'haeseleer, Geertrui Van de Voorde

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 -linear sets contained in a line of . We show that scattered -linear sets of rank in correspond to particular hyperbolic quadrics and that -linear clubs in are linked to subspaces of a certain -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 .
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation · Manufacturing Process and Optimization
