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

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 # = {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.