Rational pencils of cubics and configurations of six or seven points in $\mathbb{R}P^2$

Abstract

Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic a topological type (cubic, points), and combinatorial pencil the cyclic sequence of five combinatorial reducible cubics. Up to the action of the symmetric group $S_5$ on $\{2, ...6\}$, there are seven possible combinatorial pencils with node at 1. Consider now the set of six pencils obtained, making the node to be $1, ...6$. Up to the action of S_6 on ${1, ...6}$, there are four possible lists of six combinatorial pencils. Let seven points $1, ...7$ lie in general position in the plane. Up to the action of $S_7$ on $\{1, ...7\}$, there are fourteen possible lists of seven nodal combinatorial cubics passing through the seven points, with respective nodes at $1, ...7$.