# More projects

Here is a list of projects that make sense as parts of the stacks project. (For a list of algebra projects, see this post.) This list is a bit random, and I will edit it every now and then to add more items. Hopefully I’ll be able to take some off the list every now and then also. If you are interested in helping out with any of these, then it may be a good idea to email me so we can coordinate. It is not necessary that the first draft be complete, just having some kind of text with a few definitions, some lemmas, etc is already a good thing to have. Moreover, we can have several chapters about the same topic, of different levels of generality (the reason this works well is that we can use references to the same foundational material in both, so the amount of duplicated material can be limited).

• If X is a separated scheme of finite type over a field k and dim(X) ≤ 1 then X has an ample invertible sheaf, i.e., X is quasi-projective over k.
• If f : X —> S is a proper morphism of finite presentation all of whose fibres have dimension ≤ 1, then etale locally on S the morphism f is quasi-projective. This also works for morphisms of algebraic spaces.
• Cheap relative duality for projective morphisms. Start with P^n over a (Noetherian) ring and deduce as much as possible from that.
• More on divisors and invertible sheaves, Picard groups, etc.
• Serre duality on projective varieties.
• Classification of curves.
• Quot and Hilbert schemes.
• Linear algebraic groups.
• Geometric invariant theory. I think that a rearrangement of the material in the first few chapters of Mumford’s book might be helpful. In particular some of the material is very general, but other parts do not work in the same generality. Note that we already have the start of a chapter discussing the myriad possible notions of a quotient, see groupoids-quotients.pdf.
• Resolution of two dimensional schemes.
• Semi-stable reduction theorem for curves. (Is there any way to do this without using resolution of singularities of two dimensional schemes or geometric invariant theory?)
• Abstract deformation theory a la Schlessinger (but maybe with a bit of groupoids thrown in).
• Deformation theory applied to specific cases: zero-dimensional schemes, singularities, curves, abelian varieties, polarized projective varieties, coherent sheaves on schemes, objects in the derived category, etc.
• Brauer groups of schemes.
• The stack of curves and pointed curves, including Kontsevich moduli stacks in positive characteristic are algebraic stacks.
• The stack of polarized projective varieties is an algebraic stack.
• The moduli stack of polarized abelian schemes is an algebraic stack.
• The stacks of polarized K3 surfaces.
• Alterations and smoothness (as an application of moduli stacks of curves above).
• Add more here as needed.

## 9 thoughts on “More projects”

1. Johan, for the etale-local (on the base) quasi-projectivity of “curves”, do you have in mind the argument using Artin approximation, or something simpler? For the case when the base is a field, you should also include the generalization when the “curve” is only assumed to be an algebraic space. (This special case is already useful when setting up etale cohomology for algebraic spaces, by adapting the proofs given for schemes.) And roughly what do you have in mind for “linear algebraic groups”? That is already the title of 3 books, as you know.

• Actually Brian, I was going to ask you about the “families of curves question”. When the morphism is proper I see how to use Artin approximation, but in general I do not. Is there some “cheap” way to produce affine opens (etale locally on the base) by extending effective Cartier divisors from the special fibre? This doesn’t look easy… Right now I’m not sure how best to prove the statement as formulated in the post, although I am confident it is correct (for example we could use a compactification argument to reduce to the proper case). Suggestions? [Edit: This is wrong, see Jason’s comment below.]

Linear algebraic groups (Borel, Springer, Humphreys). Well, it seems to me a little bit of group schemes over a base has to be discussed in order to be able to even state the material in GIT. That’s why it is on the list.

• Johan, for the relative curve thing I had in mind to use Nagata; not sure what an easier method might be. As for sst reduction for curves, in Qing Liu’s book he proves the resolution in the special case of “fibered curves” over a Dedekind base, exploiting special features of that to bypass a lot of the complications of Lipman’s treatment of the general case (e.g., excellence does not pervade his discussion). I’ve never read his treatment, since I deduced the general case for myself many years ago from the excellent case and found the experience sufficiently draining that I don’t want to revisit it. But Akshay and I will force a grad student or postdoc to present it in our Mordell seminar this year, so maybe I’ll get inspired by that to read the details of how Liu pulls it off.

2. Johan, you mention “moduli stacks of abelian varieties”. The stacks project is meant to be self-contained (in some sense), but surely you are not going to embed a treatise on the entire theory of abelian varieties over fields in the middle, right? (To get the moduli spaces off the ground one needs a huge amount of the theory over fields, what with the whole saga of polarizations, etc.). Maybe I misunderstand the intent with that topic.

• Yes, you would have to talk a bit about polarizations of abelian schemes. Call me an optimist, but I think this is not nearly as bad as semi-stable reduction of curves. Moreover, I am not sure exactly how much material you would need in order to prove the stack of polarized abelian schemes is an algebraic stack; it should come _after_ proving that the stack of polarized varieties is an algebraic stack. So the only thing that you need to do to apply this to the stack of polarized abelian schemes, is to link the notion of a polarization of an abelian scheme in terms of a morphism to the dual, to the usual notion of a polarization.

3. Dear Johan,

I believe that a separated, finite type, flat family of curves is not necessarily quasi-projective over an ‘etale cover of the base. I thought Max Lieblich gave us both a counterexample some years ago.

Essentially you start with a flat, proper, family of curves over P^1 which is not globally projective. Then you consider the pullback by the projection from A^2 – {0} to P^1, but considered as a non-proper family over A^2, or better, just the germ of this family over Spec of the complete local ring of 0 in A^2. You can recover the original family over P^1 from this new family over Spec of the complete local ring. So I do not believe this new family can be quasi-projective (unless the original family is projective).

Best regards,
Jason

• Dear Jason, Thanks! I now remember this too… argh! The problem with my suggestion above to use a compactification is that the fibre over 0 in your example would not have dimension 1. Thanks again!

[Edit: An example of a non-projective proper normal surface Z with a flat morphism Z —> P^1 can be found in section 2.5 of the paper “On non-projective normal surfaces” by Stefan Schroer.]

4. Dear Johan,

I think another great project would be an exposition of Artin’s approximation theorem and Artin’s algebraization theorems.

Best regards,
Jason