The conjugacy locus of Cayley-Salmon lines
type: Item
instance of: scholarly article
title: The conjugacy locus of Cayley-Salmon lines
zbMATH Open document ID: 1390.14159
publication date: 2018-03-29T00:00:00Z
review text: 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.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 51N35
zbMATH DE Number: 6854821
zbMATH Keywords: conic
zbMATH Keywords: hexagrammum mysticum
zbMATH Keywords: Cayley-Salmon line
author: Jaydeep Chipalkatti
MaRDI profile type: Publication
full work available at URL: advgeom-2017-0045
OpenAlex ID: W2783862141
cites work: Linear orbits of d-tuples of points in P^1
cites work: Q3078305
cites work: On Hermite's invariant for binary quintics
cites work: On the coincidences of Pascal lines
cites work: The Pascal mysticum demystified
cites work: Twelve points in PG (5, 3) with 95040 self-transformations
cites work: Projective Geometry
cites work: Q4780265
cites work: Q4735276
cites work: Q3077977
cites work: Q4846449
cites work: Q3041329
cites work: Q5733090
cites work: Q5811706
cites work: Algorithms in invariant theory
DOI: 10.1515/ADVGEOM-2017-0045
published in: advg
description: scientific article; zbMATH DE number 6854821