In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank
r
−
1
intersects O in exactly one point.
An ovoid of
W
2
n
−
1
(
q
)
(a symplectic polar space of rank n) would contain
q
n
+
1
points. However it only has an ovoid if and only
n
=
2
and q is even. In that case, when the polar space is embedded into
P
G
(
3
,
q
)
the classical way, it is also an ovoid in the projective geometry sense.
Ovoids of
H
(
2
n
,
q
2
)
(
n
≥
2
)
and
H
(
2
n
+
1
,
q
2
)
(
n
≥
1
)
would contain
q
2
n
+
1
+
1
points.
An ovoid of a hyperbolic quadric
Q
+
(
2
n
−
1
,
q
)
(
n
≥
2
)
would contain
q
n
−
1
+
1
points.
An ovoid of a parabolic quadric
Q
(
2
n
,
q
)
(
n
≥
2
)
would contain
q
n
+
1
points. For
n
=
2
, it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid. If q is even,
Q
(
2
n
,
q
)
is isomorphic (as polar space) with
W
2
n
−
1
(
q
)
, and thus due to the above, it has no ovoid for
n
≥
3
.
An ovoid of an elliptic quadric
Q
−
(
2
n
+
1
,
q
)
(
n
≥
2
)
would contain
q
n
+
1
points.
