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 Gr\"obner basis techniques to solve multivariate polynomial equations.