Subgeometries and linear sets on a projective line

Lavrauw, Michel; Zanella, Corrado

doi:10.1016/j.ffa.2015.01.006

Cited by 14 publications

(14 citation statements)

References 11 publications

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“…The weight of a point P = B( p) of a linear set B(μ) equals dim(μ ∩ p) + 1. Following [7], a club of rank h is a linear set S of rank h such that one point of S has weight h − 1 and all others have weight 1. We define an i-club of rank h as a linear set C of rank h such that one point, called the head of C, has weight i and all others have weight 1.…”

Section: Linear Sets and Field Reductionmentioning

confidence: 99%

A linear set view on KM-arcs

Boeck

Voorde

2016

J Algebr Comb

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In this paper, we study KM-arcs of type t, i.e., point sets of size in such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular -linear set, called an i -club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type q/4. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type q/4

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Section: Linear Sets and Field Reductionmentioning

confidence: 99%

A linear set view on KM-arcs

Boeck

Voorde

2016

J Algebr Comb

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“…If a linear set L of rank n in PG(1, q t ) has size θ n−1 = (q n − 1)/(q − 1) (which is the maximum size for a linear set of rank n), then L is a scattered linear set. For generalities on the linear sets the reader is referred to [13], [14], [15], [16], and [19].…”

Section: Motivationmentioning

confidence: 99%

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

Havlicek

Zanella

2017

J Algebr Comb

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Let PG(1, E) be the projective line over the endomorphism ring E = End q (F q t ) of the F q -vector space F q t . As is well known there is a bijection Ψ : PG(1, E) → G 2t,t,q with the Grassmannian of the (t − 1)subspaces in PG(2t − 1, q). In this paper along with anyproperties 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 Ψ to a collection of pairwise skew (t − 1)-dimensional subspaces, with T ∈ 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 ∈ PG(1, E) is of pseudoregulus type if and only if there exists a projectivity ϕ of PG(1, E) such that L ϕ T = L T . Mathematics subject classification (2010): 51E20, 51C05, 51A45, 51B05. Keywords: scattered linear set; linear set of pseudoregulus type; projective line over a finite field; projective line over a ring.

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“…, 0, 1) q t . Applying κ −1 it follows that the transversal points of p Γ,ℓ 0 (Σ) are Pσ The splash of a q-order canonical subgeometry Σ of PG(t−1, q t ) on a line ℓ is the set of all intersections of ℓ (not contained in the span of a hyperplane of Σ) with the spans of the hyperplanes of Σ, and is always a linear set [15]. The relationship between tangent splashes and linear sets has been dealt with in [14].…”

Section: Characterization Of the Projecting Configurationsmentioning

confidence: 99%

On scattered linear sets of pseudoregulus type in PG(1,q)

Csajbók

Zanella

2016

Finite Fields and Their Applications

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Scattered linear sets of pseudoregulus type in PG(1, q t ) have been defined and investigated in [19,5]. 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 [20], (ii) as a set whose image under the field reduction map is the hypersurface of degree t in PG(2t − 1, q) studied in [10], (iii) as exterior splash, by the correspondence described in [15]. In particular, given a canonical subgeometry Σ of PG(t − 1, q t ), necessary and sufficient conditions are given for the projection of Σ with center a (t − 3)-subspace to be a linear set of pseudoregulus type. Furthermore, the q-order sublines are counted and geometrically described.

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Subgeometries and linear sets on a projective line

Cited by 14 publications

References 11 publications

A linear set view on KM-arcs

A linear set view on KM-arcs

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

On scattered linear sets of pseudoregulus type in PG(1,q)

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