ON A PAPER OF S. T. JENSEN

V. B. MEHTA AND V. SRINIVAS

The paper [3] of S. T. Jensen contains the following theorem (loc. cit., Theorem 2.1):

Let k be an algebraically closed field of characteristic p > 0 and let Y be a projective variety over k. Suppose N is a finite commutative k-group scheme acting freely on Y and that at least one of the following 3 conditions are satisfied:

1) N is a reduced cyclic group;

2) N is either local-reduced or reduced-local;

3) N is local of height ≤ 1.

Then, putting X = Y/N, there exists an exact sequence of commutative k-group schemes

0 → D(N) → Pic_{X/k} → (Pic_{Y/k})^{N} → 0, (0.1)

where (Pic_{Y/k})^{N} denotes the N-invariant subrgoup scheme of Pic_{Y/k}. In the above, D(N) denotes the Cartier dual of N; “local-reduced” refers to a local group scheme with reduced Cartier dual, and “reduced local” has a similar meaning.

In this note, we would like to point out that there is an error in the Theorem, by means of a counterexample, which is based on a construction of Igusa ([2], last paragraph). We also show that Jensen’s result is true if N is a reduced cyclic group. Using this we give an alernate proof of a theorem of Bombieri and Mumford ([1], Theorem 2).

The counter-example is based on the following.

Lemma 0.1. Let A be a connected commutative group-scheme over k, an algebraically closed field of characteristic p, and let G be a finite reduced p-group acting on A via group scheme automorphisms. Then A^{G}, the fixed sub-group scheme, has a non-zero connected component of the identity (i.e. a non-zero Lie algebra).

Proof. Let A_{1} denote the height 1 subgroup scheme of A associated to the p-Lie algebra L of A. Then G also acts on A_{1} and on L. But as G is a p-group, the subspace of invariants L^{G} is non-zero. Hence A_{1}^{G} is also non-zero, and it is a connected subgroup scheme of A^{G}.

Example 0.2. (Igusa [2]

Let E be an elliptic curve in characteristic 2 which contains a non-trivial point t of order 2 (i.e. E is ordinary). Put Z = E × E × E, and define 2 involutions σ_{1}, σ_{2} on Z, as follows:

The second author is suported by a Swarnajanthi Fellowship of the D.S.T..

σ_{1}(x, y, z) = (x + t, −y, −z),
σ_{2}(x, y, z) = (−x, y + t, −z + t).

It is easy to check that σ_{1}, σ_{2} are commuting involutions on Z, which define a free action of G = ℤ/2ℤ × ℤ/2ℤ on Z. Put X = Z/G. If Ê is the dual of E, then

Pic_{Z/k}^{0} = Ê × Ê × Ê,

and one can check that

(Pic_{Z/k}^{0})^{G} = μ_{2}^{⊕3} × (ℤ/3ℤ)^{⊕3} = _{2}Pic_{Z/k}^{0}.

So, by Jensen’s exact sequence (0.1) stated above, there would be an exact sequence of k-group schemes

0 → μ_{2}^{⊕2} → Pic_{X/k}^{0} → μ_{2}^{⊕3} → 0
∥
D(G)

Hence Pic_{X/k}^{0} is infinitesimal of multiplicative type. Since D(G) ⊂ Pic_{X/k}^{0}, we have a surjection on Cartier duals D(Pic_{X/k}^{0}) ⟶ G. Let Y → X be the étale covering corresponding to the reduced p-group D(Pic_{X/k}) = H, say. By Jensen’s sequence (0.1) there is an exact sequence

0 → D(H) → Pic_{X/k} → Pic_{Y/k}^{H} → 0.

As D(H) → Pic_{X/k}^{0} is an isomorphism, Pic_{Y/k}^{0} is a finite reduced group scheme over k. By lemma 0.1 this implies Pic_{Y/k}^{0} = 0, and hence H^{1}(Y, O_{Y}) = 0. But as there is a natural surjection H → G → 0, the covering Y → X factors through Z, so that Y is an étale cover of an abelian variety. This contradicts that H^{1}(Y, O_{Y}) = 0.

Example 0.3. In a letter to us, M. Raynaud has indicated the following example: let G be the quaternion group of order 8 and let Y be a smooth projective hypersurface in ℙ_{k}^{3}, over an algebraically closed field k of characteristic 2, on which G acts freely. Let X = Y/G. We first note that if H is an abelian subgroup of G, then Jensen’s exact sequence 0.1 would give Pic_{(Y/H)/k}^{0} ≅ D(H), the Cartier dual of H. So if Z = Y/(ℤ/2ℤ), where ℤ/2ℤ is the centre of G, we would have Pic_{Z/k} ≅ μ_{2}. The quotient group G/(ℤ/2ℤ) ≅ ℤ/2ℤ × ℤ/2ℤ acts freely on Z, say with quotient X. Since any action of ℤ/2ℤ × ℤ/2ℤ on μ_{2} is trivial, we woul have an exact sequence from (0.1)

0 → μ_{2} × μ_{2} → Pic_{X/k}^{0} → μ_{2} → 0.

This implies Pic_{X/k}^{0} is a k-group scheme of order 8 and of multiplicative type. So there exists a Galois étale covering T → X with group D(Pic_{X/k}^{0}), which is a reduced, commutative group of order 8. This contradicts the fact that π_{1}^{alg}(X) = G.

We now show that Jensen’s exact sequence (0.1) is valid in a special case; this special case has been used in some citations of Jensen’s theorem in the literature (see [4], [6]).

Proposition 0.4. Let Y be a projective variety over an algebraically closed field k of characteristic p and let G be a finite cyclic p-group acting freely on Y. Let X = Y/G. Then we have an exact sequence of group schemes

0 → D(G) → Pic_{X/k} → Pic_{Y/k}^{G} → 0.

Proof. We have to check that the associated sequence of f.p.p.f. sheaves is exact, that is, we have to show that

1) for any k-scheme S, there is an exact sequence of groups

0 → D(G)(S) → Pic_{X/k}(S) → Pic_{Y/k}^{G}(S)

2) if S is a connected k-scheme and L ∈ Pic(Y ×_{k} S) is G-invariant, then there exists an f.p.p.f. cover T → S such that the pull-back line bundle L_{T} ∈ Pic(Y ×_{k} T) descends to X ×_{k} T.

The proof of 1) is straightforward. To prove 2), let G be the theta group associated to the G-invariant line bundle L on Y ×_{k} S. So G is a group scheme over S such that for any S-scheme T, a T -valued point of G is a pair (g_{T} , φ_{T}), where g_{T} is a section of the constant group scheme G_{T} = G ×_{k} T → T, and φ_{T} : g_{T}^{∗} L_{T} ⟶^{≅} L_{T} is an isomorphism of line bundles. Thus we have an exact sequence of S-group schemes

0 → 𝔾_{m,S} → G → G_{S} → 0.

Since L is G_{S}(S) = G-invariant, where G ≅ ℤ/nℤ, we can find an f.p.p.f. cover S′ → S and α ∈ G(S′) mapping to a generator of G ⊂ G_{S}(S′). Since multiplication n : 𝔾_{m,S} → 𝔾_{m,S} is an f.p.p.f. cover, there is an f.p.p.f. cover T → S′, with T connected, and β ∈ 𝔾_{m,T} such that nβ = nα_{T} Then α_{T} − β ∈ G(T) has order n and maps to a generator of G. Since 𝔾_{m,S} ⊂ G is a central subgroup, we deduce that G(T) ≅ 𝔾_{m,S}(T) × G_{S}(T), and so L_{T} descends to a line bundle on X × T.

Remark 0.5. A non-singular projective surface X over a field k of characteristic 2 is called a singular Enriques surface if K_{X} ≅ O_{X}, β_{2}(X) = 10, χ(O_{X}) = 1 and the Frobenius map is bijective on H^{1}(X, O_{X}). Under these conditions, Mumford and Bombieri [1] prove that X has an étale covering Y of degree 2 such that Y is a K3 surface. A proof using proposition 0.4 is as follows.

Since H^{1}(X, O_{X}) is 1-dimensional and the Frobenius map is bijective on H^{1}(X, O_{X}), we must have Pic_{X/k}^{0} = μ_{2^{n}} for some n ≥ 1. Let Y → X be the associated D(μ_{2^{n}}) = ℤ/2^{n} cover. By proposition 0.4 there is an exact sequence

0 → μ_{2^{n}} → Pic_{X/k}^{0} → (Pic_{Y/k}^{0})^{Z/2^{n}ℤ} → 0

It follows that (Pic_{Y/k}^{0})^{Z/2^{n}ℤ} = 0, and so H^{1}(Y, O_{Y}) = 0 by lemma 0.1. Since K_{Y} ≅ O_{Y}, we must have χ(O_{Y}) = 2. But χ(O_{Y}) = 2^{n}χ(O_{X}), hence n = 1.

References

[1] E. Bombieri, D. Mumford, Enriques’ classification of surfaces in char. p. III, Invent. Math. 35 (1976) 197-232.

[2] J.-I. Igusa, On the strcuture of a certain class of Kähler varieties, Amer. J. Math. 76 (1954).

[3] S. T. Jensen, Picard schemes of quotients by finite commutative group schemes, Math. Scand. (1978).

[4] R. Miranda, Non-classical Godeaux surfaces in characteristic 5, Proc. A. M. S. 91 (1984).

[5] M. Raynaud, Letter to the authors.

[6] W. E. Lang, Quasi-elliptic surfaces in characteristic three, Ann. Sci. Ecole Norm. Sup., 4 Serie, 12 (1979) 473-500.

School of Mathematics, TIFR, Homi Bhabha Road, Bombay-400005, India

  E-mail address: vikram@math.tifr.res.in, srinivas@math.tifr.res.in