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Intersection theorem

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This article is about projective geometry. For a result on tensor products of modules, see Homological conjectures in commutative algebra.

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 } {\displaystyle \{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 } {\displaystyle \{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}}
  • Incidences (in addition to obvious ones such as ( A , A B ) {\displaystyle (A,AB)} ): { ( 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 ) } {\displaystyle \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}}

The implication is then ( R , P Q ) {\displaystyle (R,PQ)} —that point R is incident with line PQ.

Famous examples

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 {\displaystyle P=\mathbb {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 {\displaystyle \mathbb {P} _{2}D} if and only if D is a field; it corresponds to the identity ∀ a , b ∈ D , a ⋅ b = b ⋅ a {\displaystyle \forall a,b\in D,\quad a\cdot b=b\cdot a} .
  • Fano's axiom (which states a certain intersection does not happen) holds in P 2 D {\displaystyle \mathbb {P} _{2}D} if and only if D has characteristic ≠ 2 {\displaystyle \neq 2} ; it corresponds to the identity a + a = 0.

References

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