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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.
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