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.
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