2 JOHN BAMBERG AND NICOLA DURANTE

there is an elegant model of H(q) which begins with geometric structures

lying in PG(3,q), and it is equivalent to the model provided by Cameron

and Kantor [6, Appendix]:

Theorem 1.1 (Cameron and Kantor (paraphrased) [6]).

Let (p, σ) be a point-plane anti-flag of PG(3,q) and let Ω be a set of

q(q2

−

1)(q2

+ q + 1) parabolic

congruences3

each having axis not incident with p

or σ, but having a pencil of lines with one line incident with p and another

incident with σ. Suppose that for each pencil L with vertex not in σ and

plane not incident with p, there are precisely q + 1 elements of Ω containing

L, whose union are the lines of some linear complex (i.e., the lines of a

symplectic geometry W(3,q)). Then the following incidence structure Γ is

isomorphic to the split Cayley hexagon H(q).

Points: (a) Lines of PG(3,q).

(b) Pencils with a vertex not in σ and plane not incident with p.

Lines: (i) Pencils with a vertex in σ and plane through p.

(ii) Elements of Ω.

An element of type (a) is incident with an element P of type (i) if is an element

of P. If C is an element of type (ii), then is incident with C if is the axis of C.

Elements of type (i) and (b) are never incident. The containment relation defines

incidence between elements of type (b) and (ii).

The central result of this note is a unitary analogue of this model.

Theorem 1.2. Let O be a Hermitian curve of

H(3,q2)

and let Ω be a set

of Baer subgenerators with a point in O, such that every point of

H(3,q2)\O

is on q + 1 elements of Ω spanning a Baer subplane. Then the following

incidence structure Γ is a generalised hexagon of order (q, q).

Points: (a) Lines of

H(3,q2).

(b) Aﬃne points of

H(3,q2)\O.

Lines: (i) Points of O.

(ii) Elements of Ω.

Incidence: Inclusion or inherited incidence.

3A

pencil of lines of PG(3, q) refers to the set of lines passing through a point, lying

on a plane. A set of

q2

+ q + 1 lines concurrent with a common line , no two of which

meet in a point not on , is called a parabolic congruence, and the line is its axis. The

image of a parabolic congruence under the Klein correspondence yields a 3-dimensional

quadratic cone of

Q+(5,

q), and vice-versa (see [10, p. 30]), and so a parabolic congruence

is a union of q + 1 pencils sharing a line.