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 $e$.