5.1 Basic Definitions and Problems

Our consideration of real algebraic surfaces will be based on definitions similar to the definitions that we used in the case of curves. In particular, by a real algebraic surface of degree $m$ in the 3-dimensional projective space we shall mean a real homogeneous polynomial of degree $m$ in four variables considered up to a constant factor.

Obvious changes adapt definitions of sets of real and complex points, singular points, singular and nonsingular curves and rigid isotopy to the case of surfaces in $\mathbb{R}P^3$. Exactly as in the case of curves one formulates the topological classification problem (cf. 1.1.A above):

5.1.A. Topological Classification Problem.   Up to homeomorphism, what are the possible sets of real points of a nonsingular real projective algebraic surface of degree $m$ in $\mathbb{R}P^3$?

However, the isotopy classification problem 1.1.B splits into two problems:

5.1.B. Ambient Topological Classification Problem.   Classify up to homeomorphism the pairs $(\mathbb{R}P^3,\mathbb{R}A)$ where $A$ is a nonsingular real projective algebraic surface of degree $m$ in $\mathbb{R}P^3$?

5.1.C. Isotopy Classification Problem.   Up to ambient isotopy, what are the possible sets of real points of a nonsingular a nonsingular real projective algebraic surface of degree $m$ in $\mathbb{R}P^3$?

The reason for this splitting is that, contrary to the case of projective plane, there exists a homeomorphism of $\mathbb{R}P^3$ non-isotopic to the identity. Indeed, 3-dimensional projective space is orientable, and the mirror reflection of this space in a plane reverses orientation. Thus the reflection is not isotopic to the identity. However, there are only two isotopy classes of homeomorphisms of $\mathbb{R}P^3$. It means that the difference between 5.1.B and 5.1.C is not really big. Although the isotopy classification problem is finer, to resolve it, one should add to a solution of the ambient topological classification problem an answer to the following question:

5.1.D. Amphichirality Problem.   Which nonsingular real algebraic surfaces of degree $m$ in $\mathbb{R}P^3$ are isotopic to its own mirror image?

Each of these problems has been solved only for $m\le 4$. The difference between 5.1.B and 5.1.C does not appear: the solutions of 5.1.B and 5.1.C coincide with each other for $m\le 4$. (Thus Problem 5.1.D has a simple answer for $m\le 4$: any nonsingular real algebraic surface of degree $\le 4$ is isotopic to its mirror image.) For $m\le 3$ solutions of 5.1.A and 5.1.B also coincide, but for $m=4$ they are different: there exist nonsingular surfaces of degree 4 in $\mathbb{R}P^3$ which are homeomorphic, but embedded in $\mathbb{R}P^3$ in a such a way that there is no homeomorphism of $\mathbb{R}P^3$ mapping one of them to another. The simplest example is provided by torus defined by equation

$$(x_1^2+x_2^2+x_3^2+3x_0^2)^2-16(x_1^2+x_2^2)x_0^2=0$$

and the union of one-sheeted hyperboloid and an imaginary quadric (perturbed, if you wish to have a surface without singular points even in the complex domain)

Similar splitting happens with the rigid isotopy classification problem. Certainly, it may be transferred literally:

5.1.E. Rigid Isotopy Classification Problem.   Classify up to rigid isotopy the nonsingular surfaces of degree $m$.

However, since there exists a projective transformation of $\mathbb{R}P^3$, which is not isotopic to the identity (e.g., the mirror reflection in a plane) and a real algebraic surface can be nonisotopic rigidly to its mirror image, one may consider the following rougher problem:

5.1.F. Rough Projective Classification Problem.   Classify up to rigid isotopy and projective transformation the nonsingular surfaces of degree $m$.

Again, as in the case of topological isotopy and homeomorphism problem, the difference between these two problems is an amphichirality problem:

5.1.G. Rigid Amphichirality Problem.   Which nonsingular real algebraic
surfaces of degree $m$ in $\mathbb{R}P^3$ are rigidly isotopic to its mirror image?

Problems 5.1.E, 5.1.F and 5.1.G have been solved also for $m\le 4$. For $m\le 3$ the solutions of 5.1.E and 5.1.F coincide with each other and with the solutions of 5.1.A, 5.1.B and 5.1.C. For $m\le 2$ all these problems belong to the traditional analytic geometry. The solutions are well-known and can be found in traditional textbooks on analytic geometry. The case $m=3$ is also elementary. It was studied in the nineteenth century. The solution is associated with names of Schläfli and Klein. The case $m=4$ is really difficult. Although the first attempts of a serious attack were undertaken in the nineteenth century, too, and among the attackers we see D. Hilbert and K. Rohn, the complete solutions of all classification problems listed above were obtained only in the seventies and eighties. In higher degrees even the most rough problems, like the Harnack problem on the maximal number of components of a surface of degree $m$ are still open.