The goal of this post, which is just a bit of fun between more serious posts/projects, is to discuss some examples of algebro-geometric objects over which have no models over smaller subfields and explain how moduli theory can help clarify their discovery in certain situations.
This post was born out of a very simple question I had at the beginning of my second year of graduate school:
Main question: “What is an explicit example of a variety over which has no model over ?”
Let us be slightly more rigorous about what we mean. Namely, we say that a variety over is defined over if there exists a variety over and an isomorphism .
This sounds like a fairly simple problem. For example, one is tempted to make the claim that
is not defined over since its defining equation in involves non-rational coefficients. But, of course, obviously as a -variety one has that
by the isomorphism and the latter is obviously defined over .
So, one is quickly led to trying more complicated examples involving equations of a more sophisticated form. For example, one might try equations of the form
where or, in other words, elliptic curves. This seems promising since if and are both irrational then there’s no obvious way to ‘fix this’. Indeed, if we try to ‘suck’ the into , as we sucked the into in our previous example, we see that we’ll be forced to complicate the coefficients of and . It’s then not at all clear that somehow you can make a change of variables that simultaneously makes all coefficients rational.
Of course, this is not a proof, and some complicated and surprising isomorphisms exist. For example, certainly looks like it should not be defined over at first glance but, in fact, one has an isomorphism
and the latter, as complicated as it is, is evidently defined over .
Given the above, it then seems less obvious how to actually find an example of a variety over which provably has no model, let alone an example where satisfies specified properties (e.g. being some fixed dimension, smoothness, affineness, etc.).
The goal of this post is to try and explain a general geometric methodology by which one might try to answer the main question. We will then give some additional examples of objects without models which doesn’t fit quite as nicely into this geometric framework, but which are still interesting.
We now develop a geometric framework in which finding examples of varieties with no models can be made to be methodological.
It is first useful to widen our gaze and consider the general situation in which our main question naturally sits. Namely, let us consider the structure that the objects in our original have. Namely, we had
- A family of objects over (varieties over )
- A family of objects over (varieties over ).
- A notion of isomorphism between objects in either situation.
- A pullback map which associates to an object over an object over (the fiber product of curves).
Of course, the structure above was just part of the more genral structure where instead of objects only over or we have objects over for any -scheme (namely -schemes).
We see then that the framework in which to discuss the generalization of our main question is likely the following. Let us define a moduli problem over to be a contravariant functor
- is the category of schemes over .
- is the category of groupoids.
Here by a groupoid we mean a small category (i.e. a category whose collection of objects is a set) where every morphism in is an isomorphism. A morphism of groupoids is just a functor between the groupoids.
Let’s unpack this definition a bit. The data of a moduli problem over is the following:
- For every -scheme a groupoid , which is just a collection of objects ‘over ‘ together with a notion of isomorphism between them.
- For every map of -schemes a functor (which we call the pullback functor)..
This is precisely the structure we observed our main question having above.
Remark 1: It is probably useful to put our definition into the context of more rigorous mathematics. Namely, what we’re calling a moduli problem over is what is known (e.g. as in [Olsson, Chapter 3]) as a split category fibered in groupoids over . These are simpler objects to understand than general categories fibered in groupoids as explained in [Olsson, §3.4] but, as seen in loc. cit., essentially every category fibered in groupoids is equivalent to a split one (although it is not true that it itself must be split). Thus, we shall always confuse the two notions. We will see what this means more concretely below in Remark 3.
Let us give some examples of moduli problems over
Example 2: We start with the moduli problem that was underlying our main question. Namely, let us define
where we set to be the category of -schemes with morphisms being isomorphisms of -schemes. The pullback functor is the usual fiber product of schemes/morphisms. More explicitly, if we have a morphism of -schemes then the functor
is defined on objects by the fibered product and on morphisms by the pullback of morphisms to the fibered product.
Remark 3: We see in the above where, in reality, it is more natural to work with categories fibered in groupoids with no particular splitting as mentioned in Remark 1. Namely, the fiber product is only defined up to isomorphism. So, to really make a moduli problem in our sense, one has to choose for all morphisms and -schemes over an explicitly chosen fibered product in a way that is compatible with composition of morphisms in . Again, we will ignore this subtelty both here and in the future examples.
Example 4: Let and be integers. For a -scheme we call a morphism a genus curve over if it is smooth, proper, and for every point of the -variety is a geometrically connected curve of genus . By an -marked curve of genus over we mean a genus curve over together with -marked sections of . By a morphism of -marked
we mean a morphism of genus curves over such that for all .
We can then define a moduli problem over
where for an -scheme we set to be the groupoid of -marked genus curves over with morphisms being isomorphisms. For a morphism we define the pullback map
to be as expected:
is on the -factor and on the -factor.
Example 5: Let us define
to associate to a -scheme the groupoid of affine -schemes with morphisms being usual isomorphisms of -schemes. The pullback morphisms is the usual fiber product of affine schemes.
Example 6: Let
be a presheaf on . Then, evidently is a moduli problem over since any set can be thought of as a groupoid where the only morphisms are the identity morphisms.
Example 7: We give an even more special case of the last example. Namely, if is any -scheme, then we can think of as a moduli problem over by identifying with its contravariant functor
which is reasonable to do by Yoneda’s lemma (e.g. see [Stacks, Tag001P])
Example 8: Let be an integer. We then define the modular curve with full level structure to be the moduli problem
where is the category of pairs where is an elliptic scheme (which is a relative version of elliptic curves—see [KM, Chapter 2] or [Hida1, Chapter 2]) and
is an isomorphism of group schemes which we call a trivialization. The isomorphisms in this category are isomorphisms of elliptic schemes that preserve the trivialization in the obvious way. The pullback map is the obvious one. Let us note, as essentially shown in [KM, Chapter 2] and [Hida1,Chapter 2], one has that is essentially identified with the moduli problem from Example 4.
There is a natural generalization of to higher abelian varieties denoted by for any integers and so that . To define this would take us too far astray, but see cite [Hida2, §6.3.2] for details (note that in the notation of [Hida2, §6.3.2] our notation is , , and ).
Example 9: We denote by
the moduli problem which associates a -scheme the category or finite type affine group schemes over with morphisms being isomorphisms of -group schemes. The pullback map is the obvious one. We also have a submoduli problem of consisting of those affine group schemes which are reductive (e.g. see [Conrad1, Definition 3.1.1]).
Example 10: Let be a (connected) reductive group over (e.g. see [Milne, §19.b]). We then define a presheaf
which assigns to a -scheme the set of maximal tori in (e.g. see [Conrad1, §3.2]).
We define a morphism of moduli problems over to just be a a natural transformation of functors . There is the obvious notion of an isomorphism then between moduli problems over which means that for all -schemes the functor is an isomorphism of categories. We call an equivalence if is an equivalence of categories for all .
We end this subsection by making a definition that is clearly useful given that our main question is about isomorphism classes. Namely, note that if is a groupoid then the set
is a set which we call the connected components of — aptly named since if we think of as being a directed multigraph then is the set of connected components in the graph theoretic sense.
We note then that associated to any moduli problem over is a presheaf
which we call the component presheaf of . Let us note that if if is a morphism of moduli problems over then we get an induced morphism . Moreover, if is a presheaf then since has no non-identity morphsims for all , we see that . Combining these two, we see that any morphism , where the former is a moduli problem over and the latter is a presheaf, factors uniquely through the obvious map .
The generalized main question and fine moduli scheme
So, we have formalized a generalization of the objects in the main problem in the last section but, as of now, this all seems like empty verbiage. What is the point of the above if it doesn’t help us solve the main question? Well, it does. Or, perhaps more to the point, it more easily allows us to create a framework in which our main problem can be systematically solved.
Before we do that though, let us try and formalize what the extension of the main question to the general situation would be. A moment’s thought shows that, with our above notational choices, we can formulate it as follows:
Generalized main question: Let be a moduli problem over . Then, is the map
Let’s warm-up for our approach to understanding the generalized main question by thinking about the problem in a very specific case. Namely, as we saw in Example 7 it’s true that any scheme over over defines a moduli problem over . It turns out that we can leverage geometry then to answer the generalized main question quite easily for (assuming minor conditions).
Proposition 11: Let be a scheme locally of finite type over which is of positive dimension. Then, the map is not surjective.
Proof: Let be any affine open subset of . Evidently then one has that the map
is surjective. Since has an affine open subscheme of the same dimension, we may assume without loss of generality that is affine.
By Noether normalization (e.g. see [Stacks, Tag00OY]) we know that there exists a finite surjective map where . Note then that we have a commuting diagram
Now, the right vertical map is surjective since is surjective and, by assumption, the top horizonal map is surjective. Evidently this then implies that the bottom horizontal map is surjective. But, this is a contradiction.
Of course, this gives a fairly definitive answer to the generalized main question in the case when is a scheme. In fact, it’s to our benefit to put a little more thought into this statement. Namely, let us say that a moduli problem over has a fine moduli scheme if there exists a scheme over and an equivalence . Note that by Yoneda’s lemma such a scheme is unique up to isomorphism.
We can then deduce from Proposition 11 the following:
Corollary 12: Let be a moduli problem over which posseses a fine moduli scheme which is locally of finite type over and of positive dimension. Then, the map
is not surjective.
We now consider how this interacts with the examples from above by discussing those which possess a fine moduli scheme:
Example 13: Let and be integers. It is then a theorem of Mumford (e.g. see [Hida2, Theorem 6.20]) that , the moduli problem from Example 8, has a fine moduli scheme which is quasi-projective, smooth, and of dimension (for this latter fact see the description of the complex points of after (A4) on [Hida2, Pg. 277]). From this and Corollary 12 we deduce a negative answer to the generalized main question for the moduli problem for .
Example 14: Let be a reductive group over . Then, the moduli problem classifying maximal tori in , as in Example 10, has a fine moduli scheme space which is smooth, quasi-affine, and of positive dimension of is not a torus. Indeed, by [Conrad1, Theorem 3.2.6] the fine moduli scheme exists and is isomorphic to for any maximal torus of . Now, if were zero-dimensional it must be trivial since is smooth (by loc. cit.) and geometrically connected (since is, see [Milne, Theorem 2.37(c)]). But, this then implies that and, since is connected, that . But, this implies that (e.g. combine [Milne, Corollary 17.39(a)] and [Milne, 17.84]). Thus, combining this with Corollary 12 we see that we have a negative answer to the generalized main problem for .
Of course, the problem is that ‘most’ interesting moduli problems are not going to have a fine moduli scheme. In fact, there is a good way to think about why this is a case. One can easily note two things about schemes which makes them special in the pantheon of moduli problems over
- They have ‘gluing properties’ that a general moduli problem over needn’t have. More rigorously, any presheaf which has a fine moduli scheme is a sheaf for the étale or fppf topologies on (e.g. see [Olsson, §2.1-§2.2] and [Olsson, Theorem 4.1.2]).
- They have no non-trivial automorphisms. Namely, for any presheaf on and any -scheme we have that is a set and so, in particular, for any object of one has that is trivial (where this automorphism group is in terms of morphisms in the category ).
In particular, from 2. we see that any moduli problem over which has objects with non-identity automorphisms is bound to not have a fine moduli scheme.
Remark 15: What we’re describing above might be closer to what is called an ‘algebraic space’. See Remark 17 below.
That said, one might hope to ‘rigidify’ a moduli problem over so that it actually obtains a fine moduli scheme or, at least, so that it possesses property 2. from above. Namely, if one has an object then for any extra structure on the set of automorphisms preserving this extra structure certainly is going to be smaller than the set of all automorphisms of . So, adding extra structure (i.e. rigidifying) should cut down on the automorphism groups of objects in our moduli problem.
For example, has no fine moduli scheme since, for example, condition 2. fails for —an elliptic curve has non-trivial automorphisms (e.g. the inversion map ). But, one can think of for as a rigidification of since it’s essentially the data of an object of and an extra piece of information given by the isomorphism . Now, for (as implicitly stated in Example 13) one has that the rigidified objects of have no automorphisms and, in fact, possess fine moduli schemes.
Of course, this is not particularly useful for solving our main question since even if one can rigidify a moduli problem over so that it has a fine moduli scheme, all that will tell us is that there are objects of which don’t have models over with the extra rigidifying structure. Of course, this doesn’t preclude that these objects do have models over without this extra structure.
Coarse moduli spaces
So, we saw in the last section that one has a fairly concrete way of showing that the generalized main question has a negative answer if our moduli problem has a fine moduli scheme. We also saw that at the end of the last section that even if we can somehow modify a moduli problem to have a fine moduli scheme that this is not, in general, good enough to answer the genearlized main question. So, what do we do?
One obvious guess is that instead of trying to force Proposition 11 to apply directly to a moduli problem by assuming that has a fine moduli scheme, that we should try to adapt Proposition 11 to work in more generality.
Of course, we used geometry in the proof of Proposition 11 and so we will need to have some sort of theory of geometry for moduli spaces. But, as it stands, moduli spaces are so general as to make this not really tenable. So, a first step would be single out those moduli problems that are sufficiently geometric. Doing this, and carrying out the development of geometry in those cases, is precisely the modern theory of (algebraic/Artin) stacks as initiated by Deligne and Mumford in [DM] and further developed by many people.
We shall not need the theory of algebraic stacks explicitly but we mention very briefly the two extra major conditions that one imposes on a moduli problem to make them an algebraic stack. The first is that is a a ‘sheaf’ for the étale (equivalently, for algebraic stacks, the fppf topology—see [LMB, Corollaire 10.7]). Of course, since has values in groupoids, not sets, this is more complicated than the idea of being a presheaf being a sheaf for the étale topology and in fact making sense of what the sheaf condition means in this context is exactly what the ‘stack’ portion of the moniker ‘algebraic stack’ is referring to—see [Olsson, Chapter 4] for details. The second condition is that while is not a scheme, it at least needs to be covered by a scheme in a sense we shall not make precise here.
Remark 16: Several remarks are in order here. First, we only stated the two extra ‘major’ conditions to be an algebraic stack. There is one extra condition which is necessary—see [Olsson, Definition 8.1.4] for details. For more information on stacks see [Olsson], [LMB], and [Stacks, Tag0ELS] for general theory and [Bertin] for a more leisurely introduction.
Now the main reason that the theory of algebraic stacks is not explicitly useful for us here is that in fact (at least to the author’s knowledge) while it is incredibly rich theory, it is not sufficient to repeat the argument in Proposition 11.
This brings us to the other obvious guess for trying to escape the yoke of fine moduli schemes. Namely, for a moduli probelm we can ask whether or not there exists a scheme which may not be equivalent to , but ‘best approximates’ (amongst schemes) in a precise sense.
Let us try to be more rigorous. Namely, suppose that is a moduli problem over . We then say that a -scheme is a coarse moduli scheme for if there exists a morphism of moduli problems
- The map is initial amongst maps to -schemes.
- The map is a bijection whenever is an algebraically closed field.
Note that by condition 2. the space is uniquely determined up to isomorphism by these conditions.
Remark 17: A remark is in order about the relationship between the above, which is non-standard terminology, and the notion of coarse moduli space which exists in the literature (e.g. [Olsson, Definition 11.1.1]).
The category of -schemes is not very well-behaved within the category of moduli spaces over (e.g. a presheaf which étale locally is isomorphic to a scheme need not be isomorphic to a scheme). For this reason one generally enlarges the category of schemes to the category of algebraic spaces. Intuitively these are presheaves which are étale locally schemes (e.g. see [Stacks, Tag0BGQ]) or they are also algebraic stacks with no automorphisms (e.g. see [Stacks, Tag04SZ]).
One then defines a coarse moduli space by replacing every instance of ‘scheme’ with ‘algebraic space’ in the definition of coarse moduli scheme. Note that if is a coarse moduli space of a moduli problem which is also a scheme, then it is a coarse moduli scheme in the sense above. The converse is not clear to me (is there a non-scheme algebraic space which has an initial map to schemes?), but we will never really contend with this issue here.
One definitive advantage of working with coarse moduli spaces opposed to coarse moduli schemes is the huge generality in which they exist. This is typified by the Keel–Mori theorem (e.g. see [Olsson, Theorem 11.1.2] and [Conrad2]).
Note that since is a scheme it’s a presheaf and thus as mentioned before we have a factorization
and we denote this factorized arrow by as well.
Now, as intimated the existence of a coarse moduli scheme is actually a strong enough to answer the generalized main question in the negative. More rigorously:
Proposition 18: Suppose that is a moduli problem over which admits a coarse moduli scheme which is a positive-dimensional scheme locally of finite type over . Then, the map
is not surjective.
Proof: Let us note that we have a commuting diagram
Since the right vertical map is bijective since is a coarse moduli scheme of , we see that if the top horizontal map is surjective then so is the bottom horizontal map. But, this then contradicts Proposition 11. The exact same argument works to show that the map is not surjective.
Remark 19: One can replace Proposition 18 by a generalization where one only assumes that admits a coarse moduli space (in the sense of Remark 17) which has a positive dimensional open subscheme of finite type over . This is, I believe, automatically the case if is separated, locally of finite type, and is of positive dimension (in the sense of [Stacks,Tag04N6])—indeed one uses [Stacks, Tag0BAS] and the definition of dimension.
So, we see that the existence of a coarse moduli schemes (which is sufficiently well-behaved) is enough to guarntee a negative answer to our generalized main question. Of course, this is only useful if, unlike the case of fine modulie schemes, more moduli problems have coarse moduli schemes. The main example of this will be discussed in the next section.
But, before we go on we note that we can actually make our solution to the generalized main question for a moduli problem with coarse moduli scheme more constructive. Namely, let us say that an element of is -rational if is in . Then, evidently we see that if is in the image of the map then is -rational. So, any which is not -rational cannot be in the image. This will be the main technique we will use to find objects with no model—find objects which are not -rational.
The main question for curves
We now try to leverage our discussion of coarse moduli schemes from the last section to answer more explicitly the generalized main question for the moduli problems . It turns out that the nature of this discussion varies greatly depending on whether , , or . So, we handle these cases separately.
The case of genus one
We start with the case of genus because it has both an interesting answer, but also is likely to be more familiar to those with a basic background in algebraic geometry. The thing that makes the case of genus slightly more tricky than the case of genus greater than is that doesn’t apparently have a coarse moduli scheme if one looks through the literature. The reason for this is quite simple: is not an algebraic stack, and thus it’s not common to talk about coarse moduli spaces for such objects. But, we’ll see here that this doesn’t matter much.
We begin by noting that while , the moduli problem of genus curves, doesn’t apparently have a coarse moduli scheme, the closely related moduli problem of elliptic curves does:
Proposition 20: The moduli problem has as a coarse moduli space.
Proof: For a full rigorous proof see [Olsson, Theorem 13.1.15].
In fact, the map
realizing is the -invariant. See loc. cit. for more details, but let us suffice it to say that if one has an extension of then every elliptic curve over is isomorphic to one of the form
for some such that (e.g. see [Silverman, §III.1] and [Silverman, §III.3]). Then,
(see again [Silverman, §III.1] and compare this with [Olsson, Theorem 13.1.15]).
From this, we deduce the following:
Corollary 21: Let be an elliptic curve over . If has a model over as an elliptic curve, then .
In fact, one can show that
so that the composition
is surjective for all field extensions of . Using this, one can actually upgrade the only if statement in Corollary 21 to an if and only if statement.
So, how does this answer our question about ? Well, I claim that we can use the -invariant to build a coarse moduli scheme for . How? Well, the key observation is that we actually have a contravariant morphism of moduli problems
given on a -scheme as the functor
sending a curve of genus over to its Jacobian which is, by definition, the identity component (e.g. see [BLR, Chapter 8] and [BLR, Theorem 1 in §9.3]). The Jacobian is evidently an element of since it’s a smooth group scheme over whose fibers are cruves of genus (as can be seen from [Silverman, Proposition 3.4 in §III.3]) and which is proper (e.g. see [EGAIV Troisième partie, Corollaire 15.7.10]).
Now, what makes the map so useful is the following:
Proposition 22: For any algebraically closed extension over the induced map
is a bijection.
Proof: Suppose that and are two genus curves over . Choose base points (which is possible since is algebraically closed). Note then that there are isomorphisms of group schemes (e.g. see [Silverman, Proposition 3.4 in §III.3]). Thus, if then . Thus, is injective.
To see that the map is surjective we again note that for any elliptic curve over one has an isomorphism of group varieties (again by [Silverman, Proposition 3.4 in §III.3]).
From this, we deduce the following:
Proposition 23: The composition
is a coarse moduli scheme.
Proof: To see that this map is initial, note that we have a natural map of moduli problems
which just forgets about the marked point. Suppose that we have a map of moduli problems
for a -scheme . Note then that by Proposition 20 there exists a unique morphism such that
We then claim that
Indeed, since the map to factors through the component presheaf it suffices to note that for any genus curve over one has that
But, let us note that since is smooth, there exists an étale cover such that has a section. Note then that this implies that is isomorphic to as follows from [KM, Theorem 2.1.2]. So, then we see that has image in given by
since is an elliptic curve. But, this implies that and have the same image under the map which implies they’re equal since this map is injective since is a sheaf.
The claim that induces a bijection on -points for an algebraically closed extension of follows by combining Proposition 20 and Proposition 22.
Thus, combining everything above, we have the following corollary:
Corollary 24: Let be a genus curve over . Then, is defined over if and only if is rational.
Thus we have answered our generalized mai question for , and we will have answered our main question as soon as we exhibit a genus curve over with irrational -invariant. But, for example, follows from our discussion above we have that the genus curve we know that the curve
has -invariant and thus is not defined over .
The case of genus
We now move onto the case of genus curves. This is, in some sense, simpler than the genus case since there is a well-known explicit description of the coarse moduli scheme for in the literature (whereas, as we remarked in the last section, there is none for since it’s not an algebraic stack and thus not usually considered). But, it is also harder than the genus case since the coarse moduli scheme is more complicated.
In particular, we start with the following:
Theorem 25 ([Igusa, Theorem 2]): There is a morphism
where is given degree . Moreover, this morphism realizes as the coarse moduli scheme of .
Let us explain in more detail what this coarse moduli space is. Namely, we are thinking of as a graded ring where each is given degree . With this definition one sees that is homogenous of degree and thus we can form the quotient ring
which has an inherited grading. We can then talk about the proj construction of this graded ring. We then note that since is homogenous of degree and thus we can consider the non-vanishing locus in
which is an affine -variety with coordinate ring
(i.e. the degree elements of the localization of at ).
Let us denote this scheme which is the coarse moduli scheme by for notation convenience. The coordinates of are called the Igusa invariants. Let us note that, by definition, we can realize as a locally closed subscheme of the weighted projective space:
In particular, we see that we can describe as the set of equivalence classes of quintuples such that
- Each is in .
- One has that .
- One has that .
where is equivalent to if there exists some in such that for all .
Thus, since is evidently finite type over and of positive dimension we know that the answer is negative for the generalized main question for .
In fact, to find a genus curve over not defined over we need only find a such that quintuple
of Igusa invariants is not equivalent to rational qunituple. Unfortunately, the actual definitions of the Igusa invariants is pretty complicated (e.g. see [BSSVY, §4.1]). That said, there is an implimentation of the Igusa in Magma (e.g. see [Magma]) and using this one can show that if is the normalization of
which one can quickly check is not equivalent to a rational quintuple. Thus, is not defined over .
The case of genus greater than
The case of is essentially the same as in the case but is much less explicit. Namely, we have the following well-known result:
Theorem 26: Let be an integer. Then, has a coarse moduli space which is a smooth affine -scheme of dimension .
Proof: See [MFK, Proposition 5.4].
From this, we deduce from Proposition 18 that for each there exists a genus curve over which is not defined over . Of course, it’s more difficult to give an explicit example in general since the explicit descriptions of , and thus the -rational objects in are more difficult to describe.
The case of genus
We end with the case of genus curves. This case, for our particular generalized main question, is not so interesting. Namely, we recall the following basic theorem:
Proposition 27: Let be a field and a smooth proper geometrically connected curve over of genus . If , then .
Proof: See [Liu, Proposition 4.1 in §7.4.1].
From this we see that is a singleton and thus the answer is trivially affirmative to the generalized main question for .
In particular, this would suggest that would, if it has a coarse moduli space, have one of dimension . Indeed, one can in fact show that is a coarse moduli space (e.g. see [Hartshorne, Proposition 25.1]).
In this final section we’d like to sew up the cases of the generalized main question for the examples we discussed above. Note that Example 2, Example 4, Example 7, Example 8, Example 10 are covered by the discussion above. Thus, it reall suffices to discuss, Example 9, Example 5, and .
We would like to give at least one example of an affine variety over which has no model over . There is no moduli theoretic technique I am aware of for the production of such an example (I’m not even aware any well-behaved moduli spaces of affine varieties with no extra structure!), so our construction will be more ad hoc.
The key fact is that there exists smooth proper geometrically connected curves over such that is trivial. In fact, there exist explicit examples of such curves of genus for each (e.g. see [Poonen2]).
Remark 28: One can extend the construction below to the case of genus curves with a little more work, using the objects constructed in [Poonen3]. We leave it to the reader to flesh this out.
Let be such a curve and let be an element of not contained in . Then, is an affine open subscheme of (e.g. see [Youcis, Lemma 13] for a proof of this well-known result).
We then have the following claim:
Proposition 29: The smooth integral affine curve does not have a model over .
Proof: Suppose that has a model over . Let be a smooth compactification of . We claim that is isomorphic to . Indeed, since is projective one knows that the set of isomorphism classes of varieties over such that is classified by the cohomology set
(e.g. see [Poonen1, Theorem 4.5.2]). But, since is trivial this implies that any such must be isomorphic to over . Since is a compactification of we know that . Thus, .
Note then that becomes isomorphic to over and so consists of a degree effective divisor or, equivalently, for some . Note now that the isomorphism
extends to an isomorphism
such that maps to . But, since is trivial we know that is the identity, and so . This is a contradiction.
Thus, we see that the generalized main question has a negative answer for the moduli problem from Example 5.
Remark 30: The argument given above is somewhat persnickety and, in particular, it’s not clear (to me at least) how to generalize it to produce examples in higher dimensions.
Finding examples of presheaves that have an affirmative answer to the generalized main questions are trivial. For example, you can take any constant presheaf. Thus, one either has an affirmative or negative answer to the generalized main question for the moduli problem in Example 6 depending on the specifically chosen presheaf.
It is not obvious but it is true that generalized main question has an affirmative answer for the moduli problem from Example 9. The main ingredients are as follows. Every reductive group over is split (in the sense of [Milne, §21.a]). It is then a non-trivial theorem that has a model over . Essentially, one constructs from and the choice of a maximal torus of a root datum (e.g. see [Milne, §21.c]). One then uses the Existence Theorem (e.g. see [Milne, Theorem 23,55]) to say that there exists a split reductive group over which has this associated root datum. One then uses the Isomorphism Theorem (e.g. see [Milne, Theorem 23.25]) to deduce that .
Interestingly though, the generalized main question for the larger moduli problem of all affine finite type group schemes has a negative answer. Indeed, by [Milne, Theorem 14.37(b)] it suffices to show that there exist nilpotent lie algebras over which don’t have models over . Examples of such objects were originally constructed by Malcev in [Malcev]. But, since this article seems difficult to obtain the reader may also see [Scheuneman, §3 Proposition 2].
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