Almost everything in the first two-thirds of the article concerning interlacings of lines, as well as everything concerning nonsingular sets of points in three-dimensional space, was published by Viro in 1985 in the note []3. Interest in this subject was stimulated by work of Kharlamov on the classification of nonsingular real projective algebraic surfaces of degree 4 up to rigid isotopy (by which one means isotopies consisting of nonsingular algebraic surfaces). Earlier, a coarser classification of such surfaces up to mirror reflections and rigid isotopies was found by Nikulin [16]; and Kharlamov, using a very complicated technique which involved passing to the complex domain, proved that certain surfaces are nonamphiheiral, in the sense that they are not rigid isotopic to their mirror images. It would be worthwhile to find an elementary proof.
Some of these surfaces decompose in the ambient three-dimensional space into a one-sheeted hyperboloid with handles and a number of separate spheres (the sum of the number of handles and the number of spheres is at most ten, and there are other restrictions, but we shall not dwell on this). From Harnack's theorem on the number of components of a plane curve it follows that every plane intersects at most three of the spheres of this surface. Hence, if we choose one point on each sphere, we obtain a nonsingular set of points whose isotopy type is determined by the surface, and a rigid isotopy of the surface corresponds to an isotopy of the set of points. Thus, if there are six or seven spheres, the surface must be nonamphiheiral. In a similar way Kharlamov proved that many other degree 4 surfaces are nonamphiheiral and completed the classification of nonsingular surfaces of degree 4 (see [17]). However, he was able to prove that certain of the surfaces are nonamphiheiral only by passing to the complex domain and using the full theory of K3-surfaces.