MONOMIAL FLOCKS AND HERDS CONTAINING A MONOMIAL OVAL

Let F be a flock of the quadratic cone D: X-2(2) = X-1 X-3, in PG(3, q ), q even, and let Pi(t): X-0 = x(t)X(1) + t(1/2)X(2) + z(t)X(3), t is an element of F (q), be the q planes defining the flock F. A flock is equivalent to a herd of ovals in PG(2, q), q even, and to a flock gen eralized quadrangle of order (q(2), g). We show that if the herd conta ins a monomial oval, this oval is the Segre oval. This implies a resul t on the existence of subquadrangles T-2(phi) in the corresponding flo ck generalized quadrangle. To obtain this result, we prove that if x(t ) and z(t) both are monomial functions of t, then the flock is either the linear, FTWKB-, or Payne P-1 flock. This latter result implies, in the even case, the classification of regular partial conical flocks, as introduced by Johnson. (C) 1998 Academic Press.