2.1 Complex Topological Characteristics of a Real Curve

According to a tradition going back to Hilbert, for a long time the main question concerning the topology of real algebraic curves was considered to be the determination of which isotopy types are realized by nonsingular real projective algebraic plane curves of a given degree (i.e., Problem 1.1.B above). However, as early as in 1876 F. Klein [Kle-22] posed the question more broadly. He was also interested in how the isotopy type of a curve is connected to the way the set $\mathbb{R}A$ of its real points is positioned in the set $\mathbb{C}A$ of its complex points (i.e., the set of points of the complex projective plane whose homogeneous coordinates satisfy the equation defining the curve).

The set $\mathbb{C}A$ is an oriented smooth two-dimensional submanifold of the complex projective plane $\mathbb{C}P^2$. Its topology depends only on the degree of $A$ (in the case of nonsingular $A$). If the degree is $m$, then $\mathbb{C}A$ is a sphere with $\frac12(m-1)(m-2)$ handles. (It will be shown in Section 2.3.) Thus the literal complex analogue of Topological Classification Problem 1.1.A is trivial.

The complex analogue of Isotopy Classification Problem 1.1.B leads also to a trivial classification: the topology of the pair $(\mathbb{C}P^2,\mathbb{C}A)$ depends only on the degree of $A$, too. The reason for this is that the complex analogue of a more refined Rigid Isotopy Classification problem 1.7.A has a trivial solution: nonsingular complex projective curves of degree $m$ form a space $\mathbb{C}NC_m$ similar to $\mathbb{R}NC_m$ (see Section 1.7) and this space is connected, since it is the complement of the space $\mathbb{C}SC_m$ of singular curves in the space $\mathbb{C}C_m (=\mathbb{C}P^{\frac12m(m+3)})$ of all curves of degree $m$, and $\mathbb{C}SC_m$ has real codimension 2 in $\mathbb{C}C_m$ (its complex codimension is 1).

The set $\mathbb{C}A$ of complex points of a real curve $A$ is invariant under the complex conjugation involution $conj:\mathbb{C}P^2\to \mathbb{C}P^2:(z_0:z_1:z_2)\mapsto (\overline z_0:\overline z_1:\overline z_2)$. The curve $\mathbb{R}A$ is the fixed point set of the restriction of this involution to $\mathbb{C}A$.

The real curve $\mathbb{R}A$ may divide or not divide $\mathbb{C}A$. In the first case we say that $A$ is a dividing curve or a curve of type I, in the second case we say that it is a nondividing curve or a curve of type II. In the first case $\mathbb{R}A$ divides $\mathbb{C}A$ into two connected pieces.³ The natural orientations of these two halves determine two opposite orientations on $\mathbb{R}A$ (which is their common boundary); these orientations of $\mathbb{R}A$ are called the complex orientations of the curve.

A pair of orientations opposite to each other is called a semiorientation. Thus the complex orientations of a curve of type I comprise a semiorientation. Naturally, the latter is called a complex semiorientation.

The scheme of relative location of the ovals of a curve is called the real scheme of the curve. The real scheme enhanced by the type of the curve, and, in the case of type I, also by the complex orientations, is called the complex scheme of the curve.

We say that the real scheme of a curve of degree $m$ is of type I (type II) if any curve of degree $m$ having this real scheme is a curve of type I (type II). Otherwise (i.e., if there exist curves of both types with the given real scheme), we say that the real scheme is of indeterminate type.

The division of curves into types is due to Klein [Kle-22]. It was Rokhlin [Rok-74] who introduced the complex orientations. He introduced also the notion of complex scheme and its type [Rok-78]. In the eighties the point of view on the problems in the topology of real algebraic varieties was broadened so that the role of the main object passed from the set of real points, to this set together with its position in the complexification. This viewpoint was also promoted by Rokhlin.

As we will see, the notion of complex scheme is useful even from the point of view of purely real problems. In particular, the complex scheme of a curve is preserved under a rigid isotopy. Therefore if two curves have the same real scheme, but distinct complex schemes, the curves are not rigidly isotopic. The simplest example of this sort is provided by the curves of degree 5 shown in Figure 8, which are isotopic but not rigidly isotopic.