Embeddings and hyperplanes of discrete geometries
European Journal of Combinatorics
The genus of the GRAY graph is 7
European Journal of Combinatorics - Special issue: Topological graph theory II
Geometrical Structure of Entangled States and the Secant Variety
Quantum Information Processing
The cubic Segre variety in PG(5, 2)
Designs, Codes and Cryptography
Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$
Designs, Codes and Cryptography
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Invariant notions of a class of Segre varieties $${\mathcal{S}_{(m)}(2)}$$ of PG(2 m 驴 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 $${\mathcal{S}_{(m)}(2)}$$ and is invariant under its projective stabiliser group $${G_{{\mathcal{S}}_{(m)}(2)}}$$ . By embedding PG(2 m 驴 1, 2) into PG(2 m 驴 1, 4), a basis of the latter space is constructed that is invariant under $${G_{{\mathcal{S}}_{(m)}(2)}}$$ 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 $${G_{{\mathcal{S}}_{(m)}(2)}}$$ -invariant geometric spread of lines of PG(2 m 驴 1, 2). This spread is also related with a $${G_{{\mathcal{S}}_{(m)}(2)}}$$ -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 $${G_{{\mathcal{S}}_{(m)}(2)}}$$ , while the points of PG(7, 2) form five orbits.