The lines of PG(4, 2) are the points on a quintic in PG(9, 2)
Journal of Combinatorial Theory Series A
European Journal of Combinatorics
Configurations of planes in PG(5,2)
Discrete Mathematics
The Quintic Grassmannian G1, 4, 2 in PG(9, 2)
Designs, Codes and Cryptography
Designs, Codes and Cryptography
The Polynomial Degree of the Grassmannian $${\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}$$
Designs, Codes and Cryptography
The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)
Designs, Codes and Cryptography
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
| Hi-index | 0.00 |
For a given hypersurface ψ in PG(n, 2), with equation Q(x) = 0, where Q is a polynomial of (reduced) degree d 1, a definition is given of the ψ-associate X # of a flat X in PG(n, 2). The definition involves the fully polarized form of the polynomial Q; in the cubic case d = 3 it reads: X # = {zn, 2) | Tx, y, z) = 0 for all x, y}, where T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing Q. Some general results, valid for any degree d and projective dimension n, are first expounded. Thereafter several choices of ψ are visited, but for each choice just a few aspects are highlighted. Despite the partial nature of the survey quite a variety of behaviours of the ψ-associate are uncovered. Many of the choices of ψ which are considered are of cubic hypersurfaces in PG(5, 2). If ψ is the Segre variety it is shown that the 48 planes external to fall into eight pairs of ordered triplets {(P 1, R 1, S 1), (P 2, R 2, S 2)} such that and . Further those lines L of PG(5, 2) which are singular, satisfying that is L # = PG(5.2), are in this case shown to form a complete spread of 21 lines. Another result of note arises in the case where ψ is the underlying 35-set of a non-maximal partial spread Σ5 of five planes in PG(5, 2), where it is shown that one plane is singled out by the property that every line is singular.