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Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In particular, the map $F \times e \stackrel{+}{\to} C$ is an isomorphism of cones.

Is there a name for this property of $F$ or $e$ (which determine one another)?

Essentially, I regard the edges $e$ like this as kind of trivial, and would like to split them off to deal with the difficult part of $C$. So maybe one could speak of a "core facet" $F$ of $C$, and let the "core of $C$" be the intersection of the "core facets". Then the map $core(C) \times \prod_{\text{core edges }e} e \stackrel{+}{\to} C$ would be an isomorphism, and $core(C)$ would have no core edges.

There is a similar, familiar construction in the theory of simplicial complexes, where a "cone vertex" $v$ of $\Delta$ is one lying in every maximal face. One can safely delete all the cone vertices, and recone on them to reconstruct $\Delta$. Obviously one doesn't want to steal this terminology directly and speak of "cone edges".

EDIT: If you prefer, you can slice the (pointed) cone with a hyperplane to get a polytope, and work with that. It's then common to say "this polytope $P$ is the cone on this facet $F$, from this vertex $e$" but I still don't know a good adjective for $F$ or $e$.

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  • $\begingroup$ Sorry I don't understand. Every facet of a (general position) simplicial cone in $\mathbb R^3$ contains all edges but one, no? $\endgroup$ – მამუკა ჯიბლაძე Dec 31 '18 at 12:46
  • $\begingroup$ That's correct; I'm not asking about simplicial cones. Think instead of the cone on a square, i.e. a pyramid with no base. $\endgroup$ – Allen Knutson Dec 31 '18 at 13:58
  • $\begingroup$ You mean a cone over this pyramid? In 4d? $\endgroup$ – მამუკა ჯიბლაძე Dec 31 '18 at 15:10
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    $\begingroup$ I'd meant a cone on a square, in 3d. That cone has no edges of the special type I'm asking about. But if you want we can consider the 4d cone on a pyramid (with its base). That cone has 5 edges, of which only the line through the apex is special in the sense I'm asking about. $\endgroup$ – Allen Knutson Dec 31 '18 at 18:17
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    $\begingroup$ Maybe you prefer the statement "$Proj(R\otimes S)$ is the join of $Proj(R)$ and $Proj(S)$", where $R$ and $S$ are the monoid algebras, and "join" means the union of the lines through the two spaces? $\endgroup$ – Allen Knutson Dec 31 '18 at 18:55

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