Rigid isotopic classification of curves of degree 6 with a nondegenerate double point. (Russian. English summary) Zbl 0762.14031

Itenberg, I. V.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 193, 72-89 (1991).

The subject of this paper is related to the first part of Hilbert’s 16th problem: topological classification of real algebraic varieties. The result is the complete rigid isotopic classification of real plane sextics with one node, namely, the author presents the classification of connected components of the set of real plane sextics with a node in the space of coefficients. The answer is given as a list of Coxeter schemes of some polyhedra. The method used here was discovered by V. V. Nikulin [Math. USSR, Izv. 22, 99-172 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 109-188 (1983; Zbl 0547.10021)]. The idea is that the double coverings of the plane branched at a sextic curve are \(K3\)-surfaces with a given polarization, hence the Torelli theorem and the surjectivity of the period map allow to describe all the components mentioned above in terms of the action of antiholomorphic involutions on the lattice of two-dimensional homology classes of a \(K3\)-surface.

Reviewer: E.I.Shustin (Kujbyshev)

MSC:

14P25	Topology of real algebraic varieties
14J28	\(K3\) surfaces and Enriques surfaces
14H45	Special algebraic curves and curves of low genus
14F35	Homotopy theory and fundamental groups in algebraic geometry

Keywords:

\(K3\)-surfaces; rigid isotopic classification of real plane sextics; Coxeter schemes