Definitions: Flock, Critical Cone, Baselines, Herd Space, Herd, Semi-Field Type Flock


Theorem 11.2.4: 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 the critical cone of F is thick.

Pf: In the dual setting, the critical cone of a Flock is represented by the largest set of lines of W' = 0 which do not pass through any of the points where lines joining two points of DF meet W' = 0. In order for the cone to be thick, there should be no set of less than [q+2/2] collinear points in W'=0 through which all the lines of DG must pass. Alternatively, no set of at least q+1 - [q+2/2] = [q+1/2] collinear points of W'=0, through which no DG line can pass. Let S be the set of points of W'=0 where lines joining two points of DF meet W'=0. By Proposition 11.2.5 [back], there are (q-1)/(pd -1) < q + 1 points in S, and so, through each point of W'=0 not in S there must pass lines which do not intersect S. A line which intersects S in at least two points must intersect it in pd + 1 points. Notice that we must have d < n/2, otherwise F is a star flock. It now follows that since pd + 1 < [pn + 1/2], any set of collinear points of S has fewer than [q+1/2] points and so the critical cone of F must be thick.


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