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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 $(x,y,z)$?

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  • $\begingroup$ Should you replace 'Cartier' with 'locally free'? The concept of being Cartier only applies to the case that $Z$ is a divisor, right? $\endgroup$
    – Rhys
    Aug 16, 2013 at 16:40
  • $\begingroup$ Perhaps this terminology is nonstandard, but I have edited the question to define "Cartier" for an arbitrary subscheme. Of course, being a Cartier subscheme at a point implies that it is locally a divisor (i.e. codimension 1 in some affine open neighborhood). $\endgroup$
    – RghtHndSd
    Aug 16, 2013 at 17:06
  • $\begingroup$ Codimension one if $X$ is locally noetherian because you need Krull's principal ideal theorem for this statement. $\endgroup$
    – Cantlog
    Aug 27, 2013 at 12:53
  • $\begingroup$ I'm more than willing to stipulate to all schemes even being Noetherian and of finite type, but what statement are you referring to? $\endgroup$
    – RghtHndSd
    Aug 27, 2013 at 23:41
  • $\begingroup$ Dear rghthndsd, I refered to your sentence "(i.e. codimension 1 in some ..." in your last comment. Please be so kind to add @Cantlong if you want me beware of your comment. $\endgroup$
    – Cantlog
    Aug 28, 2013 at 12:49

1 Answer 1

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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.

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