In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects A and B (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.
For example, Desargues' theorem can be stated using the following incidence structure:
Points:
{
A
,
B
,
C
,
a
,
b
,
c
,
P
,
Q
,
R
,
O
}
Lines:
{
A
B
,
A
C
,
B
C
,
a
b
,
a
c
,
b
c
,
A
a
,
B
b
,
C
c
,
P
Q
}
Incidences (in addition to obvious ones such as
(
A
,
A
B
)
):
{
(
O
,
A
a
)
,
(
O
,
B
b
)
,
(
O
,
C
c
)
,
(
P
,
B
C
)
,
(
P
,
b
c
)
,
(
Q
,
A
C
)
,
(
Q
,
a
c
)
,
(
R
,
A
B
)
,
(
R
,
a
b
)
}
The implication is then
(
R
,
P
Q
)
—that point R is incident with line PQ.
Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield} D —
P
=
P
2
D
. The projective plane is then called desarguesian. A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.
Pappus's hexagon theorem holds in a desarguesian projective plane
P
2
D
if and only if D is a field; it corresponds to the identity
∀
a
,
b
∈
D
,
a
⋅
b
=
b
⋅
a
.
Fano's axiom (which states a certain intersection does not happen) holds in
P
2
D
if and only if D has characteristic
≠
2
; it corresponds to the identity a + a = 0.
