On invariant notions of Segre varieties in binary projective spaces

Abstract

Invariant notions of a class of Segre varieties ${\mathcal{S}_{(m)}(2)}$ of PG(2^m − 1, 2) that are direct products of m copies of PG(1, 2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains ${\mathcal{S}_{(m)}(2)}$ and is invariant under its projective stabiliser group ${G_{{\mathcal{S}}_{(m)}(2)}}$ . By embedding PG(2^m − 1, 2) into PG(2^m − 1, 4), a basis of the latter space is constructed that is invariant under ${G_{{\mathcal{S}}_{(m)}(2)}}$ as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a ${G_{{\mathcal{S}}_{(m)}(2)}}$ -invariant geometric spread of lines of PG(2^m − 1, 2). This spread is also related with a ${G_{{\mathcal{S}}_{(m)}(2)}}$ -invariant non-singular Hermitian variety. The case m = 3 is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under ${G_{{\mathcal{S}}_{(m)}(2)}}$ , while the points of PG(7, 2) form five orbits.