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arXiv · 1505.07144

When are the Cayley-Salmon lines conjugate?

Jaydeep Chipalkatti
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Abstract

Given six points on a conic, Pascal's theorem gives rise to a well-known configuration called the \emph{hexagrammum mysticum}. It consists of, amongst other things, twenty Steiner points and twenty Cayley-Salmon lines. It is a classical theorem due to von Staudt that the Steiner points fall into ten conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. It is shown in this paper that the C-S lines are pairwise conjugate precisely when the original sextuple is~\emph{tri-involutive}. The variety of tri-involutive sextuples turns out to be arithmetically Cohen-Macaulay of codimension two. We determine its $SL_2$-equivariant minimal resolution.

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Jaydeep Chipalkatti. 2015-05-26. When are the Cayley-Salmon lines conjugate?. https://arxiv.org/abs/1505.07144

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