Intersection theorem

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

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

References

• 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.