Definitions: Flock, Monomial Flock, Dual Setting, Baselines, Star Flock, Semi-Field Type Flock


Proposition 11.2.1: Let F = F(ta, t, tb) be a monomial semi-field type flock in PG(3,q) with q = pn for some prime p, a = pi, b = pj with 0 < i < j < n. Let d = gcd(i,j,n). Then, F is not a star flock, and there are exactly pd planes of the flock which pass through each primary baseline (including the carrier plane). Dually, all primary baselines contain exactly pd points of DF.

Pf: Under the assumption that 0 < i < j < n, it follows that the functions ta, t, and tb are linearly independent over GF(q). Thus, by Theorem 7.1 [backbutton], F is not a star flock.

In the dual setting, the points of DF have coordinates (ta, t, tb,1) for t in GF(q). Consider the line joining (0,0,0,1) and (ra, r, rb,1) for some fixed r in GF(q)\{0}. The points of this line, other than (0,0,0,1) have coordinates of the form (ra, r, rb, 1 +) as ranges over GF(q). When = -1, the point of this line is in W' = 0 and so, can not be in DF. We may thus assume that -1. For a point (sa, s, sb,1) of DF to be on this line, we must have s = r/1+, sa = ra/1+ and sb = rb/1+. These equations are satisfied if, and only if, (1+)a-1 = 1 and (1+)b-1 = 1. Therefore, the order of the element 1+ in the multiplicative group GF(q)\{0} must divide the gcd(a-1, q-1) = gcd(pi-1, pn-1) = pgcd(i,n)-1, and gcd(b-1,q-1) = pgcd(j,n)-1. Hence, the order of this element must divide gcd(pgcd(i,n)-1, pgcd(j,n)-1) = pd - 1, and so, 1+ is an element of the subfield GF(pd) of GF(q). Conversely, any element of this subfield will satisfy the conditions. Thus, there are pd -1 points of DF on this line corresponding to non-zero values of 1+, and together with (0,0,0,1) we get pd points on every primary baseline.


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