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Progress in the topology of real algebraic varieties over the last six years. (English. Russian original) Zbl 0619.14015
Russ. Math. Surv. 41, No. 3, 55-82 (1986); translation from Usp. Mat. Nauk 41, No. 3(249), 45-67 (1986).
This is an extended version of the author’s report to the Warsaw Congress [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 603-619 (1984; Zbl 0571.14012)]. It is devoted to a circle of problems popularized by Hilbert in his 16-th problem from his famous talk at the Paris (1900) congress. Traditionally, the main question here is the determination of possibly mutual positions of connected components of a nonsingular plane real algebraic curve of given degree and similar problems in higher dimensions. Now this purely real picture of a real algebraic curve is enriched by some data coming from the complex domain. This important modification of the viewpoint is due to Rokhlin. Naturally, this point of view is the one adopted in the paper. The paper gives a rather complete and up to data survey (without proofs) of known facts about nonsingular plane real algebraic curves. In particular, in the last section the author’s method of construction of curves with prescribed topological properties is sketched. This method leads a complete classification of curves of degree 7- the most famous contribution of the author to the field. Now this method became the main research tool in the field. The complex point of view hopefully can be extended from curves to higher dimensional varieties. At the moment, only the case of surfaces is fairly well understood. This theme is also presented in the paper. Also, Kharlamov’s results on the classification of surfaces of degree 4 are covered.
The flowering of this field in the last fifteen years is due mainly to the late V. A. Rokhlin. He not only contributed many important ideas and results, but also created an active team of researchers. The author is one of the leaders of this team now. His authoritative survey of this field is of primarily importance for all interested in the topology of real algebraic varieties.
Reviewer: N.V.Ivanov

MSC:
14F45 Topological properties in algebraic geometry
14-03 History of algebraic geometry
14N05 Projective techniques in algebraic geometry
14Pxx Real algebraic and real-analytic geometry
14J25 Special surfaces
14H99 Curves in algebraic geometry
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