A combinatorial characterization of classical unitals

Abstract

In this paper we give a characterization of classical unitals in terms of a configuration pattern formed by the feet of a unital U embedded in PG(2, q 2 ), q > 2. We show that a necessary and sufficient condition for U to be classical is the existence of two points p 0 , p 1 ∈ U with tangent lines L 0 and L 1 , respectively, such that for all points r ∈ L 0 \ { p 0 } and s ∈ L 1 \ { p 1 } the corresponding feet are collinear. Introduction and preliminary results.A unital or Hermitian arc in PG(2, q 2 ), q any prime power, is a set U of q 3 + 1 points such that every line of PG(2, q 2 ) meets U in either 1 or q + 1 points. We call a line a tangent or a secant of U accordingly as its intersection with U consists of 1 or q + 1 points. Through each point of U there pass q 2 secants and one tangent line, whereas through each point r / ∈ U there pass q + 1 tangents and q 2 − q secants. The feet of a point r / ∈ U are the q + 1 points of U that lie on the q + 1 tangents through r. For any prime power q the absolute points of a unitary polarity δ of PG(2, q 2 ) give an example of unital. Such a unital is called a classical unital or Hermitian curve.In 1976 Buekenhout [7] proved the existence of a parabolic unital in every translation plane of order q 2 with kernel containing GF(q). The term parabolic means that the line at infinity is a tangent of the unital. In [7] Buekenhout also showed that every derivable translation plane of order q 2 , with GF(q) in its kernel, contains a hyperbolic unital; that is, the line at infinity is a secant of the unital.We now briefly describe Buekenhout's two constructions, restricting ourselves to unitals embedded in PG(2, q 2 ). To do this, we make use of the representation of PG(2, q 2 ) in PG(4, q) due to André [1] and Bruck and Bose [8], [9]. Let PG(4, q) be a projective 4-space over the finite field GF(q), and let S be a regular spread of a fixed hyperplane Σ ∞ = PG(3, q) of PG(4, q). Then PG(2, q 2 ) can be represented by taking the points of PG(4, q) \ Σ ∞ as its affine points, the lines of S as its points at infinity, the planes of PG(4, q) \ Σ ∞ which meet Σ ∞ in a line of S as its extended affine lines, and S as its line L ∞ at infinity. If U is a nondegenerate quadric in PG(4, q) which meets Σ ∞ in a doubly-ruled quadric Q and if one of the reguli ruling this quadric Q is a regulus contained in the spread S , then U corresponds to a unital U in PG(2, q 2 ) and L ∞ is a secant line to U. Similarly, if U is a cone in PG(4, q) whose base

Feet in orthogonal-Buekenhout–Metz unitals

“…is classical (see Thas [10]). The conditions for this result have been relaxed after Thas's work, see [1] for a more recent result on this characterization. [3]).…”

“…Hence, for γ = s − λǫ, for all s ∈ GF (q) the intersection is empty and for when γ = ks − λǫ then the intersection contains q + 1 points. Note that for every s ∈ GF (q) we are able to choose such a γ, thus we get q lines through [1,0,0] intersecting O(τ P λ (U α,β )) in q + 1 points each.…”

“…Two standard examples of unitals are (1) The set of absolute points of a non-degenerate unitary polarity of Π: [4] proved that, for α, β ∈ GF (q 2 ) such that 4N (α) + (β − β) 2 is a non-square in GF (q), the set…”

“…(1) τ P (U α,β ) is contained in a line of Π, for all P ∈ Π \ U α,β if and only if U α,β is classical (see Thas [10]). The conditions for this result have been relaxed after Thas's work, see [1] for a more recent result on this characterization. (2) For U α,β non-classical.…”