On the maximum field of linearity of linear sets
Bence Csajbók, Giuseppe Marino, Valentina Pepe
TL;DR
This work investigates the maximum field of linearity for $\\mathbb{F}_q$-linear sets in $\\mathrm{PG}(r-1, q^n)$ under weight constraints. It proves that when every point has weight at least $2$ and $n \\\leq q$, there exists a divisor $d>1$ of $n$ such that the linear set coincides with the $\\mathbb{F}_{q^d}$-linear span of $U$, hence the maximum field of linearity is $\\mathbb{F}_{q^d}$ with $d$ determined by the minimal weight, and a stronger statement holds when $n \\\mid m$. The dual formulation via projective-spread representations and Grassmann embeddings shows that a linear set in $\\mathrm{PG}(r-1, q^n)$ attaining maximum linearity $\\mathbb{F}_q$ must contain a point of weight one, yielding size-bounds consequences. Additional results relate these structural findings to the size and spread-theoretic properties of linear sets, including a discussion of lower bounds and a counterexample illustrating limits of weight-based inferences. Overall, the paper links linear-set linearity, weight distributions, and spread geometry to provide new constraints and size estimates in finite geometry.
Abstract
Let $V$ denote an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. For an $m$-dimensional $\mathbb{F}_q$-subspace $U$ of $V$ assume that $\dim_q \left(\langle {\bf v}\rangle_{\mathbb{F}_{q^n}} \cap U\right) \geq 2$ for each non zero vector ${\bf v}\in U$. If $n\leq q$ then we prove the existence of an integer $1<d \mid n$ such that the set of one-dimensional $\mathbb{F}_{q^n}$-subspaces generated by non-zero vectors of $U$ is the same as the set of one-dimensional $\mathbb{F}_{q^n}$-subspaces generated by non-zero vectors of $\langle U\rangle_{\mathbb{F}_{q^d}}$. If we view $U$ as a point set of $\mathrm{AG}(r,q^n)$, it means that $U$ and $\langle U \rangle_{\mathbb{F}_{q^d}}$ determine the same set of directions. We prove a stronger statement when $n \mid m$. In terms of linear sets it means that an $\mathbb{F}_q$-linear set of $\mathrm{PG}(r-1,q^n)$ has maximum field of linearity $\mathbb{F}_q$ only if it has a point of weight one. We also present some consequences regarding the size of a linear set.