1.7 Digression: the Space of Real Projective Plane Curves

By the definition of real projective algebraic plane curves of degree $m$, they form a real projective space of dimension $m(m+3)/2$. The homogeneous coordinates in this projective space are the coefficients of the polynomials defining the curves. We shall denote this space by the symbol $\mathbb{R}C_m$. Its only difference with the standard space $\mathbb{R}P^{m(m+3)/2}$ is the unusual numbering of the homogeneous coordinates. The point is that the coefficients of a homogeneous polynomial in three variables have a natural double indexing by the exponents of the monomials:

$$a(x_0,x_1,x_2)=\sum\substack{ i,j\ge 0 i+j\le m} a_{ij}x^{m -i-j}_0x^i_1 x^j_2.$$

We let $\mathbb{R}NC_m$ denote the subset of $\mathbb{R}C_m$ corresponding to the real nonsingular curves. It is obviously open in $\mathbb{R}C_m$. Moreover, any nonsingular curve of degree $m$ has a neighborhood in $\mathbb{R}NC_m$ consisting of isotopic nonsingular curves. Namely, small changes in the coefficients of the polynomial defining the curve lead to polynomials which give smooth sections of a tubular fibration of the original curve. This is an easy consequence of the implicit function theorem; compare with 1.5.A, condition (3).

Curves which belong to the same component of the space $\mathbb{R}NC_m$ of nonsingular degree $m$ curves are isotopic--this follows from the fact that nonsingular curves which are close to one another are isotopic. A path in $\mathbb{R}NC_m$ defines an isotopy in $\mathbb{R}P^2$ of the set of real points of a curve. An isotopy obained in this way is made of sets of real points of of real points of curves of degree $m$. Such an isotopy is said to be rigid. This definition naturally gives rise to the following classification problem, which is every bit as classical as problems 1.1.Aand 1.1.B.

1.7.A Rigid Isotopy Classification Problem   Classify the nonsingular curves of degree $m$ up to rigid isotopy, i.e., study the partition of the space $\mathbb{R}NC_m$ of nonsingular degree $m$ curves into its components.

If $m\le 2$, it is well known that the solution of this problem is identical to that of problem 1.1.B. Isotopy also implies rigid isotopy for curves of degree 3 and 4. This was known in the last century; however, we shall not discuss this further here, since it has little relevance to what follows. At present problem 1.7.A has been solved for $m\le 6$.

Although this section is devoted to the early stages of the theory, I cannot resist commenting in some detail about a more recent result. In 1978, V. A. Rokhlin [Rok-78] discovered that for $m\ge 5$ isotopy of nonsingular curves of degree $m$ no longer implies rigid isotopy. The simplest example is given in Figure 8, which shows two curves of degree 5. They are obtained by slightly perturbing the very same curve in Figure 4 which is made up of two conics and a line. Rokhlin's original proof uses argument on complexification, it will be presented below, in Section ??? Here, to prove that these curves are not rigid isotopic, we use more elementary arguements. Note that the first curve has an oval lying inside a triangle which does not intersect the one-sided component and which has its vertices inside the other three ovals, and the second curve does not have such an oval--but under a rigid isotopy the oval cannot leave the triangle, since that would entail a violation of Bézout's theorem.

Figure 8:

We now examine the subset of $\mathbb{R}C_m$ made up of real singular curves.

It is clear that a curve of degree $m$ has a singularity at $(1:0:0)$ if and only if its polynomial has zero coefficients of the monomials $x^m_0,x^{m-1}_0x_1,x^{m-1}_0x_2$. Thus, the set of real projective plane curves of degree $m$ having a singularity at a particular point forms a subspace of codimension 3 in $\mathbb{R}C_m$.

We now consider the space $S$ of pairs of the form $(p,C)$, where $p\in \mathbb{R}P^2$, $C\in \mathbb{R}C_m$, and $p$ is a singular point of the curve $C$. $S$ is clearly an algebraic subvariety of the product $\mathbb{R}P^2\times \mathbb{R}C_m$. The restriction to $S$ of the projection $\mathbb{R}P^2\times \mathbb{R}C_m\to \mathbb{R}P^2$ is a locally trivial fibration whose fiber is the space of curves of degree $m$ with a singularity at the corresponding point, i.e., the fiber is a projective space of dimension $m(m+3)/2-3$. Thus, $S$ is a smooth manifold of dimension $m(m+3)/ 2-1$. The restriction $S\to \mathbb{R}C_m$ of the projection $\mathbb{R}P^2\times \mathbb{R}C_m\to \mathbb{R}C_m$ has as its image precisely the set of all real singular curves of degree $m$, i.e., $\mathbb{R}C_m\smallsetminus \mathbb{R} NC_m$. We let $\mathbb{R}SC_m$ denote this image. Since it is the image of a $(m(m+3)/2-1)$-dimensional manifold under smooth map, its dimension is at most $m(m+3)/ 2-1$. On the other hand, its dimension is at least equal $m(m+3)/ 2-1$, since otherwise, as a subspace of codimension $\ge 2$, it would not separate the space $\mathbb{R}C_m$, and all nonsingular curves of degree $m$ would be isotopic.

Using an argument similar to the proof that $\dim \mathbb{R}SC_m\le m(m+3)/2-1$, one can show that the set of curves having at least two singular points and the set of curves having a singular point where the matrix of second derivatives of the corresponding polynomial has rank $\le 1$, each has dimension at most $m(m+3)/2-2$. Thus, the set $\mathbb{R}SC_m$ has an open everywhere dense subset consisting of curves with only one singular point, which is a nondegenerate double point (meaning that at this point the matrix of second derivatives of the polynomial defining the curve has rank 2). This subset is called the principal part of the set $\mathbb{R}SC_m$. It is a smooth submanifold of codimension 1 in $\mathbb{R}C_m$. In fact, its preimage under the natural map $S\to \mathbb{R}C_m$ is obviously an open everywhere dense subset in the manifold $S$, and the restriction of this map to the preimage is easily verified to be a one-to-one immersion, and even a smooth imbedding.

There are two types of nondegenerate real points on a plane curve. We say that a nondegenerate real double point $(\xi_0:\xi_1:\xi_2)$ on a curve $A$ is solitary if the matrix of second partial derivatives of the polynomial defining $A$ has either two nonnegative or two nonpositive eigenvalues at the point $(\xi_0,\xi_1,\xi_2)$. A solitary nondegenerate double point of $A$ is an isolated point of the set $\mathbb{R}A$. In general, a singular point of $A$ which is an isolated point of the set $\mathbb{R}A$ will be called a solitary real singular point. The other type of nondegenerate real double point is a crossing; crossings were discussed in Section 1.5 above. Corresponding to this division of the nondegenerate real double points into solitary points and crossings, we have a partition of the principal part of the set of real singular curves of degree $m$ into two open sets.

If a curve of degree $m$ moves as a point of $\mathbb{R}C_m$ along an arc which has a transversal intersection with the half of the principal part of the set of real singular curves consisting of curves with a solitary singular point, then the set of real points on this curve undergoes a Morse modification of index 0 or 2 (i.e., either the curve acquires a solitary double point, which then becomes a new oval, or else one of the ovals contracts to a point (a solitary nondegenerate double point) and disappears). In the case of a transversal intersection with the other half of the principal part of the set of real singular curves one has a Morse modification of index 1 (i.e., two arcs of the curve approach one another and merge, with a crossing at the point where they come together, and then immediately diverge in their modified form, as happens, for example, with the hyperbola in the family of affine curves of degree 2 given by the equation $xy=t$ at the moment when $t=0)$.

A line in $\mathbb{R}C_m$ is called a (real) pencil of curves of degree $m$. If $a$ and $b$ are polynomials defining two curves of the pencil, then the other curves of the pencil are given by polynomials of the form $\lambda a+\mu b$ with $\lambda,\mu\in\mathbb{R}\smallsetminus 0$.

By the transversality theorem, the pencils which intersect the set of real singular curves only at points of the principal part and only transversally form an open everywhere dense subset of the set of all real pencils of curves of degree $m$.