Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7.

*(English)*Zbl 0576.14031
Topology, general and algebraic topology, and applications, Proc. int. Conf., Leningrad 1982, Lect. Notes Math. 1060, 187-200 (1984).

[For the entire collection see Zbl 0527.00015.]

In this work a new method of construction of a real plane algebraical curve with prescribed topology is introduced. By this method the constructions of curves of degree 7 are made. The new method is based on a construction that builds a new algebraic curve from several other ones. The new curve is arranged as a result of gluing of the initial curves. The construction can be interpreted as a perturbation of a curve with complicated singularities. While the classical methods of construction of curves consists in small perturbation of a singular curve having only nondegenerate singularities. The author describes a new simple construction of curves of degree 6. It is proved that a nonsingular curve of degree 6 with any possible mutual position of its ovals can be obtained by a small perturbation of the union of three ellipses tangent to one another in two points. The construction of curves of degree 6 and 7 is based on small perturbations of singular points of type \(J_{10}\).

In this work a new method of construction of a real plane algebraical curve with prescribed topology is introduced. By this method the constructions of curves of degree 7 are made. The new method is based on a construction that builds a new algebraic curve from several other ones. The new curve is arranged as a result of gluing of the initial curves. The construction can be interpreted as a perturbation of a curve with complicated singularities. While the classical methods of construction of curves consists in small perturbation of a singular curve having only nondegenerate singularities. The author describes a new simple construction of curves of degree 6. It is proved that a nonsingular curve of degree 6 with any possible mutual position of its ovals can be obtained by a small perturbation of the union of three ellipses tangent to one another in two points. The construction of curves of degree 6 and 7 is based on small perturbations of singular points of type \(J_{10}\).

Reviewer: S.V.Chmutov

##### MSC:

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

14Pxx | Real algebraic and real-analytic geometry |

14F45 | Topological properties in algebraic geometry |