On Invariant Notions of Segre Varieties in Binary Projective Spaces

Invariant notions of a class of Segre varieties $\text{Segrem}(2)$ 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 $\text{Segrem}(2)$ and is invariant under its projective stabiliser group $\text{Stab}m2$. 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 $\text{Stab}m2$ 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 $\text{Stab}m2$-invariant geometric spread of lines of PG(2^m

• 1, 2). This spread is also related with a $\text{Stab}m2$-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 $\text{Stab}32$, while the points of PG(7,

2. form five orbits.