Foundations of Projective Geometry
                   Robin Hartshorne
                      1967
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           Preface
            Thesenotesarosefromaone-semestercourseinthefoundationsofprojective
           geometry, given at Harvard in the fall term of 1966–1967.
            Wehaveapproachedthesubjectsimultaneouslyfromtwodifferentdirections.
           In the purely synthetic treatment, we start from axioms and build the abstract
           theory from there. For example, we have included the synthetic proof of the
           fundamental theorem for projectivities on a line, using Pappus’ Axiom. On the
           other hand we have the real projective plane as a model, and use methods of
           Euclidean geometry or analytic geometry to see what is true in that case. These
           two approaches are carried along independently, until the first is specialized by
           the introduction of more axioms, and the second is generalized by working over
           an arbitrary field or division ring, to the point where they coincide in Chapter 7,
           with the introduction of coordinates in an abstract projective plane.
            Throughout the course there is special emphasis on the various groups of
           transformations which arise in projective geometry. Thus the reader is intro-
           duced to group theory in a practical context. We do not assume any previous
           knowledge of algebra, but do recommend a reading assignment in abstract group
           theory, such as [4].
            There is a small list of problems at the end of the notes, which should be
           taken in regular doses along with the text.
            There is also a small bibliography, mentioning various works referred to in
           the preparation of these notes. However, I am most indebted to Oscar Zariski,
           who taught me the same course eleven years ago.
                                       R. Hartshorne
                                       March 1967
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