1.4 Curves of Degree $\le 5$

If $m\le 5$, then it is easy to see that the prohibitions in the previous subsection are satisfied only by the following isotopy types.

Table 1:

$m$	Isotopy types of nonsingular plane curves of degree $m$
1	$\langle J\rangle$
2	$\langle 0\rangle, \langle 1\rangle$
3	$\langle J\rangle, \langle J\amalg 1\rangle$
4	$\langle 0\rangle, \langle 1\rangle, \langle 2\rangle, \langle 1\langle 1\rangle\rangle, \langle 3\rangle, \langle 4\rangle$
5	$\langle J\rangle, \langle J\amalg 1\rangle, \langle J\amalg 2\rangle, \la... ...langle J\amalg 4\rangle, \langle J\amalg 5\rangle, \langle J\amalg 6\rangle$

For $m\le 3$ the absence of other types follows from 1.3.B and 1.3.C; for $m=4$ it follows from 1.3.B, 1.3.C and 1.3.D, or else from 1.3.B, 1.3.C and 1.3.E; and for $m=5$ it follows from 1.3.B, 1.3.C and 1.3.E. It turns out that it is possible to realize all of the types in Table 1; hence, we have the following theorem.

1.4.A   ISOTOPY CLASSIFICATION OF NONSINGULAR REAL PLANE PROJECTIVE CURVES OF DEGREE . An isotopy class of topological plane curves contains a nonsingular curve of degree $m\le 5$ if and only if it occurs in the $m$-th row of Table 1.

The curves of degree $\le 2$ are known to everyone. Both of the isotopy types of nonsingular curves of degree 3 can be realized by small perturbations of the union of a line and a conic which intersect in two real points (Figure 2). One can construct these perturbations by replacing the left side of the equation $cl=0$ defining the union of the conic $C$ and the line $L$ by the polynomial $cl+\varepsilon l_1l_2l_3$, where $l_i=0$, $i=1,2,3$, are the equations of the lines shown in 2, and $\varepsilon$ is a nonzero real number which is sufficiently small in absolute value.

Figure 2:

Figure 3:

Figure 4:

It will be left to the reader to prove that one in fact obtains the curves in Figure 2 as a result; alternatively, the reader can deduce this fact from the theorem in the next subsection.

The isotopy types of nonempty nonsingular curves of degree 4 can be realized in a similar way by small perturbations of a union of two conics which intersect in four real points (Figure 3). An empty curve of degree 4 can be defined, for example, by the equation $x^4_0+x^4_1+x^4_2=0$.

All of the isotopy types of nonsingular curves of degree 5 can be realized by small perturbations of the union of two conics and a line, shown in Figure 4. $\qedsymbol$

For the isotopy classification of nonsingular curves of degree 6 it is no longer sufficient to use this type of construction, or even the prohibitions in the previous subsection. See Section 1.13.