Intersection theorem


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 and . The "theorem" states that, whenever a set of objects satisfies the incidences, then the objects and 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 :
The implication is then —that point is incident with line.

Famous examples

Desargues' theorem holds in a projective plane if and only if is the projective plane over some division ring —. 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 satisfies the intersection theorem if and only if the division ring satisfies the rational identity.