In this chapter we introduce ‘tangents’ at a point p on an affine curve f via the intersection numbers of Chapter 6. Intuitively, a tangent at p is a line l through p which has higher contact with f than one would expect. The most general case (in some sense) is when p is a simple point on f, and there is a unique tangent line l given by an explicit formula; this provides the content of Section 7.2. Normally the tangent l at a simple point p has contact of order 2 with f at p, but when it is ≥ 3, we have very special points on f called ‘flexes’, which play a potentially important role in understanding the geometry of a curve. In the remainder of the chapter, we give a number of examples illustrating how to find the tangents to affine curves at singular points p.

Generalities about Tangents

Let p be a point of multiplicity m on an algebraic curve f in K2. Then automatically I(p, f, l) ≥ m for every line l through p. A tangent line (or just tangent) to f at p is a line l through p for which I(p, f, l) ≥ m + 1. We say that two curves f, g are tangent at a point p of intersection when there exists a line l which is both a tangent to f at p, and a tangent to g at p: two curves f, g are tangent when there exists an intersection point p at which they are tangent.