On Invariant Notions of Segre Varieties in Binary Projective Spaces

Hans Havlicek (1) , Boris Odehnal (1) , Metod Saniga (2)

1 TUW - Institut fuer Diskrete Mathematik und Geometrie
2 Astronomical Institute of the Slovak Academy of Sciences

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 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 $\Stab{m}{2}$-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.

Mots clés: Segre Variety; Binary Projective Space

Domaines: Géométrie algébrique [math.AG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Hans Havlicek, Boris Odehnal, Metod Saniga. On Invariant Notions of Segre Varieties in Binary Projective Spaces. Designs, Codes and Cryptography, 2012, pp.343-356. ⟨10.1007/s10623-011-9525-x⟩. ⟨hal-00494384⟩