Absolute points of correlations of $$PG(3,q^n)$$ P G ( 3 , q n )
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Absolute points of correlations of PG(3, qn ) Giorgio Donati1 · Nicola Durante1 Received: 14 November 2019 / Accepted: 4 August 2020 © The Author(s) 2020
Abstract The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of PG(2, q n ), have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space PG(3, q n ). As an application we show that, for q even, some of these sets are related to the Segre’s (2h + 1)-arc of PG(3, 2n ) and to the Lüneburg spread of PG(3, 22h+1 ). Keywords Sesquilinear forms · Correlations · Polarities Mathematics Subject Classification 51E20 · 05B25
1 Introduction and preliminary results 1.1 Sesquilinear forms and correlations Let V and W be two F-vector spaces, where F is a field. A map f : V −→ W is called semilinear or σ -linear if there exists an automorphism σ of F such that f (v + v ) = f (v) + f (v ) and
f (av) = a σ f (v)
for all vectors v ∈ V and all scalars a ∈ F. If σ is the identity map, then f is a usual linear map.
B
Nicola Durante [email protected] Giorgio Donati [email protected]
1
Dipartimento di Matematica e Applicazioni “Caccioppoli”, Università di Napoli “Federico II”, Complesso di Monte S. Angelo - Edificio T, via Cintia, 80126 Napoli, Italy
123
Journal of Algebraic Combinatorics
Let V be an F-vector space with finite dimension d. A map , : (v, v ) ∈ V × V −→ v, v ∈ F is a sesquilinear form or a semibilinear form on V if it is a linear map on the first argument and it is a σ -linear map on the second argument, that is: v + v , v = v, v + v , v , v, v + v = v, v + v, v , av, v = av, v , v, av = a σ v, v , for all v, v , v ∈ V , a ∈ F and σ an automorphism of F. If σ is the identity map, then , is an usual bilinear form. If B = (e1 , e2 , . . . , ed+1 ) is an ordered basis of V , then for x, y ∈ V we have x, y = X t AY σ , where A = (ei , e j ) is the associated matrix to the sesquilinear form with respect to the ordered basis B, X and Y are the columns of the coordinates of x, y w.r.t. B. The term sequi comes from the Latin, and it means one and a half. For every subspace S of V , put S := {y ∈ V : x, y = 0 ∀x ∈ S}, S ⊥ := {y ∈ V : y, x = 0 ∀x ∈ S}. Both S and S ⊥ are subspaces of V . The subspaces V and V ⊥ are called the right and the left radical of , and will be also denoted by Radr (V ) and Radl (V ), respectively. Proposition 1.1 The right and the left radicals of a sesquilinear form of a vector space V have the same dimension. −1
Proof Indeed, dimV = d + 1 − rk(B), where B = (aiσj ) and dimV ⊥ = d + 1
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