Plane Algebraic Curves
Pages: 730 | DjVu | 6.2 MB

Chapter 1 is a very exciting tour of algebraic curves through history with lots of examples and pictures. There is a wealth of examples already from Greek days. The two main systematising tools are analytic geometry (chapter 2) and projective geometry (chapter 3). Analytic geometry imposes a hierarchy on algebraic curves according to the degree of their equation. The pivotal case is the cubics. Newton carried out an algebraic classification of cubics into 72 cases, but he also noted that any cubic could be obtained from one of only five standard cases by central projection. A real theory of projective geometry, however, would have to wait until the 19th century and the realisation that the complex projective plane, realised by homogenous coordinates, is the proper setting for this theory. Thus we now have a clear algebraic basis for our theory so we now leave the historical perspective and start to develop the theory more systematically (chapters 4-6). With the basic machinery of polynomial algebra we can now treat properly some concepts we had seen in special cases (reducibility of curves, singular points) and some theorems we had only felt (Bézout's theorem, point-line duality). (Already here the authors sneak in some non-standard material (singularities as torus knots, topological proof of Bézout's theorem) to harmonise with the more modern theory later in the book.) In chapter 7 we see applications, in particular the showcase theorem on the configuration of the inflections points of a cubic, and the role of cubics elsewhere in mathematics (the addition-of-points operation makes them abelian manifolds and they are also the tori on which elliptic functions live). These were the nonsingular cubics.