2.5 Digression: Oriented Topological Plane Curves

Consider an oriented topological plane curve, i. e. an oriented closed one-dimensional submanifold of the projective plane, cf. 1.2.

A pair of its ovals is said to be injective if one of the ovals is enveloped by the other.

An injective pair of ovals is said to be positive if the orientations of the ovals determined by the orientation of the entire curve are induced by an orientation of the annulus bounded by the ovals. Otherwise, the injective pair of ovals is said to be negative. See Figure 24. It is clear that the division of pairs of ovals into positive and negative pairs does not change if the orientation of the entire curve is reversed; thus, the injective pairs of ovals of a semioriented curve (and, in particular, a curve of type I) are divided into positive and negative. We let $\Pi^+$ denote the number of positive pairs, and $\Pi^-$ denote the number of negative pairs.

Figure 24:

The ovals of an oriented curve one-sidedly embedded into $\mathbb{R}P^2$ can be divided into positive and negative. Namely, consider the Möbius strip which is obtained when the disk bounded by an oval is removed from $\mathbb{R}P^2$. If the integral homology classes which are realized in this strip by the oval and by the doubled one-sided component with the orientations determined by the orientation of the entire curve coincide, we say that the oval is negative, otherwise we say that the oval is positive. See Figure 25. In the case of a two-sided oriented curve, only the non-outer ovals can be divided into positive and negative. Namely, a non-outer oval is said to be positive if it forms a positive pair with the outer oval which envelops it; otherwise, it is said to be negative. As in the case of pairs, if the orientation of the curve is reversed, the division of ovals into positive and negative ones does not change. Let $\Lambda^+$ denote the number of positive ovals on a curve, and let $\Lambda^-$ denote the number of negative ones.

Figure 25:

To describe a semioriented topological plane curve (up to homeomorphism of the projective plane) we need to enhance the coding system introduced in 1.2. The symbols representing positive ovals will be equipped with a superscript $+$, the symbols representing negative ovals, with a superscript $-$. This kind of code of a semioriented curve is complete in the following sense: for any two semioriented curves with the same code there exists a homeomorphism of $\mathbb{R}P^2$, which maps one of them to the other preserving semiorientations.

To describe the complex scheme of a curve of degree $m$ we will use, in the case of type I, the scheme of the kind described here, for its complex semiorientation, equipped with subscript I and superscript $m$ and, in the case of type II, the notation used for the real scheme, but equipped with subscript II and superscript $m$.

It is easy to check, that the coding of this kind of the complex scheme of a plane projective real algebraic curve describes the union of $\mathbb{R}P^2$ and the complex point set of the curve up to a homeomorphism mapping $\mathbb{R}P^2$ to itself.

In these notations, the complex schemes of cubic curves shown in Figure 17 are $\langle J\rangle_{II}^3$ and $\langle J\amalg 1^-\rangle_I^3$.

The complex schemes of quartic curves realized in Figure 18 are $\langle 0\rangle_{II}^4$, $\langle 1\rangle_{II}^4$, $\langle 2\rangle_{II}^4$, $\langle 1\langle 1^-\rangle\rangle_{I}^4$, $\langle 3\rangle_{II}^4$, $\langle 4\rangle_{I}^4$.

The complex schemes of quintic curves realized in Figure 19 are $\langle J\rangle_{II}^5$, $\langle J\amalg 1\rangle_{II}^5$, $\langle J\amalg 2\rangle_{II}^5$, $\langle J\amalg 1^-\langle 1^-\rangle\rangle_I^5$, $\langle J\amalg 3\rangle_{II}^5$, $\langle J\amalg 4\rangle_{II}^5$, $\langle J\amalg1^+\amalg3^-\rangle_{I}^5$ $\langle J\amalg 5\rangle_{II}^5$, $\langle J\amalg 3^+\amalg^-\rangle_{I}^5$.

In fact, these lists of complex schemes contain all schemes of nonsingular algebraic curves for degrees 3 and 5 and all nonempty schemes for degree 4. To prove this, we need not only constructions, but also restrictions on complex schemes. In the next two sections restrictions sufficient for this will be provided.