Theta divisors and the Frobenius morphism

David A. Madore

Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following M. Raynaud, that the sheaf of locally exact differentials in characteristic p > 0 has a theta divisor, and that the generic curve in (any) genus g ≥ 2 and (any) characteristic p > 0 has a cover that is not ordinary (and which we explicitely construct).

1 Theta divisors for vector bundles

Let k be an algebraically closed field and X a smooth proper connected curve over Spec k having genus g. We assume throughout that g ≥ 2.

If E is a vector bundle (i.e. a locally free invertible sheaf) of rank r and degree d over X, we define its slope to be λ = d/r. The Riemann-Roch formula gives the Euler-Poincaré characteristic of E:

χ(X, E) = h^{0}(X, E) − h^{1}(X, E) = r(λ − (g − 1))

In particular for λ = g − 1 (the critical slope) we have χ(X, E) = 0; moreover, it is still true for any invertible sheaf L of degree 0 over X that χ(X, E ⊗ L) = 0, in other words, h^{0}(X, E ⊗ L) = h^{1}(X, E ⊗ L). Under those circumstances, it is natural to ask the following question:

Question 1.1. Suppose E has critical slope. Then for which invertible sheaves L of degree 0 (if any) is it true that h^{0}(X, E ⊗ L) = 0 (and consequently also h^{1}(X, E ⊗ L) = 0)? Is this true for some L, for many L, or for none?

We start with a necessary condition. Suppose there were some subbundle F ↣ E having slope λ(F) > λ(E) = g − 1. Then we would have χ(X, F ⊗ L) > 0, hence h^{0}(X, F ⊗ L) > 0. Now H^{0}(X, F ⊗ L) ↣ H^{0}(X, E ⊗ L), so this implies h^{0}(X, E ⊗ L) > 0. So if we are to have h^{0}(X, E ⊗ L) > 0 for some L, this must not happen, and we say that E is semi-stable:

Definition 1.2. A vector bundle E over X is said to be stable (resp. semistable) iff for every sub-vector-bundle F of E (other than 0 and E) we have λ(F) < λ(E) (resp. λ(F) ≤ λ(E)).

Another remark we can make bearing some relation with question (1.1) is that, by the semicontinuity theorem, if we let L vary on the jacobian of X, the functions h^{0}(X, E ⊗L) and h^{1}(X, E ⊗L) are upper semicontinuous. This means that they increase on closed sets. In particular, if h^{0}(X, E⊗L) = 0 (the smallest possible value) for some L, then this is true in a whole neighborhood of L, that is, for almost all L. We then say that this holds for a general invertible sheaf of degree 0 and we write h^{0}(X, E ⊗ L_{gen}) = 0.

Now introduce the jacobian variety J of X and let ℒ be the (some) Poincaré sheaf (universal invertible sheaf of degree 0) on X ×_{Spec k} J. We aim to use ℒ to let L vary and provide universal analogues for our formulæ. Let

\xymatrix{X ×_{Spec k} J \ar[d]^{f} \\ J}

be the second projection.

Consider the sheaf E⊗ℒ (by this we mean the twist by ℒ of the pullback of E to X ×_{Spec k} J). Our interest is mainly in the higher direct image Rf_{∗}(E ⊗ℒ), which incorporates information about H^{i}(X, E ⊗L) (and much more).

To be precise, we know that there exists a complex 0 → M^{0} ⟶^{u} M^{1} → 0 of vector bundles on J that universally computes the R^{i}f_{∗}(E ⊗ ℒ), in the sense that the i-th cohomology group (i = 0, 1) of the complex is R^{i}f_{∗}(E⊗ℒ) and that this remains true after any base change J′ → J. (One particularly important such base change, of course, is the embedding of a closed point {L} in J.)

Now M^{0} and M^{1} have the same rank, say s (because the Euler-Poincaré characteristic of E is 0). So we can consider the determinant of u, ⋀^{s} M^{0} ⟶^{det u} ⋀^{s} M^{1}, or rather

O_{J} ⟶^{det u} ⋀^{s} M^{1} ⊗ ( ⋀^{s} M^{0})^{⊗−1}

Exactly one of the following two things happens:

• Either the determinant det u is zero (identically). In this case, u is nowhere (i.e. on no fiber) invertible, it always has a kernel and a cokernel: we have h^{0}(X, E ⊗ L) > 0 (and of course h^{1}(X, E ⊗ L) > 0) for all L.

• Or det u is a nonzero section of the invertible sheaf ⋀^{s} M^{1} ⊗ ( ⋀^{s} M^{0})^{⊗−1} on J and it defines a positive divisor θ_{E} on J, whose support is precisely the locus of L such that h^{0}(X, E ⊗ L) > 0. We then have h^{0}(X, E ⊗ L_{gen}) = 0.

In other words, precisely in the case where h^{0}(X, E ⊗ L_{gen}) = 0 we can define a positive divisor θ_{E} on J which tells us “where the bundle E has cohomology”. We call this divisor the theta divisor of the vector bundle E. And we will use the expression “to admit a theta divisor” as synonymous for h^{0}(X, E ⊗ L_{gen}) = 0. For example, a vector bundle that admits a theta divisor is semi-stable (but the converse is not true, cf. [1]).

2 Enters the Frobenius morphism

We now assume that the base field k has characteristic p > 0. We then have a relative Frobenius morphism

\xymatrix{X \ar[rr]^{π} \ar[dr] &  & X_{1} \ar[dl] \\  & Spec k}

which is obtained by factoring the absolute Frobenius morphism through the pullback to X of the Frobenius on k (more descriptively, π has the effect, in projective space, of raising the coordinates to the p-th power, while X_{1} is the curve obtained by raising to the p-th powers the coefficients in the equations defining X).

The curve X_{1} has the same genus g as X. The morphism π is flat, finite and purely inseparable of degree p. From it we deduce a morphism O_{X_{1}} → π_{∗}O_{X} (of O_{X_{1}}-modules), which is mono because π is surjective. Call B_{1} the cokernel, so that we have the following short exact sequence:

0 → O_{X_{1}} → π_{∗}O_{X} → B_{1} → 0  (1)

Now the sheaf B_{1} can be viewed in a different way: if we call d the differential O_{X} → Ω_{X}^{1} (between sheaves of ℤ-modules) then π_{∗}(d) is O_{X_{1}}-linear and has O_{X_{1}} as kernel. Consequently, B_{1} can also be seen as the image of π_{∗}(d), hence its name of sheaf of locally exact differentials. As a subsheaf of the locally free sheaf π_{∗}Ω_{X}^{1}, it is itself a vector bundle.

In the short exact sequence (1) above, the vector bundles O_{X_{1}} and π_{∗}O_{X} have rank 1 and p respectively, so that B_{1} has rank p−1. On the other hand, since O_{X_{1}} and π_{∗}O_{X} each have Euler-Poincaré characteristic g − 1, we have χ(X_{1}, B_{1}) = 0, or in other words, λ(B_{1}) = g − 1 (the critical slope), and what we have said in the previous section applies to the sheaf B_{1}.

More precisely, we have the following long exact sequence in cohomology, derived from (1):

k \underrightarrow{}k
∥  ∥
0 → H^{0}(X_{1}, O_{X_{1}}) → H^{0}(X, O_{X}) → H^{0}(X_{1}, B_{1}) →
→ H^{1}(X_{1}, O_{X_{1}}) → H^{1}(X, O_{X}) → H^{1}(X_{1}, B_{1}) → 0

Here the first arrow is an isomorphism as shown. Consequently, the arrow H^{1}(X_{1}, O_{X_{1}}) → H^{1}(X, O_{X}) is also an isomorphism if and only if h^{0}(X_{1}, B_{1}) = 0, or, what amounts to the same, h^{1}(X_{1}, B_{1}) = 0. This is again the same as saying that B_{1} has a theta divisor (something which we will see is always true) and that it does not go through the origin.

Definition 2.1. When the equivalent conditions mentioned in the previous paragraph are satisfied, we say that the curve X is ordinary.

3 The sheaf of locally exact differentials has a theta divisor

In this section we prove the following result due to M. Raynaud ([1]):

Theorem 3.1. If X is a smooth projective connected curve over an algebraically closed field k of characteristic p > 0 and B_{1} is the sheaf of locally exact differentials on X_{1}, as introduced above, then we have h^{0}(X_{1}, B_{1} ⊗ L_{gen}) = 0, i.e. the vector bundle B_{1} admits a theta divisor (in particular, it is semi-stable).

Thus we can state the fact that a curve is ordinary simply by saying that the theta divisor of B_{1} does not go through the origin.

To start with, introduce the jacobians J and J_{1} of X and X_{1} respectively. Then J_{1} is the Frobenius image of J, and we have a relative Frobenius morphism F: J → J_{1} that is purely inseparable of degree p^{g}; it corresponds to taking the norm on invertible sheaves of degree 0 — or, if we prefer using points, it takes O_{X}(Σn_{i}x_{i}) to O_{X_{1}}(Σn_{i}π(x_{i})). On the other hand, we also have the Verschiebung morphism in the other direction V : J_{1} → J, which corresponds to pulling back by π — or again, it takes O_{X_{1}}(Σn_{i}x_{i}) to O_{X}(Σpn_{i}π^{−1}(x_{i})). The Verschiebung map also has degree p^{g}. The composite of the Verschiebung and Frobenius morphisms, in any direction, is the raising to the p-th power.

We will show something more precise than just saying that B_{1} has a theta divisor: we will actually show that this theta divisor does not contain all of ker V in the neighborhood of 0. However, we will see from actual equations that it “almost” does.

If L_{1} is an invertible sheaf of degree 0 on X_{1} (that is, a k-point of J_{1}), the short exact sequence (1) becomes, after tensoring by L_{1}:

0 → L_{1} → π_{∗}π^{∗}L_{1} → B_{1} ⊗ L_{1} → 0  (2)

Now let ℒ_{1} be the Poincaré bundle on X_{1} ×_{Spec k} J_{1}. The universal analogue of (2) above is

0 → ℒ_{1} → (π × 1_{J_{1}})_{∗}(π × 1_{J_{1}})^{∗}ℒ_{1} → B_{1} ⊗ ℒ_{1} → 0

But by the definition of the Verschiebung, the sheaf (π × 1_{J_{1}})^{∗}ℒ_{1} is also (1_{X} × V)^{∗}ℒ so that the exact sequence can be written as

0 → ℒ_{1} → (π × 1_{J_{1}})_{∗}(1_{X} × V)^{∗}ℒ → B_{1} ⊗ ℒ_{1} → 0

We now introduce projections as designated on the following diagram:

[figure: X × J; f; X × J_{1}; 1_{X}×V; π×1_{J_{1}}; g; □; X_{1} × J_{1}; f_{1}; J; J_{1}; V]  (3)

Now we want to calculate the R(f_{1})_{∗} of this. For one thing, looking at the diagram (3) above, we see that R(f_{1})_{∗}(π×1_{J_{1}})_{∗}(1_{X}×V)^{∗}ℒ is Rg_{∗}(1_{X}×V)^{∗}ℒ, and by base change (note that the morphism V is flat), this is V^{∗}Rf_{∗}ℒ. Thus we have the following distinguished triangle, in the derived category of the category of sheaves on J_{1}:

\xymatrix{ & R(f_{1})_{∗}(B_{1} ⊗ ℒ_{1}) \ar[dl]_{+1} \\ R(f_{1})_{∗}ℒ_{1} \ar[rr]^{a} &  & V^{∗}Rf_{∗}ℒ \ar[ul]}  (4)

And the corresponding long exact sequence of cohomology is

0 → (f_{1})_{∗}(B_{1} ⊗ ℒ_{1}) → R^{1}(f_{1})_{∗}ℒ_{1} →^{a} V^{∗}R^{1}f_{∗}ℒ → R^{1}(f_{1})_{∗}(B_{1} ⊗ ℒ_{1}) → 0

(the two first terms cancel). We want to show that a is generically invertible.

To start with, consider a minimal resolution of Rf_{∗}ℒ in the neighborhood of the origin. It has the form

O_{J,0} →^{u′} O_{J,0}^{g}

where u′(1) = (x_{1}, …, x_{g}) is a system of parameters around 0. Indeed, this last statement is the same as saying, if u is the transpose of u′, that the image of u is the maximal ideal of the regular local ring O_{J,0}, and this is easy because {0} is the largest closed subscheme of Spec O_{J,0} on which ℒ is trivial.

Now apply what we have just proven to J_{1} on the one hand, and to J on the other, but pulling back by V, we find the following resolution for the arrow a in triangle (4):

\xymatrix{R \ar[r]^{a_{0}} \ar[d]_{u′} & R \ar[d]^{v′} \\ R^{g} \ar[r]^{a_{1}} & R^{g}}  (5)

where we have written R = O_{J_{1},0}, and where u′(1) = (x_{1}, …, x_{g}) is a system of parameters of J_{1} around 0 and v′(1) = (y_{1}, …, y_{g}) is a regular sequence that gives an equation of ker V around 0. Now of course a_{0} is just an element of R, and it is invertible because modulo the maximal ideal of R (that is, at the origin) the arrow a is just the identity on k. So we can assume that a_{0} is the identity. What we want to prove is that det a_{1} is not zero (of course, it is invertible precisely when the curve is ordinary).

Consider the diagram (5) and its transpose (i.e. its image by the functor Hom_{R}(·, R)), and complete them both by adding the Koszul complex on either column. That is, consider the diagrams:

[figure: 0; 0; R; 1; u′; R; v′; R^{g}; a_{1}; R^{g}; ∧^{g}R^{g}; det a_{1}; ∧^{g}R^{g}; M′; h′; N′; 0; 0; and; 0; 0; ∧^{g}R^{g}; det a_{1}; ∧^{g}R^{g}; R^{g}; a_{1}^{∨}; v; R^{g}; u; R; 1; R; N; h; M; 0; 0]

in which we have written u, v and a_{1}^{∨} for the transposes of u′, v′ and a_{1} respectively and M and N for the cokernels of u and v respectively. Since u′(1) and v′(1) are regular sequences, M and N are modules of finite length, and the Koszul complex is a resolution of them: the columns of both diagram are exact. We have M′ = Ext_{R}^{g}(M, R) and N′ = Ext_{R}^{g}(N, R) (since we have taken a resolution, transposed it, and shifted in g degrees). But since the Koszul complex is autodual (that is, the left column of the right diagram is the same as the right column of the left diagram, and vice versa), M and M′ are the same and so are N and N′. Finally, it is known that (R being a regular local ring) the functor Ext_{R}^{g}(·, R) is dualizing on modules of finite length. Now h is surjective as is seen on the diagram on the right, so that its image h′ by the functor in question is injective. Hence det a_{1} is nonzero, what we wanted.

We can be more precise than this. As we have seen, M is isomorphic to k, and N to the local ring of ker V at 0: the support of θ_{B_{1}} swallows everything in ker V around the origin but just one k. (Incidentally, X is ordinary if and only if the support of θ_{B_{1}} does not contain the origin, so we recover the known fact that X is ordinary if and only if the local ring of ker V at the origin is k, i.e. V is étale.)

4 Constructing a non ordinary cover

We now present another result of M. Raynaud’s ([2]), namely the fact that a finite étale cover of an ordinary curve is not necessarily ordinary, even when the base curve is generic. In fact, we obtain a cover Y ↠ X such that the image of the map J(X_{1}) ↣ J(Y_{1}) on the jacobians is completely contained in the support of the theta divisor of B_{1} on Y_{1} — and in particular 0 is, so that Y is not ordinary. The construction is sufficiently general to apply to the generic curve (for a given genus g ≥ 2 and characteristic p). We also get estimations on the Galois group of Y over X; a theorem of Nakajima states that an abelian cover of the generic curve is ordinary, so we have to work with non abelian groups if we want a non ordinary cover — however, we will see that a nilpotent group can suffice.

We start with a few generalities on representations of the fundamental group of curves. We refer to [2] for details. If ρ: π_{1}(X) → GL(r, k) is a representation of the fundamental group of X in k-vector spaces of rank r, and ρ has open kernel (or, which amounts to the same, ρ is continuous and has finite image), then ρ defines a locally constant étale sheaf in k-vector spaces of rank r on X, written 𝕍_{ρ} (very succintly, 𝕍_{ρ} can be obtained as follows: find a Galois cover Y ↠ X whose Galois group factors through the kernel of ρ, then make π_{1}(X)/ ker ρ act on Y × k^{r} componentwise, and take the fixed points of that action). Tensoring 𝕍_{ρ} by O_{X} gives a Zarisky sheaf V_{ρ} which is locally free of rank r and has degree 0, i.e. a vector bundle of rank r and slope 0 on X. Among the functoriality properties of V_{ρ} cited in [2], we will need the fact that if Y ↠^{a} X is finite étale and ρ is a representation of π_{1}(Y) as above then a_{∗}V_{ρ} is precisely V_{ρ′}, where ρ′ is the representation of π_{1}(X) induced by ρ.

We say that a representation ρ as above has a theta divisor (respectively, is ordinary) if and only if the sheaf V_{1,ρ} ⊗ B_{1} on X_{1} has a theta divisor (respectively, has a theta divisor that does not go through the origin), V_{1,ρ} being the bundle V_{ρ} as above constructed on X_{1}. Thus, we have seen that the trivial representation has a theta divisor, and it is ordinary precisely when the curve X is ordinary. The existence of a theta divisor for B shows that if L is a general invertible sheaf of finite order n prime to p then the representation ρ of rank 1 associated to it is ordinary.

Theorem 4.1. Let k be an algebraically closed field of characteristic p > 0, and let X be the generic curve of any genus g ≥ 2 over Spec k. Then there exists a Galois cover of X with solvable Galois group of order prime to p that is not ordinary.

Let X be as stated, and let J be its jacobian. If we choose a base point on X then we get a map S^{g−1}X → J from the (g − 1)-th symmetric power of X, whose image defines a positive divisor on J, called the classical theta divisor, and written Θ. Let N = O_{J}(Θ) be the invertible sheaf defined by Θ. We can assume that N is symmetric (i.e. that if ι: J → J is the inverse map then ι^{∗}N = N) and we will do so.

Let n be a positive integer that is prime to p, and denote by α the multiplication by n map on J, which is étale of degree n^{2g}. Call A the kernel of α, the (étale) set of points of J whose order divides n. Because we have chosen N symmetric, we have α^{∗}N = N^{⊗n^{2}}.

We recall (cf. [3]) that the kernel H(N^{⊗n}) is the subgroup of closed x in J such that T_{x}^{∗}N^{⊗n} ≅ N^{⊗n} (where T_{x} denotes translation by x). This kernel is obviously A. Now in [3], D. Mumford defines another, more interesting, group associated to an invertible sheaf on an abelian variety. In our case, it is the group

𝒢 (N^{⊗n}) = {(x, φ) | φ: N^{⊗n} ˜→T_{x}^{∗}N^{⊗n}}

with multiplication defined in the obvious way. There is a short exact sequence

1 → k^{×} → 𝒢 (N^{⊗n}) → H(N^{⊗n}) → 1

and in fact k^{×} is precisely the center of 𝒢 (N^{⊗n}). The commutator of two elements of 𝒢 (N^{⊗n}) is an element of k^{×} and it depends only on the class in H(N^{⊗n}) of the two elements. Thus, the commutator defines a skewsymmetric biadditive form ⟨·, ·⟩: A × A → k^{×}. It is moreover shown in [3] that this form is non degenerate.

Let B a maximal totally isotropic subgroup of A for the form we have just defined. So B has order n^{g}, and C = A/B has order n^{g}. We factor α as follows

\xymatrix{J \ar[dr]^{β} \ar[dd]_{α} \\  & J′ = J/B \ar[dl]^{γ} \\ J}

where β has kernel B and γ has kernel (identified with) C.

Because B is isotropic, by a result in [3], the sheaf N^{⊗n} descends to an invertible sheaf M on J′, i.e. a sheaf such that N^{⊗n} = β^{∗}M. And M is a principal polarization on J′. Now note that γ^{∗}N and M^{⊗n} have the same pullback (namely N^{⊗n^{2}}) by β; if n is odd we can choose M to be symmetric, so that γ^{∗}N and M^{⊗n} coincide. We will now suppose this to be the case.

If L is an invertible sheaf on J that is algebraically equivalent to 0 (that is, a closed point of J^{∨}) then we have γ_{∗}(M ⊗ γ^{∗}L) ≅ (γ_{∗}M) ⊗ L, so that h^{0}(J′, M ⊗ γ^{∗}L) = h^{0}(J, (γ_{∗}M) ⊗ L). Now the point is that M ⊗ γ^{∗}L is a principal polarization on J, so this number is 1. In particular h^{0}(J, (γ_{∗}M) ⊗ L) > 0, and this implies that for any invertible sheaf L of degree 0 on X we have h^{0}(X, F ⊗ L) > 0, where F is the restriction of γ_{∗}M to X. This is a good first step, but we need to twist F by an invertible sheaf having the right degree to compensate for the slope of F (since the sheaves V_{ρ} have slope zero).

We now calculate the slope of F. Its rank is n^{g}. Introduce the curves Y and Z that are inverse image of X by γ and α respectively, thus:

\xymatrix{Z \ar[dr]^{β} \ar[dd]_{α} \\  & Y \ar[dl]^{γ} \\ X}

The degree of N restricted to X is well-known: it is g. Pulling this back by α, we see that the degree of N^{⊗n^{2}} restricted to Z is gn^{2g}, and that of N^{⊗n}|Z is gn^{2g−1}. Descending to Y, we see that the degree of M|Y is gn^{g−1}. So the slope of F is finally g/n.

Now assume that g divides n, i.e. that g/n = d, the slope of F, is an integer. The degree of N|X is g = nd, so there exists an invertible sheaf P of degree d on X such that N|X = P^{⊗n}. Let L′ = (M|Y) ⊗ γ^{∗}P^{⊗−1}, which is an invertible sheaf of degree zero. Its inverse image L′′ = β^{∗}L′ is such that L^{′′⊗n} is trivial on Z, so that the order of L′′ divides n (in fact, it is exactly n, but we won’t need this). If E = γ_{∗}L′ then E = F ⊗ P^{⊗−1}, which is an invertible sheaf of degree 0 on X and satisfies h^{0}(X, E ⊗ L_{d,gen}) > 0 for a general invertible sheaf L_{d,gen} of degree d on X.

Now L′′ is of order dividing n, so there is a cyclic covering of degree n Z′′ ↠ Z which trivializes it. It is Z′′ that we will prove not to be ordinary (under certain numerical conditions at least). The invertible sheaf L′ of degree 0 corresponds to an abelian representation of π_{1}(Y) that factors trough π_{1}(Z′′), and when we induce that representation to π_{1}(X) we see that E = γ_{∗}L′ is of the form V_{ρ} for some representation ρ of π_{1}(X) that factors through π_{1}(Z′′) (and which can actually be described: see [2]).

All these constructions were performed on X and J. They could equally well have been performed on X_{1} and J_{1}. We now consider E as a sheaf of X_{1}. We have seen h^{0}(X, E⊗L_{d,gen}) > 0 and we wish to have h^{0}(X, E⊗B_{1}⊗L_{gen}) > 0. We are therefore done if we can show that B_{1} contains an invertible subsheaf of degree d.

But A. Hirschowitz claims in [4] and proves in [5] that a general bundle of rank r_{0} and slope λ_{0} contains a subbundle of rank r′ and slope λ′ (the quotient having rank r′′ = r_{0} − r′ and slope λ′′) if λ′′ − λ′ ≥ g − 1. If we are looking for r′ = 1 and λ′ = d, with r_{0} = p − 1 and λ_{0} = g − 1 (the numerical values of B_{1}), so r′′ = p − 2 and λ′′ = [(g − 1)(p − 1) − d]/(p − 2), this condition is satisfied iff (g − 1 − d)(p − 1)/(p − 2) ≥ g − 1, that is iff d ≤ \frac{g−1}{p−1}. By deforming and specializing to B_{1}, we see that if this inequality is satisfied then B_{1} contains an invertible sheaf of degree d.

Finally, we have shown that if p and g are such that there exists a positive odd integer n, prime to p, dividing g, and satisfying \frac{g}{n} ≤ \frac{g−1}{p−1} then the generic curve X of genus g in characteristic p has a covering that is not ordinary. This is not always the case, but we can always reduce to that case by first taking a cyclic cover X′ of degree m prime to p of X (X′ then has genus g′ = 1 + m(g − 1)), and apply the result to X′. Here are the details:

• If p is odd, take m even, not multiple of p and large enough so that g′ = 1 + m(g − 1) ≥ p.

– If p does not divide g′, then n = g′ works (it is odd because m is even, it is prime to p, and \frac{g′}{n} = 1 ≤ \frac{g′−1}{p−1} because g′ ≥ p).

– If p does divide g′ then we double m and this is no longer the case, so we are reduced to the previous point.

• If p = 2, write g = 2^{r}s with s odd.

– If s ≥ 3, take m = 1, n = s. (Then n is odd, and \frac{g}{n} = 2^{r} ≤ g − 1.)

– If s = 1 then g = 2^{r}.

∗ If r ≥ 2, take m = 3, n = g′/2 = 3 × 2^{r−1} − 1. (Then n is

odd, and \frac{g′}{n} = 2 ≤ g′ − 1.)

∗ If r = 1 so g = 2 and we take m = 5, g′ = 6, n = 3.

Finally, we note that our final covering was constructed as a composite Z′′ ↠ Y ↠ X′ ↠ X of coverings all of which are abelian: it is therefore solvable.

References

[1] M. Raynaud, “Sections des fibrés vectoriels sur une courbe”, Bull. Soc. math. France, 110 (1982), p. 103–125.

[2] M. Raynaud, “Revêtements des courbes en caractéristique p > 0 et ordinarité”, forthcoming.

[3] D. Mumford, “On the Equations Defining Abelian Varieties, I”, Inventiones mathematicæ, 1 (1966), p. 287–354.

[4] A. Hirschowitz, “Problèmes de Brill-Noether en rang supérieur”, C. R. Acad. Sci, 307 (1988), p. 153–156.

[5] A. Hirschowitz, “Problèmes de Brill-Noether en rang

supérieur”, http://math.unice.fr/~ah/Brill/, on the

World Wide Web.

David Madore

c/o École Normale Supérieure de Paris 45 rue d’Ulm 75 230 PARIS CEDEX 05 FRANCE

           david.madore@ens.fr