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 $(x,y,z)$?

  • $\begingroup$ 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


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .