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

Abstract

Abstract.We complete the classification of flocks of the quadratic cone in PG(3, q) for q < 29, by showing by computer that there are exactly 8 flocks of the quadratic cone in PG(3,19), 18 flocks of the quadratic cone in PG(3,23), 12 flocks of the quadratic cone in PG(3,25), 14 flocks of the quadratic cone in PG (3,27), and 28 flocks of the quadratic cone in PG(3,29), up to equivalence.

A hemisystem of a nonclassical generalised quadrangle

“…If the flock is linear, that is all planes of the flock share a common line, then the resulting generalised quadrangle is isomorphic to H(3, q 2 ). There exist nonlinear flocks, and so nonclassical flock quadrangles, and these have been classified for all q 37, q = 32 (see [6], [11], [2]). For instance, if q is a prime power congruent to 2 modulo 3, there exist non-linear flock quadrangles known as the Fisher-ThasWalker-Kantor generalised quadrangles.…”

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

“…When L is a line, the plane is a conical flock plane and there is a group that induces A 4 on L modulo the kernel homology group; there is a group isomorphic to A 4 on the associated spread (the associated BLT set admits S 4 wr Z 2 ). In this setting, it is known there is a unique spread, the Penttila-Royle spread, admitting A 4 (see [27]). When L is a Baer subplane and A 4 leaves a regulus invariant, there is a derived flock admitting a group of order q(q + 1).…”

“…(G|L)/Z(G|L) is isomorphic to A 4 or S 4 when q = 23 or 47. When q = 23, there is a unique conical flock of this type due to Penttila and Royle (see [33], [27]). The group of order q + 1 = 24 is an extension of A 4 .…”

Classification of 8-dimensional rank two commutative semifields

“…Moreover, R2CS are equivalent to semifield flocks of a quadratic cone in a 3-dimensional projective space. We refer to the introduction of [21] for an excellent historical overview of the theory of flocks in finite geometry. Consequently, R2CS are also equivalent to translation ovoids of Q(4, q), the parabolic quadric in 4-dimensional projective space.…”