1.5 The Classical Method of Constructing Nonsingular Plane Curves

All of the classical constructions of the topology of nonsingular plane curves are based on a single construction, which I will call classical small perturbation. Some special cases were given in the previous subsection. Here I will give a detailed description of the conditions under which it can be applied and the results.

We say that a real singular point $\xi=(\xi_0:\xi_1:\xi_2)$ of the curve $A$ is an intersection point of two real transversal branches, or, more briefly, a crossing,² if the polynomial $a$ defining the curve has matrix of second partial derivatives at the point $(\xi_0,\xi_1,\xi_2)$ with both a positive and a negative eigenvalue, or, equivalently, if the point $\xi$ is a nondegenerate critical point of index 1 of the functions $\{x\in \mathbb{R}P^2\vert x_i\ne 0\}\to \mathbb{R}\:x\mapsto a(x)/x_i\deg a$ for $i$ with $\xi_i\ne 0$. By Morse lemma (see, e.g. [Mil-69]) in a neighborhood of such a point the curve looks like a union of two real lines. Conversely, if $\mathbb{R}A_1,\dotsc,\mathbb{R}A_k$ are nonsingular mutually transverse curves no three of which pass through the same point, then all of the singular points of the union $\mathbb{R} A_1\cup\cdots\cup \mathbb{R}A_k$ (this is precisely the pairwise intersection points) are crossings.

1.5.A Classical Small Perturbation Theorem (see Figure 5)   Let $A$ be a plane curve of degree $m$ all of whose singular points are crossings, and let $B$ be a plane curve of degree $m$ which does not pass through the singular points of $A$. Let $U$ be a regular neighborhood of the curve $\mathbb{R}A$ in $\mathbb{R}P^2$, represented as the union of a neighborhood $U_0$ of the set of singular points of $A$ and a tubular neighborhood $U_1$ of the submanifold $\mathbb{R}A\smallsetminus U_0$ in $\mathbb{R}P^2\smallsetminus U_0$.

Then there exists a nonsingular plane curve $X$ of degree $m$ such that:

(1) $\mathbb{R}X\subset U$.

(2) For each component $V$ of $U_0$ there exists a homeomorphism $h\:V\to D^1\times D^1$ such that $h(\mathbb{R}A\cap V)=D^1\times 0\cup 0\times D^1$ and $h(\mathbb{R}X\cap V)=\{(x,y)\in D^1\times D^1\vert xy=1/2\}$.

(3) $\mathbb{R}X\smallsetminus U_0$ is a section of the tubular fibration $U_1\to \mathbb{R}A\smallsetminus U_0$.

(4) $\mathbb{R}X\subset\{(x_0:x_1:x_2)\in\mathbb{R}P^2\vert a(x_0,x_1,x_2)b (x_0,x_1,x_2)\le 0\}$, where $a$ and $b$ are polynomials defining the curves $A$ and $B$.

(5) $\mathbb{R}X\cap \mathbb{R}A=\mathbb{R}X\cap \mathbb{R}B=\mathbb{R}A\cap \mathbb{R}B$.

(6) If $p\in \mathbb{R}A\cap \mathbb{R}B$ is a nonsingular point of $B$ and $\mathbb{R}B$ is transversal to $\mathbb{R}A$ at this point, then $\mathbb{R}X$ is also transversal to $\mathbb{R}A$ at the point.

There exists $\varepsilon>0$ such that for any $t\in (0,\varepsilon]$ the curve given by the polynomial $a+tb$ satisfies all of the above requirements imposed on $X$.

Figure 5:

It follows from (1)-(3) that for fixed $A$ the isotopy type of the curve $\mathbb{R}X$ depends on which of two possible ways it behaves in a neighborhood of each of the crossings of the curve $A$, and this is determined by condition (4). Thus, conditions (1)-(4) characterize the isotopy type of the curve $\mathbb{R}X$. Conditions (4)-(6) characterize its position relative to $\mathbb{R}A$.

We say that the curves defined by the polynomials $a+tb$ with $t\in (0,\varepsilon]$ are obtained by small perturbations of $A$ directed to the curve $B$. It should be noted that the curves $A$ and $B$ do not determine the isotopy type of the perturbed curves: since both of the polynomials $b$ and $-b$ determine the curve $B$, it follows that the polynomials $a-tb$ with small $t>0$ also give small perturbations of $A$ directed to $B$. But these curves are not isotopic to the curves given by $a+tb$ (at least not in $U)$, if the curve $A$ actually has singularities.

Proof. Proof of Theorem 1.5.A We set $x_t=a+tb$. It is clear that for any $t\ne 0$ the curve $X_t$ given by the polynomial $x_t$ satisfies conditions (5) and (6), and if $t>0$ it satisfies (4). For small $\vert t\vert$ we obviously have $\mathbb{R} X_t\subset U$. Furthermore, if $\vert t\vert$ is small, the curve $\mathbb{R}X_t$ is nonsingular at the points of intersection $\mathbb{R}X_t\cap \mathbb{R}B=\mathbb{R}A\cap \mathbb{R}B$, since the gradient of $x_t$ differs very little from the gradient of $a$ when $\vert t\vert$ is small, and the latter gradient is nonzero on $\mathbb{R}A\cap \mathbb{R}B$ (this is because, by assumption, $B$ does not pass through the singular points of $A)$. Outside $\mathbb{R}B$ the curve $\mathbb{R}X_t$ is a level curve of the function $a/b$. On $\mathbb{R}A\smallsetminus \mathbb{R}B$ this level curve has critical points only at the singular points of $\mathbb{R}A$, and these critical points are nondegenerate. Hence, for small $t$ the behavior of $\mathbb{R}X_t$ outside $\mathbb{R}B$ is described by the implicit function theorem and Morse Lemma (see, for example, [Mil-69]); in particular, for small $t\ne 0$ this curve is nonsingular and satisfies conditions (2) and (3). Consequently, there exists $\varepsilon>0$ such that for any $t\in (0,\varepsilon]$ the curve $\mathbb{R}X_t$ is nonsingular and satisfies (1)-(6). $\qedsymbol$