In the previous two sections we discovered that a knowledge on topology of the complexification gives restriction on topology of real part of the curve under consideration. More detailed topological information on complexification can be obtained using geometric constructions involving auxiliary curves. They use Bézout theorem. Therefore they cannot be applied to flexible curves. Here we consider first the simplest of arguments of that sort, and then obtain some special results on curves of low degrees (up to 8) which, together with forthcoming constructions will be useful in solution of some classification problems.
We will use the simplest auxiliary curves: lines. Consideration of a pencil of lines (the set of all lines passing through a point) and intersection of a curve with lines of this pencil can be thought of as a study of the curve by looking at it from the common point of the lines. However, since imaginary lines of the pencil can be included into this study and even real lines may intersect the curve in imaginary points, we have a chance to find out something on the complex part of the curve.