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On subspaces defining linear sets of maximum rank

Valentina Pepe

TL;DR

This work investigates when two ${\mathbb F}_q$-subspaces $U$ and $W$ of an ${\mathbb F}_{q^n}$-vector space $V$ define the same ${\mathbb F}_q$-linear set $L_U$ in ${\rm PG}(r-1,q^n)$, focusing on the maximum-rank case $rn-n$. It develops a geometric-algebraic framework using the Desarguesian spread, cyclic representation, Grassmann embedding, and Dickson matrices to translate $L_U=L_W$ into equal principal minors of associated Dickson matrices (for $r=2$). The main results provide a complete classification for $r=2$: either $W$ is a scalar multiple of $U$ or of $U^{\perp}$, or $L_U=L_W$ corresponds to pseudoregulus-type or generalized pseudoregulus-type linear sets under certain conditions on $n$, with further structure described for higher rank via cone decompositions. The findings advance understanding of when subspace data equivalently defines a linear set, with implications for finite geometry constructions and related combinatorial objects.

Abstract

Let $V$ denote an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. Let $U$ and $W$ be $\mathbb{F}_q$-subspaces of $V$, $L_U$ and $L_W$ the projective points of $\mathrm{PG}\,(V,q^n)$ defined by $U$ and $W$ respectively. We address the problem when $L_W=L_U$ under the hypothesis that $U$ and $W$ have maximum dimension, i.e., $\dim_{\mathbb{F}_q} W=\dim_{\mathbb{F}_q}U=$ $rn-n $, and we give a complete characterization for $r=2$.

On subspaces defining linear sets of maximum rank

TL;DR

This work investigates when two -subspaces and of an -vector space define the same -linear set in , focusing on the maximum-rank case . It develops a geometric-algebraic framework using the Desarguesian spread, cyclic representation, Grassmann embedding, and Dickson matrices to translate into equal principal minors of associated Dickson matrices (for ). The main results provide a complete classification for : either is a scalar multiple of or of , or corresponds to pseudoregulus-type or generalized pseudoregulus-type linear sets under certain conditions on , with further structure described for higher rank via cone decompositions. The findings advance understanding of when subspace data equivalently defines a linear set, with implications for finite geometry constructions and related combinatorial objects.

Abstract

Let denote an -dimensional -vector space. Let and be -subspaces of , and the projective points of defined by and respectively. We address the problem when under the hypothesis that and have maximum dimension, i.e., , and we give a complete characterization for .
Paper Structure (4 sections, 28 theorems, 72 equations)

This paper contains 4 sections, 28 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. The geometric and algebraic setting
  3. Dickson matrices with equal corresponding principal minors
  4. Proof of the main results

Key Result

Theorem 1

Let $V=\mathbb F_{q^n}^r$, $U$ and $W$ be two $(rn-n)$-dimensional vector spaces of $V$ such that $L_U=L_W$ and let $\mathbb F_q$ be the maximum field of linearity for $L_U=L_W$. Then, for every point $P \in L=L_U=L_W$, we have that $w_{L_U}(P)=w_{L_W}(P)$. If $r=2$ and $\perp$ be the polarity assoc

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Corollary 1
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 41 more