On invariant notions of Segre varieties in binary projective spaces

Invariant notions of a class of Segre varieties S(m)(2) of PG(2m − 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 GS(m)(2). By embedding PG(2 m − 1, 2) into PG(2m − 1, 4), a basis of the latter space is constructed that is invariant under GS(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 GS(m)(2)-invariant geometric spread of lines of PG(2 m − 1, 2). This spread is also related with a GS(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 GS(3)(2), while the points of PG(7, 2) form five orbits. Mathematics Subject Classification (2010): 51E20, 05B25, 15A69