On the Characterization of certain Sets of Points in Finite Projective Geometry of Dimension Three

Let PG {d, q) be the projective geometry of dimension d over the finite field GF(q) of q elements. A subset of points of PG{d,q) is said to be of type (1, n, q + l) if every line meets it in 1, n or q +1 points. In [6], Tallini Scafati classified all subsets of type (I, n, q+l) in PG(d, q) for d ^ 2, q > 4, except for the case n = \q + l. This case was completed by Hirschfeld and Thas, (see [2] and [3]), except for a certain set Kt in PG (3, q), q even. The problem for d = 3, q = 4 was solved by Hirschfeld and Hubaut, (see [1]), and in [5], Sherman gave an algebraic solution to the remaining problem of sets of type (1, 3, 5) in PG(d, 4) for d ^ 4. In this note, the set Kx of Hirschfeld and Thas is shown to be the projection of a non-singular quadric of PG (4, q). This confirms the conjecture made at the end of [3]. We shall assume the results and notation of [2] and [3]. In this paper q is a power of two.