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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.

##### MSC:
 14H45 Special algebraic curves and curves of low genus 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14P25 Topology of real algebraic varieties