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

We construct the new semifield flock of PG(3, 243) associated with the Penttila Williams translation ovoid of Q(4, 243) and we study the associated generalized quadrangle and its translation dual. Academic Press

A complete characterization of the flocks of Q + (3, q) is given. As an application, it follows that if q is odd, q # 11, 23, 59, there exist no maximal exterior sets of Q + (2n -1, q) for n>2.

We review some constructions related with desarguesian spreads of projective spaces of finite dimension over a skewfield.

Three new examples of non-linear flocks of the non-singular ruled quadric Q +(3, q) of PG(3, q) are given. INTRODUCTIONA class of non-linear flocks of the non-singular ruled quadric Yg = Q + (3, q) of PG(3, q), q odd, was described by J. A. Thas in [9]. In a recent paper [10], Thas conjectured that a flock of ~ is either linear or of the type he introduced in [9]. In this note, three new examples of non-linear flocks of the non-singular ruled quadric ~ of PG(3, q) (q = 11, 23, 59) are given, which do not belong to the class of the non-linear flocks obtained by Thas in [9]. Remember that each flock of the non-singular ruled quadric Q + (3, q), q even, is linear ([9, p. 83]).Let q be a prime power pS, p odd. A spread Y of PG(3, q) is a partition of the points of PG(3, q) into mutually skew lines. Let n(5o) denote the translation plane associated with the spread 5O in the usual way (see, for instance, [3, pp. 219-221]).A regulus of PG(3, q) is a 1-regulus according to Dembowski ([3, p. 220]), i.e. a set ~ of lines of PG(3, q) such that: (1) INI = q + 1; (2) any two distinct lines of are skew; (3) ira line I meets three distinct elements of N, then I c~ m ¢ @ for all m ~ N.Let a, b be distinct lines of a spread 5O of PG(3, q); 5O is called (a, b)-regular if, given any line c e 5 ° such that a ¢ c ~ b, then the regulus N(a, b, c) of PG(3, q) containing a, b and c satisfies ~(a, b, c) ~ 5C Let d¢ ° = Q+(3, q) be a non-singular ruled quadric ofPG(3, q); a partition of Yg into disjoint irreducible conics is aflock of Yg.A flock ~ of ~ is called linear if there exists an exterior line I (i.e. I has no points in common with d/g) such that the q + 1 planes through I intersect ~ in the conics of the flock ~-; I is the axis of the flock.