Tetrads of lines spanning PG(7, 2)

Our starting point is a very simple one, namely that of a set L4 of four mutually skew lines in PG(7, 2). Under the natural action of the stabilizer group G(L4) < GL(8, 2) the 255 points of PG(7, 2) fall into four orbits ω1, ω2, ω3, ω4, of respective lengths 12, 54, 108, 81. We show that the 135 points ∈ ω2 ∪ ω4 are the internal points of a hyperbolic quadric H7 determined by L4, and that the 81-set ω4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G81 ∼= (Z3) of G(L4). There are 40 subgroups ∼= (Z3) of G81, and each such subgroup H < G81 gives rise to a decomposition of ω4 into a triplet {RH ,RH ,R′′ H} of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S3(2) in PG(7, 2). This ties in with the recent finding that each S = S3(2) in PG(7, 2) determines a distinguished Z3 subgroup of GL(8, 2) which generates two sibling copies S ′,S ′′ of S. MSC2010: 51E20, 05B25, 15A69