Monomial Flocks and Herds Containing a Monomial Oval

Let F be a flock of the quadratic cone Q: X 2=X1X3 , in PG(3, q), q even, and let 6t : X0 =xt X1 + t X2 +zt X3 , t # Fq , 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 generalized quadrangle of order (q, q). We show that if the herd contains a monomial oval, this oval is the Segre oval. This implies a result on the existence of subquadrangles T2 (O) in the corresponding flock generalized quadrangle. To obtain this result, we prove that if xt and zt both are monomial functions of t, then the flock is either the linear, FTWKB-, or Payne P1 flock. This latter result implies, in the even case, the classification of regular partial conical flocks, as introduced by Johnson. 1998