# Histories

So I’ve been playing around with tracing the history of lemmas, etc in the Stacks project over time. I’ve written a bunch of scripts to extract this information out of the git logs. Here a pdf file showing what happened to Nakayama’s lemma over time. Enjoy!

PS: Feel free to improve on the current version of Nakayama’s lemma in the Stacks project and to submit it for inclusion, so we can add more stages to its history!

# Dilatations

We clarify the discussion in this post resulting in a generalization of a result of Mike Artin.

Let X be a Noetherian algebraic space. Let T ⊂ X be a closed subspace. Let us denote X/T the formal completion of X along T (Tag 0AIX). Let W —> X/T be a rig-etale morphism of formal algebraic spaces, which means that

1. W —> X/T is representable by algebraic spaces, i.e., it is an adic morphism of formal algebraic spaces Tag 0AQ2),
2. W —> X/T is locally of finite type, i.e., it is etale locally on affine formal algebraic pieces given by a continuous ring map A —> B which is topologically of finite type Tag 0ALL),
3. these ring maps A —> B have a completed cotangent complex whose cohomology groups are annihilated by an ideal of definition of A, for more details see Tag 0ALP and Tag 0AQK.

These three conditions correspond to condition (i) of Definition 1.7 of Artin’s paper “Formal Moduli: II”. The first result is that

Given W —> X/T rig-etale there exists a morphism of algebraic spaces Y —> X which is an isomorphism over X – T and whose completion Y/T —> X/T is isomorphic to W —> X/T.

In fact, Theorem 0ARB tells us we obtain an equivalence of categories.

This theorem does not often produce separated morphisms Y —> X if we start with a random W —> X/T. A typical thing that happens can be seen by starting with X equal to the affine line, T = {0} and W two copies of the completion of X at 0. Then the resulting Y is the affine line with 0 doubled.

Thus to get Artin’s theorem on dilatations we need to impose conditions on W —> X/T guaranteeing that Y —> X is separated or even proper. To do this we will use the notion of a rig-surjective morphism W’ —> W of locally Noetherian formal algebraic spaces W, W’ defined by requiring adic morphisms Spf(R) —> W with R a cdvr to lift to Spf(R’) —> W’ for some extension of cdvr R ⊂ R’ (Tag 0AQP). Let’s say an adic morphism W —> W’ of locally Noetherian formal algebraic spaces is a rig-monomorphism if the diagonal morphism is rig-surjective. In this language the conditions (ii) and (iii) from Definition 1.7 of Artin’s paper have the following interpretations:

1. If W —> X/T as above is separated and a rig-monomorphism, then Y —> X is separated (Tag 0ARW),
2. If W —> X/T as above is proper, a rig-monomorphism, and rig-surjective, then Y —> X is proper (Tag 0ARX).

The second statement recovers exactly Artin’s theorem on dilatations.

One typically applies the result to construct modifications Y —> X (Tag 0AD7) by taking the complete local ring A of X at a closed point x and setting W equal to the formal completion of the blow up of A at an ideal I ⊂ A which is locally principal on the puctured spectrum of A. Here the funny situation occurs that we can first read Tag 0ARX backwards over Spec(A) to conclude that W —> X/x has the required properties and then forwards to conclude that Y —> X is proper. In other situations it may not be that easy to verify the assumptions needed for the application of the theorem and it would behoove us to prove a few lemmas that help with this task.

# Spaces are fpqc sheaves

Although already mentioned in the previous post I’d like to point out once again that we have, thanks to Ofer Gabber, now a proof that algebraic spaces satisfy the sheaf condition for fpqc coverings, answering one of the questions in this post. Also thanks go to Bhargav Bhatt who sent me an email (on 9/13/13) explaining Gabber’s argument. With small comments/corrections/clarifications added over the weekend, I hope most of the kinks I introduced have been ironed out.

To read more, start here. As usual comments and suggestions for improvements are welcome. Enjoy!

# Update

Since the last update of June 2013 we have added the following material:

1. Algebraicity of stack of coherent sheaves Tag 09DS
2. Epp’s Theorem Tag 09F9 and Tag 09II
3. Reduced fibre theorem Tag 09IL
4. New chapter on fields Tag 09FA
5. Perfect generator for DQCoh(X) for schemes Tag 09IS
6. Perfect generator for DQCoh(X) for algebraic spaces Tag 09IY
7. New chapter on differential graded algebra Tag 09JD
8. Section on Noetherian schemes of dimension 1 Tag 09N7
9. Finite set of codimension 1 points on separated scheme are in an affine Tag 09NN
10. Section on curves Tag 0A22
11. Versions of Avramov’s result Tag 09Q7 and Tag 09QF
12. Obtaining triangulated categories Tag 09P5 written by John Yu and Yifei Zhao
13. Rickard’s theorem Tag 09S5
14. Section on compact objects Tag 09SM
15. Section on generators in triangulated categories Tag 09SI
16. DQCoh = D(QCoh) for Noetherian schemes Tag 09TI
17. DQCoh = D(QCoh) for Noetherian spaces Tag 09TH
18. Final exam commutative algebra Fall 2013 Tag 09TV
19. Lots of stuff about pseudo-coherence
20. Cohomology of locally compact spaces Tag 09V0
21. Proper base change in topology 09V4
22. New chapter on simplicial spaces Tag 09VI
23. Cohomological descent for proper hypercoverings in topology Tag 09XA
24. Section on henselian pairs Tag 09XD
25. Finite cover by a scheme Tag 0ACX
26. Partitions and stratifications Tag 09XY
27. Bunch of stuff about limits of sites and spaces and cohomology
28. Gabber’s result on Henselian pairs Tag 09Z8
29. Theorem of formal functiosns via derived completion Tag 0A0H
30. Lots of edits to the chapter on etale cohomology culminating in a proof of the proper base change theorem Tag 095S
31. Cohomological dim <= Krull dim for spectral spaces Tag 0A3C
32. Equivalence beteen torsion and derived complete modules Tag 0A6V
33. Local duality a la Grothendieck Tag 0A81
34. Brown representability for triangulated categories Tag 0A8E
35. Final exam algebraic geometry Spring 2014 Tag 0AAL
36. Bunch of stuff about twisted inverse image, dualizing complexes, upper shriek functors
37. New mostly empty chapter on resolution of surfaces Tag 0ADW
38. Algebraic spaces are schemes in codimension 1 Tag 0ADD
39. Extremely careful discussion of points on fibres of morphisms of algebraic fibres and when this implies the morphism is quasi-finite at the points Tag 0AC0
40. Finite groupoids Tag 0AB8
41. fppf descent data for spaces over spaces are effective Tag 0ADV
42. New chapter on pushouts of algebraic spaces Tag 0AHT including a discussion of something called formal glueing which should really have another name
43. Regular local rings are UFDs Tag 0AFW
44. Approximation for henselian pairs Tag 0AH4

This brings us up to July 1 of this year. At this point I decided to work out what happens with dilatations in Artin’s paper “Formal Moduli II”. In fact, I wrote a blog post trying to figure out what could be true. Anyway, it turns out that Artin’s theorem on dilatations holds without assuming the base Noetherian algebraic space is excellent and in the proof we don’t need to use Artin’s criteria. See Theorem Tag 0ARB and combined this with Lemma Tag 0ARX to obtain Artin’s theorem — I hope to discuss what these results mean in more detail in a later blog post. Here are some related and not so related changes to the Stacks project.

1. A chapter on formal algebraic spaces. Our approach is somehow a combination of all approaches in the literature: EGA, McQuillan, Yasuda, Beilinson-Drinfeld, Abbes, Fujiwara-Kato, and Knutson. Namely, we define a formal algebraic space as an fppf sheaf on (Sch) which \’etale locally looks like an Ind-scheme whose transition morphisms are thickenings. For the Stacks project this approach turns out to be exactly the right one and many of the earlier results on algebraic spaces can be brought to bear to get ahead fairly quickly.
2. A followup chapter discussing the notion of rig-etale (homo)morphisms ending with a proof of the algebraization theorem mentioned above. These rig-etale maps are used by Artin in his paper mentioned above to describe formal modifications and these (as well as the similarly defined rig-smooth ring maps) are the ring maps studied in Elkik’s famous paper.
3. Adjoint functor theorem Tag 0AHM
4. Remarkable result by Yasuda that every qcqs formal algebraic space is a filtered colimit of algebraic spaces along thickenings Tag 0AJD
5. Non flat completions Tag 0AL8
6. QCoh not abelian for formal algebraic spaces Tag 0ALF
7. Ofer Gabber answered completely this question see Tag 0APW
8. Algebraic spaces are fpqc sheaves by Ofer Gabber Tag 03W8

Enjoy!

# 150 contributors

Just finished working through the recent comments left on the Stacks project site and because there were a couple of new contributors we are now up to 150 contributors. Go us!

Anyway, please continue helping out by finding mistakes, etc, thank you very much. Also, we are going to have a party(!) when we reach 5000 pages, so please help: submit a new chapter, add an omitted proof, provide an alternative argument, or just randomly submit a piece of material you think may be useful, so we can get there more quickly. Thanks!

# Copying

Just a heads up for those people who are taking parts or all of the Stacks project and doing new and exciting things with it. Please make sure you comply with the license that the Stacks project is under. Thanks!

Also, if you receive texts based on the Stacks project in your inbox, send us an email. Maybe in the future we can have a hall of shame or something.

Over the summer I wrote up a bit of material laying out a (very general) theory of formal algebraic spaces for the Stacks project. The idea is to work initially with very general objects and then for later results impose those conditions that make the arguments work (similarly to what is done for algebraic spaces and algebraic stacks in the Stacks project). As is often the case when you work through a new subject some natural very basic questions arise which I am unable to answer.

This paragraph is for motivation only and you can skip it. Let X be a scheme and let Z ⊂ X be a closed subset. The completion of X along Z is the functor which associates to a scheme T the set of morphisms f : T —> X such that f(T) ⊂ Z set theoretically. My question is whether one has “countably indexed => adic*” for such a completion.

In terms of algebra this means the following. Let A be a ring and let I ⊂ A be a radical ideal. Assume there is a countable family

I ⊃ J_1 ⊃ J_2 ⊃ J_3 ⊃ …

of ideals with V(I) = V(J_n) such that for every ideal I ⊃ J with V(I) = V(J) we have J ⊃ J_n for some n. In other words, the partially ordered set of closed subschemes of \Spec(A) supported on Z = V(I) has a countable cofinal subset. Let’s write A^* = \lim A/J_n as a topological ring endowed with the limit topology.

Is there a finitely generated ideal 𐌹 ⊂ A^* such that the powers of 𐌹 form a fundamental system of open neighbourhoods of 0? In other words, is A^* an adic topological ring which has a finitely generated ideal of definition?

Now that I state it like this, it seems this cannot possibly be true. But I haven’t found a counter example. Have you?

PS: I love gothic letters… 𐌰 𐌱 𐌲 𐌳 𐌴 𐌵 𐌶 𐌷 𐌸 𐌹 𐌺 𐌻 𐌼 𐌽 𐌾 𐌿 𐍀 𐍁 𐍂 𐍃 𐍄 𐍅 𐍆 𐍇 𐍈 𐍉 𐍊

Edit Sept 12, 2014. Just got a note from Gabber where he shows that the answer is yes when I is the radical of a countably generated ideal and that there is a counter example in general.

# Special thanks to correction_bot

Hello again. This a quick post to let you know that I worked through your recent comments and updated the Stacks project website with the latest version. So get back there and find more mistakes. Thank you all!

# Stats for Newton Polygons

In the last few days I tried (unsuccessfully) to find some “new” supersingular surfaces by computation. Please read the previous post to see why one might want to find these surfaces. Anyway, one of the things that I have to show for this are some distributions of Newton Polygons (NPs) in the data. Here is an example:

 3616 2, 2, 1, 1, 1, 0, 0 302 2, 3/2, 3/2, 1, 1/2, 1/2, 0 46 2, 1, 1, 1, 1, 1, 0 24 5/3, 5/3, 5/3, 1, 1/3, 1/3, 1/3 6 3/2, 3/2, 1, 1, 1, 1/2, 1/2 5 1, 1, 1, 1, 1, 1, 1

The sequence of numbers at the top mean the following: We are looking at computations of NPs on H^2_{prim} of randomly chosen quasi-smooth surfaces over F_11 defined by an equation in weighted projective space of the form

W^2 = F(X, Y, Z)

where X, Y, Z have weights 13, 15, 19, the polynomial F is homogeneous of degree 184, and W has degree 184/2 = 92. Summing up the integers in the first column we see that we did a run of 3999 experiments and we got NP counts as shown.

The table suggests that the primitive Hodge numbers of these surfaces are h^{0, 2} = 2, h^{1, 1} = 3, and h^{2, 0} = 2 as is indeed the case. All possible NPs occur in the table, except for 4/3,4/3,4/3,1,2/3,2/3,2/3. The table suggests that the NP 2,2,1,1,1,0,0 happens generically and 2,3/2,3/2,1,1/2,1/2,0 happens in codimension 1 because 11 * 302 is almost 3616. Next, we expect the NPs 2,1,1,1,1,1,0 and 5/3,5/3,5/3,1,1/3,1/3,1/3 to happen in codimension 2. In fact, the whole table is strangely consistent with the known theory of NP jumps, except that 1,1,1,1,1,1,1 occurs too often.

Why is this strange? Well, because the equations cutting out the NP strata typically have high degree (polynomial in p) and hence we cannot expect *any* good behaviour of point counts over F_p (only when we work over F_q for a high power of p can we expect such a thing). The same happens for other experiments (see below). For smaller primes the behaviour is less regular; I think this happens because of the limited sample space.

Please let me know if you have any kind of guess as to why this should be!

PS: How did I compute these tables? To get the Frobenius polynomials I used a program I wrote a long time ago. More precisely, to replicate the results above you have checkout the double branch which computes Frobenius matrices of double covers of weighted projective planes. In each case I ran this program on random inputs repeatedly. You can find the outputs produced in this github repository.

 1679 2, 2, 1, 1, 1, 0, 0 254 2, 3/2, 3/2, 1, 1/2, 1/2, 0 32 2, 1, 1, 1, 1, 1, 0 26 5/3, 5/3, 5/3, 1, 1/3, 1/3, 1/3 8 3/2, 3/2, 1, 1, 1, 1/2, 1/2
 807 2, 2, 1, 1, 1, 0, 0 148 2, 3/2, 3/2, 1, 1/2, 1/2, 0 25 5/3, 5/3, 5/3, 1, 1/3, 1/3, 1/3 10 1, 1, 1, 1, 1, 1, 1 9 2, 1, 1, 1, 1, 1, 0
 484 2, 2, 1, 1, 1, 0, 0 202 2, 1, 1, 1, 1, 1, 0 197 2, 3/2, 3/2, 1, 1/2, 1/2, 0 74 3/2, 3/2, 1, 1, 1, 1/2, 1/2 42 1, 1, 1, 1, 1, 1, 1
 6299 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0 574 2, 2, 3/2, 3/2, 1, 1, 1, 1/2, 1/2, 0, 0 60 2, 2, 4/3, 4/3, 4/3, 1, 2/3, 2/3, 2/3, 0, 0 51 2, 5/3, 5/3, 5/3, 1, 1, 1, 1/3, 1/3, 1/3, 0 6 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0 6 7/4, 7/4, 7/4, 7/4, 1, 1, 1, 1/4, 1/4, 1/4, 1/4 2 2, 3/2, 3/2, 3/2, 3/2, 1, 1/2, 1/2, 1/2, 1/2, 0 1 5/3, 5/3, 5/3, 3/2, 3/2, 1, 1/2, 1/2, 1/3, 1/3, 1/3
 5763 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0 889 2, 2, 3/2, 3/2, 1, 1, 1, 1/2, 1/2, 0, 0 126 2, 2, 4/3, 4/3, 4/3, 1, 2/3, 2/3, 2/3, 0, 0 123 2, 5/3, 5/3, 5/3, 1, 1, 1, 1/3, 1/3, 1/3, 0 35 2, 3/2, 3/2, 3/2, 3/2, 1, 1/2, 1/2, 1/2, 1/2, 0 22 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0 16 2, 3/2, 3/2, 1, 1, 1, 1, 1, 1/2, 1/2, 0 7 7/4, 7/4, 7/4, 7/4, 1, 1, 1, 1/4, 1/4, 1/4, 1/4 6 3/2, 3/2, 3/2, 3/2, 1, 1, 1, 1/2, 1/2, 1/2, 1/2 4 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 3 5/3, 5/3, 5/3, 3/2, 3/2, 1, 1/2, 1/2, 1/3, 1/3, 1/3 1 2, 4/3, 4/3, 4/3, 1, 1, 1, 2/3, 2/3, 2/3, 0
 770 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0 138 2, 2, 3/2, 3/2, 1, 1, 1, 1/2, 1/2, 0, 0 36 2, 5/3, 5/3, 5/3, 1, 1, 1, 1/3, 1/3, 1/3, 0 30 2, 2, 4/3, 4/3, 4/3, 1, 2/3, 2/3, 2/3, 0, 0 8 7/4, 7/4, 7/4, 7/4, 1, 1, 1, 1/4, 1/4, 1/4, 1/4 8 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0 5 2, 3/2, 3/2, 3/2, 3/2, 1, 1/2, 1/2, 1/2, 1/2, 0 3 2, 3/2, 3/2, 1, 1, 1, 1, 1, 1/2, 1/2, 0 1 2, 4/3, 4/3, 4/3, 1, 1, 1, 2/3, 2/3, 2/3, 0
 431 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0 239 2, 2, 3/2, 3/2, 1, 1, 1, 1/2, 1/2, 0, 0 131 2, 2, 4/3, 4/3, 4/3, 1, 2/3, 2/3, 2/3, 0, 0 82 2, 5/3, 5/3, 5/3, 1, 1, 1, 1/3, 1/3, 1/3, 0 47 2, 3/2, 3/2, 3/2, 3/2, 1, 1/2, 1/2, 1/2, 1/2, 0 22 7/4, 7/4, 7/4, 7/4, 1, 1, 1, 1/4, 1/4, 1/4, 1/4 11 5/3, 5/3, 5/3, 3/2, 3/2, 1, 1/2, 1/2, 1/3, 1/3, 1/3 2 8/5, 8/5, 8/5, 8/5, 8/5, 1, 2/5, 2/5, 2/5, 2/5, 2/5
 3909 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0 667 2, 2, 3/2, 3/2, 1, 1, 1, 1, 1/2, 1/2, 0, 0 148 2, 5/3, 5/3, 5/3, 1, 1, 1, 1, 1/3, 1/3, 1/3, 0 121 2, 2, 4/3, 4/3, 4/3, 1, 1, 2/3, 2/3, 2/3, 0, 0 62 2, 3/2, 3/2, 3/2, 3/2, 1, 1, 1/2, 1/2, 1/2, 1/2, 0 24 2, 2, 5/4, 5/4, 5/4, 5/4, 3/4, 3/4, 3/4, 3/4, 0, 0 24 7/4, 7/4, 7/4, 7/4, 1, 1, 1, 1, 1/4, 1/4, 1/4, 1/4 12 2, 3/2, 3/2, 4/3, 4/3, 4/3, 2/3, 2/3, 2/3, 1/2, 1/2, 0 8 5/3, 5/3, 5/3, 3/2, 3/2, 1, 1, 1/2, 1/2, 1/3, 1/3, 1/3 6 2, 3/2, 3/2, 1, 1, 1, 1, 1, 1, 1/2, 1/2, 0 4 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0 2 7/6, 7/6, 7/6, 7/6, 7/6, 7/6, 5/6, 5/6, 5/6, 5/6, 5/6, 5/6 2 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 2 2, 4/3, 4/3, 4/3, 1, 1, 1, 1, 2/3, 2/3, 2/3, 0 2 5/3, 5/3, 5/3, 4/3, 4/3, 4/3, 2/3, 2/3, 2/3, 1/3, 1/3, 1/3 2 5/3, 5/3, 5/3, 1, 1, 1, 1, 1, 1, 1/3, 1/3, 1/3 2 2, 7/5, 7/5, 7/5, 7/5, 7/5, 3/5, 3/5, 3/5, 3/5, 3/5, 0 1 8/5, 8/5, 8/5, 8/5, 8/5, 1, 1, 2/5, 2/5, 2/5, 2/5, 2/5 1 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2
 586 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0 189 2, 2, 3/2, 3/2, 1, 1, 1, 1, 1/2, 1/2, 0, 0 44 2, 2, 4/3, 4/3, 4/3, 1, 1, 2/3, 2/3, 2/3, 0, 0 35 2, 5/3, 5/3, 5/3, 1, 1, 1, 1, 1/3, 1/3, 1/3, 0 34 2, 3/2, 3/2, 3/2, 3/2, 1, 1, 1/2, 1/2, 1/2, 1/2, 0 33 2, 2, 5/4, 5/4, 5/4, 5/4, 3/4, 3/4, 3/4, 3/4, 0, 0 18 7/4, 7/4, 7/4, 7/4, 1, 1, 1, 1, 1/4, 1/4, 1/4, 1/4 13 2, 3/2, 3/2, 4/3, 4/3, 4/3, 2/3, 2/3, 2/3, 1/2, 1/2, 0 8 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2 7 2, 3/2, 3/2, 1, 1, 1, 1, 1, 1, 1/2, 1/2, 0 6 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0 4 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 2 5/3, 5/3, 5/3, 4/3, 4/3, 4/3, 2/3, 2/3, 2/3, 1/3, 1/3, 1/3 2 3/2, 3/2, 3/2, 3/2, 1, 1, 1, 1, 1/2, 1/2, 1/2, 1/2 2 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
 819 2, 2, 1, 1, 1, 1, 0, 0 145 2, 3/2, 3/2, 1, 1, 1/2, 1/2, 0 15 2, 4/3, 4/3, 4/3, 2/3, 2/3, 2/3, 0 12 5/3, 5/3, 5/3, 1, 1, 1/3, 1/3, 1/3 3 2, 1, 1, 1, 1, 1, 1, 0 3 1, 1, 1, 1, 1, 1, 1, 1 2 3/2, 3/2, 3/2, 3/2, 1/2, 1/2, 1/2, 1/2
 583 2, 2, 1, 1, 1, 1, 0, 0 90 2, 3/2, 3/2, 1, 1, 1/2, 1/2, 0 24 5/3, 5/3, 5/3, 1, 1, 1/3, 1/3, 1/3 20 2, 4/3, 4/3, 4/3, 2/3, 2/3, 2/3, 0 2 2, 1, 1, 1, 1, 1, 1, 0 1 1, 1, 1, 1, 1, 1, 1, 1
 64 2, 2, 1, 1, 1, 1, 0, 0 20 2, 1, 1, 1, 1, 1, 1, 0
 560 2, 2, 1, 1, 1, 1, 1, 1, 0, 0 248 2, 3/2, 3/2, 1, 1, 1, 1, 1/2, 1/2, 0 52 2, 4/3, 4/3, 4/3, 1, 1, 2/3, 2/3, 2/3, 0 49 3/2, 3/2, 3/2, 3/2, 1, 1, 1/2, 1/2, 1/2, 1/2 44 5/3, 5/3, 5/3, 1, 1, 1, 1, 1/3, 1/3, 1/3 22 2, 5/4, 5/4, 5/4, 5/4, 3/4, 3/4, 3/4, 3/4, 0 5 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 4 4/3, 4/3, 4/3, 1, 1, 1, 1, 2/3, 2/3, 2/3 3 2, 1, 1, 1, 1, 1, 1, 1, 1, 0
 244 2, 1, 1, 1, 1, 1, 0 88 3/2, 3/2, 1, 1, 1, 1/2, 1/2 30 1, 1, 1, 1, 1, 1, 1

# 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.