Projective geometry: an introduction

211 Pages·2006·0.962 MB·English

by  Rey Casse

Projective Geometry
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Projective Geometry: An Introduction
Rey Casse
Discipline of Pure Mathematics, The University of Adelaide
1
3
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To my grandchildren
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Foreword
ProjectiveGeometryisatextbook.Itaimstobe‘student-friendly’,withoutsac-
rificing mathematical rigour. It is suitable for students in their third or fourth
year at the university. Such students would have already acquired a certain
mathematical maturity; they would understand the need for precise mathe-
matical statements, and they would expect a beautiful piece of mathematics
in return as a reward. Geometry happens to be full of unexpected and beauti-
ful results. That is why it has attracted so much interest throughout the ages;
indeed, current research in geometry suggests that it is very much alive, and
doing well.
This textbook is based on lectures given at the University of Adelaide by
Dr Sue Barwick, Dr Catherine Quinn, and myself. Chapter 1 describes the
assumed knowledge. Ideally, the reader would be familiar with linear algebra,
elementary group theory, partial differentiation, and finite fields. A reader with
less background knowledge can proceed with the study, provided there is a
readiness to accept as valid the results listed in Chapter 1; at some stage,
these results need to be properly investigated. As far as geometry is concerned,
the only assumption made is some elementary coordinate geometry. Chapter
2 introduces Desargues’ Theorem, which turns out to be of major importance.
In Chapter 3, r-dimensional projective spaces are defined by means of nine
axioms, which initially may seem like a formidable set of axioms. It is not
necessary to commit all the axioms to memory; however, special attention
should be paid to axiom A8 (called the ‘dimension axiom’ in this textbook,
but known elsewhere as the ‘Grassmann identity’). One becomes conversant
with these axioms by working out various cases in various projective spaces,
and Example 3.9 gives in detail the situation in three-dimensional projective
spaces.
One soon learns that Desargues’ Theorem is valid in all projective spaces
of dimension greater than 2. As a result, two areas of study offer themselves.
The first area, covered mainly in Chapters 3, 5, and 6, relates to projective
spaces of dimension 2 (also known as non-Desarguesian projective planes) in
which Desargues’ Theorem is not valid. The remaining chapters deal with the
(Desarguesian) projective spaces defined over some field. Within this overall
frame,thesubjectisdevelopedinatraditionalway,namelysometheory,followed
by worked examples and exercises.
Noattempthasbeenmadetocompileabibliography.Foraverycomprehen-
sive bibliography, see for example J.W.P. Hirschfeld’s book ‘Projective Geome-
tries Over Finite Fields’.
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Acknowledgements
I am greatly indebted to a number of people.
The first draft was read by Dr S. Barwick, Dr M. Brown, Mr D. Butler, Dr
S. Cox, Prof G. Ebert, Dr A. Glen, Dr D. Glynn, Dr W.-A. Jackson, Dr M.
Oxenham, Dr D. Parrott, Dr M. Paterson, Dr C. Quinn, Prof T. Penttila, Dr J.
Tuke, Prof P. Wild, to whom I am most grateful for a number of constructive
criticisms, suggestions, and improvements.
Specialthanksgotothreecolleaguesofmine:DrS.Barwick,DrW.-A.Jackson,
andDrC.Quinn,specifically.
• Dr S. Barwick and Dr C. Quinn for giving me permission to use a number
of problems set in their courses;
• DrS.BarwickandDrW.-A.Jacksonforteachingmetheintricaciesoflatex
and xfig, for latexing a few chapters, and for reformating the last draft;
• Dr S. Barwick without whose unflinching and generous support this text-
book would not have seen the light of day.
Lastly, I wish to thank the personel at Oxford University Press involved in
the production of this book, and the Department of Pure Mathematics of the
University of Adelaide for their helpful co-operation.
L. R. A. CASSE