Real algebraic plane curves: Constructions with controlled topology.

*(English. Russian original)*Zbl 0732.14026
Leningr. Math. J. 1, No. 5, 1059-1134 (1990); translation from Algebra Anal. 1, No. 5, 1-73 (1989).

From author’s abstract: This is a survey of the topology of real plane curves, concentrating on the constructive aspects of this theory, i.e., the problem of constructing curves of a given degree with a prescribed arrangement of its components. A large part of the paper is concerned with introductory material - the formulation of the basic problems and the history of the early development of the subject - so that the exposition is essentially self-contained. A detailed presentation is given of the technique of perturbing singular curves with controlled variation of the topology. The author plans to publish the final part of the survey in the following issues.

Curves with complicated singularities are perturbed, namely semi- quasihomogeneous singularities, and dissipations of such singularities are classified.

At the end of the paper counterexamples are constructed to Ragdale’s conjecture for a non-singular curve of even degree \(m: p\leq (3m^ 2- 6m+8)/8\) and \(n\leq (3m^ 2-6m)/8\) where p denotes the number of even ovals (which envelop an even number of other ovals) and n denotes the number of the other ones.

Curves with complicated singularities are perturbed, namely semi- quasihomogeneous singularities, and dissipations of such singularities are classified.

At the end of the paper counterexamples are constructed to Ragdale’s conjecture for a non-singular curve of even degree \(m: p\leq (3m^ 2- 6m+8)/8\) and \(n\leq (3m^ 2-6m)/8\) where p denotes the number of even ovals (which envelop an even number of other ovals) and n denotes the number of the other ones.

Reviewer: F.Broglia (Pisa)

##### MSC:

14P25 | Topology of real algebraic varieties |

14H20 | Singularities of curves, local rings |

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

14F45 | Topological properties in algebraic geometry |