Intersection theorem explained

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 (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 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:

\{A,B,C,a,b,c,P,Q,R,O\}

• Lines:

\{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}

• Incidences (in addition to obvious ones such as

(A,AB) ): \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\} The implication is then (R,PQ) —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 (skewfield) —

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 satisfies the intersection theorem if and only if the division ring satisfies the rational identity.

• Pappus's hexagon theorem holds in a desarguesian projective plane

P _{2} D if and only if is a field; it corresponds to the identity \forall a,b\in 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 has characteristic ≠ 2 ; it corresponds to the identity .

References

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