On invariant notions of Segre varieties in binary projective spaces (original) (raw)

Invariant notions of a class of Segre varieties 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 S (m) (2) and is invariant under its projective stabiliser group G 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 S (m) (2) as well. Such a basis can be split into two subsets of an odd and even parity 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 S (m) (2)-invariant geometric spread of lines of PG(2 m − 1, 2). This spread is also related with a G 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 S (3) (2) , while the points of PG(7, 2) form five orbits.