Tetrads of lines spanning $\operatorname*{PG}(7,2)$

Abstract

Our starting point is a very simple one, namely that of a set $\mathcal{L} _{4}$ of four mutually skew lines in $\operatorname*{PG}(7,2).$ Under the natural action of the stabilizer group $\mathcal{G}(\mathcal{L}_{4} )<\operatorname*{GL}(8,2)$ the $255$ points of $\operatorname*{PG}(7,2)$ 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 $\mathcal{H}_{7}$ determined by $\mathcal{L}_{4},$ and that the $81$-set $\omega_{4}$ (which is shown to have a sextic equation) is an orbit of a normal subgroup $\mathcal{G} _{81}\cong(Z_{3})^{4}$ of $\mathcal{G}(\mathcal{L}_{4}).$ There are $40$ subgroups $\cong(Z_{3})^{3}$ of $\mathcal{G}_{81},$ and each such subgroup $H<\mathcal{G}_{81}$ gives rise to a decomposition of $\omega_{4}$ into a triplet $\{\mathcal{R}_{H},\mathcal{R}_{H}^{\prime},\mathcal{R}_{H} ^{\prime\prime}\}$ of $27$-sets. We show in particular that the constituents of precisely $8$ of these $40$ triplets are Segre varieties $\mathcal{S} _{3}(2)$ in $\operatorname*{PG}(7,2).$ This ties in with the recent finding 225-239 --- that each $\mathcal{S}=\mathcal{S}_{3}(2)$ in $\operatorname*{PG} (7,2)$ determines a distinguished $Z_{3}$ subgroup of $\operatorname*{GL} (8,2)$ which generates two sibling copies $\mathcal{S}^{\prime},\mathcal{S} ^{\prime\prime}$ of $\mathcal{S}.$