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


Proposition 11.2.2: 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 through every line of intersection of two planes of F there pass exactly pd planes of F. Dually, all lines containing two points of DF contain exactly pd points of DF.

Pf: We shall work in the dual setting. To prove this proposition, we will exhibit a group of collineations of PG(3,q) which stabilize the plane W'=0 and the set DF which acts doubly transitively on DF. Thus, any two distinct points of DF can be mapped to any other pair of distinct points, so any line containing two points of DF can be mapped to any other line containing two points of DF. Therefore, every line containing two points of DF must contain the same number of DF points, which by Proposition 11.2.1 [back] is pd.

Consider the group G of collineations generated by the matrices:

where alpha,beta in GF(q), alphanot equal0. Clearly, the collineations of G stabilize W' = 0. Since (ta,t,tb,1)M(alpha,beta) = ((alphat+beta)a,(alphat+beta),(alphat+beta)b,1), we see that the set DF is stabilized by G. Finally, to show that G acts doubly transitively on DF, we need only show that two specific points can be mapped to any two distinct points of DF. The collineation which maps (0,0,0,1) to (ta,t,tb,1) and (1,1,1,1) to (sa,s,sb,1) with tnot equals is given by M(s-t,t).


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