Plane Algebraic Curves

In this chapter we introduce some basic notions on plane algebraic curves, we derive some fundamental properties of algebraic curves, and we outline the general working environment of the book. This chapter consists of five sections. In Sect. 2.1, we present the basic notions on curves distinguishing between affine and projective curves. Section 2.2 is devoted to polynomial and rational functions. The material of this section is presented for the more general case of varieties (i.e., irreducible algebraic sets), and will play an important role in subsequent sections. In Sect. 2.3 we focus again on the case of plane curves. The study of the intersection of curves leads to the notion of multiplicity of intersection and to Bézout's theorem. Section 2.4 is devoted to the study of linear systems of curves. We will see in the following chapters that this notion is crucial for solving the problem of parametrizing a rational curve. The chapter ends with Sect. 2.5 where we show how to locally parametrize a curve around a point of the curve by means of Puiseux series.