Digression on Real Plane Algebraic Curves

The word curve is known to be one of the most ambiguous in mathematics. Thus it makes sense to specify the type of curve to be considered. The curves to be considered here are real algebraic plane curves, i. e. plane curves which are defined by equations $f=0$, where $f$ is a polynomial over the field $\mathbb{R}$ of real numbers. The constructions of curves, which we consider below, can be described as constructions of real polynomials $f(x,y)$ of a given degree such that the curves $f(x,y)=0$ are positioned in a complicated (for this degree) way in the plane $\mathbb{R}^2$.

However, for many reasons we prefer projective curves. To a reader who does not like (i. e. is not familiar with) the projective plane, we suggest the following motivations and definitions.

It was probably Isaac Newton [10] who first observed that a curve $f(x,y)=0$ in the plane $\mathbb{R}^2$ is a more complicated object (e. g., to classify) than the cone generated by it in $\mathbb{R}^3$. If $m$ is the degree of $f$, then the cone is defined by the equation $z^mf(x/z,y/z)=0$. Newton [10] found 99 classes of curves of degree 3 on $\mathbb{R}^2$, but at the end of his text noted that curves of all 99 classes can be obtained as plane sections of only 5 cubic cones.

In the nineteenth century this observation and similar ones led to the notion of the projective plane and the idea that it is simpler to study curves in the projective plane than in the affine plane.

The real projective plane $\mathbb{R}P^2$ can be defined as the set of lines in $\mathbb{R}^3$ passing through the origin $(0,0,0)$. The line passing through $(0,0,0)$ and $(x_0,x_1,x_2)$ is denoted by $(x_0\colon x_1\colon x_2)$; the numbers $x_0$, $x_1$, $x_2$ are called homogeneous coordinates of $(x_0:x_1:x_2)$.

A cone in $\mathbb{R}^3$ with vertex $(0,0,0)$ can be viewed as a collection of lines lying on it. Since it is a one-parameter collection, it can be thought of as a curve in the projective plane. An equation $F(x,y,z)=0$, where $F$ is a homogeneous real polynomial, defines a cone in $\mathbb{R}^3$ with vertex $(0,0,0)$ and hence a curve in the projective plane $\mathbb{R}P^2$.

Take a curve on $\mathbb{R}^2$ defined by an equation $f(x,y)=0$ of degree $m$, shift it with its plane to the plane $z=1$ in $\mathbb{R}^3$ and consider lines passing through it and the origin $(0,0,0)$. These lines lie on the cone $z^mf(x/z,y/z)=0$ and fill it besides its intersection with the plane $x=0$. The corresponding curve on $\mathbb{R}P^2$ is called the projective completion of the affine curve $f(x,y)=0$. A study of real algebraic curves in the affine plane $\mathbb{R}^2$ is splits naturally into a study of their projective completions and an investigation of the position of the completions with respect to the line of infinity which is just the difference $\mathbb{R}P^2\smallsetminus \mathbb{R}^2$.

A curve (at least, an algebraic curve) is something more than just the set of points which belong to it. It is only slightly less than its equation: equations differing by a constant factor define the same curve. Modern algebraic geometry provides a lot of ways to define algebraic curve. Since we want to be as understandable as possible, we accept the following definition, which at first glance seems to be overly algebraic.

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

A point $(x_0\colon x_1\colon x_2)\in \mathbb{R}P^2$ is called a (real) singular point of the curve defined by a polynomial $F$ if the first partial derivatives of $F$ vanish in $(x_0,x_1,x_2)$ (vanishing of the derivatives implies vanishing of the homogeneous polynomial: by the Euler formula $deg(F)\cdot F(x_0,x_1,x_2)=\sum_ix_i\frac{\partial F}{\partial x_i}(x_0,x_1,x_2)$). A curve is said to be (real) nonsingular if it has no real singular points. The set $\mathbb{R}A$ of real points of a nonsingular real projective plane curve $A$ is a smooth closed one-dimensional submanifold of the projective plane. Then $\mathbb{R}A$ is a union of disjoint circles smoothly embedded in $\mathbb{R}P^2$. A circle can be positioned in $\mathbb{R}P^2$ either one-sidedly, like a projective line, or two-sidedly, like a conic. A two-sided circle is called an oval. An oval divides $\mathbb{R}P^2$ into two parts. The part homeomorphic to a disk is called the interior of the oval. Two ovals can be situated in two topologically distinct ways: each may lie outside the other one--i.e., each is in the outside component of the complement of the other--or else one of them is in the inside component of the complement of the other--in that case, we say that the first is the inner oval of the pair and the second is the outer oval. In the latter case we also say that the outer oval of the pair envelopes the inner oval. The topological type of the pair $(\mathbb{R}P^2, \; \mathbb{R}A)$ is defined by the scheme of disposition of the ovals of $\mathbb{R}A$. This scheme is called the real scheme of curve $A$.

In 1900 D. Hilbert [7] included the following question in the 16-th problem of his famous list: what real schemes can be realized by curves of a given degree? The complete answer is known now only for curves of degree not greater than 7.