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The ψ-associate X # of a flat X in PG(n, 2)

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Abstract

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 # = {z ∈ PG(n, 2) | T(x, y, z) = 0 for all x, yX}, 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 \({\mathcal{S}} _{1,2,2} \subset {\rm PG}(5, 2)\) it is shown that the 48 planes external to \({\mathcal{S}}_{1,2,2}\) fall into eight pairs of ordered triplets {(P 1, R 1, S 1), (P 2, R 2, S 2)} such that \(\psi^{\text{c}} =P_{1}\cup R_{1}\cup S_{1}\cup P_{2}\cup R_{2}\cup S_{2}\) and \(P_{i}^{\#}=R_{i}, R_{i}^{\#}=S_{i}, S_{i}^{\#}=P_{i}, i=1, 2\) . 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 \(W \in \Sigma_{5}\) is singled out by the property that every line \(L\subset W\) is singular.

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Correspondence to Ron Shaw.

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Communiacted by G. Lunardon.

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Shaw, R. The ψ-associate X # of a flat X in PG(n, 2). Des. Codes Cryptogr. 45, 229–246 (2007). https://doi.org/10.1007/s10623-007-9116-z

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  • DOI: https://doi.org/10.1007/s10623-007-9116-z

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