Aspects of the Segre variety S_{1,1,1}(2)

TL;DR

This paper explores the structure and symmetries of the Segre variety S_{1,1,1}(2) in PG(7,2), revealing invariant groups, spreads, and polynomial functions related to its geometric and algebraic properties.

Contribution

It identifies a distinguished Z_3-subgroup and G_S-invariant spread, and classifies all low-degree G_S-invariant polynomials on PG(7,2).

Findings

S determines a G_S-invariant spread of 85 lines.

S, S', S'' form triplets sharing the same Z_3-subgroup.

All 15 G_S-invariant polynomials of degree <8 are classified.

Abstract

We consider various aspects of the Segre variety S := S_{1,1,1}(2) in PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N {\rtimes} Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{-1} = Z, for all A in G_S, and in consequence S determines a G_S-invariant spread of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{1,1,1}(2) in PG(7,2) come along in triplets {S,S',S"} which share the same distinguished Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant polynomial functions on PG(7,2) which have degree < 8, and their relation to the five G_S-orbits of points in PG(7,2).