Terminology for blowups in algebraic geometry

Question

This is a partial duplicate of this Stack Exchange question which unfortunately got no answer.

All schemes are Noetherian and of finite type, although they need not be normal.

With $Z \subset X$ a closed subscheme, consider the blow up $\pi: \operatorname{Bl}_X(Z) \rightarrow X$. $Z$ is called the center and the strict transform of a subscheme $W \subset X$ with $W \not\subset Z$ is then the Zariski closure of $\pi^{-1}(W \setminus Z)$.

In my research, I must analyze some certain blowups. In doing so, I need to talk about certain subschemes which I feel should have a standard name but I have not been able to find. They are as follows, with my (nonstandard) terminology.

1. The "proper center": The closed subscheme of $X$ where $\pi$ is not an isomorphism (i.e. where the subscheme $Z$ is not Cartier).
2. The "true center": The closed subscheme of the proper center where the fiber above each point has dimension greater than zero.
3. The "quasi-finite center": The complement of the true center inside the proper center.

Are there accepted names for these? Furthermore, I also need to calculate the strict transform not with respect to the center, but rather the proper center. Is there a name for this as well?

Finally, I am also wondering if there is any expository reference where the strict transform is taken with respect to the proper center.

Answer 1

Here are some thoughts

1. I think this should be called the "true center", because it is where something interesting happens. Plus, if you blew up,say, the singular point of a nodal curve, then according to your definition the "true center" would be empty which doesn't sound right. As Karl suggests calling it the "true center" could be confusing, especially given your original feelings about this, so you could call it the non-Cartier center, but I think this is really the true center, so you might as well call it that.
2. This is probably the only one among the three that already has a name. I believe classically this locus was called the fundamental points of the map. You can find this terminology for example on page 156 in Mumford's Algebraic Geometry. I originally suggested that another reasonable name would be the exceptional center, but after rightndsd's comment I agree that it is not a great name as it suggests that it is the image of the exceptional locus. So, how about fundamental locus? Its preimage could be the prefundamental locus.
3. I'd be quite surprised if this locus did indeed have a name. This should definitely not be called a center, since it is generally not closed. Actually I don't see a reason for this to have a name. In what situation would you need this as a set other than say that a point lies in it? Given the increased complexity induced by having to distinguish between the other loci, you could just refer to this as the difference of the above two sets. Or if you really want a name you could call these the non-fundamental points of the non-Cartier center or the non-exceptional locus of the true center, or other combinations of the above.
   ADDED OK, how about the residual locus (of the true center)? Or if you adopt fundamental locus for the above, then this could be the non-fundamental locus...
4. I also don't think that anyone used a different name for the strict transform with respect to the non-Cartier center, but it feels like this should be the true strict transform. You could also just call it the strict transform with respect to the non-Cartier center.

Answer 2

Here are some of the terms I have seen.

1. The image of the exceptional set(/locus) is probably the most explanatory. The word center is also often used for the image of a single irreducible component of the exceptional locus so you might run into confusion if you call it the true center. I've also seen this called the discriminant.

I don't think I've ever seen people give names to 2 and 3, but I've seen number 2. show up.

EDIT: As pointed out in the comments, I was surprised to learn that the strict transform is taken with respect to $Z$ (or rather $X \setminus Z$) instead of depending simply on $\pi$. In birational geometry the actual ideal you are blowing up doesn't seem to matter, people just care about the map. In that setting, the strict transform is also often called the birational transform. I've also seen the proper transform.