The torsion component of the Picard scheme

This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there.

Main Theorem. Let \(S\) be a noetherian local ring and let \(G\) be a finite flat commutative group scheme over \(S\). There exists a smooth projective scheme \(X\) over \(S\) with geometrically connected \(3\)-dimensional fibers such that \(\mathbf{Pic}_{X/S}^{\tau} \cong G\). If the Cartier dual \(G^\vee\) is etale (i.e., \(G\) is of multiplicative type), then we may take \(X\) to have \(2\)-dimensional fibers instead. Moreover, \(X\) can be taken to be the quotient of a complete intersection under the free action of \(G^\vee\).

A major motivation for the Main Theorem is that it can be used to construct examples of smooth projective schemes over a DVR of equicharacteristic \(p\) such that the Hodge numbers of the special and generic fibers are not equal; for this phenomenon, see the final section. This answers a question asked in the Stanford Number Theory Learning Seminar in a way not easily findable in the literature.

The basic construction of \(X\) as in the Main Theorem will be very similar to the one given in Cotner’s previous post, but there are a few major simplifications owing to the use of descent techniques, as well as some technical difficulties to overcome coming from the new relative setting. For example, there are some small intermediate arguments with algebraic spaces owing to the fact that certain Picard functors are not obviously representable (though all the relevant ones for us are a posteriori representable). We also remove all mention of Igusa’s theorem from the previous post. Our basic strategy in the general case is as follows. (See the rest of the post for notation and definitions.)

  1. Show that any complete intersection of dimension \(\geq 3\) has trivial \(\mathbf{Pic}^\tau\).
  2. Show that if \(Y \to X\) is a \(G\)-torsor, then the pullback map \(\mathbf{Pic}_{X/S}^\tau \to \mathbf{Pic}_{Y/S}^\tau\) has kernel \(G^\vee\), the Cartier dual of \(G\).
  3. Construct a projective space \(P\) over \(S\) on which \(G^\vee\) acts, freely outside of a codimension \(\geq 4\) closed subset.
  4. Use Bertini’s theorem to slice the quotient \(P/G^\vee\) by hypersurfaces to obtain a smooth projective scheme \(X\) of dimension \(3\) so that the pullback \(Y\) of \(X\) in \(P\) is a complete intersection on which \(G^\vee\) acts freely.
  5. Conclude that \(Y \to X\) is a \(G^\vee\)-torsor and thus \(\mathbf{Pic}_{X/S}^\tau \cong (G^\vee)^\vee \cong G\) by points 1 and 2.

1. Complete intersections and Picard schemes

Definition. Let \(S\) be a scheme and let \(X\) be a closed subscheme of \(\mathbf{P}_S^n\). We say that \(X\) is a complete intersection of dimension \(d\) if it is a flat finitely presented \(S\)-scheme with fibers of pure dimension \(d\) which are complete intersections.

Lemma 1.1. Let \(X\) be a complete intersection of dimension \(d \geq 1\) over a field \(k\) and let \(N \geq 0\) be an integer. We have \(H^i(X, \mathcal{O}_X(-N)) = 0\) for all \(1 \leq i \leq d-1\), \(H^0(X, \mathcal{O}_X) = k\), and \(H^0(X, \mathcal{O}_X(-N)) = 0\) for all \(N \geq 1\). In particular, \(X\) is geometrically connected. Moreover, if \(X\) is smooth and \(d \geq 2\) then \(H^0(X, \Omega_{X/k}^1) = 0\).

Proof. We induct on the codimension \(c = n - d\) of \(X\) inside \(P = \mathbf{P}_k^n\). If \(c = 0\), then this follows from the familiar computations of the cohomology of projective space. In general, by the definition of a complete intersection there is some complete intersection \(Z \subset P\) and hyperplane \(H \subset P\) of degree \(m\) such that \(X = Z \cap H\) and moreover there is a short exact sequence

\[ 0 \to \mathcal{O}_Z(-m) \to \mathcal{O}_Z \to i_* \mathcal{O}_X \to 0, \]

where \(i: X \to Z\) is the natural inclusion. Tensoring by \(\mathcal{O}_Z(-N)\) and passing to cohomology gives exact sequences

\[ H^i(Z, \mathcal{O}_Z(-N)) \to H^i(X, \mathcal{O}_X(-N)) \to H^{i+1}(X, \mathcal{O}_Z(-m-N)). \]

By induction, if \(1 \leq i \leq c\) then the two outside terms vanish. If \(i = 0\), then we have the exact sequence

\[ 0 \to H^0(Z, \mathcal{O}_Z(-N)) \to H^0(X, \mathcal{O}_X(-N)) \to H^1(Z, \mathcal{O}_Z(-m-N)) = 0. \]

If \(N = 0\), this shows \(H^0(X, \mathcal{O}_X) = k\). If \(N \geq 1\), then again \(H^0(X, \mathcal{O}_X(-N)) = 0\). Since \(H^0(X, \mathcal{O}_X) = k\), it follows from the Stein factorization that \(X\) is geometrically connected.

Now suppose that \(X\) is smooth and \(d \geq 2\). To compute \(H^0(X, \Omega_{X/k}^1)\), recall first the Euler exact sequence for \(n\)-dimensional projective space:

\[ 0 \to \Omega_{P/k}^1 \to \mathcal{O}_P(-1)^{n+1} \to \mathcal{O} \to 0. \]

From this, we see that \(H^0(P, \Omega_{P/k}^1) = 0\). In general, if a smooth \(X\) is the intersection of hypersurfaces whose degrees sum to \(N\), then we look at the conormal exact sequence

\[ 0 \to \mathcal{O}_{X}(-N) \to i^*\Omega_{P/k}^1 \to \Omega_{X/k}^1 \to 0 \]

for \(i: X \to P\) the natural inclusion. Passing to cohomology gives the short exact sequence

\[ H^0(X, i^* \Omega_{P/k}^1) \to H^0(X, \Omega_{X/k}^1) \to H^1(X, \mathcal{O}_{X}(-N)) \]

and it suffices by the above to show that \(H^0(X, i^*\Omega_{P/k}^1) = 0\). This now follows from the short exact sequence

\[ 0 \to \Omega_{P/k}^1(-N) \to \Omega_{P/k}^1 \to i_* i^* \Omega_{P/k}^1 \to 0 \]

since \(H^0(P, \Omega_{P/k}^1) = 0\) and \(H^1(P, \Omega_{P/k}^1(-N)) = 0\). This completes the cohomology calculations.

Recall the definition of the Picard functor: if \(X \to S\) is a morphism of schemes then we define \(\mathbf{Pic}_{X/S}\) to be the fppf sheafification of the functor sending an \(S\)-scheme \(T\) to \(\mathrm{Pic}(X_T)\). The formation of this sheaf commutes with base change on S, i.e., \(\mathbf{Pic}_{X/S} \times_S T = \mathbf{Pic}_{X_T/T}\) for all \(S\)-schemes \(T\). In this generality, this functor is essentially useless, but Grothendieck and Artin proved the following remarkable representability theorems, see FGA Explained, Theorem 9.4.8, and Artin, Algebraization of Formal Moduli: I, Theorem 7.3.

Theorem 1.2. (i) (Grothendieck) Suppose that \(X \to S\) is flat, finitely presented, projective Zariski-locally on \(S\), with geometrically integral fibers. Then \(\mathbf{Pic}_{X/S}\) is represented by a separated, locally finitely presented \(S\)-scheme. If \(X(S) \neq \emptyset\), then \(\mathbf{Pic}_{X/S}(T) \cong \mathrm{Pic}(X_T)/\mathrm{Pic}(T)\) naturally for all \(S\)-schemes \(T\).

(ii) (Artin) If \(f: X \to S\) is flat, proper, finitely presented, and cohomologically flat in dimension \(0\) (i.e., \(f_* \mathcal{O}_X \cong \mathcal{O}_S\) holds universally), then \(\mathbf{Pic}_{X/S}\) is a quasi-separated algebraic space locally of finite presentation over \(S\).

Algebraic spaces only intervene for us in a rather technical way, as one can see in the proof of Lemma 1.3; ultimately everything we discuss will be a scheme, but to apply geometric methods we will need to know a priori that \(\mathbf{Pic}_{X/S}\) is an algebraic space.

There are two further important notions for us: the identity component \(\mathbf{Pic}_{X/S}^0\) and the torsion component \(\mathbf{Pic}_{X/S}^\tau\). The identity component is defined as the subfunctor of \(\mathbf{Pic}_{X/S}\) whose set of \(T\)-points consists of those \(T\)-points \(\varphi\) of \(\mathbf{Pic}_{X/S}\) such that for every \(t \in T\) there exists an algebraically closed extension \(K\) of \(k(t)\), a connected \(S\)-scheme \(C\), points \(x, y \in C\) such that \(K\) is an extension of \(k(x)\), and a \(C\)-point \(\psi\) of \(\mathbf{Pic}_{X/S}\) such that \(\varphi_t = \psi_x\) in \(\mathbf{Pic}_{X/S}(K)\) and \(\psi_y = 0\). Note that if \(\mathbf{Pic}_{X/S}\) is representable then this is the same as the set-theoretic union \(\mathbf{Pic}_{X/S}^0 = \bigcup_{s \in S} \mathbf{Pic}_{X_s/k(s)}^0\), where \(\mathbf{Pic}_{X_s/k(s)}^0\) is the identity component of the locally finite type \(k(s)\)-group scheme \(\mathbf{Pic}_{X_s/k(s)}\). The torsion component is defined as \(\mathbf{Pic}_{X/S}^{\tau} = \bigcup_{n \geq 1} [n]^{-1}(\mathbf{Pic}_{X/S}^0)\), where \([n]\) denotes the multiplication by \(n\) map; this is also a subgroup functor of \(\mathbf{Pic}_{X/S}\).

For our purposes, \(\mathbf{Pic}_{X/S}^0\) will not be a very useful functor: in general, when \(\mathbf{Pic}_{X/S}\) has non-reduced geometric fibers, \(\mathbf{Pic}_{X/S}^0\) need not be an open subscheme of \(\mathbf{Pic}_{X/S}\), though it is on fibers. (We will see examples of this phenomenon later.) However, the morphism of functors \(\mathbf{Pic}_{X/S}^\tau \to \mathbf{Pic}_{X/S}\) is always representable by open immersions when \(X\) is proper over \(S\); if \(X\) is projective over \(S\) then this morphism is also representable by closed immersions. For both of these assertions, see SGA6, Exp. XIII, Thm. 4.7.

Lemma 1.3. Let \(X\) be a complete intersection of dimension \(d \geq 2\) over a scheme \(S\). If \(X\) is smooth or \(d \geq 3\) then \(\mathbf{Pic}_{X/S}^\tau = 0\). (In particular, \(\mathbf{Pic}_{X/S}^\tau\) is representable.)

Proof. First suppose \(S = \mathrm{Spec} \, k\) for an algebraically closed field \(k\). By Theorem 1.2(ii) and Lemma 1.1, \(\mathbf{Pic}_{X/k}\) is a quasi-separated algebraic space locally of finite type over \(k\). By Lemma 4.2 of Artin’s paper cited above, it follows in this case that in fact \(\mathbf{Pic}_{X/k}\) is a \(k\)-group scheme. (Note that quasi-separatedness is part of the definition of an algebraic space in Artin’s paper.) As \(T_0 \mathbf{Pic}_{X/k} \cong H^1(X, \mathcal{O}_X)\), we see that \(\mathbf{Pic}_{X/k}^0 = 0\) by Lemma 1.1. So it suffices to show that \(\mathrm{Pic}(X)\) has no torsion. If \(d \geq 3\), then a Lefschetz theorem (see SGA2, Exp. XII, Cor. 3.7, and note that no smoothness hypotheses on \(X\) are necessary) states that \(\mathrm{Pic}(X) \cong \mathrm{Pic}(\mathbf{P}^n) \cong \mathbf{Z}\), so indeed \(\mathrm{Pic}(X)\) is torsion-free in this case. Now suppose \(X\) is smooth. Recall that \(H^1_{\mathrm{fppf}}(X, \mu_n) \cong \mathrm{Pic}(X)[n]\) for all \(n \in \mathbf{Z}\) since \(k\) is algebraically closed. If \(p\) does not divide \(n\) then \(\mu_n \cong \mathbf{Z}/n\mathbf{Z}\), so we have

\[ H^1_{\mathrm{fppf}}(X, \mu_n) \cong \mathrm{Hom}(\pi_1(X), \mathbf{Z}/n\mathbf{Z}). \]

By a Lefschetz theorem (see SGA2, Exp. XII, Cor. 3.5, and note again the lack of smoothness hypotheses), we have \(\pi_1(X) \cong \pi_1(P) = 0\), so that this Hom set is trivial. To prove that there is no \(p\)-torsion, note that there is an exact sequence of Zariski sheaves

\[ 0 \to \mathcal{O}_X^* \to \mathcal{O}_X^* \to \Omega_{X/k}^1 \]

where the first map is given by \(x \mapsto x^p\) and the second map is given by \(f \mapsto \frac{df}{f}\). The \(p\)th power map is clearly injective since \(X\) is reduced, and the composition of the two maps is evidently \(0\). Exactness in the middle is more involved: the idea is to use normality and local freeness of \(\Omega^1\) to reduce to proving that for a finitely generated field extension \(L/k\) and \(f \in L\), \(df = 0\) implies that \(f\) is a \(p\)th power. After checking this for purely transcendental extensions, one uses the existence of a separating transcendence basis to deduce the general case. Since the complete argument is rather long, we omit it.

Now let \(L \subset \Omega_{X/k}^1\) be the image of the map \(\mathcal{O}_X^* \to \Omega_{X/k}^1\) in the exact sequence above, so there is a corresponding short exact sequence for \(L\) and we obtain, passing to cohomology, an exact sequence

\[ H^0(X, L) \to \mathrm{Pic}(X) \to \mathrm{Pic}(X), \]

where the right map is multiplication by \(p\). Since \(H^0(X, \Omega_{X/k}^1) = 0\) by Lemma 1.1, it follows that \(\mathrm{Pic}(X)[p] = 0\), and indeed \(\mathbf{Pic}_{X/k}^\tau = 0\).

Now we work in the case of general \(S\). Using Lemma 1.1, a simple argument using cohomology and base change shows that \(X \to S\) is cohomologically flat in dimension \(0\), so \(\mathbf{Pic}_{X/S}\) is an algebraic space by Theorem 1.2(ii) and \(\mathbf{Pic}_{X/S}^\tau\) is also an algebraic space since \(\mathbf{Pic}_{X/S}^\tau \to \mathbf{Pic}_{X/S}\) is representable by open immersions. Since formation of \(\mathbf{Pic}_{X/S}^\tau\) commutes with arbitrary base change, we see that \((\mathbf{Pic}_{X/S}^\tau)_s = 0\) for all \(s \in S\) by the first paragraph. Thus the identity section \(S \to \mathbf{Pic}_{X/S}^\tau\) is an isomorphism on fibers, and since \(S\) is trivially flat over itself, the fibral isomorphism theorem (see EGA IV, Cor. 17.9.5) implies that \(\mathbf{Pic}_{X/S}^\tau = 0\). (Note that the fibral isomorphism criterion holds for a morphism from a scheme to an algebraic space: morphisms may be checked to be isomorphisms after etale base change, so this follows immediately from the fact that algebraic spaces admit etale covers which are schemes and the relative diagonal of an algebraic space is representable.) This completes the proof.

Question. The above proof shows that all complete intersections in dimension \(2\) (smooth or not) have trivial \(\mathbf{Pic}^0\) and no \(\ell\)-torsion for any prime \(\ell \neq p\). Do there exist (non-smooth) complete intersections \(X\) of dimension \(2\) in characteristic \(p\) such that \(\mathrm{Pic}(X)[p] \neq 0\)?

Lemma 1.4. Let \(G\) be a commutative finite locally free group scheme over a scheme \(S\) and let \(\pi: Y \to X\) be an fppf \(G\)-torsor, where \(Y\) is cohomologically flat in dimension \(0\) and \(X\) satisfies the hypotheses of the Theorem. The pullback map \(\pi^*: \mathbf{Pic}_{X/S} \to \mathbf{Pic}_{Y/S}\) has kernel isomorphic to \(G^\vee\), the Cartier dual of \(G\). The same is therefore true of \(\pi^*: \mathbf{Pic}_{X/S}^\tau \to \mathbf{Pic}_{Y/S}^\tau\).

Proof. We work with the presheaf \(\mathrm{Pic}_{X/S}\) defined by \(\mathrm{Pic}_{X/S}(T) = \mathrm{Pic}(X_T)\) for all \(S\)-schemes \(T\). There is a pullback map \(\pi^*: \mathrm{Pic}_{X/S} \to \mathrm{Pic}_{Y/S}\), and we claim that it has kernel \(G^\vee\). After fppf sheafification, this easily gives the result. To prove this claim, it suffices by base change on \(S\) to show that the kernel of the map \(\mathrm{Pic}(X) \to \mathrm{Pic}(Y)\) can be canonically identified with the group \(G^\vee(S)\), and we will do this below.

Recall that fppf descent for quasicoherent sheaves says that a line bundle on \(X\) is equivalent to the data of a line bundle \(\mathcal{L}\) on \(Y\) along with an isomorphism \(\varphi: \pi_1^* \mathcal{L} \to \pi_2^* \mathcal{L}\), where \(\pi_1, \pi_2: Y \times_X Y \to Y\) are the two canonical projections, satisfying the cocycle condition \(\pi_{23}^*\varphi \circ \pi_{12}^*\varphi = \pi_{13}^* \varphi\), where \(\pi_{ij}: Y \times_X Y \times_X Y \to Y \times_X Y\) is the projection onto the \(i\) and \(j\) coordinates. Given a line bundle \(\mathcal{M}\) on \(X\), we obtain a datum of this form (a descent datum) via taking \(\mathcal{L} = \pi^* \mathcal{M}\) and using the canonical isomorphism of functors \(\pi_1^* \circ \pi^* \cong \pi_2^* \circ \pi^*\). So a line bundle on \(X\) which becomes trivial after pullback to \(Y\) is the same as the datum of an isomorphism \(\pi_1^* \mathcal{O}_Y \to \pi_2^* \mathcal{O}_Y\) satisfying the cocycle condition. Since \(\pi_1^* \mathcal{O}_Y \cong \pi_2^* \mathcal{O}_Y \cong \mathcal{O}_{Y \times_X Y}\) canonically, this is the same as an automorphism of \(\mathcal{O}_{Y \times_X Y}\), i.e., an element of \(\mathbf{G}_m(Y \times_X Y)\), satisfying the cocycle condition. We will show below that the cocycle condition can be described more concretely in terms of \(G\).

Since \(\pi\) is a \(G\)-torsor, there is a canonical isomorphism \(G \times_S Y \to Y \times_X Y\) given functorially by \((g, y) \mapsto (y, gy)\). There is also an identification \(G \times_S G \times_S Y \cong Y \times_X Y \times_X Y\) given functorially by \((g, h, y) \mapsto (y, hy, ghy)\). Under these identifications, the maps \(\pi_{12}\), \(\pi_{23}\), and \(\pi_{13}\) are identified with maps \(G \times_S G \times_S Y \to G \times_S Y\) via

\[ \pi_{12}(g, h, y) = (h, y), \,\, \pi_{13}(g, h, y) = (gh, y), \,\, \pi_{23}(g, h, y) = (g, hy). \]

Suppose that \(\varphi \in \mathbf{G}_m(G \times_S Y)\) is such that the cocycle condition holds. Functorially, this means

\[ \varphi(g, hy)\varphi(h, y) = \varphi(gh, y) \]

for all \(g, h \in G(T)\) and \(y \in Y(T)\), where \(T\) ranges over the category of \(S\)-schemes. Since \(Y\) is cohomologically flat in dimension \(0\), we have \(\mathbf{G}_m(G \times_S Y) \cong \mathbf{G}_m(G)\) naturally via pullback of units, so \(\varphi\) may be regarded as a morphism \(\varphi: G \to \mathbf{G}_m\), and the cocycle condition is precisely saying that \(\varphi\) is a group homomorphism, i.e., \(\varphi \in G^\vee(S)\). This completes the proof.

2. Group actions

Let \(S\) be a scheme, \(G\) a finite locally free \(S\)-group scheme, and \(X\) a separated \(S\)-scheme. An action of \(G\) on \(X\) is an \(S\)-morphism \(G \times X \to X\) such that \(G(T) \times X(T) \to X(T)\) is a group action for every \(S\)-scheme \(T\). (As usual, this is equivalent to the commutativity of various diagrams like those in the ordinary definition of a group action.)

Definition. If \(T\) is an \(S\)-scheme and \(x \in X(T)\), then the stabilizer of \(x\) in \(G\) is the functor \(\mathrm{Stab}_{G_T}(x)\) sending a \(T\)-scheme \(U\) to \(\{g \in G(U): g x_U = x_U\}\). The free locus of the action is the functor \(\mathrm{Free}_{G,X/S}\) sending an \(S\)-scheme \(T\) to \(\{x \in X(T): \mathrm{Stab}_{G_T}(x) = 0\}\).

Lemma 2.1. If \(G\) acts on \(X\) and \(x \in X(T)\) for some \(S\)-scheme \(T\), then \(\mathrm{Stab}_{G_T}(x)\) is representable by a closed \(T\)-subgroup scheme of \(G_T\), and for any \(T\)-scheme \(U\) we have \(\mathrm{Stab}_{G_U}(x_U) = \mathrm{Stab}_{G_T}(x)_U\). The functor \(\mathrm{Free}_{G,X/S}\) is represented by an open subscheme of \(X\), and \(\mathrm{Free}_{G,X/S} \times_S T = \mathrm{Free}_{G_T,X_T/T}\) for every \(S\)-scheme \(T\).

Proof. The claims about base change are simple from the functorial definition, and will be omitted. For the first representability claim, note that there is a Cartesian diagram

\[\begin{tikzcd} \mathrm{Stab}_{G_X}(\id_X) \arrow[r] \arrow[d] & G \times_S X \arrow[d, ``\alpha''] \\ X \arrow[r, ``\Delta''] & X \times_S X \end{tikzcd}\]

where \(\alpha\) denotes the action map \(\alpha(g, x) = (x, gx)\). Since \(X\) is separated, \(\Delta\) is a closed embedding, and it follows by base change that \(\mathrm{Stab}_{G_X}(\mathrm{id}_X)\) is a closed \(X\)-subgroup of \(G \times_S X\), hence in particular a finite \(X\)-scheme. Now if \(x: T \to X\) is any \(S\)-morphism, we see that there is a Cartesian diagram

\[\begin{tikzcd} \mathrm{Stab}_{G_T}(x) \arrow[r] \arrow[d] & \mathrm{Stab}_{G_X}(\id_X) \arrow[d] \\ T \arrow[r, ``x''] & X \end{tikzcd}\]

so that \(\mathrm{Stab}_{G_T}(x)\) is a finite \(T\)-group scheme. The morphism \(\mathrm{Stab}_{G_T}(x) \to G_T\) is a proper monomorphism, hence a closed embedding, and we obtain the claim.

Recall that if \(H \to S\) is a finite morphism, then the function \(s \mapsto \rk H_s\) is an upper semicontinuous function: this follows directly from Nakayama’s lemma. If \(H\) is moreover an \(S\)-group scheme then \(\mathrm{rk} \, H_s \geq 1\) for all \(s \in S\) (since there exists a section \(S \to H\)), and \(H\) is trivial if and only if the fiber \(H_s\) is trivial for all \(s \in S\) (apply the fibral isomorphism criterion to the identity section). It follows from these considerations applied to \(H = \mathrm{Stab}_{G}(\id_X)\) that if \(U = \{x \in X: \mathrm{Stab}_{G_{k(x)}}(x) = 0\}\) then \(U\) is an open subscheme of \(X\) and \(U\) represents \(\mathrm{Free}_{G,X/S}\). This proves the lemma.

We will also need the following Theorem proved in SGA3, Exp. V, Thm. 4.1 and Rem. 5.1. With notation as in the first paragraph of this section, recall that a morphism of \(S\)-schemes \(Y \to X\) is a \(G\)-torsor if \(G\) acts on \(Y\) by \(X\)-automorphisms and the natural map \(G \times_S Y \to Y \times_X Y\) given by \((g, y) \mapsto (y, gy)\) is an isomorphism.

Theorem 2.2. If \(X\) is a quasi-projective \(S\)-scheme and \(G\) is a finite locally free \(S\)-group scheme acting on \(X\) then the ringed space quotient \(X/G\) exists as a quasi-projective \(S\)-scheme, and the natural morphism \(\pi: X \to X/G\) is finite and open. If \(G\) acts freely on \(X\), then \(\pi\) is a \(G\)-torsor and \(X/G\) represents the fppf sheaf quotient for the equivalence relation on \(X\) defined by the action of \(G\). In particular, in this latter case the formation of the quotient \(X/G\) commutes with all base change on \(S\).

The relevance of quasi-projectivity in this statement is that it implies that any finite collection of points \(x_1, \dots, x_n \in X\) lying over an affine open of \(S\) all lie in an affine open of \(X\). This permits one, using some nontrivial formal arguments, to reduce to the case that \(S\) and \(X\) are affine, in which case one proves that \(X/G\) can be formed as the spectrum of the ring of invariants for the action of \(G\) on \(\mathcal{O}(X)\). If \(U \subset X\) is an open \(G\)-stable subscheme then \(U/G\) exists as an open subscheme of \(X/G\). If \(V \subset X/G\) is an open subscheme then also \(\pi^{-1}(V)/G \cong V\) naturally.

If \(G\) acts freely on \(X\) then many properties of \(X/G\) follow from properties of \(X\). For example, if \(X\) is flat, then \(X/G\) is flat: in general, elementary commutative algebra arguments show that it is true that flatness may be checked on any faithfully flat cover. Moreover, if \(X\) is smooth then \(X/G\) is also smooth: this follows from the fact that “smooth = locally finite presentation + flat with geometrically regular fibers”, reduction to the Noetherian setting, and Matsumura, Commutative Ring Theory, Thm. 23.7. It is simple to see that properness of \(X\) is equivalent to that of \(X/G\), so that \(X/G\) is projective whenever \(X\) is projective.

3. Construction

Let \(S\) be a noetherian local ring and let \(G\) be a finite flat commutative group scheme of rank \(r > 1\) over \(S\), as in the Main Theorem. For any integer \(n \geq 1\) there is a natural action of \(G^\vee\) on the projective space \(P = \mathbf{P}((\mathcal{O}((G^{\vee})^n))^*)\).

Lemma 3.1. The free locus \(\mathrm{Free}_{G^\vee, P/S}\) is an open subset of \(P\) with complementary codimension \(r^n - r^{n-1}\) on fibers.

Proof. We will only address the case \(n = 3\), since that is sufficient for our purposes, but the proof is entirely similar for all other \(N\). By Lemma 2.1, it is enough to show this when \(S = \mathrm{Spec} \, k\) is the spectrum of an algebraically closed field. For a field \(K\), we have

\[ P(K) \cong (K[(G^\vee)^3] - \{0\})/K^* \]

naturally, and a point \(x \in P(K)\) has nonzero stabilizer if and only if, lifting \(x\) to some \(K\)-morphism \(\varphi: (G^\vee)^3 \to \mathbf{A}^1\), there is a \(K\)-homomorphism \(\eta: G^\vee \to \mathbf{G}_m\) such that we have

\[ \varphi(g+g_1, g+g_2, g+g_3) = \eta(g)\varphi(g_1, g_2, g_3) \]

for all \(K\)-schemes \(T\) and all \(g, g_1, g_2, g_3 \in G^\vee(T)\). Note that \(\varphi\) satisfies this transformation property under left \(G^\vee\)-translation if and only if the \(K\)-morphism \(\psi: (G^\vee)^3 \to \mathbf{A}^1\) given functorially by \(\psi(g_1, g_2, g_3) = \eta(-g_1)\varphi(g_1, g_2, g_3)\) is left \(G^\vee\)-invariant. A \(K\)-morphism \(\psi: (G^\vee)^3 \to \mathbf{A}^1\) is left \(G\)-invariant if and only if it factors through the map \((G^\vee)^3 \to (G^\vee)^2\) defined by \((g_1, g_2, g_3) \mapsto (g_1 - g_2, g_1 - g_3)\), as is easily seen by functorial reasoning.

For every character \(\eta \in \mathrm{Hom}_{k-\mathrm{gp}}(G^\vee, \mathbf{G}_m)\), there is a closed embedding \(i_{\eta}: \mathbf{P}(k[(G^\vee)^2]^*) \to P\) induced by the map \(k[(G^\vee)^2] \to k[(G^\vee)^3]\) sending \(\varphi\) to \((g_1, g_2, g_3) \mapsto \eta(g_1)\varphi(g_1 - g_2, g_1 - g_3)\), and the considerations in the previous paragraph show that

\[ \mathrm{Free}_{G^\vee,P/k}(k) = P(k) - \bigcup_{\eta \in \mathrm{Hom}_{k-\mathrm{gp}}(G^\vee, \mathbf{G}_m)} i_{\eta}(\mathbf{P}(k[(G^\vee)^2]^*))(k). \]

Note that the set \(\mathrm{Hom}_{k-\mathrm{gp}}(G^\vee, \mathbf{G}_m)\) is finite, being the set of \(k\)-points of the finite \(k\)-group \((G^\vee)^\vee \cong G\). Thus \(\bigcup_{\eta \in \mathrm{Hom}_{k-\mathrm{gp}}(G^\vee, \mathbf{G}_m)} i_{\eta}(\mathbf{P}(k[(G^\vee)^2]^*))\) is a closed subset of \(P\), and since \(k\) is algebraically closed it follows that the above equality on \(k\)-points is actually an equality of open subschemes, i.e.,

\[ \mathrm{Free}_{G^\vee,P/k} = P - \bigcup_{\eta \in \mathrm{Hom}_{k-\mathrm{gp}}(G^\vee, \mathbf{G}_m)} i_{\eta}(\mathbf{P}(k[(G^\vee)^2]^*)). \]

Since \(\mathbf{P}(k[(G^\vee)^2]^*)\) has dimension \(r^2 - 1\) and \(P\) has dimension \(r^3 - 1\), it follows that the closed complement of \(\mathrm{Free}_{G^\vee,P/k}\) has codimension \(r^3 - r^2\), as desired. This completes the proof.

By Theorem 2.2, the ringed space quotient \(P/G^{\vee}\) exists as a projective \(S\)-scheme, so because \(S\) is local it is an \(S\)-closed subscheme of some projective space \(Q = \mathbf{P}_S^N\). By the discussion directly following Theorem 2.2, \(U := \mathrm{Free}_{G^\vee,P/k}/G^\vee\) is a smooth open subscheme of \(P/G^\vee\); its closed complement \(Z\) has fibers of codimension \(r^n - r^{n-1}\) because dimension is insensitive to finite surjective maps. By Bertini’s theorem [see Poonen, Bertini theorems over finite fields, or Gabber, On space filling curves and Albanese varieties for a proof over finite fields], if \(r^{n-1} \leq t \leq r^n - 1\) then there exists \(t\) hypersurfaces \(\ov{H_1}, \dots, \ov{H_t}\) in the special fiber \(Q_s\) such that \((P/G^\vee)_s \cap \bigcap_{i=1}^t \ov{H_i}\) is a smooth integral \((r^n - t - 1)\)-dimensional closed subscheme of \(Q_s\) disjoint from the image of \(Z_s\). Lift each \(\ov{H_i}\) to a hypersurface \(H_i\) in \(Q\) and let \(X = (P/G^\vee) \cap \bigcap_{i=1}^t H_i\). Let \(Y\) be the (schematic) preimage of \(X\) in \(P\).

Lemma 3.2. The scheme \(X\) is \(S\)-smooth with equidimensional fibers of dimension \(r^n - t - 1\).

Proof. We prove first that \(X\) is \(S\)-flat. Note that the special fiber \(X_s\) is integral of dimension \(r^n - t - 1\), and Chevalley’s theorem on upper-semicontinuity of fiber dimension (see EGA IV, 13.1.3 for a proof for general \(S\)) implies that in fact every fiber of \(X\) over \(S\) has dimension at most \(r^n - t - 1\). But every fiber of \(P/G^\vee\) over \(S\) is integral of dimension \(r^n - 1\), and intersection with a hyperplane can cut the dimension down by at most \(1\); this follows from Krull’s height theorem. Since every zerodivisor in a noetherian local ring \(A\) is contained in a minimal prime ideal, if \(A\) is equidimensional and \(f\) is a nonunit then \(f\) is a nonzerodivisor if and only if \(\dim A/f = \dim A - 1\). By induction on \(t\), using the above considerations and the fact that \(P/G^\vee\) has integral fibers, we see that \(X\) has equidimensional fibers of dimension \(r^n - t - 1\). If each \(H_i\) is defined locally on \(P/G^\vee\) by some function \(f_i\), it follows that \((f_1, \dots, f_t)\) is a regular sequence since this may be checked on fibers over \(S\). So if \(U = \mathrm{Spec} \, A \subset P/G^\vee\) is an affine open on which \(H_1\) is defined by \(f_1\), there is an exact sequence \(0 \to A \to A \to A/f_1 \to 0\), where the first map is given by multiplication by \(f_1\). This map is injective after base change to any residue field of \(S\) (as we noted above), so that \(A/f_1\) is flat over \(S\): we see that \(\mathrm{Tor}_1^S(A/f_1, k(s')) = 0\) for all \(s' \in S\), so flatness follows from Stacks Project, 00M5 and a Zorn’s lemma-style argument. These same considerations show by induction on \(n\) that \(X\) is \(S\)-flat.

Now recall that the \(S\)-smooth locus in \(X\)is open, and \(X\) is \(S\)-smooth at all points of \(X_s\) since \(X\) is \(S\)-smooth at a point \(x\) if and only if it is flat over \(S\) at \(x\) and \(x\) is smooth in its fiber. Since \(X\) is proper, it follows immediately that \(X\) is smooth over \(S\), completing the proof of the Lemma.

Now we take \(n = 3\) in Lemmas 3.1 and 3.2. Note that \(Y\) is cut out of \(P\) by the preimages of the \(H_i\) in \(P\), and it is easy to see from Lemma 3.2 that \(Y\) is a complete intersection of dimension \(r^3 - t - 1\). So if \(t = r^3 - 4\) then \(X\) is smooth and projective over \(S\) with \(3\)-dimensional geometrically integral fibers and \(Y\) is a complete intersection of dimension \(3\). By Lemma 1.3 we see that \(\mathbf{Pic}_{Y/S}^\tau = 0\), and it follows from Lemma 1.4 that \(\mathbf{Pic}_{X/S}^\tau \cong G\). If \(G^\vee\) is etale and \(t = r^3 - 3\), then \(Y\) is smooth, being a \(G^\vee\)-torsor over the smooth \(X\), so the same argument shows again that \(\mathbf{Pic}_{X/S}^\tau \cong G\). This completes the proof of the Main Theorem.

Question. If \(G\) is not of multiplicative type, then does there necessarily exist a \(2\)-dimensional smooth projective \(X\) with \(\mathbf{Pic}_{X/S}^\tau \cong G\)? (This is related to the previous question: if the answer to this question is “no”, then the construction above will yield a \(2\)-dimensional complete intersection with nontrivial \(p\)-torsion in its Picard group.) What is certainly true is that the above argument can be modified in a simple way to show that if \(G\) is connected over a field \(k\) then there does exist a \(2\)-dimensional smooth projective \(X\) with \(\mathbf{Pic}_{X/k}^0 \cong G\).

4. Jumping Hodge numbers

We now give two examples of “pathologies” we can deduce from the Main Theorem. Recall that if \(i, j \geq 0\) are integers then the \((i, j)\) Hodge number of a proper scheme \(X\) over a field \(k\) is defined by \(h^{i, j}(X) = \dim H^j(X, \Omega_{X/k}^i)\). Classically (i.e., over the complex numbers), the Hodge numbers satisfy various magical properties: for example, if \(X\) is smooth and projective then one always has \(h^{i, j}(X) = h^{j, i}(X)\). As mentioned in Cotner’s previous post, taking \(S = \mathrm{Spec} \, k\) for a field \(k\) of characteristic \(p\) and \(G = \mu_p\), we find an example of a variety in characteristic \(p\) which does not satisfy Hodge symmetry: namely, \(h^{1, 0} = 0\) and \(h^{0, 1} = 1\). Another magical property of Hodge numbers over \(\mathbf{C}\) (or, more generally, over a field of characteristic \(0\)) is that they are constant in smooth projective families; this is proved via analytic methods. In the following two examples, we will see that this fails away from equicharacteristic \(0\).

Example. Jumping Hodge numbers in mixed characteristic \((0, p)\). Let \(S\) be a DVR of mixed characteristic \((0, p)\) and \(G = \mu_p\), so that the generic fiber \(G_\eta\) is etale and the special fiber \(G_s\) is connected. Let \(X\) be as in the Main Theorem, so \(X\) is smooth and projective over \(S\), and \(\mathbf{Pic}_{X/S}^\tau \cong G\). Recall that for any scheme \(Y\) over a field \(k\) such that \(\mathbf{Pic}_{Y/k}\) is representable, we have \(H^1(Y, \mathcal{O}_Y) \cong T_0 \mathbf{Pic}_{Y/k}\). Thus we have \(h^{0,1}(X_\eta) = 0\) and \(h^{0,1}(X_s) = 1\), i.e., “the Hodge number jumps”. Notice also that \(\mathbf{Pic}_{X/k}^0\) has special fiber \(\mu_p\) and generic fiber \(0\), so that this is not open in \(\mathbf{Pic}_{X/k}\).

Example. Jumping Hodge numbers in equicharacteristic \(p\) Let \(R\) be a DVR of equicharacteristic \(p\) and let \(S/R\) be a totally ramified generically separable extension of degree \(n\). Let \(G'\) denote the Weil restriction \(R_{S/R} \mathbf{G}_m\), so that \(G'\) has special fiber \(G'_s \cong \mathbf{G}_m \times \mathbf{G}_a^{n-1}\) and generic fiber \(G'_\eta \cong \mathbf{G}_m^n\). Let \(G^\vee\) denote the kernel of Frobenius on \(G'\), so that \(G\) is a finite flat commutative group scheme over \(R\) with special fiber \(G^\vee_s \cong \mu_p \times \alpha_p^{n-1}\) and generic fiber \(G^\vee_\eta \cong \mu_p^n\). So if \(G\) is the Cartier dual of \(G^\vee\) then we have \(G_s \cong \mathbf{Z}/p\mathbf{Z} \times \alpha_p^{n-1}\) and \(G_\eta \cong (\mathbf{Z}/p\mathbf{Z})^n\) Let \(X\) be the smooth projective \(R\)-scheme whose existence guaranteed by the Main Theorem, so that \(\mathbf{Pic}_{X/k}^\tau \cong G\). So as in the previous section we have \(h^{0,1}(X_\eta) = 0\) and \(h^{0,1}(X_s) = n-1\).

Notify of
1 Comment
Most Voted
Newest Oldest
Inline Feedbacks
View all comments
David Benjamin Lim
4 months ago

Insane post.