On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$

TL;DR

This paper investigates properties and characterizations of scattered linear sets of pseudoregulus type in projective lines over finite fields, providing new conditions, descriptions, and counting methods for these geometric objects.

Contribution

It introduces new characterizations and conditions for scattered linear sets of pseudoregulus type, including projections from canonical subgeometries and geometric descriptions of sublines.

Findings

Necessary and sufficient conditions for projections to be of pseudoregulus type

Descriptions of $q$-order sublines within these sets

Methods to count and geometrically interpret sublines

Abstract

Scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$ have been defined and investigated in [G. Lunardon, G. Marino, O. Polverino, R. Trombetti: Maximum scattered linear sets of pseudoregulus type and the Segre Variety ${\cal S}_{n,n}$. J. Algebr. Comb. 39 (2014), 807--831.; G. Donati, N. Durante: Scattered linear sets generated by collineations between pencils of lines. J. Algebr. Comb. 40 (2014), 1121-1134]. The aim of this paper is to continue such an investigation. Properties of a scattered linear set of pseudoregulus type, say $L$, are proved by means of three different ways to obtain $L$: (i) as projection of a $q$-order canonical subgeometry [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004), 663-669], (ii) as a set whose image under the field reduction map is the hypersurface of degree $t$ in $\mathrm{PG}(2t-1,q)$ studied in [M. Lavrauw, J. Sheekey, C. Zanella: On embeddings of minimum dimension of $\mathrm{PG}(n,q)\times \mathrm{PG}(n,q)$. Des. Codes Cryptogr. 74 (2015), 427-440], (iii) as exterior splash, by the correspondence described in [M. Lavrauw, J. Sheekey, C. Zanella: On embeddings of minimum dimension of $\mathrm{PG}(n,q)\times \mathrm{PG}(n,q)$. Des. Codes Cryptogr. 74 (2015), 427-440]. In particular, given a canonical subgeometry $\Sigma$ of $\mathrm{PG}(t-1,q^t)$, necessary and sufficient conditions are given for the projection of $\Sigma$ with center a $(t-3)$-subspace to be a linear set of pseudoregulus type. Furthermore, the $q$-order sublines are counted and geometrically described.