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On Frobenius-destabilized rank-2 vector bundles over curves

Let X be a smooth projective curve of genus g \geq 2 over an algebraically closed field k of characteristic p>0. Let M_X be the moduli space of semistable rank-2 vector bundles over X with trivial determinant. The relative Frobenius map F: X \to X_1 induces by pull-back a rational map V: M_{X_1} \to M_{X}. In this paper we show the following results. 1) For any line bundle L over X, the rank-p vector bundle F_*L is stable. 2) The rational map V has base points, i.e., there exist stable bundles E over X_1 such that F^* E is not semistable. 3) Let B \subset M_{X_1} denote the scheme-theoretical base locus of V. If g=2, p>2 and X ordinary, then B is a 0-dimensional local complete intersection of length {2/3}p(p^2 -1) and the degree of V equals {1/3}p(p^2 +2).


Introduction
Let X be a smooth projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic p > 0. Denote by F : X → X 1 the relative k-linear Frobenius map. Here X 1 = X × k,σ k, where σ : Spec(k) → Spec(k) is the Frobenius of k (see e.g. [R] section 4.1). We denote by M X , respectively M X 1 , the moduli space of semistable rank-2 vector bundles on X, respectively X 1 , with trivial determinant. The Frobenius F induces by pull-back a rational map (the Verschiebung) Here [E] denotes the S-equivalence class of the semistable bundle E. It is shown [MS] that V is genericallyétale, hence separable and dominant, if X or equivalently X 1 is an ordinary curve. Our first result is Theorem 1 Over any smooth projective curve X 1 of genus g ≥ 2 there exist stable rank-2 vector bundles E with trivial determinant, such that F * E is not semistable. In other words, V has base points.
Note that this is a statement for an arbitrary curve of genus g ≥ 2 over k, since associating X 1 to X induces an automorphism of the moduli space of curves of genus g over k. The existence of Frobenius-destabilized bundles was already proved in [LP2] Theorem A.4 by specializing the so-called Gunning bundle on a Mumford-Tate curve. The proof given in this paper is much simpler than the previous one. Given a line bundle L over X, the generalized Nagata-Segre theorem asserts the existence of rank-2 subbundles E of the rank-p bundle F * L of a certain (maximal) degree. Quite surprisingly, these subbundles E of maximal degree turn out to be stable and Frobenius-destabilized.
In the case g = 2 the moduli space M X is canonically isomorphic to the projective space P 3 k and the set of strictly semistable bundles can be identified with the Kummer surface Kum X ⊂ P 3 k associated to X. According to [LP2] Proposition A.2 the rational map V : P 3 k P 3 k is given by polynomials of degree p, which are explicitly known in the cases p = 2 [LP1] and p = 3 [LP2]. Let B be the scheme-theoretical base locus of V , i.e., the subscheme of P 3 k determined by the ideal generated by the 4 polynomials of degree p defining V . Clearly its underlying set equals supp B = {E ∈ M X 1 ∼ = P 3 k | F * E is not semistable} and supp B ⊂ P 3 k \ Kum X 1 . Since V has no base points on the ample divisor Kum X 1 , we deduce that dim B = 0. Then we show Theorem 2 Assume p > 2. Let X 1 be an ordinary curve of genus g = 2. Then the 0dimensional scheme B is a local complete intersection of length 2 3 p(p 2 − 1).
Since B is a local complete intersection, the degree of V equals deg V = p 3 − l (B) where l (B) denotes the length of B (see e.g. [O1] Proposition 2.2). Hence we obtain the Corollary Under the assumption of Theorem 2 deg V = 1 3 p(p 2 + 2).
The underlying idea of the proof of Theorem 2 is rather simple: we observe that a vector bundle E ∈ supp B corresponds via adjunction to a subbundle of the rank-p vector bundle F * (θ −1 ) for some theta characteristic θ on X (Proposition 3.1). This is the motivation to introduce Grothendieck's Quot-Scheme Q parametrizing rank-2 subbundles of degree 0 of the vector bundle F * (θ −1 ). We prove that the two 0-dimensional schemes B and Q decompose as disjoint unions B θ and Q η where θ and η vary over theta characteristics on X and p-torsion points of JX 1 respectively and that B θ and Q 0 are isomorphic, if X is ordinary (Proposition 4.6). In particular since Q is a local complete intersection, B also is.
In order to compute the length of B we show that Q is isomorphic to a determinantal scheme D defined intrinsically by the 4-th Fitting ideal of some sheaf. The non-existence of a universal family over the moduli space of rank-2 vector bundles of degree 0 forces us to work over a different parameter space constructed via the Hecke correspondence and carry out the Chern class computations on this parameter space.
The underlying set of points of B has already been studied in the literature. In fact, using the notion of p-curvature, S. Mochizuki [Mo] describes points of B as "dormant atoms" and obtains, by degenerating the genus-2 curve X to a singular curve, the above mentioned formula for their number ( [Mo] Corollary 3.7 page 267). Moreover he shows that for a general curve X the scheme B is reduced. In this context we also mention the recent work of B. Osserman [O2], [O3], which explains the relationship of supp B with Mochizuki's theory.

Acknowledgments
We would like to thank Yves Laszlo and Brian Osserman for helpful discussions and for pointing out several mistakes in a previous version of this paper. We also thank Adrian Langer for some advice with references. §1 Stability of the direct image F * L.
Let X be a smooth projective curve of genus g ≥ 2 over an algebraically closed field of characteristic p > 0 and let F : X → X 1 denote the relative Frobenius map. Let L be a line bundle over X with for some integer d. Applying the Grothendieck-Riemann-Roch theorem to the morphism F , we obtain Lemma 1.1 The slope of the rank-p vector bundle F * L equals The following result will be used in section 3.
Proposition 1.2 If g ≥ 2, then the vector bundle F * L is stable for any line bundle L on X.
Proof. Suppose that the contrary holds, i.e., F * L is not stable. Consider its Harder-Narasimhan filtration we denote by E any proper semistable subbundle of the same slope. Then clearly In case r = rk E > p−1 2 , we observe that the quotient bundle is also semistable and that its dual Q * is a subbundle of (F * L) * . Moreover, by relative duality ) and by assumption rk Q * ≤ p−r ≤ p−1 2 . Hence, replacing if necessary E and L by Q * and L −1 ⊗ ω ⊗1−p X , we may assume that E is semistable and r ≤ p−1 2 . Now, by [SB] Corollary 2, we have the inequality where µ max (F * E) (resp. µ min (F * E)) denotes the slope of the first (resp. last) graded piece of the Harder-Narasimhan filtration of F * E. The inclusion E ⊂ F * L gives, by adjunction, a nonzero map Combining this inequality with (1) and using Lemma 1.1, we obtain which simplifies to This is a contradiction, since we have assumed r ≤ p−1 2 and therefore 2r−1 p < 1. Remark 1.3 We observe that the vector bundles F * L are destabilized by Frobenius, because of the nonzero canonical map F * F * L → L and clearly µ(F * F * L) > deg L. For further properties of the bundles F * L, see [JRXY] section 5.
Remark 1.4 In the context of Proposition 1.2 we mention the following open question: given a finite separable morphism between smooth curves f : Y → X and a line bundle L ∈ Pic(Y ), is the direct image f * L stable? For a discussion, see [B]. §2 Existence of Frobenius-destabilized bundles.
Let the notation be as in the previous section. We recall the generalized Nagata-Segre theorem, proved by Hirschowitz, which says Theorem 2.1 For any vector bundle G of rank r and degree δ over any smooth curve X and for any integer n, 1 ≤ n ≤ r − 1, there exists a rank-n subbundle E ⊂ G, satisfying where ǫ is the unique integer with 0 ≤ ǫ ≤ r − 1 and ǫ + n(r − n)(g − 1) ≡ nδ mod r.
Proof of Theorem 1. We apply Theorem 2.1 to the rank-p vector bundle F * L on X 1 and n = 2, where L is a line bundle of degree g − 1 + d on X, with d ≡ −2g + 2 mod p: There exists a rank-2 vector bundle E ⊂ F * L such that (4) µ Note that our assumption on d was made to have ǫ = 0. Now we will check that any E satisfying inequality (4) is stable with F * E not semistable.
(i) E is stable: Let N be a line subbundle of E. The inclusion N ⊂ F * L gives, by adjunction, a nonzero map F * N → L, which implies (see also [JRXY] Proposition 3.2(i)) Comparing with (4) we see that deg N < µ (E).
(ii) F * E is not semistable. In fact, we claim that L destabilizes F * E. For the proof note that Lemma 1.1 implies Together with (4) this gives µ(E) > deg L p and hence This implies the assertion, since by adjunction we obtain a nonzero map F * E → L.
Replacing E by a subsheaf of suitable degree, we may assume that inequality (4) is an equality. In that case, because of our assumption on d, µ (E) is an integer, hence deg E is even. In order to get trivial determinant, we may tensorize E with a suitable line bundle. This completes the proof of Theorem 1. §3 Frobenius-destabilized bundles in genus 2.
From now on we assume that X is an ordinary curve of genus g = 2 and the characteristic of k is p > 2. Recall that M X denotes the moduli space of semistable rank-2 vector bundles with trivial determinant over X and B the scheme-theoretical base locus of the rational map which is given by polynomials of degree p.
First of all we will show that the 0-dimensional scheme B is the disjoint union of subschemes B θ indexed by theta characteristics of X.
Proposition 3.1 (a) Let E be a vector bundle on X 1 such that E ∈ supp B. Then there exists a unique theta characteristic θ on X, such that E is a subbundle of F * (θ −1 ). (b) Let θ be a theta characteristic on X. Any rank-2 subbundle E ⊂ F * (θ −1 ) of degree 0 has the following properties Proof: (a) By [LS] Corollary 2.6 we know that, for every E ∈ supp B the bundle F * E is the nonsplit extension of θ −1 by θ, for some theta characteristic θ on X (note that Ext 1 (θ −1 , θ) ∼ = k). By adjunction we get a homomorphism ψ : E → F * (θ −1 ) and we have to show that this is of maximal rank. Suppose it is not, then there is a line bundle N on the curve X 1 such that ψ factorizes as E → N → F * (θ −1 ). By stability of E we have deg N > 0. On the other hand, by adjunction, we get a nonzero homomorphism F * N → θ −1 implying p · deg N ≤ −1, a contradiction. Hence where T is a torsion sheaf supported on an effective divisor. Varying π, we obtain a family of bundles ker π ⊂ E ′ of dimension > 0 and det ker π = O X 1 . This would imply (see proof of Theorem 1) dim B > 0, a contradiction. Finally, since θ is the maximal destabilizing line subbundle of F * E, it is unique.
(b) We observe that inequality (4) holds for the pair E ⊂ F * (θ −1 ). Hence, by the proof of Theorem 1, E is stable and F * E is not semistable.
Let ϕ : . The homomorphism ϕ is surjective, since otherwise F * E would contain a line subbundle of degree > 1, contradicting [LS], Satz 2.4. Hence we get an exact sequence On the other hand, let N denote a line bundle on X 1 such that E ⊗N has trivial determinant, i.e. N −2 = det E. Applying [LS] Corollary 2.6 to the bundle F * (E ⊗N) we get an exact sequence for some theta characteristicθ. By uniqueness of the destabilizing subbundle of maximal degree of F * E, this exact sequence must coincide with (6) up to a nonzero constant. This implies that By adjunction we have the equality dim Hom (E, F * Finally, suppose that there exists a rank-2 subbundle E ′ ⊂ F * (θ −1 ) with deg E ′ ≥ 1. Then we can consider the kernel E = ker π of a surjective morphism π : E ′ → T onto a torsion sheaf with length equal to deg E ′ . By varying π and after tensoring ker π with a suitable line bundle of degree 0, we construct a family of dimension > 0 of stable rank-2 vector bundles with trivial determinant which are Frobenius-destabilized, contradicting dim B = 0. This proves (iv).
It follows from Proposition 3.1 (a) that the scheme B decomposes as a disjoint union where θ varies over the set of all theta characteristics of X and Tensor product with a 2-torsion point α ∈ JX 1 [2] ∼ = JX [2] induces an isomorphism of B θ with B θ⊗α for every theta characteristic θ. We denote by l(B) and l(B θ ) the length of the schemes B and B θ . From the preceding we deduce the relations Let θ be a theta characteristic on X. We consider the functor Q from the opposite category of k-schemes to the category of sets defined by where π X 1 : X 1 × S → X 1 denotes the natural projection and σ ∼ = σ ′ for quotients σ and σ ′ if and only if there exists an isomorphism λ : Grothendieck showed in [G] (see also [HL] section 2.2) that the functor Q is representable by a k-scheme, which we denote by Q. A k-point of Q corresponds to a quotient σ : F * (θ −1 ) → G, or equivalently to a rank-2 subsheaf E = ker σ ⊂ F * (θ −1 ) of degree 0 on X 1 . By the same argument as in Proposition 3.1 (a), any subsheaf E of degree 0 is a subbundle of F * (θ −1 ). Since by Proposition 3.1 (b) (iv) the bundle E has maximal degree as a subbundle of F * (θ −1 ), any sheaf G ∈ Q(S) is locally free (see [MuSa] or [L] Lemma 3.8).
Hence taking the kernel of σ induces a bijection of Q(S) with the following set, which we also denote by Q(S) Because of the projection formula, tensor product with a p-torsion point β ∈ JX 1 [p] red induces an isomorphism of Q η with Q η⊗β . So the scheme Q is a principal homogeneous space for the group JX 1 [p] red and we have the relation since X 1 is assumed to be ordinary. Moreover, by Proposition 3.1 we have the set-theoretical equality supp Q 0 = supp B θ .
Proof: Assertion (a) follows from the preceding remarks and dim B = 0. By [HL] Proposition 2.2.8 assertion (b) follows from the equality dim [E] and Hom denotes the sheaf of homomorphisms.
Let N X 1 denote the moduli space of semistable rank-2 vector bundles of degree 0 over X 1 . We denote by N s X 1 and M s X 1 the open subschemes of N X 1 and M X 1 corresponding to stable vector bundles. Recall (see [La1] Theorem 4.1) that N s X 1 and M s X 1 universally corepresent the functors (see e.g. [HL] Definition 2.2.1) from the opposite category of k-schemes of finite type to the category of sets defined by We denote by E the equivalence class of the vector bundle E for the relation ∼.

Consider the determinant morphism
and denote by det −1 (0) the scheme-theoretical fibre over the trivial line bundle on X 1 . Since N s X 1 universally corepresents the functor N s X 1 , we have an isomorphism M s
In the sequel we need the following relative version of Proposition 3.1 (b)(ii). By a k-scheme we always mean a k-scheme of finite type.

Lemma 4.3 Let S be a connected k-scheme and let E be a locally free sheaf of rank-2 over
Proof: First we note that by flat base change for π X 1 : X 1 × S → X 1 , we have an isomorphism We know by the proof of Proposition 3.1 (b) that the fibre ϕ (x,s) over any closed point (x, s) ∈ X × S is a surjective k-linear map. Hence ϕ is surjective by Nakayama and we have an exact sequence 0 −→ L −→ (F × id S ) * E −→ π * X (θ −1 ) −→ 0, with L locally free sheaf of rank 1. By [K] section 5, the rank-2 vector bundle (F × id S ) * E is equipped with a canonical connection where ω X denotes the canonical line bundle of X. The first fundamental form of the connection ∇ is an O X×S -linear homomorphism . The restriction of ψ ∇ to the curve X ×{s} ⊂ X ×S for any closed point s ∈ S is an isomorphism (see e.g. proof of [LS] Corollary 2.6). Hence the fibre of ψ ∇ is a k-linear isomorphism over any closed point (x, s) ∈ X × S. We conclude that ψ ∇ is an isomorphism, by Nakayama's lemma and because L is a locally free sheaf of rank 1.
We obtain the second assertion of the lemma, since

Proposition 4.4
We assume X ordinary.
(a) The forgetful morphism To show that i is a closed embedding at e ∈ Q, it is enough to show that the differential (di) e : T e Q → T [E] N X 1 is injective. The Zariski tangent spaces identify with Hom (E, G) identifies with the differential (di) e . Now since the bundle E is stable, we have k ∼ = Hom (E, E). By Proposition 3.1 (b) we obtain that the map Hom(E, E) → Hom(E, F * (θ −1 )) is an isomorphism. Thus (di) e is injective.
where the last map is the isogeny given by the Verschiebung on JX 1 , i.e. V (L) = F * L for L ∈ JX 1 . The morphism α is induced by the natural transformation of functors α : Q ⇒ JX, defined by . Using Lemma 4.3 this immediately implies that α factors through the inclusion of the reduced point {O X } ֒→ JX. Hence the image of Q under the composite morphism det •i is contained in the kernel of the isogeny V , which is the reduced scheme JX 1 [p] red , since we have assumed X ordinary. Taking connected components we see that the image of Q 0 under det •i is the reduced In order to compare the two schemes B θ and Q 0 we need the following lemma.

Lemma 4.5
(1) The closed subscheme B ⊂ M s X 1 corepresents the functor B which associates to a kscheme S the set Proof: We denote by M X 1 the algebraic stack parametrizing rank-2 vector bundles with trivial determinant over X 1 . Let M ss X 1 denote the open substack of M X 1 parametrizing semistable bundles and M unst X 1 the closed substack of M X 1 parametrizing non-semistable bundles. We will use the following facts about the stack M X 1 .
• The pull-back of O P 3 (1) by the natural map M ss X 1 → M X 1 ∼ = P 3 extends to a line bundle, which we denote by O(1), over the moduli stack M X 1 and Pic(M X 1 ) = Z · O(1). Moreover there are natural isomorphisms H 0 (M X 1 , O(n)) ∼ = H 0 (M P 3 , O P 3 (n)) for any positive integer n (see [BL] Propositions 8.3 and 8.4).
• The closed subscheme M unst X 1 is the base locus of the linear system |O(1)| over the stack M X 1 . This is seen as follows: we deduce from [S] Theorem 6.2 that M unst X 1 is the base locus of the linear system |O(n)| for some integer n. Since |O(n)| is generated by symmetric products of n sections in |O(1)|, we obtain that M unst X 1 is the base locus of |O(1)|. We need to compute the fibre product functor B = B × M s X 1 M s X 1 . Let V : M X 1 → M X denote the morphism of stacks induced by pull-back under the Frobenius map F : X → X 1 . Let S be a k-scheme and consider a vector bundle E ∈ M s X 1 (S). Since the subscheme B is defined as base locus of the linear system V * |O P 3 (1)|, we obtain that E ∈ B(S) if and only if E lies in the base locus of V * |O(1)| -here we use the isomorphism |O P 3 (1)| ∼ = |O(1)| -, or equivalently V(E) = (F × id S ) * E lies in the base locus of |O(1)|, which is the closed substack M unst X 1 .
By [Sh] section 5 the substack M unst,1 X 1 of M unst X 1 parametrizing non-semistable vector bundles having a maximal destabilizing line subbundle of degree 1 is an open substack of M unst X 1 . By [LS] Corollary 2.6 the vector bundle V(E) lies in M unst,1 X 1 (S). We then consider the universal exact sequence defined by the Harder-Narasimhan filtration over M unst,1 with L and M locally free sheaves over X × S such that deg L |X×{s} = − deg M |X×{s} = 1 for any s ∈ S. This proves (1).
As for (2), we add the condition that the family E is Frobenius-destabilized by the thetacharacteristic θ.

Proposition 4.6 There is a scheme-theoretical equality
Proof: Since B θ and Q 0 corepresent the two functors B θ and Q 0 it will be enough to show that there is a canonical bijection between the set B θ (S) and Q 0 (S) for any k-scheme S. We recall that Q 0 (S) = {E ֒→ π * X 1 (F * (θ −1 )) | E locally free sheaf over X 1 × S of rank 2, π * X (F * (θ −1 ))/E locally free, det E ∼ = O X 1 ×S }/ ∼ = Note that the property det E ∼ = O X 1 ×S is implied as follows: by Proposition 4.4 (b) we have det E ∼ = π * S L for some line bundle L over S and by Lemma 4.3 we conclude that L = O S .
First we show that the natural map Q 0 (S) −→ M s X 1 (S) is injective. Suppose that there exist E, E ′ ∈ Q 0 (S) such that E = E ′ , i.e. E ′ ∼ = E ⊗ π * S (L) for some line bundle L on S. Then by Lemma 4.3 we have two inclusions Therefore the two sets Q 0 (S) and B θ (S) are naturally subsets of M s X 1 (S).
Hence it remains to show that B θ (S) ⊂ Q 0 (S). Consider a sheaf E with E ∈ B θ (S)see Lemma 4.5 (2). As in the proof of Lemma 4.3 we consider the canonical connection ∇ on which is surjective on closed points (x, s) ∈ X × S. Hence we can conclude that ψ ∇ is an isomorphism. Moreover taking the determinant, we obtain Hence its classifying morphism Φ L⊗L : S −→ Pic 2 (X) factorizes through the inclusion of the reduced point {ω X } ֒→ Pic 2 (X). Moreover the composite map of Φ L with the duplication map [2] Φ L⊗L : S Φ L −→ Pic 1 (X) [2] −→ Pic 2 (X) coincides with Φ L⊗L . We deduce that Φ L factorizes through the inclusion of the reduced point {θ} ֒→ Pic 1 (X). Note that the fibre [2] −1 (ω X ) is reduced, since p > 2. Since Pic 1 (X) is a fine moduli space, there exists a line bundle N over S such that By adjunction the morphism σ gives a nonzero morphism j : . We now show that j is injective. Suppose it is not. Then there exists a subsheafẼ 0 ⊂ π * X 1 (F * (θ −1 )) and a surjective map τ : E 0 →Ẽ 0 . Let K denote the kernel of τ . Again by adjunction we obtain a map α : Here τ * denotes the map (F ×id S ) * τ . Since σ is surjective, α is also surjective. We denote by M the kernel of α. The induced map τ : π * X (θ) = ker σ → M is surjective, because τ * is surjective. Moreover the first fundamental form of the canonical connection∇ on (F × id S ) * Ẽ 0 induces an O X×S -linear homomorphism ψ∇ : M → π * X (θ) and the composite map θ) coincides with the first fundamental form of ∇ of (F × id S ) * E 0 , which is an isomorphism. Therefore τ is an isomorphism too. So τ * is an isomorphism and (F × id S ) * K = 0. We deduce that K = 0.
In order to show that E 0 ∈ Q 0 (S), it remains to verify that the quotient sheaf π X 1 (F * (θ −1 ))/E 0 is flat over S. We recall that flatness implies locally freeness because of maximality of degree. But flatness follows from [HL] Lemma 2.1.4, since the restriction of j to X 1 × {s} is injective for any closed s ∈ S by Proposition 3.1 (a).
Combining this proposition with relations (7) and (8), we obtain In this section we introduce a determinantal subscheme D ⊂ N X 1 , whose length will be computed in the next section. We also show that D is isomorphic to Grothendieck's Quotscheme Q. We first define a determinantal subschemeD of a variety JX 1 × Z covering N X 1 and then we show thatD is a P 1 -fibration over anétale cover of D ⊂ N X 1 .
Since there does not exist a universal bundle over X 1 ×M X 1 , following an idea of Mukai [Mu], we consider the moduli space M X 1 (x) of stable rank-2 vector bundles on X 1 with determinant O X 1 (x) for a fixed point x ∈ X 1 . According to [N1] the variety M X 1 (x) is a smooth intersection of two quadrics in P 5 . Let U denote a universal bundle on X 1 × M X 1 (x) and denote considered as a rank-2 vector bundle on M X 1 (x). Then the projectivized bundle is a P 1 -bundle over M X 1 (x). The variety Z parametrizes pairs (F z , l z ) consisting of a stable vector bundle F z ∈ M X 1 (x) and a linear form l z : F z (x) → k x on the fibre of F z over x. Thus to any z ∈ Z one can associate an exact sequence uniquely determined up to a multiplicative constant. Clearly E z is semistable, since F z is stable, and det E z = O X 1 . Hence we get a diagram (the so-called Hecke correspondence) with ϕ(z) = [E z ] and π(z) = F z . We note that there is an isomorphism ϕ −1 (E) ∼ = P 1 and that π(ϕ −1 (E)) ⊂ M X 1 (x) ⊂ P 5 is a conic for any stable E ∈ M s X 1 . On X 1 × Z there exists a "universal" bundle, which we denote by V (see [Mu] (3.8)). It has the property Let L denote a Poincaré bundle on X 1 × JX 1 . By abuse of notation we also denote by V and L their pull-backs to X 1 × JX 1 × Z. We denote by π X 1 and q the canonical projections We consider the map m given by tensor product Note that the restriction of m to the stable locus m s : JX 1 × M s X 1 −→ N s X 1 is anétale map of degree 16. We denote by ψ the composite map Let D ∈ |ω X 1 | be a smooth canonical divisor on X 1 . We introduce the following sheaves over The next proposition is an even degree analogue of [LN] Theorem 3.1 .
Proposition 5.1 (a) The sheaves F 0 and F 1 are locally free of rank 4p and 4p − 4 respectively and there is an exact sequence Proof: We consider the canonical exact sequence over X 1 ×JX 1 ×Z associated to the effective divisor π * By Proposition 1.2 the rank-p vector bundle F * (θ −1 ) is stable and since This implies R 1 q * (L * ⊗ V * ⊗ π * X 1 (F * (θ −1 ) ⊗ ω X 1 )) = 0. By the base change theorems the sheaf F 1 is locally free. Taking direct images by q (note that q * (L * ⊗ V * ⊗ π * X 1 F * (θ −1 )) = 0 because it is a torsion sheaf), we obtain the exact sequence It follows from the proof of Proposition 3.1 (a) that any nonzero homomorphism L ⊗ E −→ F * (θ −1 ) is injective. Moreover by Proposition 3.1 (b) (iii) for any subbundle L ⊗ E ⊂ F * (θ −1 ) we have dim Hom(L ⊗ E, F * (θ −1 )) = 1, or equivalently dim H 1 (L * ⊗ E * ⊗ F * (θ −1 )) = 5. Using the base change theorems we obtain the following series of equivalences (L, z) ∈ suppD ⇐⇒ rk γ (L,z) Finally we clearly have the equality supp ψ(D) = supp Q. Since dim Q = 0 and since the fibers of the morphism ϕ over stable vector bundles are projective lines, we deduce that dimD = 1. This proves part (a).
Part (b) follows from Porteous' formula, which says that the fundamental class δ ∈ H 10 (JX 1 × Z, Z l ) of the determinantal subschemeD is given (with the notation of [ACGH], p.86) by Let M be a sheaf over a k-scheme S. We denote by We now define the 0-dimensional subscheme D ⊂ N s X 1 , which is supported on supp Q. Consider a bundle E ∈ N s X 1 with dim Hom(E, F * (θ −1 )) ≥ 1 or equivalently dim H 1 (E * ⊗ F * (θ −1 )) ≥ 5. The GIT-construction of the moduli space N s X 1 realizes N s X 1 as a quotient of an open subset U of a Quot-scheme by the group PGL(N) for some N. It can be shown (see e.g. [La2] section 3) that U is a principal PGL(N)-bundle for theétale topology over N s X 1 . Hence there exists anétale neighbourhood τ : U → U of E over which the PGL(N)-bundle is trivial, i.e., admits a section. The universal bundle over the Quot-scheme restricts to a bundle E over X 1 × U. Choose a point E ∈ U over E. Since τ isétale, it induces an isomorphism of the local rings O U ,E and O U,E . We simply denote this ring by O E . Consider the scheme structure at E defined by the ideal Fitt 4 [R 1 π U * (E * ⊗ π * X 1 F * (θ −1 ))]. This also defines a scheme structure at E ∈ U, which does not depend on the choice of theétale neighbourhood. Note that there is a "universal" bundle E over X 1 × Spec(O E ).

Lemma 5.2 There is a scheme-theoretical equalitỹ
Proof: We consider a point E ∈ suppD = suppQ and denote by Z E the fibre ψ −1 (Spec(O E )) and by ψ E : Z E → Spec(O E ) the morphism obtained from ψ after taking the base change Spec(O E ) → N s X 1 . The lemma now follows because the formation of the Fitting ideal and taking the higher direct image R 1 π Spec(O E ) * commutes with the base change ψ E (see [E] Corollary 20.5 and [Ha] Proposition 12.5), i.e.
Lemma 5.3 The subscheme D ⊂ N s X 1 corepresents the functor which associates to any k-scheme S the set Proof: This is an immediate consequence of the definition of D and the fact that N s X 1 universally corepresents the functor N s X 1 .
Lemma 5.4 Let S be a k-scheme and E a sheaf over X 1 ×S with E ∈ N s X 1 (S). We suppose that the set-theoretical image of the classifying morphism of E Proof: Since the set-theoretical support of im Φ E is a point, there exists an Artinian ring A such that Φ E factorizes through the inclusion Spec(A) ֒→ N s X 1 . By the argument, which we already used in the definition of D, there exists a universal bundle E 0 over X 1 × Spec(A). So we have shown property (1). As for (2), we consider the ideal I ⊂ A defined byĨ = ker(O Spec(A) → ϕ * O S ), whereĨ denotes the associated O Spec(A) -module. If I = 0, we replace A by A/I and we are done.
Proposition 5.5 There is a scheme-theoretical equality D = Q.
Proof: We note that D(S) and Q(S) are subsets of N s X 1 (S) (the injectivity of the map Q(S) → N s X 1 (S) is proved similarly as in the proof of Proposition 4.5). Since D and Q corepresent the two functors D and Q, it will be enough to show that the set D(S) coincides with Q(S) for any k-scheme.
We first show that D(S) ⊂ Q (S). Consider a sheaf E with E ∈ D (S). For simplicity we denote the sheaf E * ⊗ π * X 1 (F * (θ −1 )) by H. By [Ha] Theorem 12.11 there is an isomorphism Since we have assumed Fitt 4 [R 1 π S * H] = 0, we obtain dim H 1 (X 1 × {s}, H |X 1 ×{s} ) ≥ 5, or equivalently dim H 0 (X 1 × {s}, H |X 1 ×{s} ) ≥ 1, i.e., the vector bundle E |X 1 ×{s} is a subsheaf, hence subbundle, of F * (θ −1 ). This implies that the set-theoretical image of the classifying map Φ E is contained in suppQ. Taking connected components of S, we can assume that the image of Φ E is a point. Therefore we can apply Lemma 5.4: there exists a locally free sheaf E 0 over X 1 × S 0 such that E ∼ (id X 1 × ϕ) * E 0 . For simplicity we write H 0 = E * 0 ⊗ π * X 1 (F * (θ −1 )). In particular H = (id X 1 × ϕ) * H 0 . Since the projection π S 0 : X 1 × S 0 → S 0 is of relative dimension 1, taking the higher direct image R 1 π S 0 * commutes with the (not necessarily flat) base change ϕ : S → S 0 ([Ha] Proposition 12.5), i.e., there is an isomorphism Since the formation of Fitting ideals also commutes with any base change (see [E] Corollary 20.5), we obtain We deduce by [E] Proposition 20.8 that the sheaf R 1 π S 0 * H 0 is a free A-module of rank 5. By [Ha] Theorem 12.11 (b) we deduce that there is an isomorphism Again by Proposition 3.1 (b) (iii) we obtain dim π S 0 * H 0 ⊗ k(s 0 ) = 1. In particular the O S 0module π S 0 * H 0 is not zero and therefore there exists a nonzero global section i ∈ H 0 (S 0 , π S 0 * H 0 ) = H 0 (X 1 × S 0 , E * 0 ⊗ π * X 1 F * (θ −1 )). We pull-back i under the map id X 1 × ϕ and we obtain a nonzero section ). Now we apply Lemma 4.3 and we continue as in the proof of Proposition 4.6. This shows that E ∈ Q (S).
We now show that Q(S) ⊂ D (S). Consider a sheaf E ∈ Q(S). The nonzero global section j ∈ H 0 (X 1 × S, H) = H 0 (S, π S * H) determines by evaluation at a point s ∈ S an element α ∈ π S * H ⊗ k(s). The image of α under the natural map coincides with j |X 1 ×{s} which is nonzero. Moreover since dim H 0 (X 1 × {s}, H |X 1 ×{s} ) = 1, we obtain that ϕ 0 (s) is surjective. Hence by [Ha] Theorem 12.11 the sheaf R 1 π S * H is locally free of rank 5. Again by [E] Proposition 20.8 this is equivalent to Fitt 4 [R 1 π S * H] = 0 and Fitt 5 [R 1 π S * H] = O S and we are done. §6 Chern class computations.
In this section we will compute the length of the determinantal subscheme D ⊂ N X 1 by evaluating the Chern class c 5 (F 0 − F 1 ) -see Proposition 5.1 (b).
Let l be a prime number different from p. We have to recall some properties of the cohomology ring H * (X 1 × JX 1 × Z, Z l ) (see also [LN]). In the sequel we identify all classes of H * (X 1 , Z l ), H * (JX 1 , Z l ) etc. with their preimages in H * (X 1 × JX 1 × Z, Z l ) under the natural pull-back maps.
Then we have for dimensional reasons and noting that H 5 (M X 1 (x), Z l ) = 0, that the following classes are all zero: (16) f 2 , ξ 3 1 , α 4 , ξ 1 f, ξ 2 f, αξ 2 , αΛ, Θ 2 Λ, Θ 3 . Finally, Z is the P 1 -bundle associated to the vector bundle U x on M X 1 (x). Let H ∈ H 2 (Z, Z l ) denote the first Chern class of the tautological line bundle on Z. We have, using the definition of the Chern classes c i (U) and (11) The proof of Theorem 2 is now straightforward. It suffices to combine Corollary 4.7, Proposition 5.5 and Proposition 6.2 to obtain the length l (B).
The fact that B is a local complete intersection follows from the isomorphism B θ = Q 0 (Proposition 4.6) and Proposition 4.1. §8 Questions and Remarks.
(1) Is the rank-p vector bundle F * L very stable, i.e. F * L has no nilpotent ω X 1 -valued endomorphisms, for a general line bundle?