Theorem 11.2.1: A flock F of a non-empty cone is of semi-field type if, and only if, it has a normalized representation as F(f(t),t,h(t)) with f and h additive functions over GF(q).
If q is prime, then the only additive functions over GF(q) are linear or the zero function. Since we require that f(0) = 0, it follows by Proposition 8.1.6 that the flock must be linear, which we record as:
Corollary 11.2.1: If q is a prime, then any semi-field type flock of a non-empty cone is linear.
It does not require much more effort to show that:
Corollary 11.2.2: If q = p2 for some prime p, then any semi-field type flock of a non-empty cone is a star flock.
This corollary was proved for quadratic cones by both Lunardon [Lu??] and Thas [Th93b] and we note how easily the result follows from a consideration of the herd space.
Star flocks of the form F = F(t,g(t),0) where g is a pe-linearized polynomial also provide examples of semi-field type flocks.
A monomial semi-field type flock is a semi-field type flock F = F(f(t),t,h(t)) where f and h are monomial additive functions, that is, if q = pn for some prime p, then f(t) = Ata and h(t) = Btb where a = pi and b = pj, 0i,j < n. Clearly, if either of i or j is 0, or if i = j, or if either of A or B is 0, then the flock is a star flock.
By passing to a projectively equivalent flock , we can assume that A = B = 1 and that a < b. To examine these flocks in more detail we will turn once again to the dual setting and assume that the semi-field type flock is not a star flock, so, not all the points of DF lie in a plane.
We first consider how primary and secondary baselines are represented in the dual setting. A primary baseline is the intersection of the plane W = 0 with one of the other planes of the flock, so in the dual setting this is a line joining the point (0,0,0,1) with one of the other points of DF. A secondary baseline is the projection into W = 0 of the line of intersection of two flock planes (which is not in W = 0) from the vertex V. Thus, it is the intersection of the plane determined by V and the intersection line, with W = 0. This plane in the dual setting is a point in W' = 0 which is on a line determined by two points of DF which does not pass through (0,0,0,1), and the secondary baseline is therefore the line joining this point with (0,0,0,1).
In a semi-field type flock, all secondary baselines coincide with primary baselines, so the secondary baselines in the dual setting all contain at least one point of DF other than (0,0,0,1). Monomial semi-field type flocks possess a great deal of uniformity, and the structure of the set of DF points can be determined, as the following propositions show.
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.
Proposition 11.2.2: With the notation of the above proposition, 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.
Proposition 11.2.3: With the notation of the above propositions, in the dual setting, every three non-collinear points of DF is contained in an affine plane of points of DF of order pd.
These properties permit us to use a remarkable result of F. Buekenhout, which states:
Theorem 11.2.2: A non-trivial linear space with more than three points on some line is either an affine plane or the point-line design of an affine space, AG(n,q), for some n and q if and only if every triangle is contained in an affine plane. [Bu69]
This result is sharp in that if all lines contain either 2 or 3 points then there are linear spaces satisfying the condition of the theorem which are not affine spaces. In a linear space containing points, lines and planes, if two lines are in a common plane and do not meet we say that they are parallel. This relation between lines is clearly reflexive and symmetric. In a non-trivial linear space in which every triangle is contained in an affine plane, if the relation of parallelism is an equivalence relation (i.e., it is also transitive) then the linear space is an affine space.
Proposition 11.2.4: With the notation of the above propositions, in the dual setting, the relation of parallelism on the lines containing at least two points of DF is an equivalence relation.
Consequently, we have:
Theorem 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), and n' = n/d. Then the points DF and lines joining pairs of them are the points and lines of an AG(n',pd).
The structural properties of the points of DF permit us to determine the configuration of points where the primary lines of the dual flock intersect W' = 0. In turn, this will allow us to examine the possibilities for the dual generator lines DG which must avoid these points.
Using a theorem of J.M.N. Brown [Br88] concerning embeddings of higher dimensional projective spaces into lower dimensional ones, we can show:
Proposition 11.2.5: Using the notation of Theorem 11.2.3, the configuration of points in W' = 0 which are the intersections of the lines of DF with W' = 0 is an embedded PG(n'-1,pd). Furthermore, if n 3d, then this configuration is not contained in a line. Also, the points of W' = 0 off of the three lines X' = 0, Y' = 0 and Z' = 0 can be partitioned into disjoint copies of the embedded PG(n'-1,pd).
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 wide.
We will now turn our attention to monomial semi-field flocks of quadratic cones. In order to show that a flock of a wide cone is a flock of a quadratic cone, we must be able to find a conic contained in the set of points associated to the herd of the flock. The only proper star flocks in this category are, by Theorem 11.1.6 , the Kantor K1 type flocks, in odd characteristic. The other known examples, to be defined below, are in characteristic 3, the family of Ganley type flocks and the sporadic Penttila-Williams flock in PG(3,35). That there are no semi-field type flocks of quadratic cones in even characteristic is a well known result of N. Johnson.
Theorem 11.2.5: All semi-field type flocks of quadratic cones in even characteristic are linear. [Jo87]
However, we note that there are semi-field type flocks (monomial) of wide cones in even characteristic which we will investigate below. Johnson's result then implies that these flocks will give rise to more conic blocking sets in even characteristic.
For q = 3n, a Ganley type flock is one of the form F = F(t3,t,t9). For q = 3, this is a linear flock (t3 = t9 = t), and for q = 9 it is a proper star flock (t9 = t) , which is a Kantor K1 flock.
Proposition 11.2.6: A Ganley type flock is a flock of a quadratic cone.
For q = 35, the Penttila-Williams flock is given by F = F(t3,t,t27).
Proposition 11.2.7: The Penttila-Williams flock in PG(3,35) is a flock of a quadratic cone.