The conjugacy locus of Cayley-Salmon lines
A hexagon is inscribed into a conic \(\mathcal K\), if the intersection points of opposite sides are incident with a straight line, the \textit{Pascal line}. Permuting the hexagon vertices produces 60 Pascal axes that intersect, three at a time, in 20 \textit{Steiner points} and in 60 \textit{Kirkman points}. Three Kirman points and one Steiner point are incident with a total of 20 \textit{Cayley-Salmon lines}. The starting point of this article is the observation that a combinatorial duality between these points and lines does not extend to a geometric duality: While the Steiner points fall into ten pairs conjugate with respect to \(\mathcal K\), this is not generally the case for Cayley-Salmon lines. The main result states such a pairing for Cayley-Salmon lines is possible (and compatible with the combinatorial structure) if and only if the original sextuple of points is projectively equivalent to the set \(\{0,1,\infty,p,\frac{p-1}{p}, \frac{1}{1-p}\}\) and \(p \in \mathbb R\cup \{\infty\}\). Sextuples with this property are termed \textit{tri-symmetric} by the author. For the proof, he identifies the projective plane with quadratic forms in two variables and encodes geometric relations via transvectants. In this way, the statement is reduced to the vanishing of certain homogeneous polynomials. Several further resulting pertain to the geometric, combinatorial and algebraic structure of tri-symmetric sextuples. They possess, in general, precisely three collinear involution centers and can be grouped into two triples that have equal Desargues centers with respect to their respective polar triangles. The Zariski closure \(\Omega\) of the subset of tri-symmetric sextuples in \((P^1)^6\) is arithmetically Cohen-Macaulay. Its dimension is four and its degree is 16.
scientific article; zbMATH DE number 6854821
scientific article