On the coincidence of Pascal lines

TL;DR

This paper characterizes special configurations of six points on a conic where Pascal lines coincide, revealing two main geometric components and employing computational algebraic techniques.

Contribution

It provides a complete synthetic classification of sextuples with coincident Pascal lines, identifying two key geometric configurations and using Gr"obner basis methods.

Findings

Identifies two irreducible components of sextuples with coincident Pascal lines.

Characterizes sextuples in involution and ricochet configurations.

Uses Gr"obner basis techniques to analyze polynomial equations.

Abstract

Let ${\mathcal K}$ denote a smooth conic in the complex projective plane. Pascal's theorem says that, given six points $A,B,C,D,E,F$ on ${\mathcal K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. This defines the Pascal line of the array $\left[ \begin{array}{ccc} A & B & C \\ F & E & D \end{array} \right]$, and one gets sixty such lines in general by permuting the points. In this paper we consider the variety $\Psi$ of sextuples $\{A, \dots, F\}$, for which some of these Pascal lines coincide. We show that $\Psi$ has two irreducible components: a five-dimensional component of sextuples in involution, and a four-dimensional component of the so-called `ricochet configurations'. This gives a complete synthetic characterisation of points in $\Psi$. The proof relies upon Gr\"obner basis techniques to solve multivariate polynomial equations.