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

TL;DR

This paper classifies the topological types of pencils of cubics passing through six or seven points in the real projective plane, revealing the combinatorial structures and symmetries involved.

Contribution

It provides a complete classification of combinatorial pencils of cubics with nodes at specified points, considering symmetries and general position configurations.

Findings

Seven possible combinatorial pencils with node at a fixed point.

Four possible lists of six pencils when varying the node among six points.

Fourteen possible lists of seven nodal cubics passing through seven points.

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 $1, ...7$.