Aspects of the Segre variety S-1,S-1,S-1(2)
We consider various aspects of the Segre variety S := S-1,S-1,S-1(2) in PG(7, 2), whose stabilizer groupGS < GL(8, 2) has the structure N x Sym(3), whereN := GL(2, 2) x GL(2, 2) xGL(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. GS, and in consequence S determines a GS-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties S-1,S-1,S-1(2) in PG(7, 2) come along in triplets {S, S', S ''} which share the same distinguished Z3-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).
segre variety s-1,s-1,s-1(2) invariant polynomials line-spread