Scattered linear sets generated by collineations between pencils of lines

Abstract

Let A and B be two points of PG(2, q n ), and let be a collineation between the pencils of lines with vertices A and B. In this paper, we prove that the set of points of intersection of corresponding lines under is either the union of a scattered GF(q)-linear set of rank n + 1 with the line AB or the union of q − 1 scattered GF(q)-linear sets of rank n with A and B. We also determine the intersection configurations of two scattered GF(q)-linear sets of rank n + 1 of PG(2, q n ) both meeting the line AB in a GF(q)-linear set of pseudoregulus type with transversal points A and B.

Exterior splashes and linear sets of rank 3

“…By [12], all scattered linear sets of rank 3 are projectively equivalent. Scattered linear sets were introduced in [5], and have recently been studied in [11,12,13,15].…”

A Carlitz type result for linearized polynomials

“…Then by (13) either b 1 = 0 or b 4 = 0. In the former case putting together Equations (14), (15), (16) we get N(a i ) = N(b i ) for i ∈ {2, 3, 4} and hence there exists λ ∈ F * q 5 such that g(x) = f (λx)/λ. If a 1 = b 4 = 0, then inĝ(x) the coefficients of x q is zero thus applying the previous result we get g(x) =f (µx)/µ, where µ = λ −1 .…”

“…Following [22] and [16] a maximum scattered F q -linear set L U of rank n in PG(1, q n ) is of pseudoregulus type if it is PΓL(2, q n )-equivalent to L f with f (x) = x q or, equivalently, if there exists ϕ ∈ GL(2, q n ) such that For the proof of the previous result see also [20]. The known pairwise non-equivalent families of q-polynomials over F q n which define maximum scattered linear sets of rank n in PG(1, q n ) are Remark 5.3.…”

On the equivalence of linear sets

“…q 5 is a collineation, as well as the map ϕ 2 : PG(4, q) → Σ 2 similarly defined. By [14,3], the projections p Γ, ℓ (Σ 1 ) and p Γ, ℓ (Σ 2 ) coincide, and are a linear set of pseudoregulus type, say L = { (λ, λ q , 0, 0, 0) q 5 : λ ∈ F * q 5 }.…”