Projective geometries PG(2,qρν) with subgeometry PG(2,qν), ρ>2

When ρ is composite, the design has more structure: this aspect is discussed here.

Following Carmichael, split the lines of the main geometry into 3 classes:

  • lines containing qν+1 points of the sub-geometry; called secants (denoted by s). Note that if a line of the main geometry contains 2 points of the sub-geometry it must contain qν+1 of them, by the rules of projective geometry. There are q2ν+qν+1 of these.
  • lines containing just one point of the sub-geometry: call these tangents (t). There are (q2ν+qν+1)(qρν-qν) of these.
  • lines containing no points of the sub-geometry: called exterior lines (x), (q2ρν+qρν+1) - (q2ν+qν+1)(qρν-qν+1) in number (simplifying to q3ν(q(ρ-1)ν-1)(q(ρ-2)ν-1)). When ρ=2, these do not exist.

The points can likewise be put into 3 classes, as follows:

  • the points of the sub-geometry, or arc, referred to as C0: these are q2ν+qν+1 in number
  • points of the main geometry, not belonging to the sub-geometry, but which occur on the secants: C1. There are (q2ν+qν+1)(qρν-qν) of these. Note that each such point can occur just once on the secants (otherwise a pair of lines would have 2 points in common).
  • points of the main geometry which do not occur on the secants. These are denoted by C2, and there are (q2ρν+qρν+1) - (q2ν+qν+1)(qρν-qν+1) or q3ν(q(ρ-1)ν-1)(q(ρ-2)ν-1) of these. When ρ=2, these do not exist.

The next bits of information refer to the numbers of each of one class of point on each class of line, and vice versa, using the matrix layout of Greig. The row borders refer to the point classes, and the column borders to the line classes. The entries in the body comprise a pair of numbers: the first is the number of points on a line, in the given categories, and the second gives the number of lines on a point, in the given categories.

Number of points per line, and number of lines per point, class by class
    q2ν+qν+1 (q2ν+qν+1)(qρν-qν) q3ν(q(ρ-1)ν-1)(q(ρ-2)ν-1)
    s t x
q2ν+qν+1 C0 qν+1,qν+1 1,qρν-qν 0,0
(q2ν+qν+1)(qρν-qν) C1 qρν-qν,1 q2ν,q2ν q2ν+qν+1,qρν-q2ν
q3ν(q(ρ-1)ν-1)(q(ρ-2)ν-1) C2 0,0 qρν-q2ν,q2ν+qν+1 qρν-q2ν-qν,qρν-q2ν-qν

The complete design above is a BIBD with λ=1.

An informal proof of the above is straightforward (and follows Carmichael). The number of secants and of points in C0 is given by the parameters of the sub-geometry, as are the number of points of the sub-geometry in those secants and their replication there. The number of C1 points (resp tangents), is the number of points (lines) in the sub-geometry times the difference in the replication (block size) between the full geometry and the sub-geometry. There is obviously just one point of the sub-geometry in a tangent, and none in the exterior lines. The number of exterior lines, and of C2 points, is given by difference. There are no C2 points on the secants, by definition, and the number of C1 points there is given by the difference in block sizes. These points can occur once only in the secants, otherwise two lines would have two points in common, as they already have one point in common in the sub-geometry.

So that gives numbers of all the constituent elements, and the first row and column of the matrix proper. The key to the rest of the matrix lies in the central element. Each point of the sub-geometry occurs (qρν-qν) times in the tangents, and has already occurred with all the other sub-geometry points and with (qν+1)(qρν-qν) points of C1. So there are q2ν(qρν-qν) such points left, giving q2ν per line. This is the maximum, since each tangent must have one point in common with each of the secants; qν+1 of these will have the same point of the sub-geometry as the given tangent, leaving q2ν points from C1.

Conversely, each point of C1 must occur with the q2ν points of the sub-geometry not in the secant it occurred in.

The remaining entries in the matrix are then obtained by difference, though they are all of some interest in their own right.

Thus, consider the number of points of C1 in an exterior line: each exterior line must have one point in common with each of the secants, in particular with the C1 points of the secants (since these are the only points in common). So there must be q2ν+qν+1 of them. Note that each of these transversals can have at most one point in common with each of the partial transversals represented by the C1 points of the tangents.

Given then the number of C2 points in the tangents and exterior lines, what is the relationship between them? For the moment, forget the C1 points, and suppose that the C2 points of the exterior line are S,T,...,U. These points cannot occur together in the tangents, and each occur q2ν+qν+1 times in the tangents: so that accounts for (q2ν+qν+1)(qρν-q2ν-q2ν) lines. This leaves (q2ν+qν+1)q2ν lines with which our given exterior line must have a point in common, not any of S,T,...,U. So this is where the C1 points must come back in: each of the (q2ν+qν+1) C1 points occurs q2ν times in the tangents, thus accounting for the remaining lines. In other words, given any exterior line, the tangents can be partitioned into sets according to the point held in common with the given exterior line.

A similar argument applies the other way round, obviously.

  1. The design obtained by dropping the points of the sub-geometry (but retaining the remainder of each of the secants and tangents, and all the exterior lines), is a PBD with (q2ρν+qρν-q2ν-qν) points and block sizes in {qρν+1, qρν, qρν-qν}.
  2. One could then add back one of the points of the sub-geometry (and its lines) to give a PBD with block sizes in {qρν+1, qρν, qρν-qν, qρν-qν+1}.
  3. One could go further, by adding back sets of such points, no 3 collinear (say a {0,1,2}-arc of the sub-geometry) to give PBDs with block sizes in {qρν+1, qρν, qρν-qν, qρν-qν+1, qρν-qν+2}.
  4. Also, one could "spike" to include a full line of sub-geometry points, giving a PBD with (q2ρν+qρν-q2ν+1) points with block sizes in {qρν+1, qρν, qρν-qν, qρν-qν+1}.
  5. Dropping a further point, on the other hand, gives rise to two different designs depending on whether it's from C1 or C2.
    • Dropping a point from C1, and all its lines, then the result is a GDD with block sizes in {qρν+1, qρν, qρν-qν} and one group of size (qρν-qν-1), q2ν groups of size (qρν-1), and (qρν-q2ν) groups of size qρν.
    • Dropping a point from C2, and all its lines, however, gives a GDD with block sizes in {qρν+1, qρν, qρν-qν}, while there are (q2ν+qν+1) groups of size (qρν-1) and (qρν-q2ν-qν) groups of size qρν.
  6. The design obtained by dropping all of the lines associated with the sub-geometry, the secants, as well as the points of the sub-geometry, is a GDD with the C2 points in groups of size 1, and the C1 points in q2ν+qν+1 groups of size (qρν-qν). The block sizes are in {qρν+1, qρν}.
  7. Going back to the original PBD, and dropping a line and all its points from it, gives another PBD. This is the Euclidean version of the geometry - the line dropped is the "line at infinity". Now it makes a difference as to which sort of line is deleted - in other words, how the line at infinity meets the sub-geometry.
    • If the line deleted is (what is left of) a secant, then the number of points left is (q2ρν-q2ν) points and the block sizes are in {qρν+1, qρν, qρν-1,qρν-qν}.
    • If the line deleted is (what is left of) a tangent, then the number of points left is (q2ρν-q2ν-qν) and the block sizes are in {qρν+1, qρν, qρν-1,qρν-1,qρν-qν,qρν-qν-1}.
    • Finally, if an exterior line is deleted, the number of points left is (q2ρν-qν-1), and the block sizes are in {qρν+1, qρν, qρν-1,qρν-qν}.
  8. Again, one can delete a point of any type from each of these to get a variety of GDDs.
  9. Conversely, one can add back points of the sub-geometry to get more PBDs.

A completely different set of PBDs can be obtained by taking the points of C0 and C1, or C0 and C2, or C1 or C2 individually (with all the appropriate lines). These can then be augmented, as was the PBD above, by adding back single points, or spiking to include several collinear points. Many of these are given in the applet here: this does not consider whether ρ is composite.


Last Updated on 11/07/2004
By D.H.Rees