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

The conjectured structure previously given here was incorrect.

The following are examples of simple cases: even the smallest realistic example (ρ=9) would require enormous computing capacity.

PG(2,24)
(a) Geometry PG(2,2) within PG(2,22) within PG(2,24)
    7 14 84 168
    s t1 t2 x
7 C0 3,3 1,2 1,12 0,0
14 C1 2,1 4,4 0,0 1,12
84 C20 12,1 0,0 4,4 6,12
168 C21 0,0 12,1 12,6 10,10
PG(2,34)
(a) Geometry PG(2,3) within PG(2,32) within PG(2,34)
    13 78 936 5616
    s t1 t2 x
13 C0 4,4 1,6 1,72 0,0
78 C1 6,1 9,9 0,0 1,72
936 C21 72,1 0,0 9,9 12,72
6936 C22 0,0 72,1 72,12 69,69
PG(2,26)
(a) Geometry PG(2,2) within PG(2,22) within PG(2,26)
    7 14 420 840 840
    s t1 t2 x1 x2
7 C0 3,3 1,2 1,60 0,0 0,0
14 C1 2,1 4,4 0,0 1,60 0,0
420 C20 60,1 0,0 4,4 6,12 7,48
840 C21 0,0 60,1 12,6 10,10 14,48
2880 C22 0,0 0,0 48,7 48,14 44,44
PG(2,26)

Note that tangents and external lines are in a different order to the usual, as are point classes.

(a) Geometry PG(2,2) within PG(2,23) within PG(2,26)
    7 42 24 392 2352 1344
    s t1 x1 t2 x2 x3
7 C0 3,3 1,6 0,0 1,56 0,0 0,0
42 C10 6,1 4,4 7,4 0,0 1,56 0,0
24 C20 0,0 4,7 2,2 0,0 0,0 1,56
392 C11 56,1 0,0 0,0 4,4 6,36 7,24
2352 C21 0,0 56,1 0,0 36,6 38,38 35,20
1344 C22 0,0 0,0 56,1 24,7 20,35 22,22


Last Updated on 03/10/2005
By D.H.Rees