Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface

Claire Voisin Institut de mathématiques de Jussieu, CNRS,UMR 7586

1 Introduction

If C is a smooth projective curve of genus g and K_{C} is its canonical bundle, the theorem of Noether asserts that the multiplication map

μ_{0} : H^{0}(C, K_{C}) ⊗ H^{0}(C, K_{C}) → H^{0}(C, K_{C}^{⊗2})

is surjective when C is non hyperelliptic.

The theorem of Petri concerns then the ideal I of C in its canonical embedding, assuming C is not hyperelliptic. It says that I is generated by its elements of degree 2 if C is neither trigonal nor a plane quintic.

In [7], M. Green introduced and studied the Koszul complexes

⋀^{p+1} H^{0}(X, L)⊗H^{0}(X, L^{q−1}) →^{δ} ⋀^{p} H^{0}(X, L)⊗H^{0}(X, L^{q}) →^{δ} ⋀^{p−1} H^{0}(X, L)⊗H^{0}(X, L^{q+1})

for X a variety and L a line bundle on X. Denoting by K_{p,q}(X, L) the cohomology at the middle of the sequence above, one sees immediately that the surjectivity of the map μ_{0} is equivalent to K_{0,2}(C, K_{C}) = 0, and that if this is the case, the ideal I is generated by quadrics if and only if K_{1,2}(C, K_{C}) = 0. On the other hand, C being non hyperelliptic is equivalent to the fact that the Clifford index Cliff C is strictly positive, where

CliffC := Min{d − 2r, ∃L ∈ P ic C, d^{0}L = d, h^{0}(L) = r + 1 ≥ 2, h^{1}(L) ≥ 2}.

Similarly, C is neither hyperelliptic, nor trigonal nor a plane quintic if and only if Cliff C > 1.

Green’s conjecture on syzygies of canonical curves generalizes then the theorems of Noether and Petri as follows

Conjecture 1 [7] For a smooth projective curve C in characteristic 0, the condition Cliff C > l is equivalent to the fact that K_{l′,2}(C, K_{C}) = 0, ∀l′ ≤ l.

The interest of this formulation of Noether and Petri’s theorems is already illustrated in [9], where these theorems are given a modern proof, using geometric technics of computation of syzygies.

For our purpose, and as is done in [7], it is convenient to use the duality (cf [7])

K_{p,2}(C, K_{C}) ≅ K_{g−p−2,1}(C, K_{C})^{∗}

to reformulate the conjecture as follows

Conjecture 2 [7] For a smooth projective curve C of genus g in characteristic 0, the condition Cliff C > l is equivalent to the fact that K_{g−l′−2,1}(C, K_{C}) = 0, ∀l′ ≤ l.

If C is now a generic curve, the theorem of Brill-Noether (cf [2], [11]) implies that

Cliff C = gon(C) − 2

where the gonality gon(C) := Min {d, ∃L ∈ P ic C, d^{0}L = d, h^{0}(L) ≥ 2}, and that

gon(C) = \frac{g + 3}{2}, if g is odd,

gon(C) = \frac{g + 2}{2}, if g is even.

Hence we arrive at the following conjecture (the generic Green conjecture on syzygies of a canonical curve) :

Conjecture 3 Let C be a generic curve of genus g. Then if g = 2k + 1 or g = 2k, we have K_{k,1}(C, K_{C}) = 0.

Remark 1 The actual conjecture is K_{l,1}(C, K_{C}) = 0, ∀l ≥ k; but it is easy to prove that

K_{k,1}(C, K_{C}) = 0 ⇒ K_{l,1}(C, K_{C}) = 0, ∀l ≥ k.

Notice that in the appendix to [7], Green and Lazarsfeld prove the conjecture 1 in the direction ⇐ (i. e. they produce non zero syzygies from special linear systems.) Hence the conjecture above cannot be improved, namely, under the assumptions above, we have K_{k−1,1}(C, K_{C}) ≠ 0.

Teixidor [16] has recently proposed an approach to the conjecture 3. Her method uses a degeneration to a tree of elliptic curves and the theory of limit linear series of Eisenbud and Harris [6], adapted to vector bundles of higher rank. It is very likely that her method will lead to a proof of the generic Green conjecture.

We propose here a completely different approach, which does not prove Conjecture 3 in odd genus, but proves Green’s Conjecture 2 for generic curves C of genus g(C) and gonality gon(C) in the range

\frac{g(C)}{3} + 1 ≤ gon(C) ≤ \frac{g(C)}{2} + 1.

The inequality on the left says that the gonality has to be not too small, but the only case which is excluded by the inequality on the right is that of a generic curve of odd genus g, which has gonality \frac{g+3}{2}.

Recall from [11] that if S is a K3 surface endowed with a ample line bundle L such that L generates P ic S and L^{2} = 2g − 2, the smooth members C ∈| L | are of genus g and generic in the sense of Brill-Noether, so that in particular they have the same Clifford index as a generic curve. Hence conjecture 1 predicts that their syzygies vanish as stated in conjecture 3. This is indeed what we prove here, in the case where the genus is even. Note first that the hyperplane restriction theorem [7] says that

K_{k,1}(C, K_{C}) = K_{k,1}(S, L)  (1.1)

whenever C is a hyperplane section of a K3 surface S (note that K_{C} = L_{|C} in this case). Conjecture 3 for curves of even genus is therefore implied by

Theorem 1 The pair (S, L) being as above, with g = 2k, we have

K_{k,1}(S, L) = 0.  (1.2)

The body of the paper will be devoted to the proof of (1.2). It turns out that Theorem 1 in turn has much stronger consequences than the generic syzygy conjecture for curves of even genus, and we shall explain this now. In fact we have the following corollary :

Corollary 1 For any δ ≤ \frac{k}{2}, the generic curve of genus 2k−δ which is k+1−δgonal satisfies

K_{k,1}(C, K_{C}) = 0

or equivalently by the duality theorem

K_{k−δ−2,2}(C, K_{C}) = 0.

Notice that this result is optimal and exactly predicted by Green’s conjecture 1, since the Clifford index of such curve is less than or equal to k − 1 − δ. An easy computation shows that the pairs

g = 2k − δ, gon = k + 1 − δ, k ≥ 0, δ ≤ \frac{k}{2}

are exactly the pairs satisfying the inequalities

\frac{g}{3} + 1 ≤ gon ≤ \frac{g}{2} + 1.

Hence Green’s Conjecture is proved for generic curves of genus and gonality in this range. On the other hand, Teixidor [17] has proved the Green conjecture for generic curves of fixed gonality in the range gon ≤ \frac{g}{3}. Combining this and our corollary we see that Green’s conjecture is proved for generic curves of any fixed gonality, with the exception of generic curves of odd genus (which satisfy gon = \frac{g}{2} + 1 + \frac{1}{2}).

Proof of Corollary 1. Let (S, L) be as in theorem 1. A generic member X of | L | is k + 1-gonal. As in section 2, and following [11], it follows that there is a rank 2 vector bundle E on S with det E = L, c_{2}(E) = k + 1, and h^{0}(E) = k + 2. The zero set of a generic section of E is a generic member of a g_{k+1}^{1} of a generic curve X ∈| L |.

Now let x_{1}, …, x_{δ} be generic points of S. Beacause δ ≤ \frac{k}{2}, the space

H_{x_{·}} = H^{0}(S, E ⊗ I_{x_{1}} ⊗ … ⊗ I_{x_{δ}})

has rank at least 2. One checks that for α, β generic in this space, the curve X defined by the equation

det (α ∧ β) ∈ H^{0}(S, det E) = H^{0}(S, L)

is nodal with nodes exactly as the x_{i}’s. On the other hand, the two sections α, β generate a rank 1 subsheaf of the restriction E_{|X}. Let now

n : C → X

be the normalization. The rank 1 subsheaf introduced above induces a line subbundle

D ⊂ n^{∗}E

with two sections, and it is obvious that the moving part of this linear system on C is of degree k + 1 − δ, since the sections λα + μβ of E vanish at the x_{i}’s, so that the moving part of their zero sets is of degree k + 1 − δ. Hence C is k + 1 − δ-gonal. It remains to show that

K_{k,1}(C, K_{C}) = 0.  (1.3)

Now we have by the hyperplane restriction theorem and by theorem 1 the vanishing

K_{k,1}(X, K_{X}) = 0.  (1.4)

We prove now that this implies (1.3). Notice that there is an identification of H^{0}(C, K_{C}) with a subspace of H^{0}(X, K_{X}), namely the last space is a space of meromorphic forms on C with logarithmic singularities over the nodes satisfying the condition that the sum of the residues over each node vanishes. H^{0}(C, K_{C}) is then the subspace of forms which are regular.

From this inclusion j : H^{0}(C, K_{C}) ↪ H^{0}(X, K_{X}), we can deduce a commutative diagram of Koszul complexes

⋀^{k+1} H^{0}(C, K_{C}) → ⋀^{k} H^{0}(C, K_{C}) ⊗ H^{0}(C, K_{C}) → ⋀^{k−1} H^{0}(C, K_{C}) ⊗ H^{0}(C, K_{C}^{⊗2})
j ↓ j ↓ j ↓
⋀^{k+1} H^{0}(X, K_{X}) → ⋀^{k} H^{0}(X, K_{X}) ⊗ H^{0}(X, K_{X}) → ⋀^{k−1} H^{0}(X, K_{X}) ⊗ H^{0}(X, K_{X}^{⊗2}).

We claim that this induces an inclusion

j : K_{k,1}(C, K_{C}) ↪ K_{k,1}(X, K_{X}).

Indeed, consider in general the Koszul differential

δ : ⋀^{l} H^{0}(Y, L) → H^{0}(Y, L) ⊗ ⋀^{l−1} H^{0}(Y, L).

Then if

∧ : H^{0}(Y, L) ⊗ ⋀^{l−1} H^{0}(Y, L) → ⋀^{l} H^{0}(Y, L)

is the wedge product map, one has

∧ ∘ δ = ±lId.  (1.5)

Consider now the inclusion

j : H^{0}(C, K_{C}) ⊗ ⋀^{k} H^{0}(C, K_{C}) → H^{0}(X, K_{X}) ⊗ ⋀^{k} H^{0}(X, K_{X}).

Let α ∈ H^{0}(C, K_{C}) ⊗ ⋀^{k} H^{0}(C, K_{C}) such that δα = 0 and j(α) = δβ. Then (1.5) gives

j(α) = δβ = ±\frac{1}{k + 1}δ(∧ ∘ δβ)

= ±\frac{1}{k + 1}δ(∧(j(α))).

But ∧(j(α)) = j(∧α), so that this implies by injectivity of j that α = ±\frac{1}{k+1}δ(∧α). Hence α is in fact exact. Hence our claim is proven.

In the missed case of a generic curve of odd genus, we have the following corollary :

Corollary 2 Let C be a generic curve of genus g = 2k − 1; then

K_{k,1}(C, K_{C}) = 0.

(Notice that the generic Green conjecture predicts in fact that K_{k−1,1}(C, K_{C}) = 0.)

Proof of Corollary 2. The K3 surface S being as above, let X be a member of | L | with exactly one node as singularity. Let C be the normalization of X. Then the genus of C is equal to 2k − 1.

We have as before an inclusion

H^{0}(C, K_{C}) ↪ H^{0}(X, K_{X})  (1.6)

which induces by the same argument as in the proof of corollary 1 an inclusion

K_{k,1}(C, K_{C}) ↪ K_{k,1}(X, K_{X}).

The hyperplane restriction theorem can be applied to X ⊂ S, and together with the vanishing (1.2), it gives

K_{k,1}(X, K_{X}) = 0.

Hence K_{k,1}(C, K_{C}) = 0.

We conclude this introduction with a sketch of the main ideas in the proof of theorem 1. The very starting point is the following observation : denote by S^{[l]} the Hilbert scheme parametrizing 0-dimensional length l subschemes of S. Let I_{l} ⊂ S × S^{[l]} be the incidence subscheme and

\begin{matrix}I_{l} & →^{π_{l}} & S^{[l]} \\ q ↓ \\ S \end{matrix}

be the incidence correspondence. Let

E_{L} := R^{0}π_{l}_{∗}q^{∗}L

and L_{l} := det E_{L}. Then we have Fact. K_{l−1,1}(S, L) = 0 if and only if

H^{0}(I_{l}, π_{l}^{∗}L_{l}) = π_{l}^{∗}H^{0}(S^{[l]}, L_{l}).

Our strategy will be then to construct a subvariety Z of S^{[k+1]}, such that

H^{0}(Z̃, π_{l}^{∗}L_{l}) = π_{l}^{∗}H^{0}(Z, L_{l})

where Z̃ := π_{l}^{−1}(Z), and the restriction map

H^{0}(I_{l}, π_{l}^{∗}L_{l}) → H^{0}(Z̃, π_{l}^{∗}L_{l})

is injective.

As in the papers [11], [8], the key role in constructing our variety Z and verifying the conditions above will be played by the Lazarsfeld-Mukai vector bundle on S associated with minimal degree base-point free linear systems on smooth members of | L |.

Terminology. In this paper, we shall say that a Zariski open subset U ⊂ X is large if the complementary closed subset Z = X − U has codimension non smaller that 2 in X. In the considered cases, the variety X will be normal, and we will use freely the fact that for a line bundle L on X

H^{0}(X, L) ≅ H^{0}(U, L_{|U})

for U a large open subset of X.

2 Strategy of the proof

We start with the following observation : Let X be a smooth projective variety. Denote by X_{curv}^{[k]} the Hilbert scheme parametrizing curvilinear 0-dimensional subschemes of X of length k. X_{curv}^{[k]} is smooth, and if X is a curve or a surface, it is a large open set in the Hilbert scheme X^{[k]} which is smooth.

Let

I_{k} →^{π_{k}} X_{curv}^{[k]}
q ↓
X

be the incidence correspondence. For a line bundle L on X denote by E_{L} the vector bundle on X_{curv}^{[k]} defined by E_{L} = R^{0}π_{k}_{∗}q^{∗}L, and let

L_{k} := det E_{L}.

We have

Lemma 1 There is a natural isomorphism

K_{k,1}(X, L) ≅ H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1})/π_{k+1}^{∗}H^{0}(X_{curv}^{[k+1]}, L_{k+1}).

In particular, K_{k,1}(X, L) = 0 is equivalent to

H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) = π_{k+1}^{∗}H^{0}(X_{curv}^{[k+1]}, L_{k+1}).

Proof. Recall that K_{k,1}(X, L) is the cohomology at the middle of the sequence

⋀^{k+1} H^{0}(X, L) →^{δ} H^{0}(X, L) ⊗ ⋀^{k} H^{0}(X, L) →^{δ} H^{0}(X, L^{⊗2}) ⊗ ⋀^{k−1} H^{0}(X, L).(2.7)

Now note that there is a natural morphism

τ : I_{k+1} → X × X_{curv}^{[k]}  (2.8)

which to (x, z), x ∈ Supp z associates (x, z′), where z′ is the residual scheme of x in z. This morphism is well defined because we are working with curvilinear schemes.

One shows easily that τ identifies I_{k+1} to a large open subset of the blowup of X × X_{curv}^{[k]} along the incidence subscheme I_{k}. (Indeed, away from I_{k} the inverse τ^{−1} of τ is given by

τ^{−1}((x, z)) = x ∪ z, x ∈ X, z ∈ X_{curv}^{[k]}.)

Furthermore, if D ⊂ I_{k+1} is the exceptional divisor one has

π_{k+1}^{∗}L_{k+1} = τ^{∗}(L ⊠ L_{k})(−D).  (2.9)

This isomorphism is obtained by studying the natural morphism of vector bundles over I_{k+1}

π_{k+1}^{∗}E_{L} → τ^{∗}(pr_{1}^{∗}E ⊕ pr_{2}^{∗}E_{L}),

which at a point z̃ such that τ(z̃) = (x, z′) identifies to the restriction map

H^{0}(L_{|z̃}) → L_{|x} ⊕ H^{0}(L_{|z′}).

It is immediate to see that the cokernel of this morphism is supported on the exceptional divisor D of τ and is of rank 1 on D.

It follows that

H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1})
= Ker (H^{0}(X, L) ⊗ H^{0}(X_{curv}^{[k]}, L_{k}) →^{rest} H^{0}(I_{k}, L ⊠ L_{k}_{|I_{k}})).  (2.10)

We now apply the description above to I_{k} : we note that denoting by p_{i}, i = 1, 2, the compositions of the projections with the inclusion I_{k} ↪ X × X_{curv}^{[k]}, we have

p_{2} = π_{k}, p_{1} = pr_{1} ∘ τ,

where

τ : I_{k} → X × X_{curv}^{[k−1]}

is defined as in (2.8). Hence applying formula (2.9), we get

L ⊠ L_{k}_{|I_{k}} = τ^{∗}(L^{2} ⊠ L_{k−1})(−D),

where D is now the exceptional divisor of the blowing-down morphism τ : I_{k} → X × X_{curv}^{[k−1]}. So we conclude that there is a natural inclusion

i : H^{0}(I_{k}, L ⊠ L_{k}_{|I_{k}}) ⊂ H^{0}(X, L^{⊗2}) ⊗ H^{0}(X_{curv}^{[k−1]}, L_{k−1}).

Hence we have constructed from formula (2.10) an exact sequence

0 → H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) →^{j} H^{0}(X, L) ⊗ H^{0}(X_{curv}^{[k]}, L_{k})

→^{i ∘ rest} H^{0}(X, L^{⊗2}) ⊗ H^{0}(X_{curv}^{[k−1]}, L_{k−1}).

Next, it is a standard fact that the natural map

⋀^{l} H^{0}(X, L) → H^{0}(X_{curv}^{[l]}, L_{l})  (2.11)

induced by the evaluation map

H^{0}(X, L) ⊗ O_{X_{curv}^{[k]}} → E_{L}

are isomorphisms for any l. To check this, one considers the large open subset U^{[l]} of X_{curv}^{[l]}, made of subschemes z which have at most one point of multiplicity 2 as singularity. This set U^{[l]} has the following description : one considers inside X^{l} the large open set X_{0}^{l} made of l-uples (x_{1}, …, x_{l}), where at most two of the x_{i}’s coincide. Inside X_{0}^{l}, one blows-up the generalized diagonal ∪_{i≠j}{x_{i} = x_{j}}, which is smooth there, and one takes the quotient of the resulting variety X̃_{0}^{l} by the action of the symmetric group S_{l}. By the same argument as above one finds that if

r : X̃_{0}^{l} → U^{[l]}

is the quotient map, one has

r^{∗}L_{l} ≅ L ⊠ … ⊠ L(−E),

where E is the exceptional divisor of the blowing-up map τ′ : X̃_{0}^{l} → X_{0}^{l}. It follows that one has an identification

H^{0}(X_{curv}^{[l]}, L_{l}) = H^{0}(U^{[l]}, L_{l}) ≅ H^{0}(X̃_{0}^{l}, τ′^{∗}(⊠_{i=1}^{i=l}L)(−E))^{S_{l}}.

The last space is a subspace of

H^{0}(X_{0}^{l}, ⊠_{i=1}^{i=l}L)^{S_{l}} ≅ H^{0}(X^{l}, ⊠_{i=1}^{i=l}L)^{S_{l}} = (⊗_{i=1}^{i=l}H^{0}(X, L))^{S_{l}}.

But looking more closely at the action of S_{l} on the line bundle r^{∗}L_{k}, one verifies that the induced action of S_{l} on ⊗_{i=1}^{i=l}H^{0}(X, L) is the twisted action. Hence the invariant subspace

(⊗_{i=1}^{i=l}H^{0}(X, L))^{S_{l}}

is isomorphic to ⋀^{l} H^{0}(X, L). One verifies then that the injective map

H^{0}(X_{curv}^{[l]}, L_{l}) → ⋀^{l} H^{0}(X, L)

so obtained is a left inverse for the map (2.11).

Using the isomorphisms (2.11), the exact sequence above becomes

0 → H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) →^{j} H^{0}(X, L) ⊗ ⋀^{k} H^{0}(X, L)

→^{i ∘ rest} H^{0}(X, L^{⊗2}) ⊗ ⋀^{k−1} H^{0}(X, L),

hence it remains it remains only to show that the maps j ∘ π_{k+1}^{∗} and i ∘ rest identify via the isomorphisms (2.11) to the differentials δ of the sequence (2.7), which is easy.

We consider now a K3 surface S endowed with an ample line bundle L generating P ic S and satisfying

L^{2} = 2g − 2, g = 2k, k > 1.

We now explain our strategy to prove the vanishing

K_{k,1}(S, L) = 0.

Assume we have a subscheme T ⊂ S^{[k+1]} such that, if T̃ denotes the subvariety π_{k+1}^{−1}(T) of I_{k+1}, the following conditions are satisfied :

1. We have an isomorphism

H^{0}(T̃, π_{k+1}^{∗}L_{k+1}) = π_{k+1}^{∗}H^{0}(T, L_{k+1}).

2. The restriction map

H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) → H^{0}(T̃, π_{k+1}^{∗}L_{k+1})

is injective.

Then we claim that K_{k,1}(S, L) = 0.

Indeed we have the trace maps

tr : H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) → H^{0}(S_{curv}^{[k+1]}, L_{k+1})
tr_{T} : H^{0}(T̃, π_{k+1}^{∗}L_{k+1}) → H^{0}(T, L_{k+1})

which commute with the restriction maps and which compose to (k + 1) Id with the pull-back maps. If σ ∈ H^{0}(I_{k+1}, π^{∗}L_{k+1}), there exists by property 1 a β ∈ H^{0}(T, L_{k+1}) such that

σ_{|T̃} = π_{k+1}^{∗}β.

Then

β = \frac{1}{k + 1}tr_{T} (σ_{|T̃}) = (\frac{1}{k + 1}tr σ)_{|T} .

Hence the section

σ′ = σ − π^{∗}(\frac{1}{k + 1}T r σ)

vanishes on T̃, hence it is zero by property 2. So we have

H^{0}(I_{k+1}, π^{∗}L_{k+1}) = π^{∗}H^{0}(S_{curv}^{[k+1]}, L_{k+1})

and this proves our claim, using lemma 1.

We will have to weaken the assumptions above as follows : Suppose we have a normal scheme Z together with a morphism

j : Z → I_{k+1}

such that π ∘ j is generically one to one on its image, which is not contained in the branch locus of π_{k+1}. Suppose also that we have a normal scheme Z′ together with a proper degree k morphism π′ : Z′ → Z and a morphism j′ : Z′ → I_{k+1} satisfying the conditions that

π_{k+1} ∘ j′ = j ∘ π′

and the union j(Z) ∪ j′(Z′) is equal set theoretically to π_{k+1}^{−1}(π_{k+1} ∘ j(Z)). Finally assume there are subschemes Z′_{1} ⊂ Z′, Z_{1} ⊂ Z such that

π′_{|Z′_{1}} =: ϕ : Z′_{1} → Z_{1}

is a birational isomorphism and j ∘ ϕ = j′_{|Z′_{1}}.

(Hence roughly speaking, and up to birational maps, π_{k+1}^{−1}(π_{k+1} ∘ j(Z)) is the scheme obtained by gluing Z′ and Z along Z′_{1} ≅ Z_{1}.)

Assume now that they satisfy the following set (H) of hypotheses 1. The map

π′^{∗} : H^{0}(Z, (π_{k+1} ∘ j)^{∗}L_{k+1}) → H^{0}(Z′, (π_{k+1} ∘ j′)^{∗}L_{k+1})

is an isomorphism.

2. The restriction map

H^{0}(Z, (π_{k+1} ∘ j)^{∗}L_{k+1}) → H^{0}(Z_{1}, (π_{k+1} ∘ j)^{∗}L_{k+1}_{|Z_{1}})

is injective.

3. The restriction map

j^{∗} : H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) → H^{0}(Z, (π_{k+1} ∘ j)^{∗}L_{k+1})

is injective.

Then we claim that K_{k,1}(S, L) = 0.

Indeed by Lemma 1 we have to show that

H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) = π_{k+1}^{∗}H^{0}(S_{curv}^{[k+1]}, L_{k+1}).

Now if σ ∈ H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}), by hypothesis H1, j′^{∗}σ = π′^{∗}α for some α ∈ H^{0}(Z, (π_{k+1} ∘j)^{∗}L_{k+1}). We show now that j^{∗}σ = α. Indeed, by property H2, it suffices to show that this is true after restriction to Z_{1}, and since ϕ : Z′_{1} → Z_{1} is dominating, it suffices to show that

ϕ^{∗}(α_{|Z_{1}}) = ϕ^{∗}(j^{∗}σ_{|Z_{1}}).

But this follows from j ∘ ϕ = j′_{|Z′_{1}} and from j′^{∗}σ = π′^{∗}α, with ϕ = π′_{|Z′_{1}}. Finally it follows from the equalities α = j^{∗}σ and j′^{∗}σ = π′^{∗}α that

σ′ = σ − π^{∗}(\frac{1}{k + 1}T r σ)

vanishes along j(Z) ∪ j′(Z′). (Indeed, because we know that j(Z) is not contained in the ramification locus of π_{k+1} and π_{k+1} ∘j : Z → X_{curv}^{[k+1]} is generically one to one on its image, the map π_{k+1} restricted to j(Z) ∪ j′(Z′) is generically a degree k + 1 unramified map onto its image. On the other hand, the two equalities above say that σ_{|j(Z)∪j′(Z′)} is generically a pull-back of a section of L_{k+1} on this image. Hence the previously used trace argument applies to an open set of this image.)

Now this implies that σ′ vanishes by hypothesis H3. This concludes the proof of our claim.

We conclude this section with the description of the schemes Z, Z′ we will be considering.

Recall from [8], [11], [12], that there is a unique stable bundle E of rank 2 on S, (the Lazarsfeld-Mukai vector bundle,) which satisfies the following properties:

det E = L, c_{2}(E) = k + 1, h^{0}(E) = k + 2.

Such vector bundle is obtained by choosing a line bundle D on a generic member C of | L |, such that h^{0}(D) = 2 and deg D = k + 1. Such a line bundle exists by Brill-Noether theory, and it is generated by global sections since C does not carry a g_{k}^{1} by Lazarsfeld [11]. Then we have a vector bundle F on S defined by the exact sequence

0 → F → H^{0}(D) ⊗ O_{S} → D → 0  (2.12)

and E is defined as the dual of F. The stability of E follows from the fact that P ic S = ℤL and H^{0}(S, E(−L)) = 0. The uniqueness of such E follows then from the fact that χ(E, E′) = 2 for any other vector bundle E′ with the same numerical properties, so that either Hom (E, E′) ≠ 0 or Hom (E′, E) ≠ 0. But then by stability, E = E′.

The property h^{0}(S, E) = k + 2 follows from the sequence dual to (2.12)

0 → H^{0}(D)^{∗} ⊗ O_{S} → E → K_{C} − D → 0,  (2.13)

and from Riemann-Roch which gives h^{0}(K_{C} − D) = k.

Another way to construct the bundle E is via Serre’s construction. By Riemann-Roch the divisors D of degree k + 1 on smooth members C of | L | which satisfy h^{0}(C, D) = 2 are exactly the subschemes z of degree k + 1 on S contained in a smooth member C of | L | and satisfying the condition that the restriction map

H^{0}(S, L) → H^{0}(L_{|z})

is not surjective. Note that since the curves C are general in the sense of Brill-Noether, the corank of this map is exactly 1 and furthermore for any z′ ⊊ z the restriction map

H^{0}(S, L) → H^{0}(L_{|z′})

is surjective. Hence, since K_{S} is trivial, to such z corresponds a vector bundle E together with a section σ_{z} vanishing on z. This E is an extension

0 → O_{S} →^{σ_{z}} E →→^{∧σ_{z}} I_{z}(L) → 0.  (2.14)

Computing the numerical invariants of this bundle E, and arguing as before by stability, we see that this bundle is isomorphic to the one constructed above. Notice that each g_{k+1}^{1}, D on a smooth member C ∈| L | provides by (2.13) a rank 2 subspace of sections of E, and that the zero sets of these sections identify to the members of | D |, as subschemes of S.

It follows from the exact sequence (2.14) twisted by E that h^{0}(S, E⊗I_{z}) = 1 for any z as above. Hence the morphism

ℙ(H^{0}(S, E)) → S^{[k+1]}

which to σ associates its zero set, is in fact an embedding. One sees easily that the open set ℙ(H^{0}(S, E))_{curv} corresponding to curvilinear subschemes is large in ℙ(H^{0}(S, E)).

Let now W := π_{k+1}^{−1}(ℙ(H^{0}(S, E))_{curv}) ⊂ I_{k+1}. W is easily shown to be smooth. There is a natural morphism

ψ : W → S_{curv}^{[k]}

defined as the restriction of pr_{2}∘τ to W. This ψ can be shown to be generically of degree one on its image.

Consider the blow-up { \atop S × W} of S ×W along K := (Id, ψ)^{−1}(I_{k}). It admits a morphism { \atop (Id, ψ)} to the blow-up of S ×S_{curv}^{[k]} along I_{k}, and the later contains

I_{k+1} as a large open set. One verifies that { \atop (Id, ψ)}^{−1}(I_{k+1}) is a large open set of { \atop S × W}. This will be our scheme Z. The morphism j : Z → I_{k+1} will be simply the restriction to Z of { \atop (Id, ψ)}.

Again one can show (using now the assumption that k > 1) that the morphism π_{k+1} ∘ j : Z → S^{[k+1]} is generically of degree one on its image.

Next let π′′ : W̃ → W be the degree k cover obtained by completing the Cartesian diagram

\begin{matrix}W̃ & → & I_{k} \\ π′′ ↓ & & π_{k} ↓ \\ W & →^{ψ} & S_{curv}^{[k]} \end{matrix} .

Consider the rational map

j′ : S × W̃ − − − − > I_{k+1}

which to (s, s_{1}, w), s_{1} ∈ Supp ψ(w) associates (s_{1}, s ∪ ψ(w)). This morphism becomes well defined after blowing-up K′ := (Id, π′′)^{−1}(K) and restricting to a large open subset. Our scheme Z′ will be this large open set. The morphism π′ : Z′ → Z is the restriction to Z′ of the morphism Bl_{K′}(S×W̃) → Bl_{K}(S×Z) induced by (Id, π′′). The morphism j′ : Z′ → I_{k+1} is induced by the rational map j′ above. We have

π_{k+1} ∘ j′ = π_{k+1} ∘ j ∘ π′.

Indeed, both maps send (s, s_{1}, w), s_{1} ∈ Supp ψ(w) to s ∪ ψ(w). It is obvious that π_{k+1}^{−1}(π_{k+1}∘j(Z)) is equal to j(Z)∪j′(Z′). Indeed, the fiber over s∪ψ(w) ∈ π_{k+1} ∘ j(Z) consists in choosing one point in the scheme s ∪ ψ(w). This point may be s, in which case we are in j(Z), or has to be a point s_{1} contained in Supp ψ(w) in which case it determines a point (s_{1}, w) of W̃ over w, and we are then in j′(Z′).

Remark 2 The scheme Z is non necessarily smooth, but one can show that K is reduced, so that its singular locus is of codimension at least two in S × W. The same thing is true for Z′ and K′. If one wants to work with smooth schemes Z_{0} and Z′_{0} (so as to be exactly in the conditions (H) described above), it suffices to restrict to the blowing-ups of S × W − K_{sing} along K − K_{sing} and S × W − K′_{sing} along K′ − K′_{sing}. All what follows will be true for these subschemes.

To conclude, it remains now to construct Z_{1} and Z′_{1}. Z_{1} will be the exceptional divisor of Z (recall that Z is a large open set in Bl_{K}(S × W)). Hence Z_{1} is the inverse image under the blow-up map Z → S × W of

K = {(s, w) ∈ S × W, s ∈ Supp ψ(w)}.

We now construct a generic lifting of Z_{1} in Z′, the closure of the image of which will be Z′_{1}. By definition of Z′ as a large open set of Bl_{K′}(S × W̃), it suffices to construct a lifting of K to a component of K′ in S × W̃. But if (s, w) ∈ K, we have s ∈ Supp ψ(w) so that (s, w) identifies to an element w̃ of W̃. Our lifting sends simply (s, w) to (s, w̃).

It remains finally to see that the morphisms j′ and j ∘π′ agree on Z′_{1}. Since I_{k+1} is contained in S × S_{curv}^{[k+1]}, it suffices to prove that pr_{1} ∘ j′ and pr_{1} ∘ j ∘ π′ agree on Z′_{1} and that pr_{2} ∘ j′ and pr_{2} ∘ j ∘ π′ agree on Z′_{1}, with pr_{2} = π_{k+1} on I_{k+1}. For the first one this is obvious since both maps factor through the contraction Z′_{1} → K′, and are equal on K′ ⊂ S × W̃ to the first projection on S, as follows from the definition of the lifting K → K′.

As for the second one, it follows from the fact that, by construction, π_{k+1}∘j′ and π_{k+1} ∘ j ∘ π′ agree on Z′.

3 Proof of the assumptions H2 and H3

We start with the proof of hypothesis H2.

Proposition 1 Let

Z_{1} ⊂ Z →^{π_{k+1}∘j} S^{[k+1]}

be as in the previous section. Then the restriction map

H^{0}(Z, (π_{k+1} ∘ j)^{∗}L_{k+1}) → H^{0}(Z_{1}, (π_{k+1} ∘ j)^{∗}L_{k+1}_{|Z_{1}})

is injective.

The proof will be obtained by restricting the construction to a generic smooth member C ∈| L |. Indeed, recall that Z is a large open set in the blow-up of S × W along the incidence subscheme K = (id, ψ)^{−1}(I_{k}), where

W = {(x, σ) ∈ S × ℙ(H^{0}(S, E))_{curv}, σ(x) = 0},

and ψ : W → S^{[k]} sends (x, σ) to the residual scheme of x in V (σ). Now since k ≥ 1, the generic element z = V (σ) is supported in a pencil of elements of | L |, the generic member being smooth. It follows that a generic element of S × W is of the form (s_{1}, s_{2}, z), z = V (σ), σ(s_{2}) = 0 and there exists a smooth member C ∈| L | such that s_{1}, s_{2}, z are supported on C. Hence it suffices to prove the analogue of proposition 1 with Z replaced by Z_{C}, the proper transform of C × W_{C} in Z ⊂ Bl_{K}(S × W), where

W_{C} := {(c, σ) ∈ C × ℙ(H^{0}(S, E)), σ(c) = 0, V (σ) ⊂ C},

and Z_{1} is replaced by Z_{1,C} := Z_{1} ∩ Z_{C}.

Proposition 2 The restriction map

H^{0}(Z_{C}, (π_{k+1} ∘ j)^{∗}L_{k+1}_{|Z_{C}}) → H^{0}(Z_{1}, (π_{k+1} ∘ j)^{∗}L_{k+1}_{|Z_{1,C}})

is injective.

Proof. By the description of the bundle E given in the previous section, we note that the set

{σ ∈ ℙ(H^{0}(S, E)), V (σ) ⊂ C},

identifies by the map σ ↦ V (σ) to the disjoint union of the ℙ^{1} ⊂ C^{(k+1)} corresponding to g_{k+1}^{1}’s on C. If D is such a g_{k+1}^{1} on C, D gives a morphism of degree k + 1

ϕ_{D} : C → ℙ^{1}

or a line bundle L_{D} on C of degree k + 1 with two sections. By definition, W_{C} identifies (via ψ) to the disjoint union of copies C_{D} of C contained in C^{(k)} and indexed by the g_{k+1}^{1}’s D of C, where the map

ψ_{D} : C ≅ C_{D} → C^{(k)}

is given by

c ↦ the unique effective divisor equivalent to D − c.

Finally Z_{C} identifies to a disjoint union of surfaces Z_{C,D} isomorphic to C ×C, since the pull-back Δ_{D} to C ×C_{D} of the incidence scheme in C ×C^{(k)} is of pure codimension 1, so that the blow-up of C × C_{D} along Δ_{D} is isomorphic to C × C_{D}. Note that under this isomorphism, the intersection of Z_{1} with C × C_{D} becomes identified to Δ_{D}.

Recall now that on the large open set Z ⊂ { \atop S × W}, we have

(π_{k+1} ∘ j)^{∗}L_{k+1} = τ^{∗}(L ⊠ ψ^{∗}L_{k})(−Z_{1}).

We have L_{|C} = K_{C} and in the sequel we will use the notation H_{D} for the line bundle L_{k}_{|C_{D}}. (It will be shown that H_{D} ≡ kL_{D} but this will not be used now.) Restricting the equality above to Z_{C}, we have to show that for each D the restriction map

H^{0}(C × C, K_{C} ⊠ H_{D}(−Δ_{D})) → H^{0}(Δ_{D}, K_{C} ⊠ H_{D}(−Δ_{D})_{|Δ_{D}})

is injective. In other words we want to show that

H^{0}(C × C, K_{C} ⊠ H_{D}(−2Δ_{D})) = 0.  (3.15)

Now, since Δ_{D} is the restriction to C × C_{D} of the incidence scheme, and since C_{D} parametrizes the effective divisors of the form L_{D} − x, x ∈ C, it is clear that

Δ_{D} = (ϕ_{D}, ϕ_{D})^{−1}(diag (ℙ^{1})) − diag (C).

Hence we have

Δ_{D} ≡ L_{D} ⊠ L_{D} − diag (C)

in C × C. It follows that

K_{C} ⊠ H_{D}(−2Δ_{D}) ≡ (K_{C} − 2L_{D}) ⊠ (H_{D} − 2L_{D}) + 2diag C.

Now we have the equality

H^{0}(C, K_{C} − 2L_{D}) = 0,  (3.16)

which is proven in [11], since C is generic in S. (Indeed for a base point free pencil, | L_{D} |, the condition that the μ_{0}-map

H^{0}(C, L_{D}) ⊗ H^{0}(C, K_{C} − L_{D}) → H^{0}(C, K_{C})

is injective is equivalent by the base-point free pencil trick to the condition

H^{0}(C, K_{C} − 2L_{D}) = 0.

The equality (3.15) follows now from (3.16) and from the fact that the map H^{0}(C, 2L_{D}) → H^{0}(2L_{D}_{|2x}) is surjective for generic x in C. Hence by Riemann-Roch, H^{0}(C, K_{C} − 2L_{D}) = 0 implies H^{0}(C, K_{C} − 2L_{D} + 2x) = 0 for generic x ∈ C. It follows that

H^{0}(C × C, (K_{C} − 2L_{D}) ⊠ (H_{D} − 2L_{D}) + 2diag C) = 0,

which proves the proposition 2, and hence proposition 1 is proven.

We turn now to the proof of hypothesis H3.

Proposition 3 The morphism Z →^{j} I_{k+1} being defined as in the previous section, the pull-back map

j^{∗} : H^{0}(I_{k+1}, π_{k+1}^{∗}L_{k+1}) → H^{0}(Z, (π_{k+1} ∘ j)^{∗}L_{k+1})

is injective.

The proof proceeds in several steps, and occupies the remainder of this section. Recall that I_{k+1} is a large open set in the blow-up of S×S^{[k]} along the incidence subscheme I_{k} and that we have the following formula

π_{k+1}^{∗}L_{k+1} = τ^{∗}(L ⊠ L_{k})(−D),

where D is the exceptional divisor and τ is the blowing-up map. Since Z is a large open set in the proper transform of this blowing-up under the morphism (Id, ψ) : S × W → S × S^{[k]}, it suffices to prove

Proposition 4 The restriction map

ψ^{∗} : H^{0}(S^{[k]}, L_{k}) → H^{0}(W, ψ^{∗}L_{k})

is injective.

In order to prove this proposition, we first show

Lemma 2 Denoting by π : W → ℙ(H^{0}(S, E)) the restriction of the morphism π_{k+1} : I_{k+1} → S^{[k+1]}, we have the formula

ψ^{∗}L_{k} = π^{∗}O_{ℙ(H^{0}(S,E))}(k).

Proof. By definition, ψ^{∗}L_{k} = det ψ^{∗}E_{L,k}, where the bundle E_{L,k} has for fiber H^{0}(L_{|z}) at a point z ∈ S^{[k]}. Now, if z ∈ W, the scheme z′ = ψ(z) has length k, hence the restriction map

H^{0}(S, L) → H^{0}(L_{|z′})

is surjective. On the other hand if z′′ = π(z), we have z′ ⊂ z′′ and the restriction map

H^{0}(S, L) → H^{0}(L_{|z′′})

is not surjective. Hence we have

H^{0}(S, L ⊗ I_{z′}) = H^{0}(S, L ⊗ I_{z′′}),

and the fiber of ψ^{∗}E_{L,k} at z is canonically isomorphic to H^{0}(S, L)/H^{0}(S, L ⊗ I_{π(z)}). Hence we have

ψ^{∗}L_{k} = −π^{∗}detF,

where the bundle F on ℙ(H^{0}(S, E)) is the bundle with fiber H^{0}(S, L ⊗ I_{z_{σ}}) at σ, z_{σ} = V (σ). Now recall that for each σ we have the exact sequence

0 → O_{S} →^{σ} E →^{∧σ} I_{z_{σ}}(L) → 0.

This induces the exact sequence

0 →< σ >→ H^{0}(S, E) →^{∧σ} H^{0}(S, I_{z_{σ}}(L)) → 0.

We conclude immediately from this that F fits into the exact sequence

0 → O_{ℙ(H^{0}(S,E))}(−2) → H^{0}(S, E) ⊗ O_{ℙ(H^{0}(S,E))}(−1) → F → 0.

Since rank H^{0}(S, E) = k + 2, it follows that det F = O_{ℙ(H^{0}(S,E))}(−k).

It follows from this lemma that we have a natural inclusion

S^{k}H^{0}(S, E)^{∗} ↪ H^{0}(W, ψ^{∗}L_{k}).  (3.17)

(It will be proven in the next section that this inclusion is in fact an isomorphism, but we shall not need this here.)

Our strategy to prove proposition 4 will be first to construct an isomorphism

H^{0}(S^{[k]}, L_{k}) = ∧^{k}H^{0}(S, L) ≅ S^{k}H^{0}(S, E)^{∗}  (3.18)

and then to show that composed with the inclusion (3.17), it is equal, up to a coefficient, to the pull-back map ψ^{∗}.

Construction of the isomorphism (3.18). We note first that the determinant map

det : ⋀^{2} H^{0}(S, E) → H^{0}(S, det E) = H^{0}(S, L)

does not vanish on any element of rank 2. Indeed, such an element of rank 2 is given by a subspace W of rank 2 of H^{0}(S, E), and if its determinant would vanish this would imply that W generates a rank 1 subsheaf of E with at least two sections. But since P ic S is generated by L and H^{0}(S, E(−L)) = 0 this is impossible.Hence det provides a morphism

d : G_{2} → ℙ(H^{0}(S, L)),

where G_{2} is the Grassmannian of rank two vector subspaces of H^{0}(S, E), or dually a base-point free linear system

K := H^{0}(S, L)^{∗} →^{d^{∗}} H^{0}(G_{2}, L) = ∧^{2}H^{0}(S, E)^{∗},

where L is the Plücker polarization on G_{2}. Notice that since rank K = 2k + 1, and dim G_{2} = 2k, d^{∗} has to be injective. Since K is base-point free, we have the exact Koszul complex on G_{2}

0 → ⋀^{2k+1} K ⊗ L^{−(2k+1)} → … → K ⊗ L^{−1} → O_{G_{2}} → 0.

We can now tensor this sequence with S^{k}E, where the rank 2 vector bundle E on G_{2} is dual to the tautological rank two subbundle and satisfies H^{0}(G_{2}, S^{k}E) = S^{k}H^{0}(S, E)^{∗}.

This provides the exact complex

0 → ⋀^{2k+1} K ⊗ L^{−(2k+1)} ⊗ S^{k}E → … → K ⊗ L^{−1} ⊗ S^{k}E → S^{k}E → 0. (3.19)

In this complex K^{.}, the term S^{k}E is put in degree 0. The hypercohomology ℍ^{0}(G_{2}, K^{·}) vanishes. Now we have a spectral sequence

E_{1}^{p,q} = H^{q}(G_{2}, K^{p}) ⇒ ℍ^{p+q}(G_{2}, K^{·}).

It is obvious for degree reasons that all differentials d_{r} starting from the term E_{r}^{0,0} vanish. On the other hand the terms E_{1}^{p,q} with p + q = −1 are of the form

H^{q}(G_{2}, ⋀^{q+1} K ⊗ L^{−q−1} ⊗ S^{k}E).

Using the proposition 9 proven in the appendix, we see that these terms are all 0, except for

E_{1}^{−k−1,k} = H^{k}(G_{2}, ⋀^{k+1} K ⊗ L^{−k−1} ⊗ S^{k}E),

which is equal to ⋀^{k+1} K. It follows that there is only one non zero differential which arrives in some E_{r}^{0,0}, namely

d_{k+1} : E_{k+1}^{−k−1,k} → E_{k+1}^{0,0}.

This implies that

E_{k+1}^{0,0} = E_{1}^{0,0} = H^{0}(G_{2}, S^{k}E) = S^{k}H^{0}(S, E)^{∗}

and that the differential d_{k+1} above is surjective, since the spectral sequence abuts to 0. Hence we have build a surjective map d_{k+1} from a subquotient of E_{1}^{−k−1,k} = ⋀^{k+1} K to S^{k}H^{0}(S, E)^{∗}. Since

dim ⋀^{k+1} K = dim S^{k}H^{0}(S, E)^{∗} = C_{2k+1}^{k+1}

this subquotient must in fact be equal to ⋀^{k+1} K and the map d_{k+1} has to be an isomorphism. Finally, since rank K = 2k + 1,

⋀^{k+1} K = (⋀^{k} K)^{∗} = ⋀^{k} H^{0}(S, L).

Hence we have constructed our isomorphism

d_{k+1} : ⋀^{k} H^{0}(S, L) → S^{k}H^{0}(S, E)^{∗}.

To conclude the proof of proposition 4, it remains only to show :

Proposition 5 The map d_{k+1} constructed above identifies up to a coefficient to the map

ψ^{∗} : H^{0}(S^{[k]}, L_{k}) ≅ ⋀^{k} H^{0}(S, L) → H^{0}(W, ψ^{∗}L_{k}),

which takes values in S^{k}H^{0}(S, E)^{∗} ⊂ H^{0}(W, ψ^{∗}L_{k}).

Proof. First of all it is clear that ψ^{∗} takes values in π^{∗}H^{0}(ℙ(H^{0}(S, E)), O(k)) = S^{k}H^{0}(S, E)^{∗}. Indeed, this map is the pull-back map associated to the morphism

W → Grass(k + 1, H^{0}(S, L))
z ↦ H^{0}(S, L ⊗ I_{z′}), z′ = ψ(z).

But as mentioned in the proof of lemma 2, this morphism factors through π : W → ℙ(H^{0}(S, E)). More precisely, we noticed in the proof of lemma 2 that the restriction map

ψ^{∗} : ∧^{k}H^{0}(S, L) → S^{k}H^{0}(S, E)^{∗}

corresponds to the morphism

ℙ(H^{0}(S, E)) → Grass(k + 1, H^{0}(S, L))
σ ↦ det(σ ∧ H^{0}(S, E)).

But this morphism is the composition of the morphism

β : ℙ(H^{0}(S, E)) → Grass(k + 1, ⋀^{2} H^{0}(S, E))

σ ↦ σ ∧ H^{0}(S, E).

and of the rational map

det : Grass(k + 1, ⋀^{2} H^{0}(S, E)) → Grass(k + 1, H^{0}(S, L))

induced by the determinant map det : ⋀^{2} H^{0}(S, E) → H^{0}(S, L).

Next, we note that, with the same spectral sequence argument, and replacing K = H^{0}(S, L)^{∗} ⊂ ⋀^{2} H^{0}(S, E)^{∗} by the base point free linear system K′ = ⋀^{2} H^{0}(S, E)^{∗} on G_{2}, we could have constructed more generally a surjective map

D_{k+1} : ⋀^{k+1}(⋀^{2} H^{0}(S, E)^{∗}) → S^{k}H^{0}(S, E)^{∗},

whose restriction to ⋀^{k+1} K is equal to d_{k+1}.

Hence proposition 5 will follow from the following

Lemma 3 The maps D_{k+1} and β^{∗} from ⋀^{k+1}(⋀^{2} H^{0}(S, E)^{∗}) to S^{k}H^{0}(S, E)^{∗} coincide up to a coefficient.

Proof. We could argue by Sl(k + 2)-equivariance. A more direct way to prove this is to note the following : If W ⊂ ⋀^{2} H^{0}(S, E)^{∗} is a rank k + 1 vector subspace in general position, it defines a codimension k + 1 complete intersection subvariety G_{W} of G_{2}. Consider the incidence correspondence

P →^{f} ℙ(H^{0}(S, E))
g ↓
G_{2}

Then we have a hypersurface X_{W} = f(g^{−1})(G_{W}) of ℙ(H^{0}(S, E)), which is easily proven to be of degree k. It is clear that

H^{0}(G_{2}, S^{k}E ⊗ I_{G_{W}}) = H^{0}(ℙ(H^{0}(S, E)), O_{ℙ(H^{0}(S,E))}(k)(−X_{W})).

On the other hand, from the linear system W we can construct a Koszul complex which is a resolution of I_{G_{W}}. Hence it is clear that

D_{k+1}(⋀^{k+1} W) ⊂ H^{0}(G_{2}, S^{k}E ⊗ I_{G_{W}}).

In other words, if η is a generator of ⋀^{k+1} W, D_{k+1}(η) is a defining equation of X_{W} or 0. It remains then only to prove that β^{∗}η also vanishes on X_{W}. But by definition

X_{W} = {x ∈ ℙ(H^{0}(S, E)), ∃0 ≠ γ ∈ ℙ(H^{0}(S, E)/ < x >), x ∧ γ ⊥ W }.

This means that for x ∈ X_{W}, the composed map

W ↪ ⋀^{2} H^{0}(S, E)^{∗} → (x ∧ H^{0}(S, E))^{∗}

is not an isomorphism, hence its determinant vanishes. But this determinant is equal to β^{∗}η(x).

4 Proof of the assumption H1

Recall that we have a Cartesian diagram

Z′  →^{π′}  Z
τ′ ↓  τ ↓
S × W̃ →^{(Id,π′′)} S × W

where the vertical maps τ, τ′ are blow-ups and the degree k morphism π′′ fits into the Cartesian diagram

\begin{matrix}W̃ & → & I_{k} \\ π′′ ↓ & & π_{k} ↓ \\ W & →^{ψ} & S_{curv}^{[k]} \end{matrix} .

We have the morphisms

j′ : Z′ → I_{k+1}, j : Z → I_{k+1}

such that π_{k+1} ∘ j′ = π_{k+1} ∘ j ∘ π′ and the formula

(π_{k+1} ∘ j)^{∗}L_{k+1} = τ^{∗}(L ⊠ ψ^{∗}L_{k})(−D)

where D is the exceptional divisor of τ. Pulling-back this equality to Z′, we have

(π_{k+1} ∘ j′)^{∗}L_{k+1} = τ′^{∗}(L ⊠ (ψ ∘ π′′)^{∗}L_{k})(−D′),

where D′ is the exceptional divisor of τ′. Since D′ = π′^{−1}(D) and π′ is surjective, we conclude that in order to prove H1, that is the fact that the pull-back map

π′^{∗} : H^{0}(Z, (π_{k+1} ∘ j)^{∗}L_{k+1}) → H^{0}(Z′, (π_{k+1} ∘ j′)^{∗}L_{k+1})

is surjective, it suffices to show that the pull-back map

π′′^{∗} : H^{0}(W, ψ^{∗}L_{k}) → H^{0}(W̃ , (ψ ∘ π′′)^{∗}L_{k})

is surjective.

Now recall that we have a morphism

π : W → ℙ(H^{0}(S, E))

such that (cf lemma 2)

ψ^{∗}L_{k} = π^{∗}O_{ℙ(H^{0}(S,E))}(k).

Denoting by r := π ∘ π′′ : W̃ → ℙ((H^{0}(S, E)), we shall prove the following stronger statement

Theorem 2 The pull-back map

r^{∗} : H^{0}(ℙ(H^{0}(E)), O_{ℙ(H^{0}(E))}(k)) → H^{0}(W̃ , (ψ ∘ π′′)^{∗}L_{k})  (4.20)

is surjective.

The end of this section will be devoted to the proof of this theorem, which proceeds in several steps. In what follows, we shall use the notation H^{0}(E) for H^{0}(S, E).

Notice to begin with that W̃ is a large open set in the subscheme

W′ ⊂ { \atop S × S} × ℙ(H^{0}(E)),

where { \atop S × S} is the blow-up of S × S along the diagonal, defined as

W′ := {(x, y, η, σ), σ_{|η=0}, {x}, {y} ⊂ η}.

(Here η is a subscheme of length 2 of S, and we see elements of { \atop S × S} as elements (x, y) of S × S together with a schematic structure η of length 2 on {x} ∪ {y}.)

The map r is just the restriction to W′ of the second projection. Hence we have

H^{0}(W̃ , (ψ ∘ π′′)^{∗}L_{k}) = H^{0}(W′, pr_{2}^{∗}O_{ℙ(H^{0}(E))}(k))

and the surjectivity of (4.20) is equivalent to the condition

H^{1}({ \atop S × S} × ℙ(H^{0}(E)), pr_{2}^{∗}O(k) ⊗ I_{W′}) = 0.  (4.21)

Now notice that there is a vector bundle Ẽ_{2} on { \atop S × S} such that W′ is the zero set of a section σ of Ẽ_{2} ⊠ O_{ℙ(H^{0}(E))}(1). Indeed it suffices to take for Ẽ_{2} the vector bundle with fiber H^{0}(E_{|η}) at the point (x, y, η) of { \atop S × S}. Then the section σ takes the value τ_{|η} at the point (x, y, η, τ) of { \atop S × S} × ℙ(H^{0}(E)). One checks easily that W′ is reduced of codimension 4. Hence we have a Koszul resolution of I_{W′}

0 → ⋀^{4} Ẽ_{2}^{∗} ⊠ O(−4) → … → Ẽ_{2}^{∗} ⊠ O(−1) → I_{W′} → 0.  (4.22)

Our first goal will be to compute the cohomology groups of { \atop S × S} × ℙ(H^{0}(E)) with value in ⋀^{i} Ẽ_{2}^{∗} ⊠ O(k − i). Since k ≥ 2, i ≤ 4, O(k − i) has no higher cohomology on ℙ(H^{0}(E)) = ℙ^{k+1}. Hence we have

H^{l}({ \atop S × S} ×ℙ(H^{0}(E)), ⋀^{i} Ẽ_{2}^{∗} ⊠O(k −i)) = H^{l}({ \atop S × S}, ⋀^{i} Ẽ_{2}^{∗})⊗S^{k−i}H^{0}(S, E)^{∗}.

We have now the following proposition

Proposition 6 1. H^{2}({ \atop S × S}, Ẽ_{2}^{∗}) = pr_{1}^{∗}H^{2}(S, E^{∗}) ⊕ pr_{2}^{∗}H^{2}(S, E^{∗}) and

H^{1}({ \atop S × S}, Ẽ_{2}^{∗}) = 0.

2. H^{2}({ \atop S × S}, ⋀^{2} Ẽ_{2}^{∗}) = pr_{1}^{∗}H^{2}(S, −L) ⊕ pr_{2}^{∗}H^{2}(S, −L).

3. H^{4}({ \atop S × S}, ⋀^{4} Ẽ_{2}^{∗}) is dual to Ker (H^{0}(S, L) ⊗ H^{0}(S, L) → H^{0}(S, 2L)).

4. H^{3}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗}) = 0 and H^{4}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗}) admits as a quotient

H^{4}({ \atop S × S}, τ^{∗}(E^{∗} ⊠ (−L))(2Δ)) ⊕ H^{4}({ \atop S × S}, τ^{∗}((−L) ⊠ E^{∗})(2Δ)),

which is dual to the direct sum of two copies of

Ker (H^{0}(S, E) ⊗ H^{0}(S, L) → H^{0}(S, E ⊗ L)).

(Here Δ ⊂ { \atop S × S} is the exceptional divisor.)

Proof.

1. The bundle Ẽ_{2} fits into the exact sequence

0 → Ẽ_{2} → τ^{∗}(pr_{1}^{∗}E ⊕ pr_{2}^{∗}E) → τ′^{∗}E → 0  (4.23)

where τ : { \atop S × S} → S×S is the blowing-down map, and where τ′ : Δ → Diag S is its restriction to the exceptional divisor.

Dualizing, we get the following exact sequence

0 → τ^{∗}(pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) → Ẽ_{2}^{∗} → τ′^{∗}E^{∗} ⊗ O_{Δ}(Δ) → 0.  (4.24)

Now R^{0}τ′_{∗}O_{Δ}(Δ) = R^{1}τ′_{∗}O_{Δ}(Δ) = 0 hence the sheaf on the right has no cohomology. It follows that

H^{i}({ \atop S × S}, Ẽ_{2}^{∗}) = H^{i}({ \atop S × S}, τ^{∗}(pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E))

= H^{i}(S × S, pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E).

Since E^{∗} has no odd dimensional cohomology, nor O_{S}, it follows from Künneth formula that the same is true for pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E on S × S. Finally we have

H^{2}(S × S, pr_{1}^{∗}E^{∗}) = H^{2}(S, E^{∗})

since H^{0}(S, E^{∗}) = 0. This proves 1.

2. From (4.24) we deduce that ⋀^{2} Ẽ_{2}^{∗} has a filtration whose successive terms are

⋀^{2} τ^{∗}(pr_{1}^{∗}E^{∗}⊕pr_{2}^{∗}E^{∗}), τ^{∗}(pr_{1}^{∗}E^{∗}⊕pr_{2}^{∗}E^{∗})⊗τ′^{∗}E^{∗}⊗O_{Δ}(Δ), ⋀^{2} τ′^{∗}E^{∗}⊗O_{Δ}(2Δ).

The sheaf (pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ τ′^{∗}E^{∗} ⊗ O_{Δ}(Δ) has no cohomology, since O_{Δ}(Δ) has no cohomology along the fibers of τ′. Hence we have an exact sequence

H^{1}(Δ, ⋀^{2} τ′^{∗}E^{∗}⊗O_{Δ}(2Δ)) → H^{2}({ \atop S × S}, τ^{∗} ⋀^{2}(pr_{1}^{∗}E^{∗}⊕pr_{2}^{∗}E^{∗})) → H^{2}({ \atop S × S}, ⋀^{2} Ẽ_{2}^{∗})

→ H^{2}(Δ, ⋀^{2} τ′^{∗}E^{∗} ⊗ O_{Δ}(2Δ)) …

But since

R^{1}τ′_{∗}(2Δ_{|Δ}) = O_{S}, R^{0}τ′_{∗}(2Δ_{|Δ}) = 0,

the term on the left is equal to H^{0}(S, ⋀^{2} E^{∗}) = 0 and the term on the right is equal to H^{1}(S, ⋀^{2} E^{∗}) = 0. Hence we have

H^{2}({ \atop S × S}, ⋀^{2} Ẽ_{2}^{∗}) = H^{2}({ \atop S × S}, τ^{∗} ⋀^{2}(pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}))

= H^{2}(S × S, ⋀^{2}(pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}))

Finally

⋀^{2}(pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) = pr_{1}^{∗} ⋀^{2} E^{∗} ⊕ E^{∗} ⊠ E^{∗} ⊕ pr_{2}^{∗} ⋀^{2} E^{∗}.

The central term has no cohomology in degree 2 by Künneth formula, because H^{1}(S, E^{∗}) = H^{0}(S, E^{∗}) = 0, and we have

H^{2}(S × S, pr_{1}^{∗} ⋀^{2} E^{∗}) = H^{2}(S, ⋀^{2} E^{∗}) = H^{2}(S, −L),

where the first equality follows from Künneth formula and H^{0}(S, ⋀^{2} E^{∗}) = 0. This proves 2.

3. We have det Ẽ_{2}^{∗} = τ^{∗}((−L) ⊠ (−L))(2Δ) by the exact sequence (4.23). Hence

⋀^{3} Ẽ_{2}^{∗} = Ẽ_{2} ⊗ det Ẽ_{2}^{∗} = Ẽ_{2} ⊗ τ^{∗}((−L) ⊠ (−L))(2Δ).  (4.25)

The exact sequence (4.23) gives now the long exact sequence

H^{2}(Δ, τ′^{∗}(E(−2L))(2Δ_{|Δ})) → H^{3}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗})

→ H^{3}({ \atop S × S}, τ^{∗}((pr_{1}^{∗}E ⊕ pr_{2}^{∗}E) ⊗ ((−L) ⊠ (−L)))(2Δ)).

Since R^{0}τ′_{∗}O_{Δ}(2Δ) = 0, R^{1}τ′_{∗}O_{Δ}(2Δ) = O_{S}, the left hand side is equal to H^{1}(S, E(−2L)), which is easily seen to be 0.

Next we have K{ \atop S×S} = O{ \atop S×S}(Δ), hence

H^{3}({ \atop S × S}, τ^{∗}(pr_{1}^{∗}E ⊕ pr_{2}^{∗}E) ⊗ ((−L) ⊠ (−L)))(2Δ))

is dual to

H^{1}({ \atop S × S}, τ^{∗}((pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L))(−Δ)).  (4.26)

Writing the exact sequence

0 → τ^{∗}((pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L))(−Δ) → τ^{∗}((pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L))

→ τ^{∗}((pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L))_{|Δ} → 0

where

τ^{∗}((pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L))_{|Δ} ≅ τ′^{∗}((E^{∗} ⊕ E^{∗})(2L)),

and using the isomorphism

E^{∗} ⊗ L ≅ E,

we see that the vanishing of the cohomology group (4.26) follows from the fact that the multiplication map

H^{0}(S, E) ⊗ H^{0}(S, L) → H^{0}(S, E ⊗ L)

is surjective, which is easily checked, and from the vanishing

H^{1}(S × S, (pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ L ⊠ L) = 0.

Finally the equality (4.25) and the exact sequence (4.23) also show that

H^{4}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗}) admits

H^{4}({ \atop S × S}, τ^{∗}((pr_{1}^{∗}E ⊕ pr_{2}^{∗}E) ⊗ ((−L) ⊠ (−L)))(2Δ))

as a quotient. By Serre’s duality this space is dual to

H^{0}({ \atop S × S}, τ^{∗}((pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L))(−Δ)).  (4.27)

But this is equal to

H^{0}(S × S, (pr_{1}^{∗}E^{∗} ⊕ pr_{2}^{∗}E^{∗}) ⊗ (L ⊠ L) ⊗ I_{Diag}).

We use then the fact that

pr_{1}^{∗}E^{∗} ⊗ (L ⊠ L) = E ⊠ L

to conclude that (4.27) is equal to the sum of two copies of

Ker (H^{0}(S, E) ⊗ H^{0}(S, L) → H^{0}(S, E ⊗ L)).

4. We already noticed that

⋀^{4} Ẽ_{2}^{∗} = det Ẽ_{2}^{∗} = τ^{∗}((−L) ⊠ (−L))(2Δ).

It follows then from Serre’s duality and K{ \atop S×S} = O{ \atop S×S}(Δ) that H^{4}({ \atop S × S}, ⋀^{4} Ẽ_{2}^{∗}) is dual to

H^{0}({ \atop S × S}, τ^{∗}(L ⊠ L)(−Δ)) = Ker (H^{0}(S, L) ⊗ H^{0}(S, L) → H^{0}(S, 2L))).

Hence 4 is proven.

Coming back to the Koszul resolution of I_{W′} ⊗ pr_{2}^{∗}O(k) induced by (4.22), we see that in order to prove the vanishing (4.21), it suffices to show :

a) H^{1}({ \atop S × S} × ℙ(H^{0}(E)), Ẽ_{2}^{∗} ⊠ O(k − 1)) = 0.

b) The interior product with σ

int(σ) : H^{2}({ \atop S × S} × ℙ(H^{0}(E)), ⋀^{2} Ẽ_{2}^{∗} ⊠ O(k − 2))

→ H^{2}({ \atop S × S} × ℙ(H^{0}(E)), Ẽ_{2}^{∗} ⊠ O(k − 1))

is injective.

c) H^{3}({ \atop S × S} × ℙ(H^{0}(E)), ⋀^{3} Ẽ_{2}^{∗} ⊠ O(k − 3)) = 0.

d) The interior product with σ

int(σ) : H^{4}({ \atop S × S} × ℙ(H^{0}(E)), ⋀^{4} Ẽ_{2}^{∗} ⊠ O(k − 4))

→ H^{4}({ \atop S × S} × ℙ(H^{0}(E)), ⋀^{3} Ẽ_{2}^{∗} ⊠ O(k − 3))

is injective.

The conditions a) and c) have been established in proposition 6. We now dualize property b) as follows : by proposition 6, 1 and 2, we have

H^{2}({ \atop S × S} × ℙ(H^{0}(E)), ⋀^{2} Ẽ_{2}^{∗} ⊠ O(k − 2))

= (pr_{1}^{∗}H^{2}(S, −L) ⊕ pr_{2}^{∗}H^{2}(S, −L)) ⊗ S^{k−2}H^{0}(S, E)^{∗},

H^{2}({ \atop S × S} × ℙ(H^{0}(E)), Ẽ_{2}^{∗} ⊠ O(k − 1))

= (pr_{1}^{∗}H^{2}(S, E^{∗}) ⊕ pr_{2}^{∗}H^{2}(S, E^{∗})) ⊗ S^{k−1}H^{0}(S, E)^{∗}.

Dualizing, we get

H^{2}({ \atop S × S} × ℙ(H^{0}(E)), ⋀^{2} Ẽ_{2}^{∗} ⊠ O(k − 2))^{∗}

= (H^{0}(S, L) ⊕ H^{0}(S, L)) ⊗ S^{k−2}H^{0}(S, E),

H^{2}({ \atop S × S} × ℙ(H^{0}(E)), Ẽ_{2}^{∗} ⊠ O(k − 1))^{∗}

= (H^{0}(S, E) ⊕ H^{0}(S, E)) ⊗ S^{k−1}H^{0}(S, E).

It is then immediate to check that the transpose of the map int(σ) is the map ∧σ, so that b) translates into the condition that

∧σ : (H^{0}(S, E)⊕H^{0}(S, E))⊗S^{k−1}H^{0}(S, E) → (H^{0}(S, L)⊕H^{0}(S, L))⊗S^{k−2}H^{0}(S, E)

is surjective.

Now retracing through the isomorphisms given by proposition 6, one checks that the map ∧σ is up to sign equal to the direct sum of two copies of the composed map

μ : H^{0}(S, E) ⊗ S^{k−1}H^{0}(S, E) → H^{0}(S, E) ⊗ H^{0}(S, E) ⊗ S^{k−2}H^{0}(S, E)

→^{det⊗id} H^{0}(S, L) ⊗ S^{k−2}H^{0}(S, E).

Similarly statement d) dualizes as follows : by proposition 6, the space

H^{4}({ \atop S × S}, ⋀^{4} Ẽ_{2}^{∗} ⊠ O(k − 4)) ≅ H^{4}({ \atop S × S}, ⋀^{4} Ẽ_{2}^{∗}) ⊗ S^{k−4}H^{0}(S, E)^{∗}

is dual to

Ker (H^{0}(S, L) ⊗ H^{0}(S, L) → H^{0}(S, 2L)) ⊗ S^{k−4}H^{0}(S, E).

Next, we know by proposition 6, 4, that

H^{4}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗} ⊠ O(k − 3)) ≅ H^{4}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗}) ⊗ S^{k−3}H^{0}(S, E)^{∗}

admits a quotient which is dual to the direct sum of two copies of

Ker (H^{0}(S, E) ⊗ H^{0}(S, L) → H^{0}(S, E ⊗ L)) ⊗ S^{k−3}H^{0}(S, E)).

Denoting by Q_{E,L} := Ker (H^{0}(S, E) ⊗ H^{0}(S, L) → H^{0}(S, E ⊗ L)), Q_{L,E} := Ker (H^{0}(S, L) ⊗ H^{0}(S, E) → H^{0}(S, E ⊗ L)) and Q_{L,L} = Ker (H^{0}(S, L) ⊗ H^{0}(S, L) → H^{0}(S, 2L)), we have an inclusion

(Q_{L,E} ⊕ Q_{E,L}) ⊗ S^{k−3}H^{0}(S, E) ⊂ H^{4}({ \atop S × S}, ⋀^{3} Ẽ_{2}^{∗} ⊠ O(k − 3))^{∗}

and to prove d) it suffices to show that the map dual to int(σ) restricts on this subspace to a surjection

∧σ : (Q_{L,E} ⊕ Q_{E,L}) ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L} ⊗ S^{k−4}H^{0}(S, E).

But retracing through the isomorphisms of proposition 6 and recalling the definition of σ, one checks easily that the first component

∧σ_{1} : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L} ⊗ S^{k−4}H^{0}(S, E)

of the map above is the following composite

μ′ : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) ⊂ H^{0}(S, L) ⊗ H^{0}(S, E) ⊗ S^{k−3}H^{0}(S, E) →

H^{0}(S, L) ⊗ H^{0}(S, E) ⊗ H^{0}(S, E) ⊗ S^{k−4}H^{0}(S, E)

→^{id⊗det⊗id} H^{0}(S, L) ⊗ H^{0}(S, L) ⊗ S^{k−4}H^{0}(S, E),

which takes obviously value in Q_{L,L} ⊗ S^{k−4}H^{0}(S, E), while the second component is equal to the first composed with the permutation exchanging factors on both sides.

To conclude then that

∧σ : (Q_{L,E} ⊕ Q_{E,L}) ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L} ⊗ S^{k−4}H^{0}(S, E)

is surjective, it suffices to show that

μ′_{−} : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L}^{−} ⊗ S^{k−4}H^{0}(S, E)

and

μ′_{+} : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L}^{+} ⊗ S^{k−4}H^{0}(S, E)

are surjective, where Q_{L,L}^{+}, (resp. Q_{L,L}^{−}) are the symmetric, resp. antisymmetric part of Q_{L,L} and μ′_{+} (resp. μ′_{−}) are the composition of μ′ with the projections on the symmetric (resp. antisymmetric) part of Q_{L,L}.

In conclusion, theorem 2 will be a consequence of the following propositions Proposition 7 The composed map

μ : H^{0}(S, E) ⊗ S^{k−1}H^{0}(S, E) → H^{0}(S, E) ⊗ H^{0}(S, E) ⊗ S^{k−2}H^{0}(S, E)

→^{det} H^{0}(S, L) ⊗ S^{k−2}H^{0}(S, E)

is surjective.

Proposition 8 a) The map

μ′_{−} : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L}^{−} ⊗ S^{k−4}H^{0}(S, E)

defined above is surjective.

b) The map

μ′_{+} : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L}^{+} ⊗ S^{k−4}H^{0}(S, E)

defined above is surjective.

Proof of proposition 7. Let α, β ∈ H^{0}(S, E) and γ ∈ H^{0}(S, L) such that

γ = det(α ∧ β).

Then we observe first that if D ⊂ H^{0}(S, E) is the rank 2 vector subspace generated by α and β, we have

γ ⊗ S^{k−2}D ⊂ Im μ

since the composite

D ⊗ S^{k−1}D → D ⊗ D ⊗ S^{k−2}D → ⋀^{2} D ⊗ S^{k−2}D

is surjective.

Recall now that the map det determines a morphism

d : G_{2} → ℙH^{0}(S, L)

which is surjective and finite since both spaces are of the same dimension 2k. The fiber d^{−1}(γ) is then a finite subscheme Z_{γ} ⊂ G_{2} which is the complete intersection of a space W of hyperplane sections of the Grassmannian G_{2}.

Now by the above observation, and since d is surjective, it suffices to show that the subspaces S^{k−2}D for D ∈ Z_{γ} generate S^{k−2}H^{0}(S, E). If we dualize, this is equivalent to say that the dual map

S^{k−2}H^{0}(S, E)^{∗} → ⊕_{D∈Z_{γ}}S^{k−2}D^{∗}

is injective. But this map identifies to the restriction

H^{0}(G_{2}, S^{k−2}E) → H^{0}(Z_{γ}, S^{k−2}E_{|Z_{γ}}),

at least for a reduced Z_{γ}, which will be the case for a generic γ.

Hence it suffices to show that

H^{0}(G_{2}, S^{k−2}E ⊗ I_{Z_{γ}}) = 0.  (4.28)

Now we use the Koszul resolution

0 → ⋀^{2k} W ⊗ L^{−2k} → … → W ⊗ L^{−1} → I_{Z_{γ}} → 0

The vanishing (4.28) will then follow from the vanishing

H^{i}(G_{2}, S^{k−2}E ⊗ L^{−i−1}), i = 0, 2k − 1

which is proved in proposition 9 of the appendix.

Proof of proposition 8, a). Notice first that the natural composed map

⋀^{3} H^{0}(S, E) → ⋀^{2} H^{0}(S, E) ⊗ H^{0}(S, E)

→^{det⊗id} H^{0}(S, L) ⊗ H^{0}(S, E)

has its image contained in Q_{L,E}. Hence it suffices to show that the following composite

μ′′ : ⋀^{3} H^{0}(S, E) ⊗ S^{k−3}H^{0}(S, E) → ⋀^{2} H^{0}(S, E) ⊗ H^{0}(S, E) ⊗ S^{k−3}H^{0}(S, E)

→^{det⊗μ} H^{0}(S, L) ⊗ H^{0}(S, L) ⊗ S^{k−4}H^{0}(S, E) → ⋀^{2} H^{0}(S, L) ⊗ S^{k−4}H^{0}(S, E)

is surjective.

Now note that for α_{1}, α_{2}, α_{3} ∈ H^{0}(S, E)

μ′′(α_{1} ∧ α_{2} ∧ α_{3} ⊗ α_{3}^{k−3}) = 2(k − 3)det(α_{2} ∧ α_{3}) ∧ det(α_{1} ∧ α_{3}) ⊗ α_{3}^{k−4}.(4.29)

Fix now γ ∈ H^{0}(S, L) and consider the set of couples (α_{1}, α_{3}) such that

det(α_{1} ∧ α_{3}) = γ.

For any α_{2} and any such (α_{1}, α_{3}), we have

μ′′(α_{1} ∧ α_{2} ∧ α_{3} ⊗ α_{3}^{k−3}) = 2(k − 3)det(α_{2} ∧ α_{3}) ∧ γ ⊗ α_{3}^{k−4}.

Note that the vector α_{3} for such pairs takes arbitrary value in some of the lines D ∈ Z_{γ}, where the notations are as in the previous proposition.

Now we have the map

μ′′′ : H^{0}(S, E) ⊗ S^{k−3}H^{0}(S, E) → H^{0}(S, L) ⊗ S^{k−4}H^{0}(S, E)

analogous to μ and the formula above shows that

μ′′(α_{1} ∧ α_{2} ∧ α_{3} ⊗ α_{3}^{k−3}) = 2γ ∧ μ′′′(α_{2} ⊗ α_{3}^{k−3}).

With the same proof as in the previous proposition, one shows now that the S^{k−3}D, D ∈ Z_{γ} generate S^{k−3}H^{0}(S, E) and that μ′′′ is surjective. Hence the α_{2}⊗α_{3}^{k−3}, α_{3} ∈ D, D ∈ Z_{γ} generate H^{0}(S, E)⊗S^{k−3}H^{0}(S, E) and the μ′′′(α_{2}⊗ α_{3}^{k−3}), α_{3} ∈ D, D ∈ Z_{γ} generate by the surjectivity of μ′′′ the space H^{0}(S, L)⊗ S^{k−4}H^{0}(S, E). Hence Im μ′′ contains γ ∧ H^{0}(S, L) ⊗ S^{k−4}H^{0}(S, E), and since γ was generic, we conclude that μ′′ is surjective.

Proof of proposition 8, b). We want to prove that

μ′_{+} : Q_{L,E} ⊗ S^{k−3}H^{0}(S, E) → Q_{L,L}^{+} ⊗ S^{k−4}H^{0}(S, E)

is surjective. Denote similarly, for C a generic member of | L |,

Q_{K_{C},E} := Ker (H^{0}(C, K_{C}) ⊗ H^{0}(C, E_{|C}) → H^{0}(C, E ⊗ K_{C})),
Q_{K_{C},K_{C}}^{+} := Ker (S^{2}H^{0}(C, K_{C}) → H^{0}(C, K_{C}^{⊗2})).

Then we can define

μ′_{+,C} : Q_{K_{C},E} ⊗ S^{k−3}H^{0}(C, E_{|C}) → Q_{K_{C},K_{C}}^{+} ⊗ S^{k−4}H^{0}(C, E_{|C})

as the composite

Q_{K_{C},E} ⊗ S^{k−3}H^{0}(C, E_{|C}) ⊂ H^{0}(C, K_{C}) ⊗ H^{0}(C, E_{|C}) ⊗ S^{k−3}H^{0}(C, E_{|C})

→ H^{0}(C, K_{C}) ⊗ H^{0}(C, E_{|C}) ⊗ H^{0}(C, E_{|C}) ⊗ S^{k−4}H^{0}(C, E_{|C}) →^{id⊗det⊗id}

H^{0}(C, K_{C})⊗H^{0}(C, K_{C})⊗S^{k−4}H^{0}(C, E_{|C}) → S^{2}H^{0}(C, K_{C})⊗S^{k−4}H^{0}(C, E_{|C}).

Now the restriction map H^{0}(S, E) → H^{0}(C, E_{|C}) is an isomorphism, and the restriction map H^{0}(S, L) → H^{0}(C, K_{C}) is surjective with kernel σ, the defining equation of C. Hence the restrictions induce a surjection

Q_{L,E} → Q_{K_{C},E}

and an isomorphism

Q_{L,L}^{+} ≅ Q_{K_{C},K_{C}}^{+},

and it suffices to show that μ′_{+,C} is surjective. A fortiori it suffices to show that the composite

μ′_{C} : Q_{K_{C},E} ⊗ S^{k−3}H^{0}(C, E_{|C}) ⊂ H^{0}(C, K_{C}) ⊗ H^{0}(C, E_{|C}) ⊗ S^{k−3}H^{0}(C, E_{|C})

→ H^{0}(C, K_{C}) ⊗ H^{0}(C, E_{|C}) ⊗ H^{0}(C, E_{|C}) ⊗ S^{k−4}H^{0}(C, E_{|C})

→^{id⊗det⊗id} H^{0}(C, K_{C}) ⊗ H^{0}(C, K_{C}) ⊗ S^{k−4}H^{0}(C, E_{|C})

which takes value in Q_{K_{C},K_{C}} := Ker (H^{0}(C, K_{C})^{⊗2} → H^{0}(C, K_{C}^{⊗2})), is surjective on this last space.

Let us now consider the following diagram of exact sequences

0 → Q_{K_{C},E} ⊗ S^{k−3}H^{0}(C, E_{|C}) → H^{0}(C, K_{C}) ⊗ H^{0}(C, E_{|C}) ⊗ S^{k−3}H^{0}(C, E_{|C})
μ′_{C} ↓  id ⊗ μ_{C} ↓
0 → Q_{K_{C},K_{C}} ⊗ S^{k−4}H^{0}(C, E_{|C}) → H^{0}(C, K_{C}) ⊗ H^{0}(C, K_{C}) ⊗ S^{k−4}H^{0}(C, E_{|C})

\begin{aligned}→ & H^{0}(C, E ⊗ K_{C}) ⊗ S^{k−3}H^{0}(C, E_{|C}) & → & 0 \\ & μ_{C,K_{C}} ↓ \\ → & H^{0}(C, K_{C}^{⊗2}) ⊗ S^{k−4}H^{0}(C, E_{|C}) & → & 0 \end{aligned},

where the vertical maps μ_{C} and μ_{C,K_{C}} are defined in a way similar to μ e.g μ_{C} is the composite

H^{0}(C, E_{|C}) ⊗ S^{k−3}H^{0}(C, E_{|C}) ⊂ H^{0}(C, E_{|C}) ⊗ H^{0}(C, E_{|C}) ⊗ S^{k−4}H^{0}(C, E_{|C})

→^{det⊗id} H^{0}(C, K_{C}) ⊗ S^{k−4}H^{0}(C, E_{|C}),

and μ_{C,K_{C}} is defined similarly with a twist by K_{C}.

One checks easily the surjectivity of the multiplication maps on the left.

The proof of proposition 7 shows as well that μ_{C} is surjective, as is μ_{C,K_{C}} by the commutativity of the diagram above. Hence the surjectivity of μ′_{C} will follow by diagram chasing from the surjectivity of the induced multiplication map

H^{0}(C, K_{C}) ⊗ Ker μ_{C} → Ker μ_{C,K_{C}}.  (4.30)

In what follows we will use again the notation H^{0}(E) for H^{0}(S, E) = H^{0}(C, E_{|C}). Define the vector bundle Q on C as the kernel of the surjective composite morphism of vector bundles

S^{k−3}H^{0}(E) ⊗ E ⊂ S^{k−4}H^{0}(E) ⊗ H^{0}(E) ⊗ E →^{id⊗det} S^{k−4}H^{0}(E) ⊗ K_{C}.

Then we clearly have

Ker μ_{C} = H^{0}(C, Q), Ker μ_{C,K_{C}} = H^{0}(C, Q ⊗ K_{C})

so that the surjectivity of the map (4.30) is equivalent to the surjectivity of the multiplication map

H^{0}(C, Q) ⊗ H^{0}(C, K_{C}) → H^{0}(C, Q ⊗ K_{C}).  (4.31)

Now we proceed as follows : let σ ∈ H^{0}(S, L) be the defining equation for C. Recall the finite reduced subscheme Z_{σ} = d^{−1}(σ) ⊂ G_{2} made of the rank 2 vector subspaces D of H^{0}(S, E) such that det D = σ. For each such D, there is a line subbundle L_{D} of E on C, of degree k + 1 with two sections without common zeroes (see section 2). The space D identifies naturally to H^{0}(C, L_{D}).

Clearly the image of the inclusion

S^{k−3}H^{0}(C, L_{D}) ⊗ L_{D} ⊂ S^{k−3}H^{0}(E) ⊗ E

is contained in Q.

Let now

N := ⊕_{D∈Z_{σ}}S^{k−3}D ⊗ L_{D}.

Then by the observation above we have a morphism

α : N → Q.

The surjectivity of 4.31 will follow from the following three lemmas :

Lemma 4 The morphism α is surjective.

Denoting M := Ker α we also prove

Lemma 5 The vector bundle M is generated by its sections.

Lemma 6 The space H^{0}(C, M) is generated by the subspaces H^{0}(C, M(−x)), x ∈ C.

We explain first how these three lemmas imply our result. Using the exact sequence

0 → M → N → Q → 0

given by lemma 4, we see that the map (4.31) will be surjective if the multiplication map

H^{0}(C, N) ⊗ H^{0}(C, K_{C}) → H^{0}(C, N ⊗ K_{C})

is surjective, and H^{1}(C, M ⊗ K_{C}) = 0.

The first condition is easy to check. Indeed N is a direct sum of line bundles L_{D} corresponding to g_{k+1}^{1}’s on C, and the result is easy to prove for each of them. As for the second condition, it is equivalent to H^{0}(C, M^{∗}) = 0 by Serre’s duality. But since M is generated by sections by lemma 5, we have an inclusion

H^{0}(C, M^{∗}) ⊂ H^{0}(C, M)^{∗}.

The image of this inclusion obviously vanishes on each subspace H^{0}(C, M(−x)), hence it must be 0 since we know by lemma 6 that these subspaces generate H^{0}(C, M).

To conclude the proof of 8,b) it remains only to prove these three lemmas.

Proof of lemma 4. First of all we note that the bundle Q is generated by its sections, since there is a natural surjection

S^{k−2}H^{0}(E) ⊗ O_{C} → Q → 0.

Hence it suffices to show that the map

H^{0}(C, N) → H^{0}(C, Q)

is surjective.

But by definition

H^{0}(C, Q) = Ker (H^{0}(E) ⊗ S^{k−3}H^{0}(E) →^{μ_{C}} H^{0}(C, K_{C}) ⊗ S^{k−4}H^{0}(E))

and

H^{0}(C, N) = ⊕_{D∈Z_{σ}}D ⊗ S^{k−3}D.

Hence we need to show that the sequence

⊕_{D∈Z_{σ}}D ⊗ S^{k−3}D → H^{0}(E) ⊗ S^{k−3}H^{0}(E) →^{μ_{C}} H^{0}(C, K_{C}) ⊗ S^{k−4}H^{0}(E)

is exact at the middle. Again this will follow from a cohomological computation on the Grassmannian G_{2}. Indeed, the notations being as in the proof of Propositions 4 and 7, the sequence above dualizes as

I_{Z_{σ}}(L) ⊗ S^{k−4}H^{0}(G_{2}, E) → H^{0}(G_{2}, E) ⊗ S^{k−3}H^{0}(G_{2}, E)
→ H^{0}(E ⊗ S^{k−3}E_{|Z_{σ}}),  (4.32)

where the map

I_{Z_{σ}}(L) ⊗ S^{k−4}H^{0}(G_{2}, E) → H^{0}(G_{2}, E) ⊗ S^{k−3}H^{0}(G_{2}, E)

is composed of the inclusion

I_{Z_{σ}}(L) ⊗ S^{k−4}H^{0}(G_{2}, E) ⊂ H^{0}(G_{2}, L) ⊗ S^{k−4}H^{0}(G_{2}, E)
≅ ∧^{2}H^{0}(G_{2}, E) ⊗ S^{k−4}H^{0}(G_{2}, E)

and of the (Koszul) map

∧^{2}H^{0}(G_{2}, E) ⊗ S^{k−4}H^{0}(G_{2}, E) → H^{0}(G_{2}, E) ⊗ S^{k−3}H^{0}(G_{2}, E).

One checks easily that H^{0}(G_{2}, E)⊗S^{k−3}H^{0}(G_{2}, E) ≅ H^{0}(G_{2}, E⊗S^{k−3}E). Hence the kernel in the middle identifies to H^{0}(G_{2}, E ⊗ S^{k−3}E ⊗ I_{Z_{σ}}). Furthermore S^{k−4}H^{0}(G_{2}, E) ≅ H^{0}(G_{2}, S^{k−4}E) identifies to H^{0}(G_{2}, E ⊗ S^{k−3}E ⊗ L^{−1}) via the (Koszul) inclusion

S^{k−4}E ⊗ L = S^{k−4}E ⊗ ⋀^{2} E ⊂ E ⊗ S^{k−3}E.

Hence the exactness at the middle of the sequence 4.32 will follow from the surjectivity of the multiplication map

H^{0}(G_{2}, E ⊗ S^{k−3}E ⊗ L^{−1}) ⊗ I_{Z_{σ}}(L) → H^{0}(G_{2}, E ⊗ S^{k−3}E ⊗ I_{Z_{σ}}). (4.33)

Now let W := I_{Z_{σ}}(L). The Koszul resolution of I_{Z_{σ}}

0 → ⋀^{2k} W ⊗ L^{−2k} → … → W ⊗ L^{−1} → I_{Z_{σ}} → 0

twisted by E ⊗ S^{k−3}E shows that the surjectivity of (4.33) will hold if we know that

H^{i}(G_{2}, E ⊗ S^{k−3}E ⊗ L^{−i−1}) = 0, 1 ≤ i < 2k.

Since we have the exact sequence

0 → S^{k−4}E ⊗ L → E ⊗ S^{k−3}E → S^{k−2}E → 0,

it suffices to know that

H^{i}(G_{2}, S^{k−4}E ⊗ L^{−i}) = 0, 1 ≤ i < 2k,

and

H^{i}(G_{2}, S^{k−2}E ⊗ L^{−i−1}) = 0, 1 ≤ i < 2k.

This is proved in Proposition 9.

Proof of lemma 5. The bundles N and Q are generated by global sections. To prove that M is generated by global sections, it suffices to prove that for any x ∈ C, the restriction map H^{0}(C, N (−x)) → H^{0}(C, Q(−x)) is surjective. For each g_{k+1}^{1} L_{D} on C, denote by σ_{D,x} ∈ H^{0}(C, L_{D}) ≅ D a generator for H^{0}(C, L_{D}(−x)). We need to show the exactness of the sequence

⊕_{D∈Z_{σ}}σ_{D,x} ⊗ S^{k−3}D → H^{0}(C, E(−x)) ⊗ S^{k−3}H^{0}(E)
→^{μ_{C}} H^{0}(C, K_{C}(−x)) ⊗ S^{k−4}H^{0}(E).  (4.34)

(Indeed, by definition, ⊕_{D∈Z_{σ}}σ_{D,x} ⊗ S^{k−3}D identifies to H^{0}(C, N (−x)) and

Ker (H^{0}(C, E(−x)) ⊗ S^{k−3}H^{0}(E) →^{μ_{C}} H^{0}(C, K_{C}(−x)) ⊗ S^{k−4}H^{0}(E))

identifies to H^{0}(C, Q(−x)).)

Denote by K_{x} ⊂ H^{0}(E) the subspace H^{0}(C, E(−x)). Note that via the identification H^{0}(C, L_{D}) = D ⊂ H^{0}(E), σ_{D,x} becomes a generator of the 1dimensional vector space D ∩ K_{x}. Furthermore, K_{x} being of codimension 2 in H^{0}(E) determines a section τ_{x} ∈ ⋀^{2} H^{0}(E)^{∗} up to a coefficient. Clearly τ_{x} belongs to H^{0}(C, K_{C})^{∗} ⊂ ⋀^{2} H^{0}(E)^{∗} and identifies also to the linear form on H^{0}(C, K_{C}) defining H^{0}(C, K_{C}(−x)). Let G_{x} ⊂ G_{2} be the hyperplane section defined by τ_{x}. The scheme Z_{σ} is a complete intersection of hyperplane sections of G_{x}. The variety G_{x} admits a desingularization P_{x} →^{g} G_{x} defined as

P_{x} = {(u, Δ) ∈ ℙ(K_{x}) × G_{2}, u ∈ Δ ∩ K_{x}}.

Note that if

P →^{g} G_{2}
f ↓
ℙ(H^{0}(E))

is the incidence variety, P_{x} can also be defined as f^{−1}(ℙ(K_{x})) ⊂ P.

Each line D parametrized by Z_{σ} meets K_{x} along a one dimensional vector space, because L_{D} has no base-point, so that H^{0}(L_{D}(−x)) ≅ D ∩ K_{x} is 1dimensional. It follows that the scheme Z_{σ} can also be seen as the complete intersection in P_{x} of hypersurfaces in | g^{∗}L |.

We now dualize the sequence (4.34). The space H^{0}(C, K_{C}(−x)) admits for dual the space W ⊂ H^{0}(P_{x}, g^{∗}L) defining Z_{σ} ⊂ P_{x}. The vector space < σ_{D,x} >^{∗} identifies clearly to the fiber of the line bundle H_{x} := f^{∗}O_{ℙ(K_{x})}(1) at the point D ∈ Z_{σ}. Hence our sequence dualizes as

W ⊗ S^{k−4}H^{0}(E)^{∗} → f^{∗}H^{0}(ℙ(K_{x}), O(1)) ⊗ H^{0}(P_{x}, g^{∗}S^{k−3}E)
→ H^{0}(S^{k−3}E ⊗ H_{x}| Z_{σ}).  (4.35)

The second space in this sequence is easily shown to identify to H^{0}(P_{x}, g^{∗}S^{k−3}E⊗ H_{x}), so that the kernel at the middle is equal to

H^{0}(P_{x}, g^{∗}S^{k−3}E ⊗ H_{x} ⊗ I_{Z_{σ}}).

The first map in (4.35) is induced by the isomorphism

S^{k−4}H^{0}(E)^{∗} ≅ H^{0}(P_{x}, g^{∗}S^{k−4}E),

the multiplication

W ⊗ H^{0}(P_{x}, g^{∗}S^{k−4}E) → H^{0}(P_{x}, g^{∗}(S^{k−4}E ⊗ L) ⊗ I_{Z_{σ}})

and the composed bundle map

g^{∗}S^{k−4}E ⊗ L → g^{∗}S^{k−3}E ⊗ E → g^{∗}S^{k−3}E ⊗ H_{x},

where the last map is induced by the natural surjective map g^{∗}E → H_{x}. The exactness of (4.35) will then follow from the surjectivity of

W ⊗ H^{0}(P_{x}, g^{∗}(S^{k−3}E ⊗ L^{−1}) ⊗ H_{x}) → H^{0}(P_{x}, g^{∗}S^{k−3}E ⊗ H_{x} ⊗ I_{Z_{σ}}) (4.36)

and from the equality

H^{0}(P_{x}, g^{∗}(S^{k−3}E ⊗ L^{−1}) ⊗ H_{x}) = H^{0}(P_{x}, g^{∗}S^{k−4}E).  (4.37)

This last equality is proved as follows : on P_{x} we have the exact sequence

0 → g^{∗}L ⊗ H_{x}^{−1} → g^{∗}E → H_{x} → 0,

which gives

0 → g^{∗}S^{k−4}E ⊗ g^{∗}L ⊗ H_{x}^{−1} → g^{∗}S^{k−3}E → H_{x}^{k−3} → 0.

Tensoring this with H_{x} ⊗ L^{−1} we get

0 → g^{∗}S^{k−4}E → g^{∗}(S^{k−3}E ⊗ L^{−1}) ⊗ H_{x} → g^{∗}L^{−1} ⊗ H_{x}^{k−2} → 0.

But the right hand side has no non-zero sections since it is of negative degree on the fibers of f. Hence the equality (4.37).

Since Z_{σ} ⊂ P_{x} is the complete intersection of the space W of sections of g^{∗}L, we have a Koszul resolution of I_{Z_{σ}}, which takes the form

0 → ⋀^{2k−1} W ⊗ g^{∗}L^{−2k−1} → … → W ⊗ g^{∗}L^{−1} → I_{Z_{σ}} → 0.

We can tensor it with g^{∗}S^{k−3}E ⊗ H_{x}, and the surjectivity of the map (4.36) will follow from the following vanishing

H^{i}(P_{x}, g^{∗}(S^{k−3}E ⊗ L^{−i−1}) ⊗ H_{x}) = 0, 1 ≤ i < 2k − 1 = dim P_{x}. (4.38)

Recall now that P_{x} ⊂ P is the complete intersection of two sections of H = f^{∗}O_{ℙ(H^{0}(E))}(1), with H_{x} = H_{|P_{x}}. The vanishing (4.38) will then follow from

H^{i}(P, g^{∗}(S^{k−3}E ⊗ L^{−i−1}) ⊗ H) = 0, 1 ≤ i < 2k − 1
H^{i+1}(P, g^{∗}(S^{k−3}E ⊗ L^{−i−1})) = 0, 1 ≤ i < 2k − 1
H^{i+2}(P, g^{∗}(S^{k−3}E ⊗ L^{−i−1}) ⊗ H^{−1}) = 0, 1 ≤ i < 2k − 1.

The second equality follows immediately from the proposition 9, and the third is obvious since H^{−1} has no cohomology on the fibers of g. The first equality is proven as follows : we have

H^{i}(P, g^{∗}S^{k−3}E ⊗ L^{−i−1}) ⊗ H) = H^{i}(G_{2}, S^{k−3}E ⊗ E ⊗ L^{−i−1}),

since R^{0}g_{∗}H = E. Now we have the exact sequence on G_{2}

0 → S^{k−4}E ⊗ L → S^{k−3}E ⊗ E → S^{k−2}E → 0.

Hence the needed equality will follow from the vanishings

H^{i}(G_{2}, S^{k−4}E ⊗ L^{−i}) = 0,
H^{i}(G_{2}, S^{k−2}E ⊗ L^{−i−1}) = 0,

for 1 ≤ i < 2k − 1, which are proved in proposition (9). Hence lemma 5 is proven.

Proof of lemma 6. Let x_{1}, …, x_{2k−1} be points of C in general position. We will show that the natural map

⊕_{i}H^{0}(C, M(−x_{i})) → H^{0}(C, M)  (4.39)

is surjective.

Recall that

H^{0}(C, M) = Ker ⊕_{D∈Z_{σ}} S^{k−3}D ⊗ D → S^{k−3}H^{0}(E) ⊗ H^{0}(E).

It follows from this, using the identifications

H^{0}(E)^{∗} = H^{0}(G_{2}, E), D^{∗} ≅ E_{D}

that

H^{0}(C, M)^{∗} = Coker H^{0}(G_{2}, S^{k−3}E ⊗ E) → H^{0}(S^{k−3}E ⊗ E_{|Z_{σ}})
= H^{1}(G_{2}, S^{k−3}E ⊗ E ⊗ I_{Z_{σ}}).

Similarly

H^{0}(C, M(−x_{i})) = Ker ⊕_{D∈Z_{σ}} S^{k−3}D ⊗ σ_{D,x_{i}} → S^{k−3}H^{0}(E) ⊗ K_{x_{i}}

which, with the notations of the previous proof, dualizes to

H^{0}(C, M(−x_{i}))^{∗} = Coker (H^{0}(P_{x_{i}}, g^{∗}S^{k−3}E ⊗ H_{x_{i}}) → H^{0}(Z_{σ}, S^{k−3}E ⊗ H_{x_{i}}))

= H^{1}(P_{x_{i}}, g^{∗}S^{k−3}E ⊗ H_{x_{i}} ⊗ I_{Z_{σ}}),

where we view Z_{σ} as a subscheme of P_{x_{i}} as well. Hence we have to show that the natural map (induced by the morphism g^{∗}E → H_{x_{i}} on P_{x_{i}})

H^{1}(G_{2}, S^{k−3}E ⊗ E ⊗ I_{Z_{σ}}) → ⊕_{i}H^{1}(P_{x_{i}}, g^{∗}S^{k−3}E ⊗ H_{x_{i}} ⊗ I_{Z_{σ}}) (4.40)

is injective.

Let R ⊂ G_{2} be the curve which is the complete intersection of the sections σ_{x_{i}} ∈ H^{0}(G_{2}, L). We have first

Fact.The restriction map

H^{0}(G_{2}, S^{k−3}E ⊗ E) → H^{0}(R, S^{k−3}E ⊗ E_{|R})

is surjective.

Using the Koszul resolution of I_{D} this is obtained by application of the proposition 9.

¿From this we conclude that the restriction map

H^{1}(G_{2}, S^{k−3}E ⊗ E ⊗ I_{Z_{σ}}) → H^{1}(R, S^{k−3}E ⊗ E_{|R} ⊗ I_{Z_{σ}})

is injective.

Consider now the fibered product

R̃ := P_{x_{1}} ×_{G_{2}} × … ×_{G_{2}} P_{x_{2k−1}}.

Denote by g̃ : R̃ → R ⊂ G_{2} the natural morphism. One shows easily that the curve R̃ is isomorphic to R excepted over the intersection of R with the Grassmannian of lines in ℙ(K_{x_{i}}) for some i. Here R has nodes, which are replaced in R̃ by lines.

(This fact is obviously true set theoretically, and is proved scheme theoretically by the computation of the canonical bundles, which gives :

KR̃ = g̃^{∗}K_{R}.)

The zero set Z_{σ} is supported away of this singular locus. For each i we have a natural restriction map

H^{1}(P_{x_{i}}, g^{∗}S^{k−3}E ⊗ H_{x_{i}} ⊗ I_{Z_{σ}}) → H^{1}(R̃, g̃^{∗}S^{k−3}E ⊗ H_{x_{i}} ⊗ I_{Z_{σ}}),

since R̃ = P_{x_{1}} ×_{G_{2}} … ×_{G_{2}} P_{x_{2k−1}} admits a natural morphism to P_{x_{i}}. Next we have by the above description of R̃ an isomorphism

H^{1}(R, S^{k−3}E ⊗ E_{|R} ⊗ I_{Z_{σ}}) ≅ H^{1}(R̃, g̃^{∗}S^{k−3}E ⊗ E ⊗ I_{Z_{σ}})

and it follows that the injectivity of the map (4.40) will be a consequence of the injectivity of the map

H^{1}(R̃, g̃^{∗}S^{k−3}E ⊗ E ⊗ I_{Z_{σ}}) → ⊕_{i}H^{1}(R̃, g̃^{∗}S^{k−3}E ⊗ H_{x_{i}} ⊗ I_{Z_{σ}}) (4.41)

induced by the morphisms g̃^{∗}E → H_{x_{i}} on R̃. Recall now that Z_{σ} ⊂ R is defined by a section of L so that similarly Z_{σ} ⊂ R̃ is defined by a section of g̃^{∗}L. Hence we have

I_{Z_{σ}} ≅ g̃^{∗}L^{−1}.

Furthermore

KR̃ = g̃^{∗}K_{R} = g̃^{∗}(K_{G_{2}}_{|R} ⊗ L^{2k−1}) = g̃^{∗}L^{k−3}.

Hence the map (4.41) dualizes by Serre’s duality as the map

⊕_{i}H^{0}(R̃, g̃^{∗}S^{k−3}E^{∗} ⊗ H_{x_{i}}^{∗} ⊗ g̃^{∗}L ⊗ g̃^{∗}L^{k−3})  (4.42)
→ H^{0}(R̃, g̃^{∗}(S^{k−3}E^{∗} ⊗ E^{∗}) ⊗ g̃^{∗}L ⊗ g̃^{∗}L^{k−3})  (4.43)

given by the inclusions H_{x_{i}}^{∗} ⊂ g̃^{∗}E^{∗} on R̃. Since det E = L, we have

E^{∗} ⊗ L ≅ E,

Hence (4.42) rewrites as

⊕_{i}H^{0}(R̃, g̃^{∗}(S^{k−3}E ⊗ L) ⊗ H_{x_{i}}^{∗}) → H^{0}(R̃, g̃^{∗}(S^{k−3}E ⊗ E))  (4.44)

given by the inclusions

H_{x_{i}}^{∗} ⊗ g̃^{∗}L ⊂ g̃^{∗}E.

We want to show that (4.41) is injective, or that (4.44) is surjective. We already noticed that the restriction map

H^{0}(G_{2}, S^{k−3}E ⊗ E) = S^{k−3}H^{0}(E)^{∗} ⊗ H^{0}(E)^{∗}

→ H^{0}(R, S^{k−3}E ⊗ E) = H^{0}(R̃, g̃^{∗}(S^{k−3}E ⊗ E))

is surjective. On the other hand, consider the 2-dimensional subspace H_{x_{i}} = K_{x_{i}}^{⊥} ⊂ H^{0}(E)^{∗}. It is obvious that it parametrizes sections of

Ker (H^{0}(P_{x_{i}}, g^{∗}E) → H^{0}(P_{x_{i}}, H_{x_{i}})) = H^{0}(P_{x_{i}}, g^{∗}L ⊗ H_{x_{i}}^{∗}).

Hence the surjective map

S^{k−3}H^{0}(E)^{∗} ⊗ H^{0}(E)^{∗} → H^{0}(R̃, g̃^{∗}(S^{k−3}E ⊗ E))

sends S^{k−3}H^{0}(E)^{∗} ⊗ H_{x_{i}} in the subspace H^{0}(R̃, g̃^{∗}(S^{k−3}E ⊗ L) ⊗ H_{x_{i}}^{∗}).

Now since the x_{i}’s are generic, the spaces H_{x_{i}} generate H^{0}(E)^{∗}, hence the S^{k−3}H^{0}(E)^{∗} ⊗ H_{x_{i}}’s generate S^{k−3}H^{0}(E)^{∗} ⊗ H^{0}(E)^{∗}. Hence we have shown that (4.44) is surjective.

5 Appendix

We consider the Grassmannian G_{2} of rank 2 vector subspaces of a k + 2dimensional vector space V. Let L be the line bundle on G_{2} whose sections give the Plücker embedding. If E is the dual of the tautological subbundle S ⊂ V ⊗ O_{G_{2}}, we have L = det E. The cohomology groups H^{p}(G_{2}, L^{−q} ⊗ S^{q′}E) are described in the following proposition.

Proposition 9 For q > 0, q′ > 0, we have

H^{p}(G_{2}, L^{−q} ⊗ S^{q′}E) = 0 if p ≠ k, 2k.

Furthermore, for p = k, we have

H^{p}(G_{2}, L^{−q} ⊗ S^{q′}E) = 0 if − q + q′ + 1 < 0,

and for p = 2k, we have

H^{p}(G_{2}, L^{−q} ⊗ S^{q′}E) = 0 if − q + q′ ≥ −k − 1.

Proof. Let

\begin{matrix}P & →^{g} & G_{2} \\ f ↓ \\ ℙ(V) \end{matrix}

be the incidence variety. P is a ℙ^{1}-bundle over G_{2} and a ℙ^{k}-bundle over ℙ(V). Let H := f^{∗}O_{ℙ(V)}(1) and let L′ = g^{∗}L. Then E = R^{0}g_{∗}H and S^{q′}E = R^{0}g_{∗}(q′H). It follows that we have

H^{p}(G_{2}, L^{−q} ⊗ S^{q′}E) = H^{p}(P, −qL′ + q′H).

Next the line bundle L′ restricts to O(1) on the fibers of π. It follows from this that

K_{P} = −(k + 1)L′ − 2H,

and K_{P/ℙ(V)} = −(k + 1)L′ + kH.

Now since q > 0 we have R^{l}f_{∗}(−qL′ + q′H) = 0 for l < k and hence

H^{p}(P, −qL′ + q′H) = H^{p−k}(ℙ(V), R^{k}f_{∗}(L′ + q′H)).  (5.45)

By Serre’s duality, we have

R^{k}f_{∗}(−qL′ + q′H) = (R^{0}f_{∗}(qL′ − q′H − (k + 1)L′ + kH))^{∗}
= (R^{0}f_{∗}((q − (k + 1))L′ + (k − q′)H))^{∗}.

Now we have

R^{0}f_{∗}((q − (k + 1))L′) = 0  (5.46)

if q < k + 1, and

R^{0}f_{∗}((q − (k + 1))L′) ≅ S^{q−k−1}(Ω_{ℙ(V)}(2))  (5.47)

for q ≥ k + 1. (The isomorphism (5.47) for q − k + 1 = 1 follows from the isomorphism

H^{0}(P, L′) = H^{0}(G_{2}, L) = ⋀^{2} V^{∗} = H^{0}(ℙ(V), Ω_{ℙ(V)}(2))

and from the comparison of the kernels of the surjective evaluation maps

H^{0}(P, L′) → H^{0}(f^{−1}(x), L′)

and

H^{0}(ℙ(V), Ω_{ℙ(V)}(2)) → Ω_{ℙ(V)}(2)_{x}.)

Finally we conclude from (5.45), (5.46) and (5.47) that

1. H^{p}(P, −qL′ + q′H) = 0 for p < k.

2. H^{p}(P, −qL′ + q′H) = 0 for q < k + 1.

3. For p ≥ k, q ≥ k + 1,

H^{p}(P, −qL′ + q′H) = H^{p−k}(ℙ(V), S^{q−k−1}(T_{ℙ(V)}(−2))(q′ − k)).

To conclude, consider the Euler exact sequence

0 → O_{ℙ(V)}(−1) → V ⊗ O_{ℙ(V)} → T_{ℙ(V)}(−1) → 0.

It induces the exact sequences

0 → S^{q−k−2}V ⊗ O_{ℙ(V)}(−q + q′) → S^{q−k−1}V ⊗ O_{ℙ(V)}(−q + q′ + 1)

→ S^{q−k−1}(T_{ℙ(V)}(−1))(−q + q′ + 1) → 0.

Hence we conclude that the space H^{p}(G_{2}, L^{−q} ⊗ S^{q′}E) which by the above is also isomorphic to

H^{p−k}(ℙ(V), S^{q−k−1}(T_{ℙ(V)}(−2))(q′ − k))
= H^{p−k}(ℙ(V), S^{q−k−1}(T_{ℙ(V)}(−1))(−q + q′ + 1))

is equal to 0 for p − k ≠ 0, k (since p ≤ 2k), and that :

- for p − k = 0 it is 0 if −q + q′ + 1 < 0;

- for p − k = k it is 0 if −q + q′ ≥ −k − 1.

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