Let $X$ be a scheme and $p\in X$ a closed point. We say that $(X,p)$ is etale locally isomorphic to $(Y,q)$ if there exists an etale neighborhood of $p$ in $X$, and etale neighborhood of $q$ in $Y$, and an isomorphism between them.
Note that etale maps induce isomorphisms on tangent cones, so if $(X,p)$ is etale locally isomorphic to $(Y,q)$, then $TC_pX\cong TC_qY$. In particular, if $(Y,q)$ is itself a cone, then it must be the tangent cone to $X$ at $p$. My question is about a partial converse to this statement.
Question: When is $(X,p)$ etale locally isomorphic to $(TC_pX,0)$?
(Note: This question is a reformulation of Local intersection cohomology without the intersection cohomology, which may have scared people off.)
