On Invariant Notions of Segre Varieties in Binary Projective Spaces

TL;DR

This paper investigates invariant geometric structures of Segre varieties in binary projective spaces, revealing their connections with quadrics, spreads, and Hermitian varieties, and providing detailed analysis for the case when m=3.

Contribution

It introduces invariant notions of Segre varieties in binary projective spaces and constructs invariant bases, spreads, and varieties related to their stabilizer groups.

Findings

Existence of an invariant hyperbolic quadric containing the Segre varieties.

Construction of an invariant basis in an extended projective space.

Classification of orbits of points and lines under the stabilizer group for m=3.

Abstract

Invariant notions of a class of Segre varieties $\Segrem(2)$ 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 $\Segrem(2)$ and is invariant under its projective stabiliser group $\Stab{m}{2}$. 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 $\Stab{m}{2}$ 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 $\Stab{m}{2}$-invariant geometric spread of lines of PG(2^m - 1, 2). This spread is also related with a $\Stab{m}{2}$-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 $\Stab{3}{2}$, while the points of PG(7, 2) form five orbits.