The conjugacy locus of Cayley–Salmon lines

Abstract

Abstract Given six points on a conic, Pascal’s theorem gives rise to a configuration called the hexagrammum mysticum. It contains 20 Steiner points and 20 Cayley–Salmon lines. By a classical theorem due to von Staudt, the Steiner points fall into 10 conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. We show that the C-S lines are pairwise conjugate precisely when the original sextuple is tri-symmetric. The variety of tri-symmetric sextuples turns out to be arithmetically Cohen–Macaulay of codimension two. We determine its SL2-equivariant minimal resolution.