2010/06/30 by Hans Havlicek, Boris Odehnal, Metod Saniga +1
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Invariant (physics) #Linear subspace #Mathematical physics #Mathematics #Polynomial and algebraic computation #Projective space #Projective test #Pure mathematics #Tensor decomposition and applications #math-ph #math.AG #math.MP
paper · pdf · doi:10.1007/s10623-011-9525-x
published as Designs, Codes and Cryptography 62 (2012) 343-356 · 18 pages, 1 figure; v2 - version accepted in Designs, Codes and Cryptography
arxiv created 2011/03/31 · openalex publication_date 2011/06/03 · arxiv updated 2012/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Invariant notions of a class of Segre varieties \Segrem(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 \Segrem(2) and is invariant under its projective stabiliser group \Stabm2. By embedding PG(2m - 1, 2) into \PG(2m - 1, 4), a basis of the latter space is constructed that is invariant under \Stabm2 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 \Stabm2-invariant geometric spread of lines of PG(2m - 1, 2). This spread is also related with a \Stabm2-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 \Stab32, while the points of PG(7, 2) form five orbits.