Flocks of a quadratic cone in PG(3, q), q ≤ 8

Abstract

If ~(O) is a quadratic cone in PG(3,q), with vertex x, then a flock of c~(O) is a partition of cg(O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q2 and a generalized quadrangle of order (q2,q). Here we determine all flocks of ~(O) for q ~< 8.

Classification of flocks of the quadratic cone over fields of order at most 29

“…Since then, it has become clear that complete classification in the Miquelian Laguerre case is extremely difficult, with further constructions appearing in [10], [34], [25], [32], [9], and so attention has turned to small field orders. The previously known classification results are [41] for fields of orders 2, 3, 4 and De Clerck- Gevaert-Thas (1988) [11] for fields of orders 5, 7, 8 (these are computerfree results); Mylle (1991) [27] for the field of order 9, De Clerck-Herssens (1992) [12] for the fields of orders 11, 16, Penttila-Royle (1998) [35] for the fields of orders 13, 17, Brown-O'Keefe-Payne-Penttila-Royle [7] for the field of order 32 (these are computer-based results), see Theorem 2.6. Here we add the fields of orders 19, 23, 25, 27 and 29 to the list, finding that there are exactly 8 flocks of the quadratic cone in PG (3,19) (Corollary 4.7), 18 flocks of the quadratic cone in PG(3,23) (Corollary 5.8), 12 flocks of the quadratic cone in PG(3,25) (Corollary 6.3), 14 flocks of the quadratic cone in PG (3,27) (Corollary 7.3), and 28 flocks of the quadratic cone in PG(3,29) (Corollary 8.3), up to equivalence, with these being computer-based results.…”

“…So, by Theorem 2.4, it follows that B is classical or Fisher. (vi) [11] (xii) [7] All flocks of the quadratic cone o/PG (3,32) are linear, Fisher-Thas-Walker, or one of the two classes of Payne or Subiaco, up to equivalence.…”

“…Here the definitive work on construction is [35], where 9 BLT-sets of ß(4,23) are listed, namely, the classical, Fisher [15], Fisher-Thas-Walker [15], [43], Kantor monomial [22], De Clerck-Herssens [12] with a group of order 72, which we shall refer to as DCH72, and four new ones with groups of orders 1152, 24, 16 and 6, which we shall refer to as PR1152, PR24, PR16 and PR6. The ad hoc hypothesis P(23) used to characterise these BLT-sets B of 0(4,23) is on the stabiliser of B in PGO (5,23): that it has order divisible by 5 or is isomorphic to Z>8 χ €2 and has 2 elements with no fixed points, 2 elements with 2 fixed points, 5 elements with 22 fixed points, 3 elements with 26 fixed points and 3 elements with 530 fixed points (in the action on the points of (4,23)).…”

Ostrom-derivates

“…When q = 8, there is a unique non-linear flock due to Herssens and De Clerck [4], which is the Betten flock. Hence, we may always assume that q +1 is prime.…”

“…Note the translation planes of order 16 are determined and only the Desarguesian is a conical flock plane. Hence, we must consider when q = 2 4 . In this case, there is a unique non-linear flock due to Herssens and De Clerck [4], and the full group on the flock is cyclic of order 8.…”