A Result on Spreads of the Generalized QuadrangleT2(O), withOan Oval Arising from a Flock, and Applications

Abstract

If F is a flock of the quadratic cone K of PG(3, q), q even, then the corresponding generalized quadrangle S(F) of order (q 2 , q) has subquadrangles T 2 (O), with O an oval, of order q. We prove in a geometrical way that any such T 2 (O) has spreads S consisting of an element y ∈ O and the q 2 lines not in the plane PG(2, q) of O of q quadratic cones K x , x ∈ O − {y}, of the space PG(3, q) containing T 2 (O), where K x has vertex x, is tangent to PG(2, q) at x y and has nucleus line xn, with n the nucleus of O. We also show how the oval O can be directly constructed from the flock F.

Geometrical Constructions of Flock Generalized Quadrangles

“…Then S(F ) has at least q 3 +q 2 subquadrangles of order q; see Thas [16]. By Payne [8] any of these subquadrangles S$ is a T 2 (O) of Tits, with O an oval of PG (2, q).…”

A geometrical construction of the oval(s) associated with an α-flock

“…In this section we add coordinates to the construction of Theorem 1.4 to show that the hyperoval completion of O is the same as the hyperoval constructed from an aflock by Cherowitzo and that the spread S of T 2 ðOÞ is the same as that constructed by Brown, O'Keefe, Payne, Penttila and Royle. Note that in [8] Thas showed that in the case of a flock of a quadratic cone that the q þ 1 (flock, axis to base oval of cone) pairs gave rise to the q þ 1 herd hyperovals constructed from a flock as formalised in Theorem 1.1.…”

“…In [3] the set of q þ 1 functions defining the hyperovals as above is called a herd. In [8] Thas gave a geometrical construction of these hyperovals from the flock (although not a geometrical proof of the construction).…”

“…In Section 3 we give a generalisation of a construction in [8] that each pair (axial line of K a , flock of K a ) gives rise to an oval of PGð2; qÞ. In the case where K a is a quadratic cone it was shown in [8] that in this way a flock gives rise to the q þ 1 hyperovals of the corresponding herd, while in Section 4 we show that for a general a-flock the oval completes to the hyperoval of Theorem 1.2. In this way we have a geometric proof of Theorem 1.1 and Theorem 1.2.…”