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.
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.
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.
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 .
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.
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
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.