4 - Topology of Algebraic Curves in ℙ²(ℂ)

Summary

INTRODUCTION

The gallery of real curves in Chapter 1 presented a wide range of behavior. It was so wide, we were led to ask “Where are the nice theorems” We've already seen how broadening curves' living space to ℙ²(ℂ) can lead to more unified results, Bézout's theorem in Chapter 3 being a prime example. But what about those real curves we met in Chapter 1 having more than one connected component? Or ones having mixed dimensions? Does working in ℙ²(ℂ) perform its magic for cases like this?

Yes. In this chapter we'll see that individual curves in ℙ²(ℂ) are generally much nicer and properties more predictable than their real counterparts. For example, we will show that every algebraic curve in ℙ²(ℂ) is connected and that every irreducible curve is orientable. These are powerful theorems that help to smooth out the wrinkles in the real setting.

Most algebraic curves in ℂ² or ℙ²(ℂ) are everywhere smooth. We will make this more precise in the next chapter, but any polynomial of a given degree with “randomly chosen” real or complex coefficients defines a real 2-manifold that is locally the graph of a smooth function. Such a curve in ℙ²(ℂ) is therefore a closed manifold (a manifold having no boundary) that is orientable and thus has a topological genus.