Tetrads of lines spanning PG(7,2)

TL;DR

This paper explores the geometric and group-theoretic structure of four mutually skew lines in PG(7,2), revealing orbit decompositions, quadric properties, and connections to Segre varieties and subgroup actions.

Contribution

It provides a detailed analysis of the orbit structure and subgroup actions related to four skew lines in PG(7,2), linking geometric configurations to algebraic group properties and Segre varieties.

Findings

Points in certain orbits form the internal points of a hyperbolic quadric.

The 81-set omega_4 has a sextic equation and relates to a (Z_3)^4 subgroup.

Eight triplets of decompositions correspond to Segre varieties S_3(2).

Abstract

Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.