Projective Geometry

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 n-dimensional 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 FP2.
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. Cross-ratio preserving transformations between lines
• Cross-ratio.
• Characterizations of cross-ratio preserving transformations between straight lines.
• Cross-ratio preserving transformations between coplanar lines.
• Cross-ratio preserving transformations of a line, involutions.