5.4 Surfaces of Low Degree

Surfaces of degree 1 and 2 are well-known. Any surface of degree 1 is a projective plane. All of them are transformed to each other by a rigid isotopy consisting of projective transformations of the whole ambient space $\mathbb{R}P^3$.

Nonsingular surfaces of degree 2 (nonsingular quadrics) are of three types. It follows from the well-known classification of real nondegenerate quadratic forms in 4 variables up to linear transformation. Indeed, by this classification any such a form can be turned to one of the following:

1. $+x_0^2+x_1^2+x_2^2+x_3^2$,
2. $+x_0^2+x_1^2+x_2^2-x_3^2$,
3. $+x_0^2+x_1^2-x_2^2-x_3^2$,
4. $+x_0^2-x_1^2-x_2^2-x_3^2$,
5. $-x_0^2-x_1^2-x_2^2-x_3^2$.

Multiplication by $-1$ identifies the first of them with the last and the second with the fourth reducing the number of classes to three. Since the reduction of a quadratic form to a canonical one can be done in a continuous way, all quadrics belonging to the same type also can be transformed to each other by a rigid isotopy made of projective transformations.

The first of the types consists of quadrics with empty set of real points. In traditional analytic geometry these quadrics are called imaginary ellipsoids. A canonical representative of this class is defined by equation $x_0^2+x_1^2+x_3^2+x_4^2=0$.

The second type consists of quadrics with the set of real points homeomorphic to sphere. In the notations of the previous section this is $S$. The canonical equation is $x_0^2+x_1^2+x_2^2-x_3^2=0$.

The third type consists of quadrics with the set of real points homeomorphic to torus. They are known as one-sheeted hyperboloids. The set of real points is not contractible (it contains a line), so in the notations above it should be presented as $S_1^1$. The canonical equation is $x_0^2+x_1^2-x_2^2-x_3^2=0$.

Quadrics of the last two types (i. e., quadrics with nonempty real part) can be obtained by small perturbations of a union of two real planes. To obtain a quadric with real part homeomorphic to sphere, one may perturb the union of two real planes in the following way. Let the plane be defined by equations $L_1(x_0,x_1,x_2,x_3)=0$ and $L_2(x_0,x_1,x_2,x_3)=0$. Then the union is defined by equation $L_1(x_0,x_1,x_2,x_3)L_2(x_0,x_1,x_2,x_3)=0$. Perturb this equation adding a small positive definite quadratic form. Say, take

$$L_1(x_0,x_1,x_2,x_3)L_2(x_0,x_1,x_2,x_3)+\varepsilon (x_0^2+x_1^2+x_2^2+x_3^2)=0$$

with a small $\varepsilon>0$. This equation defines a quadric. Its real part does not meet plane $L_1(x_0,x_1,x_2,x_3)=L_2(x_0,x_1,x_2,x_3)$, since on the real part of the quadric the product $L_1(x_0,x_1,x_2,x_3)L_2(x_0,x_1,x_2,x_3)$ is negative. Therefore the real part of the quadric is contractible in $\mathbb{R}P^3$. Since it is obtained by a perturbation of the union of two planes, it is not empty, provided $\varepsilon>0$ is small enough. As easy to see, it is not singular for small $\varepsilon>0$. Of course, this can be proven explicitely, as an exercise in analytic geometry. See Figure 33

Figure 33: Two-sheeted hyperboloid as a result of small perturbation of a pair of planes.

To obtain a noncontractible nonsingular quadric (one-sheeted hyperboloid), one can perturb the same equation $L_1(x_0,x_1,x_2,x_3)L_2(x_0,x_1,x_2,x_3)=0$, but by a small form which takes both positive and negative values on the intersection line of the planes. See Figure 34.

Figure 34: One-sheeted hyperboloid as a result of small perturbation of a pair of planes.

Nonsingular surfaces of degree 3 (nonsingular cubics) are of five types. Here is the complete list of there topological types:

$$P,\quad P\amalg S,\quad P_1,\quad P_2,\quad P_3.$$

Let us prove, first, that only topological types from this list can be realized. Since the degree is odd, a nonsingular surface has to be one-sided. By 5.3.D if it is not connected, then it is homeomorphic to $P\amalg S$. By the Generalized Harnack Inequality 5.3.G, the total Betti number of the real part is at most $3^3-4\times3^2+6\times3=9$. On the other hand, the first Betti number of a projective plane with $g$ handles is $1+2g$ and the total Betti number $b_*(P_g)$ is $3+2g$. Therefore in the case of a nonsingular cubic with connected real part, it is of the type $P_g$ with $g\le 3$.

All the five topological types are realized by small perturbations of unions of a nonsingular quadric and a plane transversal to one another. This is similar to the perturbations considered above, in the case of spatial quadrics. See Figures 35 and 36.

Figure 35: Constructing cubic surfaces of types $P\amalg S$, $P$, $P_1$ and $P_2$.

Figure 36: Constructing a cubic surface of type $P_3$.

An alternative way to construct nonsingular surfaces of degree 3 of all the topological types is provided by a connection between nonsingular spatial cubics and plane nonsingular quartics. More precisely, there is a correspondence assigning a plane nonsingular quartic with a selected real double tangent line to a nonsingular spatial cubic with a selected real point on it. It goes as follows. Consider the projection of the cubic from a point selected on it to a plane. The projection is similar to the well-known stereographic projection of a sphere to plane.