Affine and Projective Space

The historical roots of algebraic geometry lie in the study of curved lines in the plane, or as we would prefer to say today, planar curves. The treatment of modern algebraic geometry offered in the present book takes a starting point which is more general and at the same time more restricted: More general in the sense that we will study geometric objects such as curves and surfaces, say, in spaces of any dimensions, and where the points, including those at infinity, are described by coordinates which are elements of a general field, not just real numbers. More special in the sense that we consider geometric objects defined by polynomial equations in the coordinates of the space in which they lie. Historically the necessity of dealing with points at infinity is one of the reasons why ordinary space had to be completed to projective space by adding points at infinity. In this first chapter we establish these foundations for our subject. The theorem of Desargues is treated in some detail, since it illustrates in a beautiful way the role played by points at infinity, the concept of duality and the interplay between different projective coordinate systems.