When are the Cayley-Salmon lines conjugate?

TL;DR

This paper characterizes when Cayley-Salmon lines are conjugate, showing it occurs exactly for tri-involutive sextuples, and explores the algebraic structure of their configuration space.

Contribution

It establishes a precise condition for Cayley-Salmon lines to be conjugate and analyzes the algebraic properties of the associated sextuple variety.

Findings

Cayley-Salmon lines are conjugate iff the sextuple is tri-involutive.

The sextuple variety is arithmetically Cohen-Macaulay of codimension two.

The $SL_2$-equivariant minimal resolution of the variety is determined.

Abstract

Given six points on a conic, Pascal's theorem gives rise to a well-known configuration called the \emph{hexagrammum mysticum}. It consists of, amongst other things, twenty Steiner points and twenty Cayley-Salmon lines. It is a classical theorem due to von Staudt that the Steiner points fall into ten conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. It is shown in this paper that the C-S lines are pairwise conjugate precisely when the original sextuple is~\emph{tri-involutive}. The variety of tri-involutive sextuples turns out to be arithmetically Cohen-Macaulay of codimension two. We determine its $SL_2$-equivariant minimal resolution.