Table of Contents Introduction ........................ 1 The Projective Plane ........................ 1 The Topology of RP 2 ........................ 3 Projective Spaces in General ........................ 4 The Riemannsphere and Complex Projective Spaces ........................ 4 Axiomatics and Finite Geometries ........................ 6 Duality and Conics ........................ 7 Exercises 1-34 ........................ 10 Crossratio ........................ 15 The -invariant ........................ 16 Harmonic Division ........................ 17 Involutions ........................ 17 Exercises 35-56 ........................ 19 Non-Singular Quadrics ........................ 22 Involutions and Non-Singular Quadrics ........................ 26 Exercises 57-76 ........................ 28 Ellipses,Hyperbolas,Parabolas and circular points at ........................ 31 The Space of Conics ∞ ........................ 32 Exercises 77-105 ........................ 37 Pencils of Conics ........................ 40 Exercises 106-134 ........................ 43 Quadric Surfaces ........................ 46 Quadrics as Double Coverings ........................ 49 Hyperplane sections and M¨obius Transformations ........................ 50 Line Correspondences ........................ 50 Conic Correspondences ........................ 51 Exercises 135-165 ........................ 53 Birational Geometry of Quadric Surfaces ........................ 56 Blowing Ups and Down ........................ 57 Cremona Transformations ........................ 58 Exercises 166-185 ........................ 60 Plane Cubics ........................ 63 Classification of Singular Cubics ........................ 63 Flexes and the Hessian ........................ 64 Weierstraß Normal Form ........................ 65 Nodal and Cuspidal Cubics ........................ 66 The Group Law ........................ 66 The Weierstraß ℘-function ........................ 68 Two Special Cubics ........................ 70 Isogenies ........................ 71 Hesse Normal Form ........................ 72 Polars to Cubics ........................ 73 The Dual Cubic ........................ 74 Exercises 186-282 ........................ 75 2 Nets of Conics ........................ 86 Jacobian Nets ........................ 88 Singular Nets ........................ 90 Higher dimensional Systems of Conics ........................ 90 The Birational map between P(S2V ) and P(S2V ∗) ........................ 92 The Tangent Locus to a Conic ........................ 93 Exercises 283-322 ........................ 95 Quadrics ........................ 99 Quadrics as Double Coverings ........................ 100 Birational Representation of Quadrics ........................ 100 Linear Subspaces on a Quadric ........................ 101 Topology of Quadrics ........................ 102 Exercises 323-346 ........................ 104 The Quadric Line Complex ........................ 107 The Real Quadric ........................ 108 Exercises 347-368 ........................ 110 Pencils of Quadrics ........................ 112 The Elliptic Quartic ........................ 112 Generic Pencils of Quadrics ........................ 113 The Geometry of Elliptic Quartics ........................ 114 Degenerate Pencils ........................ 114 Exercises 369-386 ........................ 116 Involutions on Twisted Cubics and Nets of Quadrics ........................ 119 Exercises 387-407 ........................ 121 3 PREFACE The following are the notes I wrote down for a course in Projective Geometry at Chalmers in the fall of 1989. The course was intended to start more or less from scratch, and from one small corner of the Algebraic Geometrical Universe expand gradually the horizon of the students. Thus I eschewed all ambitions of setting up a nice machinery from which the results could then follow nicely and smoothly. It is in fact my conviction that the presentation of a more or less formidable machinery may be good in hindsight but that it postpones motivation and ultimate gratification. I have hence worked rather intuitivly, and I have not been ashamed of waving my hands, especially at the end. The idea being that the students should first be confronted with the problem before they start to develop the concepts to formulate and attack it. To this effect I have included a fair amount of exercises. Many of those are routine and many are computational, some fill in gaps in proofs, while others are pure digressions, of which there are endless opportunities. One learns by doing and a prospective reader is very much encouraged to try his or her hand at most of them, or at least those that catches her or his fancy. On the other hand I have become aware of the frustrations such an approach may entail as well, and I regret many of the repetitions of standard arguments that I have employed over and over again. Nevertheless I think that the attempt is worthwhile. As to the choice of topics. I have decided to concentrate on conics, as they are quite elementary and provide a bridge between linear algebra and algebraic geom- etry. In particular I have treated linear systems of conics, which are all more or less classified throughout the course of the lectures. The final section on nets is a bit sketchy and ought to be expanded in a revised version, in particular the strat- ification of all singular nets (following C.T.C Wall) should be presented in detail.I have also digressed on quadrics in P3, a topic which one cannot pass over easily, and also perhaps less of a necessity on cubics and elliptic curves, where I have pre- sented a particularly hefty collection of exercises. This choice was motivated by the geometry of a non-singular net which is intimately connected with the discriminant cubic, although I have presented far more than is strictly necessary. This is also yet another indulgence. One has to confine oneself, yet I feel that many topics are missing and ought to be included in a revision. E.g. higher dimensional quadrics, in particular a thorough discussion of the quadric line complex. Pencils of quadric surfaces may also be included, as to further justify the lengthy digression on elliptic curves. The cubic hypersurface is another topic which is always very tempting to include. It is also a shame not to have included anything on metrics on projective planes as compact manifolds. The geometry of M¨obius transformations is discussed in some detail, but little on the geometry of PGL(3,C). Elementary toplogy is hinted at in the exercises, especially the Euler characteristics, but this being rather fundamental, it ought to be lifted out more in the open. In general one may discuss whether not many of the exercises would not deserve a fuller treatment in the text. As to arithmetic applications there are some in the text, or mainly in the ex- ercises. One could include a little bit more on finite geometries, most of it is in the exercises; and also be more systematic on elementary diophantine discussions 4 (almost all of what there is is to be found in the exercises), in particular it may not be amiss to prove the Hasse principle for Quadrics, although it may be too much of a digression. Finally connecting with the lengthy section on cubic curves, one may discuss quadratic equations, lattices, SL(2,C) and orders, as yet another aspect of quadrics. The big omission in this version is of course the pictures. I have not yet been able to produce nice pictures of what I want, nor have I figured out how to insert those in the text. When this has been done another version will appear. I have never seen a net of conics, and I definitely intend to include such pictures in the future (hoping not to be disappointed). It is also my opinion that only nice graphics, which to my knowledge is not so much available in classical geometry, would justify yet another elementary text. But basically this has been a self-indulgence, and it may just remain so. Any suggestions are of course welcome, and advice well-taken if not always followed. G¨oteborg December 20, 1989 5 Preface to the extended version To the notes of the fall 1989 I have added (in November 1990) three chapters , on Quadrics in general, the Quadratic Line complex and finally on pencils of Quadric surfaces. Yet the work is not complete, the chapter on the Quadratic line complex can easily be expanded. And there should be a chapter on nets of quadric surfaces, Whether to include a chapter on the cubic hypersurface is open, but as the intersection of two quadrics in P4 would be natural, the temptation is great. The chapter on the Quadratic Line complex easily lends itself to a discussion of quadratic complexes and Kummer surfaces, thus K-3 surfaces, Abelina surfaces and Jacobians of genus two curves naturally come into play. Those may be added at some later date. Still there is no discussion of metric properties of projective spaces, nor any continued discussion of PGL(3, C) and higher groups. This may also be included. And the illustrations are still missing (except for the title page!) G¨oteborg February 27,1991 1 Introduction Projective Geometry deals with properties that are invariant under projections. Hence angles and distances are not preserved, but collinearity is. In many ways it is more fundamental than Euclidean Geometry, and also simpler in terms of its axiomatic presentation. Projective Geometry is also ”global” in a sense that Euclidean Geometry is not. In Euclidean Geometry lines may or may not meet, if not, this is an indication that something is