Pencils of cubics with eight base points lying in convex position in $\mathbb{R}P^2$

Abstract

To a generic configuration of eight points in convex position in the plane, we associate a list consisting of the following information: for all of the 56 conics determined by five of the points, we specify the position of each remaining point, inside or outside. We prove that the number of possible lists, up to the action of $D_8$ on the set of points, is 47, and we give two possible ways of encoding these lists. A generic compex pencil of cubics has twelve singular (nodal) cubics and nine distinct base points, any eight of them determine the ninth one, hence the pencil. If the base points are real, exactly eight of these singular cubics are distinguished, that is to say real with a loop containing some base points. We call combinatorial cubic a topological type (cubic, base points), and combinatorial pencil the sequence of eight successive combinatorial distinguished cubics. Let us choose representants of the 47 orbits. In most cases, but four exceptions, a list determines a unique combinatorial pencil. We give a complete classification of the combinatorial pencils of cubics with eight base points in convex position. Up to the action of $D_8$ on the set of points, there are $43$ such pencils.