Definitions: Flock, Penttila-Williams Flock, Quadratic Cone, Semi-Field Type Flock


Proposition 11.2.7: The Penttila-Williams flock in PG(3,35) is a flock of a quadratic cone.

Pf: Let q = 35, and F = F(t3,t,t27). 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 + t27Z = 0 for t0 in GF(q). Consider the conic in W = 0 given by X2 + YZ = 0. The conic is in the complement of the baselines if, and only if, X2 + (-t2X -t26Z)Z = 0 has no non-trivial (i.e., X = Z = 0) solutions in GF(q). This quadratic equation has no non-trivial solutions provided (-t2)2 - 4(-t26) is a non-square for all t0. Now,

(-t2)2 - 4(-t26) = t4 + t26 = t4(t22 + 1).
In GF(35), the polynomial t22 + 1 takes only non-square values for all t0, thus the expression above is a non-square for all t0 and F is a flock of a quadratic cone.


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