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.
Motivation
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.
Moduli theory
We now develop a geometric framework in which finding examples of varieties with no models can be made to be methodological.
Moduli problems
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
where
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:
where
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
surjective?
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).
Namely:
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
such that
- 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
if and
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
then
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]).
Miscellanea
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 .
Example 5
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.
Example 6
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.
Example 9
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].
References
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I think I object to your claim that your curve with $j$-invariant $\pi$ lies in $\PP^2_{\overline\QQ}$, unless of course you are (somewhat confusingly vis-a-vis the Lindemann-Weierstrass theorem) using $\pi$ to denote an algebraic number.
Ah! Thank you! I originally had everything with curves over
not
, thus the confusion. Thanks for leeting me know! I’ve fixed it.
Thanks for another enlightening post. I think there are some typos :
1. Example 14 : “if G/N_G(T) were finite-dimensional it must be trivial” should be “if G/N_G(T) were zero-dimensional it must be trivial”
2. The case of genus 2 : Here J_i should be of degree i instead of 2i.
And in the end when you take C to be the normalization of
V(yz^4-x^6-x^5z-x^4z^2-x^3z^3-x^2z^4-xz^5-iz^6),
the first term should be y^2z^4 I guess ?
All fixed! Thanks very much!