Monomial Flocks and Herds Containing a Monomial Oval

AbstractLet F be a flock of the quadratic cone Q:X22=X1X3, inPG(3, q),qeven, and letΠt:X0=xtX1+t1/2X2+ztX3,t∈Fq, be theqplanes defining the flock F. A flock is equivalent to a herd of ovals inPG(2, q),qeven, and to a flock generalized quadrangle of order (q2, 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 subquadranglesT2(O) in the corresponding flock generalized quadrangle. To obtain this result, we prove that ifxtandztboth are monomial functions of t, then the flock is either the linear, FTWKB-, or PayneP1flock. This latter result implies, in the even case, the classification of regular partial conical flocks, as introduced by Johnson