In projective geometry, a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. We shall restrict ourself to the case of finite-dimensional projective spaces.
Let
K
be a field and
V
(
K
)
a vector space over
K
. A mapping
ρ
from
V
(
K
)
to
K
such that
(Q1)
ρ
(
λ
x
→
)
=
λ
2
ρ
(
x
→
)
for any
λ
∈
K
and
x
→
∈
V
(
K
)
.
(Q2)
f
(
x
→
,
y
→
)
:=
ρ
(
x
→
+
y
→
)
−
ρ
(
x
→
)
−
ρ
(
y
→
)
is a bilinear form.
is called quadratic form. The bilinear form
f
is symmetric.
In case of
char
K
≠
2
we have
f
(
x
→
,
x
→
)
=
2
ρ
(
x
→
)
, i.e.
f
and
ρ
are mutually determined in a unique way.
In case of
char
K
=
2
we have always
f
(
x
→
,
x
→
)
=
0
, i.e.
f
is symplectic.
For
V
(
K
)
=
K
n
and
x
→
=
∑
i
=
1
n
x
i
e
→
i
(
{
e
→
1
,
…
,
e
→
n
}
is a base of
V
(
K
)
)
ρ
has the form
ρ
(
x
→
)
=
∑
1
=
i
≤
k
n
a
i
k
x
i
x
k
with
a
i
k
:=
f
(
e
→
i
,
e
→
k
)
for
i
≠
k
and
a
i
k
:=
ρ
(
e
→
i
)
for
i
=
k
and
f
(
x
→
,
y
→
)
=
∑
1
=
i
≤
k
n
a
i
k
(
x
i
y
k
+
x
k
y
i
)
.
For example:
n
=
3
,
ρ
(
x
→
)
=
x
1
x
2
−
x
3
2
,
f
(
x
→
,
y
→
)
=
x
1
y
2
+
x
2
y
1
−
2
x
3
y
3
.
Definition and properties of a quadric
Below let
K
be a field,
2
≤
n
∈
N
, and
P
n
(
K
)
=
(
P
,
G
,
∈
)
the n-dimensional projective space over
K
, i.e.
P
=
{
⟨
x
→
⟩
∣
0
→
≠
x
→
∈
V
n
+
1
(
K
)
}
,
the set of points. (
V
n
+
1
(
K
)
is a (n + 1)-dimensional vector space over the field
K
and
⟨
x
→
⟩
is the 1-dimensional subspace generated by
x
→
),
G
=
{
{
⟨
x
→
⟩
∈
P
∣
x
→
∈
U
}
∣
U
2-dimensional subspace of
V
n
+
1
(
K
)
}
,
the set of lines.
Additionally let be
ρ
a quadratic form on vector space
V
n
+
1
(
K
)
. A point
⟨
x
→
⟩
∈
P
is called singular if
ρ
(
x
→
)
=
0
. The set
Q
=
{
⟨
x
→
⟩
∈
P
∣
ρ
(
x
→
)
=
0
}
of singular points of
ρ
is called quadric (with respect to the quadratic form
ρ
). For point
P
=
⟨
p
→
⟩
∈
P
the set
P
⊥
:=
{
⟨
x
→
⟩
∈
P
∣
f
(
p
→
,
x
→
)
=
0
}
is called polar space of
P
(with respect to
ρ
). Obviously
P
⊥
is either a hyperplane or
P
.
For the considerations below we assume:
Q
≠
∅
.
Example: For
ρ
(
x
→
)
=
x
1
x
2
−
x
3
2
we get a conic in
P
2
(
K
)
.
For the intersection of a line with a quadric
Q
we get:
Lemma: For a line
g
(of
P
n
(
K
)
) the following cases occur:
a)
g
∩
Q
=
∅
and
g
is called
exterior line or
b)
g
⊂
Q
and
g
is called
tangent line or
b′)
|
g
∩
Q
|
=
1
and
g
is called
tangent line or
c)
|
g
∩
Q
|
=
2
and
g
is called
secant line.
Lemma: A line
g
through point
P
∈
Q
is a tangent line if and only if
g
⊂
P
⊥
.
Lemma:
a)
R
:=
{
P
∈
P
∣
P
⊥
=
P
}
is a flat (projective subspace).
R
is called
f-radical of quadric
Q
.
b)
S
:=
R
∩
Q
is a flat.
S
is called
singular radical or
ρ
-radical of
Q
.
c) In case of
char
K
≠
2
we have
R
=
S
.
A quadric is called non-degenerate if
S
=
∅
.
Remark: An oval conic is a non-degenerate quadric. In case of
char
K
=
2
its knot is the f-radical, i.e.
∅
=
S
≠
R
.
A quadric is a rather homogeneous object:
Lemma: For any point
P
∈
P
∖
(
Q
∪
R
)
there exists an involutorial central collineation
σ
P
with center
P
and
σ
P
(
Q
)
=
Q
.
Proof: Due to
P
∈
P
∖
(
Q
∪
R
)
the polar space
P
⊥
is a hyperplane.
The linear mapping
φ
:
x
→
→
x
→
−
f
(
p
→
,
x
→
)
ρ
(
p
→
)
p
→
induces an involutorial central collineation with axis
P
⊥
and centre
P
which leaves
Q
invariant.
In case of
char
K
≠
2
mapping
φ
gets the familiar shape
φ
:
x
→
→
x
→
−
2
f
(
p
→
,
x
→
)
f
(
p
→
,
p
→
)
p
→
with
φ
(
p
→
)
=
−
p
→
and
φ
(
x
→
)
=
x
→
for any
⟨
x
→
⟩
∈
P
⊥
.
Remark:
a) The image of an exterior, tangent and secant line, respectively, by the involution
σ
P
of the Lemma above is an exterior, tangent and secant line, respectively.
b)
R
is pointwise fixed by
σ
P
.
Let be
Π
(
Q
)
the group of projective collineations of
P
n
(
K
)
which leaves
Q
invariant. We get
Lemma:
Π
(
Q
)
operates transitively on
Q
∖
R
.
A subspace
U
of
P
n
(
K
)
is called
ρ
-subspace if
U
⊂
Q
(for example: points on a sphere or lines on a hyperboloid (s. below)).
Lemma: Any two maximal
ρ
-subspaces have the same dimension
m
.
Let be
m
the dimension of the maximal
ρ
-subspaces of
Q
. The integer
i
:=
m
+
1
is called index of
Q
.
Theorem: (BUEKENHOUT) For the index
i
of a non-degenerate quadric
Q
in
P
n
(
K
)
the following is true:
i
≤
n
+
1
2
.
Let be
Q
a non-degenerate quadric in
P
n
(
K
)
,
n
≥
2
, and
i
its index.
In case of
i
=
1
quadric
Q
is called
sphere (or oval conic if
n
=
2
).
In case of
i
=
2
quadric
Q
is called
hyperboloid (of one sheet).
Example:
a) Quadric
Q
in
P
2
(
K
)
with form
ρ
(
x
→
)
=
x
1
x
2
−
x
3
2
is non-degenerate with index 1.
b) If polynomial
q
(
ξ
)
=
ξ
2
+
a
0
ξ
+
b
0
is irreducible over
K
the quadratic form
ρ
(
x
→
)
=
x
1
2
+
a
0
x
1
x
2
+
b
0
x
2
2
−
x
3
x
4
gives rise to a non-degenerate quadric
Q
in
P
3
(
K
)
.
c) In
P
3
(
K
)
the quadratic form
ρ
(
x
→
)
=
x
1
x
2
+
x
3
x
4
gives rise to a
hyperboloid.
Remark: It is not reasonable to define formally quadrics for "vector spaces" (strictly speaking, modules) over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally different from usual quadrics. The reason is the following statement.
Theorem: A division ring
K
is commutative if and only if any equation
x
2
+
a
x
+
b
=
0
,
a
,
b
∈
K
has at most two solutions.
There are generalizations of quadrics: quadratic sets. A quadratic set is a set of points of a projective plane/space, which bears the same geometric properties as a quadric: any line intersects a quadratic set in no or 1 or two lines or is contained in the set.
