Derivation of Flocks of Quadratic Cones

Abstract

In this paper a new construction (named derivation) of q flocks of the quadratic cone of PG(3,q) from a given one is defined. The known examples of flocks are studied, and a new class of flocks is found by derivation of likeable Kantor flocks. 1980 Mathematics Subject Classification (1985 Revision): 51E20. * The first two authors are partially supported by M.P.I. 1. Recently many authors have studied flocks of quadratic cones in PG(3, q) and their connection with generalized quadrangles and translation planes ([7], [8], [9], [10], [12], [13], [14], [17], [18], [19], [20], [21], [22]). A complete list of the known flocks of the quadratic cone and of the associated generalized quadrangles and translation planes is contained in the paper [10].In this paper, for any q -p e , p an odd prime, we construct q other flocks from a given one, by embedding the quadratic cone into the non-singular quadric Q(4,q) of PG(4,q).This construction method will be called derivation. It also shows that, for q odd, to each flock there corresponds a particular set of #4-1 points of (4,#) and, conversely, that to each such set of (4,#) there correspond g 4-1 flocks of the quadratic cone.Furthermore, we prove that the flocks derived from the likeable flocks of Kantor for q = 5 e and e > l, belong to an unknown class.In this note, PFO + (6 9 q) will denote the group of all collineations of PG(5,q) leaving Q + (5,q) and each of its Systems of generating planes invariant.2. Let ^0 be a quadratic cone having vertex p 0 . A flock of ^0 is a partitition of the points of ^0 \ {PQ} into q disjoint conics.Let / be an exterior line (i.e. a line not intersecting ^0). The linear flock of ^0 having axis / is the set ^0 = {/In^0|/I is a plane, Ι^Π,ρ Ό φ Π}.Brought to you by | University of Arizona Authenticated Download Date | 5/27/15 1:33 AM

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“…These examples generalise the Fisher-Thas-Walker flocks in PG(3, q) q odd, [3,13], since by [1] such a flock in PG(3, q) has BLT-set a normal rational curve on Q(4, q).…”

“…In particular, the dual setting for q even generalises [12, 1.5.3], the algebraic condition generalises [12, 1.5.5], the existence of the partial ovoid of Q + (n + 2, q) generalises [12, 1.3], the process of derivation for q odd generalises [1] and the construction of herds of caps for q even generalises [2, Theorem 1] (see also [11,Theorem 2.1]). …”

Some remarks on flocks

“…Here we consider a number of generalisations of BLT-sets of Q(4, q), q odd: to BLT-sets of H (3, q 2 ), q odd, in Section 5; to BLT-sets of T*(^), in Section 6; to BLT-sets of finite generalised quadrangles in general, in Section 2; to BLT-sets of the polar spaces Q(2n, q), q odd, of rank n > 2, in Section 7. The material in Section 2 is related to work of De Soete and Thas [4], predating the introduction of BLT-sets of Q(4, q) in Bader, Lunardon and Thas [2] by six years. While we note the advances of Shult and Thas [14] in Section 3, we restrict ourselves to BLT-sets of points, rather than sets of subspaces with the BLT-property.…”

“…In Section 4, we give new proofs of the fundamental results of Bader, Lunardon and Thas [2] relating flocks of the quadratic cone of PG (3, q), q odd, and BLT-sets of Q (4, q). In Section 5, we show that there is a unique BLT-set of H (3,9).…”

Translation Generalized Quadrangles for Which the Translation Dual Arises From a Flock

“…Then |P| = (s + 1)(st + 1) and |B| = (t + 1)(st + 1) [18]; also, s ≤ t 2 and, dually, t ≤ s 2 . There is a point-line duality for GQ's of order (s, t) for which in any definition or theorem the words "point" and "line", and the parameters s and t are interchanged.…”