1.1 Basic Definitions and Problems

A curve (at least, an algebraic curve) is something more than just the set of points which belong to it. There are many ways to introduce algebraic curves. In the elementary situation of real plane projective curves the simplest and most convenient is the following definition, which at first glance seems to be overly algebraic.

By a real projective algebraic plane curve¹ of degree $m$ we mean a homogeneous real polynomial of degree $m$ in three variables, considered up to constant factors. If $a$ is such a polynomial, then the equation $a(x_0,x_1,x_2)=0$ defines the set of real points of the curve in the real projective plane $\mathbb{R}P^2$. We let $\mathbb{R}A$ denote the set of real points of the curve $A$. Following tradition, we shall also call this set a curve, avoiding this terminology only in cases where confusion could result.

A point $(x_0:x_1:x_2)\in \mathbb{R}P^2$ is called a (real) singular point of the curve $A$ if $(x_0,x_1,x_2)\in \mathbb{R}^3$ is a critical point of the polynomial $a$ which defines the curve. The curve $A$ is said to be (real) nonsingular if it has no real singular points. The set of real points of a nonsingular real projective plane curve is a smooth closed one-dimensional submanifold of the projective plane.

In the topology of nonsingular real projective algebraic plane curves, as in other similar areas, the first natural questions that arise are classification problems.

1.1.A (Topological Classification Problem)   Up to homeomorphism, what are the possible sets of real points of a nonsingular real projective algebraic plane curve of degree $m$?

1.1.B (Isotopy Classification Problem)   Up to homeomorphism, what are the possible pairs $(\mathbb{R}P^2,\mathbb{R}A)$ where $A$ is a nonsingular real projective algebraic plane curve of degree $m$?

It is well known that the components of a closed one-dimensional manifold are homeomorphic to a circle, and the topological type of the manifold is determined by the number of components; thus, the first problem reduces to asking about the number of components of a curve of degree $m$. The answer to this question, which was found by Harnack [Har-76] in 1876, is described in Sections 1.6 and 1.8 below.

The second problem has a more naive formulation as the question of how a nonsingular curve of degree $m$ can be situated in $\mathbb{R}P^2$. Here we are really talking about the isotopy classification, since any homeomorphism $\mathbb{R}P^2\to \mathbb{R}P^2$ is isotopic to the identity map. At present the second problem has been solved only for $m\le 7$. The solution is completely elementary when $m\le 5$: it was known in the last century, and we shall give the result in this section. But before proceeding to an exposition of these earliest achievements in the study of the topology of real algebraic curves, we shall recall the isotopy classification of closed one-dimensional submanifolds of the projective plane.