Intersection theorem

Summary

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 PQ.

Famous examplesEdit

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 a + a = 0.

ReferencesEdit

  • Rowen, Louis Halle, ed. (1980). Polynomial Identities in Ring Theory. Pure and Applied Mathematics. Vol. 84. Academic Press. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505.
  • Amitsur, S. A. (1966). "Rational Identities and Applications to Algebra and Geometry". Journal of Algebra. 3 (3): 304–359. doi:10.1016/0021-8693(66)90004-4.