On the coincidence of Pascal lines

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 Gröbner basis techniques to solve multivariate polynomial equations.

22 pages, with 7 diagrams