One of the main functions of this book is to place algebraic curves in a natural setting (the complex projective plane) where they can be studied easily. For some readers, particularly those whose background is not in mathematics, this may prove to be a psychological barrier. I can only assure them that the reward is much greater than the mental effort involved. History has shown that placing algebraic curves in a natural setting provides a flood of illumination, enabling one much better to comprehend the features one meets in everyday applications.

Let me motivate this by looking briefly at the important question of understanding how two curves intersect each other. The simplest situation is provided by two lines. In general, you expect two distinct lines to meet in a single point. But there are lines (x = 0 and x = 1 for instance) which do not intersect. This really is a nuisance. It would be much nicer to say that these lines intersect at a single point, namely a ‘point at infinity’. In this chapter, we will extend the ordinary affine plane to a ‘projective plane’ by adding points at infinity, and gain much as a result.

The Projective Plane

The construction is based on the following intuition. Think of the affine plane K2 as a plane L in K3 which does not pass through the origin O, and let L0 be the parallel plane through O.