Definition (from Eisenbud-Harris' Geometry of Schemes): Let $X$ be any scheme, $Y \subset X$ a subscheme. We say that $Y$ is a Cartier subscheme in $X$ if it is locally the zero locus of a single nonzerodivisor.
Given a scheme $X$ and a closed subscheme $Z$, is there a common name for the (closed) locus where the ideal sheaf corresponding to $Z$ is not Cartier? My motivation is as follows.
Let $X$ be a scheme, $Z$ some closed subscheme, and consider the blowup of $Z$ in $X$. $Z$ is known as the "center" of the blowup, and the blowup is guaranteed to be an isomorphism on $U = X \setminus Z$. However, it is certainly possible that there is a larger open subscheme $V$ such that the blowup is an isomorphism over $V$. Indeed, where ever $Z$ is Cartier the blowup will be an isomorphism which is an open subscheme by Nakayama.
For example, take the scheme $X = \operatorname{Spec}(k[x,y,z]/(z^2-xy))$ and $Z$ corresponds to the ideal $(x,z)$. Then the blow-up is an isomorphism except at the point corresponding to $(x,y,z)$. In this case, the center of the blow up is the closed subscheme corresponding to $(x,z)$ and my question is what should we call the point corresponding to $(x,y,z)$?
This question is answered on MathOverflow. A quick summary:
In the case of a blowup, Sándor Kovács recommends calling the closed subscheme of $X$ where the map $\pi : \operatorname{Bl}_X(Z) \rightarrow X$ is not as isomorphism the "true center". There are also several other subschemes discussed there and suggested names for them.