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The conjugacy locus of Cayley–Salmon lines

  • Jaydeep Chipalkatti EMAIL logo
Published/Copyright: January 7, 2018
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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.

MSC 2010: 14N05; 51N35

Communicated by: I. Coskun


Acknowledgements

I thank the referee for a thorough reading of the manuscript and several constructive suggestions.

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Received: 2016-2-4
Revised: 2016-3-2
Published Online: 2018-1-7
Published in Print: 2018-1-26

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