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

Proposition 11.2.3: 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). In the dual setting, every three non-collinear points of DF is contained in an affine plane of points of DF of order pd.

Pf: Let K = GF(pd). Let A, B and C be three non-collinear points of DF. By the proof of Proposition 11.2.2 , we can take A = (0,0,0,1), B = (1,1,1,1) and C = (sa,s,sb,1) with s not in K (otherwise the points are collinear). Consider the set S of p2d points consisting of A and the points (ta,t,tb,1) where t = 1/u, (s-1)/u or (s+v-1)/uv with u and v in K\{0}. Notice that B and C are amongst these points. We define the lines to be the subsets of S which are collinear on a line of PG(3,q). We claim that the linear space S is an affine plane, necessarily of order pd. First, we note that all the points of S lie in the projective plane of PG(3,q) determined by A, B and C. The points of the line BC (in PG(3,q)), other than B have coordinates of the form (sa,s,sb,1) + (v-1)(1,1,1,1) = (sa+v-1, s+v-1,sb+v-1,v) for v in GF(q). With v restricted to K\{0}, these coordinates may be rewritten as ((s+v-1/v)a,(s+v-1/v),(s+v-1/v)b,1). The point of intersection, P, of BC with the plane W'=0 has coordinates ((s-1)a,s-1,(s-1)b,0). The points on the line joining A to ((s+v-1/v)a,(s+v-1/v),(s+v-1/v)b,1) have coordinates of the form ((s+v-1/v)a,(s+v-1/v),(s+v-1/v)b,1) + (u-1)(0,0,0,1) = ((s+v-1/v)a,(s+v-1/v),(s+v-1/v)b,u) for u in GF(q). Restricting u to K\{0}, gives the points of S of the form ((s+v-1/uv)a,(s+v-1/uv),(s+v-1/uv)b,1). Similarly, the points of AB in S are of the form (1/u,1/u,1/u,1) = ((1/u)a,1/u,(1/u)b,1), and the points of AP in S are of the form ((s-1/u)a,(s-1)/u,(s-1/u)b,1). Given two points of S, the line joining them in PG(3,q) contains pd points of DF by Proposition 11.2.1 , and by the proof of that proposition , they all lie in S. Hence, S contains p2d points, every two points lie on a unique line and each line contains exactly pd points. By Dembowski, Theorem 3.2.4(b) [De67] this is sufficient to show that S is an affine plane of order pd.