Definitions: Flock, Ganley type flock, Quadratic Cone, Semi-Field Type Flock

Proposition 11.2.6: A Ganley type flock is a flock of a quadratic cone.

Pf: Let q = 3n, and F = F(t3,t,t9). To show that this flock is a flock of a quadratic cone, we must find a conic in the complement of the baselines of the flock. The baselines of F are the lines in W = 0 satisfying the equations t3X + tY + t9Z = 0 for t0 in GF(q). Consider the conic in W = 0 given by X2 + mYZ -m3Z2 = 0, where m is any non-square in GF(q). The conic is in the complement of the baselines if, and only if, X2 + m(-t2X -t8Z)Z -m3Z2 = 0 has no non-trivial (i.e., X = Z = 0) solutions in GF(q). This quadratic equation has no non-trivial solutions provided (-mt2)2 - 4(-mt8 -m3) is a non-square for all t0. Now,

(-mt2)2 - 4(-mt8 -m3) = m2t4 + mt8 + m3
= m(t8 + mt4 + m2) = m(t8 -2mt4 + m2) = m(t4 - m)2,
since 1 = -2 in fields of characteristic 3. Since m is a non-square, m(t4 - m)2 is a non-square for all t in GF(q), and F is a flock of a quadratic cone.