Projective geometries PG(k,qρν) with subgeometry PG(k,qν), k>2, ρ=2
These are the Baer sub-geometries: see the CRC Handbook of Designs VI.7.8,
Beth, Jungnickel & Lenz VIII.9.13 and the references therein.
The same approach is used as when k=2, namely, to classify the points and lines, and
then to consider removing parts of the geometry to get other kinds of designs.
The starting point is a BIBD with λ=1, obtained by considering just the lines and points
of the geometry. There are (q2(k+1)ν-1)/(q2ν-1)
points, ((q2(k+1)ν-1)(q2kν-1))/((q4ν-1)(q2ν-1))
lines, (q2ν+1) points on a line, and (q2kν-1)/(q2ν-1)
lines on a point.
There are secants (s), tangents (t) and exterior lines (x) as before, but no C2 points.
(The number of lines in the sub-geometry times the difference in the line
sizes = the difference in the number of treatments, so C1 is sufficient,
for all k.) Every point, not in the sub-geometry, appears exactly
once on a secant: all its other concurrences with points of the sub-geometry
must be in tangents. So it is not possible to split the points of C1 into
those which occur on tangents, and those which do not.
From these structures can be derived various geometries of lower dimension which
have subgeometries different in kind: for example, in the 3-dimensional geometry with ρ=2,
there are two types of plane.
The first sort are planes which contain the whole subplane corresponding: these constitute the "standard" case,
which have been considered in lower dimensions, and which I shall refer to as being of type B0.
The second sort are planes which contain only one line of the subgeometry: these are said to be of type B1.
Likewise, 3-dimensional geometries may contain just 2- or 1-dimensional subgeometries, and
so forth. However, these subspaces still possess structure, since they are composed of
the secants, tangents and exterior lines defined in the full geometry. Some of these possibilities
are considered on accompanying web-pages.
The matrix showing the numbers of points per line, and vice versa, class by class is:
Number of points per line, and number of lines per point, class by class
| |
|
((q(k+1)ν-1)(qkν-1))/((q2ν-1)(qν-1)) |
((qν(q(k+1)ν-1)(qkν-1)(q(k-1)ν-1))/((q2ν-1)(qν-1)) |
((q4ν(q(k+1)ν-1)(qkν-1)(q(k-1)ν-1)(q(k-2)ν-1))/((q4ν-1)(q2ν-1)) |
| |
|
s |
t |
x |
| (q(k+1)ν-1)/(qν-1) |
C0 |
qν+1,(qkν-1)/(qν-1) |
1,(qν(qkν-1)(q(k-1)ν-1))/(q2ν-1) |
0,0 |
| qν(q(k+1)ν-1)(qkν-1)/(q2ν-1) |
C1 |
q2ν-qν,1 |
q2ν,(q2ν(q(k-1)ν-1))/(qν-1) |
q2ν+1,(q3ν(q(k-1)ν-1)(q(k-2)ν-1))/(q2ν-1) |
The complete design above is a BIBD with λ=1.
The proof is straightforward and all the elements of the table can
be written down from the definitions or after a little arithmetic. Note
that the number of exterior lines is zero if k=2.
- Dropping the points of the sub-geometry, but retaining the rest of all the lines, gives a PBD
for (qν(q(k+1)ν-1)(qkν-1))/(q(2ν-1)
points with block sizes in {q2ν - qν,q2ν,q2ν+1}.
- As before, one could then add back one or more points of the sub-geometry
to get more PBDs. In particular, if a secant is spiked to its full size
by adding back its points of the sub-geometry, a PBD is given with
qν{(q(k+1)ν-1)(qkν-1)+(q2ν-1)(qν-1))}/(q(2ν-1)
points, and block sizes in
{q2ν+1,q2ν - qν+1,q2ν}
- Alternatively, dropping a further point, now necessarily from C1,
and all its lines, would give a GDD with points and with the same block
sizes, one group of size (q2ν - qν-1),
q2ν(q(k-1)ν-1)/(qν-1)
groups of size q2ν-1, and the remaining groups
of size q2ν.
- If the whole of the secants are deleted, as well as the points of the sub-geometry, then a
GDD is obtained with blocksizes in {q2ν, q2ν+1},
and with
((q(k+1)ν-1)(qkν-1))/((q2ν-1)(qν-1))
groups of size (q2ν-qν).
An applet gives details of these designs
for specified values of q, ν and k.
Last Updated on 05/05/2004
By D.H.Rees