An example of a non-reduced Picard scheme
Let be a smooth projective connected scheme over an algebraically closed field
(experts will notice that several of these hypotheses can be weakened in what follows). Attached to
is the Picard scheme
, a locally finite type
-scheme defined functorially as sending a
-scheme
to the group
. (This uses the fact that there exists a
-point; in general one needs to sheafify with respect to a suitable Grothendieck topology.) Using smoothness of
, one can verify from the functorial definition that
satisfies the valuative criterion of properness, so its identity component
is a proper
-group scheme. If
is dimension 1, then using dimensional vanishing for coherent cohomology one can furthermore verify the infinitesimal criterion of smoothness for
, so that
is an abelian variety. Smoothness also holds if
is of characteristic 0, since in this case it is a theorem of Cartier that every
-group scheme is smooth. However, there are several examples which show that
need not be reduced even if
is a surface in positive characteristic. In this post I will describe one such example due to Serre.
First, how can one get a handle on smoothness of the Picard scheme? There are two fundamental inequalities which we will describe concerning the dimension of the Picard scheme. First,
immediate from the following more precise Lemma.
Lemma. There is a natural isomorphism , where
denotes the tangent space at the identity.
Proof. Recall the natural isomorphism . We have a split short exact sequence of abelian sheaves
, giving rise to the exact sequence of (Cech) cohomology groups
Since the short exact sequence is split, the leftmost map in this sequence is surjective and thus the lemma follows from the fact that and similarly for
. QED
If is characteristic 0 (resp.
is a curve), then Hodge symmetry (resp. Serre duality) implies that
, and it follows that
Although Hodge symmetry can fail in positive characteristic (as we will see later in our example), this inequality is always true. This is a result due to Igusa, see A Fundamental Inequality in the Theory of Picard Varieties. In fact, Igusa shows that the Albanese morphism induces an injective map
. Since
, it follows that
, yielding the above inequality. This is a deep theorem; it relies on a serious understanding of Albanese varieties to reduce to the case that
is a surface, and then resolution of singularities for surfaces to reduce to the case that
is a smooth surface; it is here that some real geometry takes place. In any case, the combination of these two results shows that it would be enough to find a smooth projective integral
-scheme
such that
Our work is now to find such an . In fact, with (substantially) more work we will show that the
we construct has the property that
, see the Remark below.
The example
With these preparations, we are ready to describe Serre’s example. We will show that if is the characteristic of
, then there is a smooth surface
in
with a free action of the group
. Grant this for now; we will construct
at the very end. Note that
exists in this case as a smooth projective variety (see SGA3, Exp. V, Thm. 4.1). We will show
and
, and it will follow immediately that
is a nontrivial finite connected
-group scheme, giving the promised example.
Lemma. .
Proof. Note that the etale cover induces an injective map
, so it suffices to show that
. Indeed, we will show that this is the case whenever
is a hypersurface inside a projective space
of dimension
. Recall the conormal exact sequence
where is the natural closed embedding and
is the quasicoherent sheaf defining it. Note that if
is a hypersurface of degree
, then
and thus
. Thus we have an exact sequence
and we see that it is enough to show and
. We’ll take these equalities in turns.
1. .
Recall the Euler exact sequence for
Taking global sections shows that . Moreover, tensoring by
and passing again to cohomology shows that
since
and
.
Consider now the exact sequence
coming from tensoring against the exact sequence defining the closed subscheme
. Taking cohomology gives the exact sequence
The two outside terms are by the above, so we see
.
2. .
We have an exact sequence
coming from tensoring against the exact sequence defining the closed subscheme
. This gives the exact sequence on cohomology
As is a projective space of dimension at least 3, the two outside terms vanish and thus so does the inner term. Since
is a closed embedding (hence in particular affine), we find
as desired. This completes the proof of the Lemma.
Lemma. .
Proof. We will first prove the weaker claim that is nonzero, in order to introduce some cool results. This is not a necessary step, but the ideas will be used later to deduce that
. Recall first the following amazing isomorphism. If
is a finite commutative
-group scheme and
is an arbitrary proper flat connected
-scheme, then
where denotes the Cartier dual of
. There is an easy-to-define natural map from left to right, but it is not obvious that it is an isomorphism; for this see Milne’s Etale Cohomology, III, Prop. 4.16, which proves a more general statement. Taking
, we see that
Note that this latter group is nonzero because admits a covering by
(namely,
!). In fact, by a Lefschetz theorem (see SGA2, Exp. XII, Cor. 3.5), since
is a hypersurface in the
-dimensional simply connected variety
, it follows that
is simply connected and thus
.
Consider now the Artin-Schreier exact sequence of etale sheaves
where the rightmost nonzero map is given by . Passing to etale cohomology gives
The leftmost map is surjective because and
is algebraically closed. Thus we have the equality
(the superscript
denoting Frobenius-invariants). In particular,
.
Now we prove the actual statement of the Lemma (using none of the preceding). Recall that the Hochschild-Serre spectral sequence of the covering
is given by
The exact sequence of low degree terms is given by
We have already seen that . Moreover,
and the action of
on this vector space is trivial since all of the functions in it are constant. Thus by familiar properties of group cohomology we have
, using that
and
is characteristic
. Thus
. QED
We have now shown while
. By our introductory remarks, it follows that
is
-dimensional but has a nontrivial tangent space at the identity. Thus indeed it is a nontrivial finite connected group scheme, as desired.
Remark. In fact, we have : recall first that a finite connected
-group is scheme-theoretically isomorphic to
for some
and
. Since
is 1-dimensional, it follows that for
we have
, and we need to show
. In any case, we have shown that there is a nontrivial
-homomorphism
, so
contains
as a closed
-subgroup. It is clear
is a finite connected group scheme, and if it is nonzero then it has a
-dimensional tangent space. Suppose for the sake of contradiction that
is nontrivial, so by our above remarks
is a connected group scheme of order
, hence isomorphic either to
or
by the classification of group schemes of order
over an algebraically closed field of characteristic
. Let
denote the corresponding order
closed subgroup, so that
is an extension of either
or
by
. In the first case,
must be either
or
(since one can show using the Kummer sequence for
that
is isomorphic to
); in the second,
is
(since one can show via similar methods that
and the latter is trivial), so in particular
contains
as a closed
-subgroup. But note
and
using in both cases that . On the other hand, we have
and
, so
cannot contain
or
as a closed
-subgroup. Moreover,
is 1-dimensional and we saw that the Frobenius on
is not identically 0 (since
), so being semi-linear it must be injective. In particular, using the exact sequence of fppf sheaves
we deduce
. Because
is self-dual it follows that
This exhausts the possibilities for , showing that
must be trivial, i.e.,
.
Question. Is there a good way to get a hold on in general? E.g., is there a good example of some
such that this is nontrivial?
It remains only to construct the surface in
with a free
-action. As before, let
be a prime number greater than or equal to
and let
. Let
be the 4-dimensional
-representation of
defined by letting the canonical generator of
act via the matrix
Note that this matrix is equal to , where
is a matrix whose fourth power is
; it follows that the
th power of this matrix is indeed
since
. This representation induces an action of
on
respecting the natural grading, and by functoriality of the
construction this gives an action of
on
whose only fixed point is
. The quotient
in the sense of locally ringed spaces exists as a scheme (see SGA3, Exp. V, Thm. 4.1), and it is projective (see SGA3, Exp. V, Rem. 5.1), hence a closed subscheme of some projective space
. Moreover,
is smooth away from the image
of
. By Bertini’s theorem, there is some hyperplane
in
not containing
such that
is smooth and integral. If
is the preimage of
in
, then
is also smooth, being a
-torsor over the smooth
, and
acts freely on
because
does not contain the unique fixed point
. This completes the construction.
I think there’s a small typo: When you discuss the Hochschild-Serre spectral sequence, the
term should be
.
With regards to your example of a scheme
with non-trivial
, would the following example work? Take
a supersingular elliptic curve. Then
is one-dimensional by Riemann-Roch. Now
and the frobenius is surjective on
. Therefore
. But now
is supersingular so
on
, and hence
.
Ben, thanks for the correction and the example. Another way to see the equality you say is by noting that elliptic curves are self-dual and supersingular elliptic curves have torsion which is a nontrivial extension of
by
(this uses some Dieudonne module calculations), so the multiplication-by-p map
admits an intermediate
cover.
The question wasn’t very well-posed; it must be very hard, for example, to give a nice characterization of those abelian varieties over
which admit
as a closed subgroup. Perhaps a better set of questions: are there any examples of a smooth projective connected variety over
with
? What about other finite group schemes
of
-power order? Are there obstructions to the existence of such
in terms of
? If
is of multiplicative type then I suspect (though I haven’t checked carefully) that you can run the same kind of construction above to construct such
(of higher dimension), but the same methods don’t seem to apply to finite unipotent group schemes like
, where the fundamental group plays no role in understanding
-covers.
There is a smooth projective connected variety with
for any connected finite commutative group scheme
. Or, more generally, there is a smooth projective connected variety with
for any finite commutative group scheme
.
Here is a rough idea of the construction:
1) Firstly consider the stack
and observe that
by faithfully flat descent. So if we apply the construction to
, we get that
.
This already gives you an example of a connected, smooth, proper Artin stack over
with
. If
is connected then we get that
is already connected, so
.
2) Now the idea is to approximate
by
where
is a complete intersection with a free action of
. If
then
. So the same argument with faithfully flat descent would show that
. Similarly, if
is connected we get
3) The only thing that is left is to construct
with an action of
as above. The construction is similar to the one you explained in your post. One considers the standard action of
on
and shows that the locus, where the action is non-free, is of codimension at least
. So you can run a similar argument as in the post to construct the desired
of dimension at least
. Then
does the job.
Of course, I skipped many details. I think I’ve worked out something similar a couple of years ago, so I can send you the notes if you want to.
The torsion component of the Picard scheme.