Projective Geometry
Budapest
Semesters in Mathematics
Lecture Notes by Balázs
Csikós
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CONTENTS
Part
1. Introduction.

The invention of perspective drawing.

The real projective space, points at infinity.

Topological structure of the real projective straight
line and plane.
Part
2. Linear spaces and the
associated projective spaces

Groups, rings, division rings and fields.

Vector spaces and their subspaces.

Basis, coordinates, dimension.

The projective space associated to a linear space.

Projective coordinate systems.

The theorems of Desargues and Pappus.
Part
3. Examples

Projective spaces over finite fields.

Complex projective spaces and the Hopf fibration.

Digression: Stereographic projection and inversion.

The stereographic image of the Hopf fibration.

Quaternions.
Part
4. The axiomatic treatment
of projective spaces

The incidence axioms of an ndimensional projective space.

The duality principle, the dual space.

Desargues' theorem and the incidence axioms.

Collineations.
Part
5. Desarguesian projective
spaces

Construction of the division ring F.

Construction of a collineation between P and FP^{2}.
Part
6. The Fundamental Theorem
of projective geometry

The Projective General Linear group.

Collineations induced by automorphisms of F.

The Fundamental Theorem of projective geometry.
Part
7. Crossratio preserving
transformations between lines

Crossratio.

Characterizations of crossratio preserving transformations
between straight lines.

Crossratio preserving transformations between coplanar
lines.

Crossratio preserving transformations of a line, involutions.