Английская Википедия:Intersection theorem

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

The implication is then <math>(R,PQ)</math>—that point Шаблон:Math is incident with line Шаблон:Math.

Famous examples

Desargues' theorem holds in a projective plane Шаблон:Math if and only if Шаблон:Math is the projective plane over some division ring (skewfield) Шаблон:Math — <math>P=\mathbb{P}_{2}D</math>. 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 Шаблон:Math satisfies the intersection theorem if and only if the division ring Шаблон:Math satisfies the rational identity.

• Pappus's hexagon theorem holds in a desarguesian projective plane <math>\mathbb{P}_{2}D</math> if and only if Шаблон:Math is a field; it corresponds to the identity <math>\forall a,b\in D, \quad a\cdot b=b\cdot a</math>.
• Fano's axiom (which states a certain intersection does not happen) holds in <math>\mathbb{P}_{2}D</math> if and only if Шаблон:Math has characteristic <math>\neq 2</math>; it corresponds to the identity Шаблон:Math.

References

• Шаблон:Cite book
• Шаблон:Cite journal