Pencils of cubics as tools to solve an interpolation problem

Abstract

Consider a convex piece of curve in the plane and four points lying on it. One wants to find a sufficient condition on these points to grant that there exists a rational cubic passing through them, and interpolating the curve in a prescribed way. We found such a condition, studying real pencils of cubics. A generic complex pencil has twelve singular (nodal) cubics and nine distinct base points; any eight of them determines the ninth one and hence the pencil. If the pencil is real, exactly eight of its singular cubics are distinguished, that is real, with a loop containing some base points. If eight of the base points lie in convex position in ${\mathbb{R}P^2}$ , one can often deduce the sequence of distinguished cubics out of some simpler combinatorial data, expressed in terms of mutual positions of points and conics.