2007 | OriginalPaper | Buchkapitel
Other Good Stuff
Erschienen in: q-Clan Geometries in Characteristic 2
Verlag: Birkhäuser Basel
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Let
S
be a GQ with parameters (
s, t
),
s ≥
1,
t ≥
1. A
spread
of
S
is a set
M
of lines that partition the points of
S
. Dually, an
ovoid
of
S
is a set
$$ \mathcal{M} $$
of points of
S
such that each line of
S
is incident with a unique point of
$$ \mathcal{O} $$
. It is easy to see that a spread must have 1 +
st
lines and an ovoid must have 1 +
st
points. For example, if a GQ
S′
of order
q
is contained as a subquadrangle in a GQ
S
with order
(q2, q)
, then each point
X
of a line
l
exterior to
S′
is on a unique line of
S′
. Hence the
q2
+1 lines of
S′
that meet
l
form a spread of
S′
said to be subtended by
l
. Spreads and ovoids of GQ have been studied a great deal and have a wide variety of connections with other geometric objects. For a general reference see J. A. Thas and S. E. Payne [TP94]. For
q
= 2e see especially [BOPPR1] and [BOPPR2]. In this section we give a very brief introduction to the material contained in these latter two papers.