The conjugacy locus of Cayley–Salmon lines
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Jaydeep Chipalkatti
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.
Acknowledgements
I thank the referee for a thorough reading of the manuscript and several constructive suggestions.
References
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Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
