## Projective Geometry I - Lecture Notes

#### 5.1 Fano's Axiom P6

**Def:** * complete quadrangle, diagonal points.*
**P6: (Fano's Axiom) The diagonal points of a complete quadrangle are not collinear.**

**Proposition 5.2**: The real projective plane satisfies P6.

*Temporary definition*: a Fano Plane

Discuss why this is temporary.

**Def**: *complete quadrilateral, diagonal lines*.

**Prop 5.4**: If \pi is a Fano plane, then so is its dual \pi*.

#### 5.2 Harmonic Points

**Def**: An ordered quadruple of distinct points A, B, C, D on a line is called a* harmonic
quadruple* if there is a complete quadrangle XYZW such that A and B are diagonal points of
the quadrangle and C and D are on the remaining two sides. H(A,B; C,D).
**Prop 5.7**: Let A,B,C be three distinct points on a line. Then (assuming P6) there is a point D
such that H(A,B; C,D). Furthermore, if P5 is assumed, this point is unique. (D is called the
4th harmonic point of A, B, C, or the harmonic conjugate of C wrt A and B.)

**Prop 5.8**: Let H(A,B; C,D). Then (assuming P5) H(C,D; A,B).

**Example**: In PG(2,3) there are only 4 points on a line, they always form a harmonic
quadruple in any order.

(note the slighting of nondesarguesian planes in the comment after this example).

**Example**: *Cross ratio* in the Euclidean plane.

**Example**: *Cross ratio for lines* in the Euclidean plane.

#### 5.3 Perspectivities and Projectivities

**Def**:* perspectivity, projectivity*
**Prop 5.14**: Let *l* be a line. Then the set of projectivities of *l* into itself forms a group, which
we will call *PJ(l)*.

**Prop 5.15**: PJ(*l*) is 3-transitive.

**Lemma 5.17**: Suppose that ABCD __^__ A'BC'D' are perspective in a projective plane
satisfying P5 and P6. If H(A,B;C,D) then H(A',B; C',D').

**Prop. 5.18**: Assuming P5 and P6, a projectivity sends harmonic points into harmonic points.

Thus PJ(*l*) is **not** 4-transitive in Desarguesian planes.