Intersection theorem

projective geometry

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:
Lines:
Incidences (in addition to obvious ones such as ):
The implication is then —that point R is incident with line .

Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield) 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 if and only if D is a field; it corresponds to the identity Fano's axiom (which states a certain intersection does not happen) holds in if and only if D has characteristic ; it corresponds to the identity 1=a + a = 0.