Riemann existence theorem and construction of real algebraic curves
by S.Yu. Orevkov

Abstract

In this paper we propose a method of construction of plane real algebraic curves given by   f( x, y ) = 0,  deg_y( f ) = 3   which has a required arrangement on the affine plane with respect to the pencil of vertical lines. The methode is base on the techniques of "dessins d'enfant".

As an illustration, we give a construction of a new complex scheme of projective M-curve of degree 7 and a construction of 66 new isotopy types of M-curves of degree 9.