Curves in $\mathbb{A}^{2}_{k}\mbox{ and in }\mathbb{P}^{2}_{k}$

Abstract

In this chapter we introduce the first interesting class of planar curves, namely the conic sections. This leads to a first discussion of singular and non singular points. Closely tied to these concepts is the notion of the tangent at a point on a curve. We then move on to a discussion of curves of higher degrees, and introduce the concepts of tangent lines, the tangent cone and the multiplicity of a point on a curve which may have singularities. A number of important examples of higher order curves are discussed. Elliptic curves are briefly discussed, this class of curves (which are certainly not ellipses) played an important role for the fruitful interplay between geometry and function theory, so central in the pathbreaking work of Niels Henrik Abel.