Tetrads of lines spanning PG(7,2)
Abstract
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 omega1, omega2, omega3 omega4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega2 omega4 are the internal points of a hyperbolic quadric H7 determined by L4; and that the 81-set omega4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G81 isomorphic to (Z3)4 of G(L4): There are 40 subgroups (isomorphic to (Z3)3) of G81; and each such subgroup H < G81 gives rise to a decomposition of omega4 into a triplet 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 Segre 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.