Semifield flocks, eggs, and ovoids of Q (4,q)

Abstract

Abstract In this article we consider the connection between semifield flocks of a quadratic cone in PG(3, q n ), eggs in PG(4n - 1, q ) and ovoids of Q (4, q n ), when q is odd. Starting from a semifield flock ℱ of a quadratic cone in PG(3, q n ),q odd, one can obtain an ovoid ## add figure 'advg.5.3.333_01.gif' ##(ℱ ) of Q (4, q n ) using the construction of Thas [J. A. Thas, Symplectic spreads in PG(3, q )inversive planes and projective planes. Discrete Math. 174 (1997), 329–336]. With a semifield flock there also corresponds a good egg ℰ of PG(4n - 1, q ) (see, e.g., [M. Lavrauw, T. Penttila, On eggs and translation generalised quadrangles. J. Combin. Theory Ser. A 96 (2001), 303–315]) and the TGQ T (ℰ ) contains at least q 3n + q 2n subquadrangles all isomorphic to Q (4, q n ) (Thas [J. A. Thas, Generalized quadrangles of order (s, s 2). I. J. Combin. Theory Ser. A 67 (1994), 140–160]). Hence by subtending one can obtain ovoids of Q (4, q n ) (consider the set of points in the subquadrangle collinear with a point not in the subquadrangle). Here we prove that all the ovoids subtended from points of type (ii) are isomorphic to ## add figure 'advg.5.3.333_01.gif' ##(ℱ ), and that in at least 2q n subGQ’s the ovoids subtended from points of type (i) are isomorphic to the ovoids subtended from points of type (ii).