In this chapter, we will establish the fascinating fact that an irreducible cubic in Pℂ2 has the natural structure of an Abelian group. This will enable us to describe some of the geometric properties of cubics in group theoretic terms. The starting point is the observation that if P, Q are simple points on an irreducible cubic F, and the line L joining them is not tangent to F at either point, then there is a naturally associated third point P✶Q on F, namely the third point where F meets L. In Section 17.2, we will extend this idea to any two simple points on F, and establish the basic properties of the operation ✶. The key technical property depends on a subtle result about pencils of cubics, called the Theorem of the Nine Associated Points; the sole object of the next section is to establish this result.

The Nine Associated Points

By Bézout's Theorem, any two cubic curves in Pℂ2 having no common component intersect in nine points, counted properly. One of the keys to understanding the geometry of cubic curves is the following result.

Lemma 17.1Let F1, F2 be cubic curves in Pℂ2having no common component, and intersecting in nine distinct points P1, …, P9. Then any cubic F which passes through P1, …, P8must pass through P9. (Theorem of the Nine Associated Points.)