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 $E={{\mathrm{End}}}_q({\mathbb F}_{q^t})$ of the ${\mathbb F}_q$-vector space ${\mathbb F}_{q^t}$. As is well known, there is a bijection $\varPsi :{{\mathrm{{PG}}}}(1,E)\rightarrow {\mathcal G}_{2t,t,q}$ 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 $T^\varPsi$ 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 $L'_T$, corresponding via $\varPsi$ to a collection of pairwise skew $(t-1)$-dimensional subspaces, with $T\in L'_T$, 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 $\varphi$ of ${{\mathrm{{PG}}}}(1,E)$ such that $L_T^\varphi =L'_T$.