Linear sets in the projective line over the endomorphism ring of a finite field

Abstract

Let $\mathrm{PG}(1,E)$ be the projective line over the endomorphism ring $E=En{d}_q({\mathbb{F}}_{q^t})$ of the ${\mathbb{F}}_q$-vector space ${\mathbb{F}}_{q^t}$. As is well known there is a bijection $\Psi :\mathrm{PG}(1,E)\to {\mathcal{G}}_{2t,t,q}$ with the Grassmannian of the $(t-1)$-subspaces in $\mathrm{PG}(2t-1,q)$. In this paper along with any ${\mathbb{F}}_q$-linear set $L$ of rank $t$ in $\mathrm{PG}(1,q^t)$, determined by a $(t-1)$-dimensional subspace $T^{\Psi }$ of $\mathrm{PG}(2t-1,q)$, a subset $L_T$ of $\mathrm{PG}(1,E)$ is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring $E$. In particular the attention is focused on the relationship between $L_T$ and the set $L_T^{\prime }$, corresponding via $\Psi$ to a collection of pairwise skew $(t-1)$-dimensional subspaces, with $T\in L_T^{\prime }$, each of which determine $L$. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set $L$ related to $T\in \mathrm{PG}(1,E)$ is of pseudoregulus type if and only if there exists a projectivity $\phi$ of $\mathrm{PG}(1,E)$ such that $L_T^{\phi }=L_T^{\prime }$.

References in corpus (2)

• Extending the Concept of Chain Geometry
• On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$