Let $X$ be a smooth complex variety, $Z\subset Y\subset X$ closed subvarieties. Let $C$ be the normal cone to $Y$ in $X$. Then $Z$ can be seen as a closed subscheme of $C$ as well.
Is there a relationship between the normal cone to $Z$ in $X$ and the normal cone to $Z$ in $C$ ? Certainly they are isomorphic if $Z\subset X$ is a regular embedding.