In this post I talked a bit about flattening of morphisms. Meanwhile I have written some more about this in the stacks project which led to a change in definitions. Namely, I have formally introduced the following terminology:
- Given a morphism of schemes X —> S we say there exists a universal flattening of X if there exists a monomorphism of schemes S’ —> S such that the base change X_{S’} of X is flat over S’ and such that for any morphism of schemes T —> S we have that X_T is flat over T if and only if T —> S factors through S’.
- Given a morphism of schemes X —> S we say there exists a flattening stratification of X if there exists a universal flattening S’ —> S and moreover S’ is isomorphic as an S-scheme to the disjoint union of locally closed subschemes of S.
Of course the definition of “having a flattening stratification” this is a bit nonsensical, since we really want to know how to “enumerate” the locally closed subschemes so obtained. Please let me know if you think this terminology isn’t suitable.
Perhaps the simplest case where a universal flattening doesn’t exist is the immersion of A^1 – {0} into A^2. Currently the strongest existence result in the stacks project is (see Lemma Tag 05UH):
If f : X —> S is of finite presentation and X is S-pure then a universal flattening S’ —> S of X exists.
Note that the assumptions hold f is proper and of finite presentation. It is much easier to prove that a flattening stratification exists if f is projective and of finite presentation and I strongly urge the reader to always use the result on projective morphisms, and only use the result quoted above if absolutely necessary.
PS: I recently received a preprint by Andrew Kresch where, besides other results, he gives examples of cases where the universal flattening exists (he call this the “flatification”) but where there does not exist a flattening stratification.
Dear Johan,
In the definition of “stratification” might it not be better to allow S’ –> S to be a monomorphism which is <etale locally> a finite disjoint union of locally closed subschemes? For instance, if S is a nodal curve, I believe you can find an example where the universal flattening is the normalization of S, but with one of the two preimages of the node removed. I would still like to think of this as a “flattening stratification”.
Best regards,
Jason
OK, I see your point, but I am not sure that is what we want. I think this is related to my comment in the blog that we really want to have a way of indexing the strata. Namely, in your example even after etale localization the “same” stratum is biting itself in its tail! Right?
This leads to the following. Any finite presentation surjective monomorphism X —> S is its own universal flattening. Suppose S is reasonable (eg a variety). Can you give an example of such a X —> S which is not etale locally on S a disjoint union of locally closed subschemes?
I guess the cone over your nodal curve example will work? (With one extra stratum, namely the vertex of the cone.)
Dear Johan,
Your example is much clearer than mine. I think my example is as follows: you start with a nodal curve $S$. Denote the normalization by $B$. Let $p$ and $q$ be the closed points of $B$ mapping to the node of $S$.
Now take $B \times \mathbf{P}^1$ with its projection to $B$. Over one of the two points, say over $p$, choose a closed point $(p,t)$ on the fiber $\{p\} \times \mathbf{P}^1$. Now blow up that point to get a surface $Y$ with a projection to $B$ which is a $\mathbf{P}^1$-bundle away from $p$, but whose fiber of $p$ is a union of two copies of $\mathbf{P}^1$.
Now glue the fiber over $q$, which is a copy of $\mathbf{P}^1$, to one of these two copies of $\mathbf{P}^1$ in the fiber over $p$ via an isomorphism (you can certainly do this glueing in the category of algebraic spaces by Artin, but I believe you can even do this in the category of schemes). Call the new surface $X$. The normalization of $X$ is $Y$. If we compose the projection from $Y$ to $B$ with the normalization map from $B$ to $S$, then this factors through the normalization map from $X$ to $Y$, i.e., there is a unique morphism fro $X$ to $C$ compatible with the map from $Y$ to $C$. And this morphism from $X$ to $S$ is the one I want to consider.
When we pull this back to $B$, we almost get $Y$, but with one extra copy of $\mathbf{P}^1$ just “stuck” to the fiber over $q$. So this fails to be flat over $q$. But I believe it is flat on the complement of $q$. So the universal flattening is $S’$, the open complement of $q$ in $B$.
Best regards,
Jason
OK, that’s beautiful! And I guess that is a counter example to what you and Martin conjecture on page 4083 of your paper about Quot? Can you tell me briefly what’s known about that question now?
Anyway, now I finally get what you are driving at! Thanks!
Dear Johan,
I have to look at it again, but I believe that conjecture is only about the case when the coarse moduli space is projective. And I thought somebody proved the conjecture (maybe David Rydh?). In the example above, the surface is not projective (because a general fiber is algebraically equivalent to one component of the special fiber).
Best regards,
Jason
D’oh! Johan is right, I am wrong. I don’t even know my own conjectures 🙁
This was discussed earlier on the Stacks Project blog (http://math.columbia.edu/~dejong/wordpress/?p=686). I think Jason’s example is identical to the example given by Kresch using Hironaka’s example of non-projective threefolds (see comment to the post 686 referred to above).
I think that Johan’s terminology is reasonable. I had not previously appreciated the difference between general monomorphism and those that étale-locally are stratifications. Perhaps the latter class has some use but I agree with Johan that the important thing is whether we have an indexing or not and this is a global question.
Every example of monomorphisms that are not étale-local stratifications are related to the fact that unibranchedness is not stable under generizations (e.g., the cone over a nodal curve is unibranch at the apex but not along the “double directrix” corresponding to the node of the curve).
Also note that even if S is normal (or even regular), there are examples of such monomorphisms if the dimension is at least 3 (embed the cone over a nodal curve in 3-space). Is S is normal of dimension 2 (or arbitrary of dimension 1) on the other hand, I suppose that every surjective monomorphism étale-locally could very well be a stratification.
Gosh, sorry I did not remember we had previously discussed this. Thanks David!
Dear David,
I never claimed the example above is original (although I do think it is a bit different from Kresch’s example). I would love to see a “serious” application for flattening stratifications (when they exist) which does not “automatically” generalize to universal flattenings which are \’etale locally stratifications. Of course since I leave “serious” and “automatically” undefined, that gives me the right to deny all examples 🙂
Best regards,
Jason