Our attempts at obtaining complete lists of curves of a given degree have been somewhat limited – apart from lines, we have only managed to list conies. However, it is curves of higher degree which proliferate in applications, and it is time for us to make progress on the next case of cubics. However, we have learnt important lessons from our studies of conies. It is easier to list in the complex case than the real case, and easier to list in the projective than the affine case. On that basis, the natural way forward is to attempt to list complex projective cubics. We will approach this classification in two steps. First, we will use the basic geometry of cubics to categorize (complex projective) cubics into nine distinct ‘geometric types’: that is the object of Section 15.1. Then in the remainder of the chapter we will show that our categorization is almost a classification up to projective equivalence.

Geometric Types of Cubics

The basic fact which allows us to categorize cubic curves in Pℂ2 into a small number of types is the following lemma.

Lemma 15.1An irreducible cubic F in Pℂ2 has at most one singular point, necessarily of multiplicity 2, so either a node or a cusp.

Proof F cannot have two distinct singular points P1, P2; for then the line L joining P1, P2 would meet F in ≥ 2 + 2 = 4 points, so have to be a component of F, contradicting irreducibility.