Shioda’s conjecture

At the Seattle workshop I mentioned in the previous post I was a mentor of a group of recent PhDs consisting of Jennifer Park, Daniel Litt, Runpu (Z)Hong, Dingxin Zhang, Sam Raskin, and Francois Greer. The topic was “supersingular surfaces in positive characteristics”. Any mistakes and/or misrepresentations in this post are mine.

Let us define a smooth projective variety over a field K of characteristic p > 0 to be supersingular if for each i the Newton slopes on the “motive” H^i(X) are all i/2. (This is just one possible choice of definition.) If the Tate conjecture is true for H^2, then for a supersingular variety over an algebraically closed field the rank of the Picard group is equal to the second betti number (!).

Motivation for the choice of topic was the recent successes (due to my colleague Maulik, as well as Liedtke, Lieblich, and Charles) on moduli of supersingular K3 surfaces. Basically, we know that supersingular K3 surfaces satisfy the Tate conjecture, are unirational, form a family of dimension 9, and that any degeneration of a supersingular K3 has potentially good reduction.

Now, let’s discuss some things one can say for other surfaces.

Degeneration of supersingular surfaces. Let K = k((t)) and let X be a supersingular surface over K. The method here, following Rudakov-Zink-Shafarevich (RZS), is to note that the formal Brauer group of any special fibre X_0 has vanishing p-divisible part (they only prove this under the assumption that H^1(X_0, O_{X_0}) = 0). This imposes a strong condition on the limits. For example, if X_0 is equal to the union of two smooth surfaces S_1, S_2 glued along transversally along a nonsingular curve C, then S_1 and S_2 are forced to have slopes 1 of the Newton polygon of H^2 and the map Pic^0(S_1) x Pic^0(S_2) —> Pic^0(C) has to be surjective.

One can make examples of this kind of degeneration, by considering a family of supersingular genus 2 curves specializing to a good curve consisting of two supersingular elliptic curves glued in a point and taking the product with another supersingular elliptic curve.

Looking at quintic surfaces in P^3, we found that applying the RZS criterion to GIT-stable limits does not always give enough information, and that it is better to consider stable limits of surfaces in the sense of birational geometry. In fact, it appears likely that supersingular quintics have potential good reduction at least for large enough primes (but this may be an empty statement — see below).

Degenerations of supersingular elliptic surfaces with a section. If (X, σ) is an elliptic surface over a curve C and if all fibres are semi-stable, then this determines a morphism C —> \bar M_{1, 1}. Hence we can use Abramovich-Vistoli(+Olsson) to take a limit in \bar M_g(\bar M_{1, 1}, degree) and see that our elliptic surface degenerates to an elliptic surface X_0 over a semi-stable limit C_0 of C, right? For example if g = g(C) = 0, then C_0 is a tree of curves and at first it looks like RZS implies that C_0 has to be irreducible: namely the glueing curves are elliptic and the component surfaces are elliptic with nonconstant j-invariant hence have trivial Pic^0. Well, this is not quite the case as in this game you have to allow the base curve to become stacky at the nodes. And then the analysis of RZS still works (we think), but now you are glueing along stacky curves whose Pic^0 may be zero (or finite).

In fact, looking at 1-parameter families of supersingular elliptic K3 surfaces (which we know exist) we proved this kind of behaviour must happen, i.e., the Abramovich-Vistoli limit must produce a reducible stacky limit curve C_0. This is not a contradiction with the previously mentioned good reduction of K3 surfaces, as what (probably) happens is that one of the irreducible components X_0 is a K3 and the others are (for example) rational elliptic surfaces. Slogan: there is a difference between limits of X as an abstract surface and limits of X as an elliptic surface.

Unirational surfaces are supersingular. Shioda gave an example of a supersingular surface with q = p_g = 0 which is not unirational. However, he also conjectured

Let X be a simply-connected surface (i. e. without connected etale covering of degree > 1) in char p > 0. Then X is unirational if and only if it is supersingular.

As far as I can tell this conjecture is still open (please let me know if this is no longer the case). To try and disprove this, we can try to find “new” examples of supersinguar surfaces. E.g., we can look for whole families of them as in the next paragraph. But, we can also look for “sporadic” surfaces (like Fermat surfaces for which the conjecture is known). In fact, I don’t know if for every p > 5 there is a supersingular quintic surface we could try the conjecture on (again, please let me know if there are examples). I also tried to find new ones by computation which I will report on in the next blog post.

In fact, I can point out some examples for which I don’t know if the conjecture holds. Let C and C’ be supersingular hyperelliptic curves in char p > 2 and let i, i’ denote the hyperelliptic involutions. Note: for p > 2 there exists a 1-parameter family of supersingular hyperelliptic curves for genus 2 and for genus 3 too (Oort). Then let X be the resolution of singularities of (C x C’)/<(i, i')>. It seems to me that \pi_1(X) = 0 and of course X is supersingular. But I don’t know how to prove that X is unirational, do you? Countably many cases of this are discussed by Shioda and others (eg when the curves C and C’ are related to Fermat curves); also if C and C’ are elliptic curves, then X is a Kummer surface and X was proved to be unirational by Shioda (a beautiful alternative non-computational proof of this was found by Katsura).

Moduli of supersingular surfaces. Suppose given a family of surfaces over a base B, for example the universal family of quintic surfaces in P^3. What one can try is look at the lower bound on the codimension of the supersingular stratum in B using the fact that Newton polygons jump in codimension 1 (if they do jump). We tried this on the first day and it turns out that for quintic surfaces in P^3 and for elliptic surfaces which are not K3 and not rational, the lower bound you get is higher than the dimension of B, i.e., it is useless. In fact, I would like to know

Are there infinitely many primes p such that there is a 1-parameter family of supersingular quintic surfaces?

In fact, I don’t know a 1-parameter family of supersingular quintic surfaces except for p = 5. Of course, we can ask the displayed question for surfaces of any given degree > 4 in P^3. Also, we can ask this for elliptic surfaces of given height > 2 (i.e., not rational or K3).

But, looking around the literature, families of supersingular surfaces seem to be hard to come by and the ones I’ve found are always families of unirational surfaces (so useless from the point of trying to address Shioda’s conjecture). Please let me know if you know of examples where unirationality (currently) isn’t known.

To finish the discussion let me mention two examples of families.

For surfaces in P^3 we can consider the Zariski surfaces

X : T_0^p = F(T_1, T_2, T_3)

where F is a general homogeneous form of degree p. Such a surface has a large number of A_{p – 1} singularities and the desingularization X is a unirational surface. If I understand well, then all the new algebraic cycles come from the resolution of the singularities.

Let q be a power of p. In Shioda’s paper one finds the family of surfaces

T_0^q T_2 + T_1^q T_3 + T_0 F(T_2, T_3) + T_1 G(T_2, T_3) = O

where F, G are degree q homogeneous without common factors. These are smooth and unirational and have 2q – 2 moduli.

Back from Seattle

Just got back from a workshop/conference organized by Max Lieblich. Thanks for all the comments you guys left on the Stacks project this week. I just finished dealing with these and I am ready for more.

One word about submitting slogans: let’s try to really think about using different language in the slogans. It doesn’t need to be entirely mathematically correct or be entirely equivalent to the actual statement. Maybe the idea is even to make the reader think a little bit about why the slogan is a slogan for the result in question. If while using the sloganerator you come across a lemma and the only slogan that comes to mind is very close to the actual statement, then you can just click on “get new tag” to see if the next one is “more fun” so to speak.

Also, we now have almost 150 contributors to the Stacks project. I need 4 more of you to leave an intelligent comment (for example) with your name. Thanks!

Stacks incubator?

Here is the idea (stolen from the cring project): collect any expository notes written by you and put them on the web under the GFDL. Why? I have several reasons:

  1. I have many times come across good expositions by younger people of (parts of) results in our field which ultimately disappear as the webpage is removed when the person moves on.
  2. It turns out to be very helpful to use an initial write-up on some topic (no matter how badly written) to start a new chapter or a new section of the Stacks project.
  3. Keep a record of past contributions in this form; this will allow people to compare the older version of material with what it transforms into in the Stacks project.
  4. Expositions may cover results from the Stacks project sketching alternative proofs, more elementary treatments, etc.
  5. Outlining more advanced material can be done in such notes, even if the necessary preliminaries aren’t yet available in the Stacks project.
  6. Lower the threshold for participation in the Stacks project.

The last one because we’ll accept any latex source that compiles whose content is suitable (is about algebraic geometry, sheaves, commutative algebra, stacks, cohomology, dgas, etc).

I haven’t yet started a repository on github for this because I want your input on the name. It seems that using “incubator” is American English. Here are some (silly) alternative names for the repository: “Stacks dump”, “Stacks raw”, “Stacks stack” or “Stacks stock”. Please leave a comment if you have a preference for one of these or if you have another (sillier) name to suggest.

Once the repository goes up we can after a while set up a web page which displays compiled (pdf) versions of these notes somewhere. We can have a big sign telling the casual visitor that these notes are just there in the hope that they will help out understanding the material.

I will try to initially populate the repository with latex files sent to me over the past 6 years some of which have already gone into the Stacks project and some of which haven’t. (If this concerns you I will email you and ask for your permission before I do so, or you can email me to remind me.) Also, please feel free to email me anything that fits the description given above (

Note, note, note: You can do whatever you want with material you write yourself; you own it. To emphasize this, I always suggest people who contribute material to the Stacks project, to keep a copy of their work on their webpage, or post it on the arxiv, or whatever. The same is true for submitting to the incubator (or whatever it will be called). If later you realize you want to turn whatever you contributed into a paper (to be published) you can certainly do so (and at your request we can even remove anything you contributed, as long as it hasn’t gone into the Stacks project yet).

Finally, if you have a friend with a nice write-up, gently suggest they consider the incubator…


Pieter Belmans and Johan Commelin have added a bunch of new features to the Stacks project website:

  1. Nested enumerations are now displayed correctly; this lemma is an example.
  2. Outside references are more visible; this lemma has one.
  3. Footnotes are now displayed as footnotes; this section has four footnotes.
  4. They have implemented a system for historical remarks; Nakayama’s lemma is currently the only tag which has a historical remark.
  5. They have implemented a system for slogans; these lemmas are examples (look at the top underneath the header and breadcrumb).
  6. They have written the sloganerator; click or read below.

Huge thanks for all the work put into this by Pieter and Johan.

What is the sloganerator? Roughly, it gives you a random result from the Stacks project and asks you to type in a slogan describing the result. Your suggestion will become a comment on the tag’s page the Stacks project website. In due time we will then add the slogan to the actual Stacks project and you slogan will become visible as in the examples above.

The idea of having “slogans” or “human readable” descriptions of the results in the Stacks project has been around for a while. See this blog post and follow the links to the older blog posts. This will hopefully give you a better idea of what this is all about.


PS: If you’d like to suggest a slogan on any given tag of the Stacks project, then please just leave a comment on the tag’s page. Similarly with outside references and historical remarks.


This is a follow up of Example wanted. There I ask for two examples.

Firstly, I ask for a Noetherian local domain A such that its completion A* has an isolated singularity and such that Spec(A) does not have a resolution of singularities.

I now think such an example cannot exist. Namely, conjecturally resolution for Spec(A*) would proceed by blowing up nonsingular centers each lying about the closed point, which would transfer over to Spec(A) thereby giving a resolution for Spec(A).

Secondly, I ask for a Noetherian local ring A and a proper morphism Y —> Spec(A*) of algebraic spaces which is an iso above the puctured spectrum U* which is NOT the base change of a similar morphism X —> Spec(A).

As Jason pointed out in the comments on the aforementioned blog post, to get such an example we have to assume that A is nonexcellent since otherwise Artin’s result on dilatations kicks in to show that X does exist. In fact we have the following:

  1. We may assume A is henselian, see Lemma Tag 0AE4
  2. It holds when A is a G-ring, see Lemma Tag 0AE5
  3. There exists a blow up Y’ —> Spec(A*) with center supported on the closed point which dominates Y and which is the base change of some X’ —> Spec(A) as above, see Lemma Tag 0AE6 and Lemma Tag 0AFK.

I’ve tried to make a counter example for non-G-rings, but failed. So now I am beginning to wonder: maybe there isn’t one? [Edit 20 October 2014: there is none as can be seen by visiting the upgraded Lemma Tag 0AE5.]

If so, then perhaps Artin’s result on dilatations (in formal moduli II) holds for Noetherian algebraic spaces without any supplementary conditions. Yes, this is a ridiculous step to take (Artin’s result is about formal algebraic spaces and a lot stronger than the question asked above), and I say this, not because I have a good reason to think this is true, but just to make it easier for you and me to make a counter example. I don’t have one, do you? [Edit 20 October 2014: there is none as can be seen by visiting this blog post.]

You should probably stop reading here, because now things become really vague. Looking at affine schemes \’etale over Y leads to the following type of question. Suppose that f : V* —> Spec(A*) is a finite type morphism with V* affine and f^{-1}(U*) —> U* \’etale. Then we can ask whether V* is the base change of a similar type of morphism V —> Spec(A). The answer is a resounding NO because for example the morphism f could be an open immersion whose complement is a closed subscheme of Spec(A*) which is not the base change of a closed subscheme of Spec(A). But suppose we only ask for a V —> Spec(A) such that the m_A-adic formal completion of V is isomorphic to the m_{A*}-adic formal completion of V*? Namely, if this question has a positive answer, then we might be able to use this to construct an X as above whose base change is Y by glueing affine pieces. I also would dearly love a counter example to this question (again it holds if A is a G-ring so a counter example would have to involve some kind of bad ring). [Edit 20 October 2014: The existence of these algebras follows from the paper by Elkik on solutions of equations over henselian rings, see this blog post.]

Anyway, any suggestions, ideas, references, etc are very welcome. Thanks!

Bibliography + references

As I’ve said before I want to add more outside references everywhere locally in the Stacks project. Here is a way you can help: follow this link to find a listing of all results in the Stacks project whose LaTeX environment is coded as a theorem or a proposition (many important results from the literature are called lemmas in the Stacks project — if you want you can complain about this too). Then look through the list to see if you know a precise reference for one of these results (what I mean is a precise theorem/proposition/lemma in a paper which is mathematically very closely related, e.g., logically equivalent, a special case, or a generalization, or overlapping in many instances, etc). Then either leave a comment on the corresponding tag’s page or send an email to stacks dot project at google mail. Thanks!

Chapter of the day

There is a new chapter entitled “Pushouts of Algebraic Spaces”. Here is a link to the introduction. It contains a section on something that is often called formal glueing of quasi-coherent modules. The name presumably comes from the fact that this can be used and is often used to glue coherent sheaves on Noetherian schemes given on an open and on the formal completion along the closed. However, the mathematics is more like what one does when studying elementary distinguished squares in the Nishnevich topology — in fact, I wonder what topology we get if we replace elementary distinguished squares by the notion in Situation Tag 0AEV? Hmm…

Lemma of the day

Let (A,I) be a henselian pair with A Noetherian. Let A^* be the I-adic completion of A. Assume at least one of the following conditions holds

  1. A → A^* is a regular ring map,
  2. A is a Noetherian G-ring, or
  3. (A,I) is the henselization (More on Algebra, Lemma 15.7.10) of a pair (B,J) where B is a Noetherian G-ring.

Given f_1, …, f_m ∈ A[x_1, …, x_n] and a_1, …, a_n ∈ A^* such that f_j(a_1, …, a_n) = 0 for j = 1, …, m, for every N ≥ 1 there exist b_1, …, b_n ∈ A such that a_i − b_i ∈ I^N and such that f_j(b_1, …, b_n) = 0 for j = 1, …, m. See Lemma Tag 0AH5.

Slogan: Approximation for henselian pairs.