Complex topological invariants of real algebraic curves on a hyperboloid and on an ellipsoid.

*(English. Russian original)*Zbl 0793.14020
St. Petersbg. Math. J. 3, No. 5, 1023-1042 (1992); translation from Algebra Anal. 3, No. 5, 88-108 (1991).

The relations between the topology of a real algebraic curve lying on a nonsingular real quadric and the topology of complexification of curve in the complexification of the quadric are studied. A real algebraic curve is of type I if it separates its complexification and of type II if it does not. A curve of type I has complex orientations, as the edge of the surfaces into which it separates its complexification. The diagram of the dispositions on the quadric of the components of a real algebraic curve along with the type of curve and, for type I curves, the complex orientations is called the complex scheme.

In the present paper the author’s results from Funct. Anal. Appl. 16, 202-204 (1983); translation from Funkts. Anal. Prilozh. 16, No. 3, 56-57 (1982; Zbl 0572.14015) and Sov. Math., Dokl. 27, 14-17 (1983); translation from Dokl. Akad. Nauk SSSR 268, 22-26 (1983; Zbl 0637.14024) are applied to the study of real algebraic curves on a hyperboloid and on an ellipsoid. We also find some new properties of the complex topological invariants of such curves and classify complex schemes for nonsingular curves of order \(m \leq 8\). We follow V. A. Rokhlin’s ideas used in Russ. Math. Surv. 33, 85-98 (1978); translation from Usp. Mat. Nauk 33, No. 5(203), 77-89 (1978; Zbl 0437.14013) for the study of complex topological characteristics of plane curves. Except for §3.7, the results are formulated only for nonsingular curves.

In the present paper the author’s results from Funct. Anal. Appl. 16, 202-204 (1983); translation from Funkts. Anal. Prilozh. 16, No. 3, 56-57 (1982; Zbl 0572.14015) and Sov. Math., Dokl. 27, 14-17 (1983); translation from Dokl. Akad. Nauk SSSR 268, 22-26 (1983; Zbl 0637.14024) are applied to the study of real algebraic curves on a hyperboloid and on an ellipsoid. We also find some new properties of the complex topological invariants of such curves and classify complex schemes for nonsingular curves of order \(m \leq 8\). We follow V. A. Rokhlin’s ideas used in Russ. Math. Surv. 33, 85-98 (1978); translation from Usp. Mat. Nauk 33, No. 5(203), 77-89 (1978; Zbl 0437.14013) for the study of complex topological characteristics of plane curves. Except for §3.7, the results are formulated only for nonsingular curves.

##### MSC:

14H45 | Special algebraic curves and curves of low genus |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14P25 | Topology of real algebraic varieties |