Spinor sheaves on singular quadrics

We define reflexive sheaves on a singular quadric Q that generalize the spinor bundles on smooth quadrics, using matrix factorizations of the equation of Q. We study the first properties of these spinor sheaves, give a Horrocks-type criterion, and show that they are semi-stable, and indeed stable in some cases.


Introduction
On smooth quadric hypersurfaces there exist certain vector bundles called spinor bundles, which play a role similar that of the tautological bundles on Grassmannians. On a (2n − 1)-dimensional quadric there is one spinor bundle, of rank 2 n−1 . On a 2n-dimensional quadric there are two, both of rank 2 n−1 . In low dimensions they coincide with other well-known bundles: On Q 1 ∼ = P 1 it is O(1). On Q 2 ∼ = P 1 × P 1 they are O(1, 0) and O(0, 1). On Q 3 , which is the Lagrangian Grassmannian LG (2,4), it is the tautological quotient bundle. On Q 4 ∼ = G(2, 4) they are the tautological quotient bundle and the dual of the tautological subbundle.
Ottaviani [12] gave geometric and representation-theoretic descriptions of spinor bundles, showed that they are stable, and applied them to moduli spaces of vector bundles on Q 5 and Q 6 . Later [13] he used them to give a Horrocks-type splitting criterion on smooth quadrics. Kapranov [7] described spinor bundles 1 via Clifford algebras and used them to show that the derived category of a smooth quadric is generated by an exceptional collection  Beȋlinson's exceptional collection in the derived category of projective space [3]. Langer [10] described them via explicit matrix factorizations of q = x 1 x 2 + · · · + x 2m−1 x 2m and q = x 2 0 + x 1 x 2 + · · · + x 2m−1 x 2m and used them to study the Frobenius morphism on smooth quadrics in characteristic p > 2.
It is these last two approaches that we generalize to singular quadrics. Given a linear space Λ on a quadric Q defined by a polynomial q, we construct a left ideal in the Clifford algebra of q, and from this a matrix factorization of q, and from this two sheaves S and T on Q, which we call spinor sheaves. They fail to be vector bundles where Λ meets the singular locus Q sing . In contrast to the one or two rigid bundles we had when Q was smooth, now we have one or two families of reflexive sheaves for each Grassmannian of linear spaces on Q sing .
We treat smooth and singular quadrics uniformly, but even on smooth quadrics our description of spinor bundles has advantages over Ottaviani's, with which it is difficult to do homological algebra, Kapranov's, with which it is difficult to do geometry, and Langer's, with which it is difficult to vary the quadric in a family.
In §2 we give the details of the construction. In §3 we describe how S and T vary with Λ. In §4 we study their dual sheaves. In §5 we describe how they restrict to a hyperplane section of Q and pull back to a cone on Q, and we prove a Horrocks-type criterion. In §6 we show that they are stable when Λ is maximal and properly semi-stable otherwise.
I was motivated to define spinor sheaves by studying a moduli problem on the complete intersection of four quadrics [1]. This work was supported in part by the National Science Foundation under grants nos. DMS-0354112, DMS-0556042, and DMS-0838210.

The Construction
Let V be a complex vector space equipped with a quadratic form q of rank at least 2, so the corresponding quadric hypersuface Q ⊂ PV is reduced. Let b be the symmetric bilinear form associated to q. Let be the Clifford algebra of q. If {v 1 , . . . , v n } is a basis of V then is a basis of Cℓ. Let W ⊂ V be an isotropic subspace, that is, one with q| W = 0, or equivalently PW ⊂ Q. Choose a basis w 1 , . . . , w m and let I be the left ideal I = Cℓ · w 1 · · · w m . Since W is isotropic, choosing a different basis just rescales the generator w 1 · · · w m by the determinant of the change-of-basis matrix, so I is independent of this choice. Since Cℓ is Z/2-graded, we can write I = I even ⊕I odd . We will always consider I in the category of graded left Cℓ-modules and maps that respect the grading.
A matrix factorization of q is a pair of N × N matrices ϕ and ψ, of polynomials in general but of linear forms in our case, such that An N × N matrix of linear forms is the same as a map O N PV (−1) → O N PV , and it will be more natural to work with the latter.
From the module I, define a map of vector bundles here we are regarding O PV (−1) as the tautological line bundle, that is, as a Then the compositions are just multiplication by q, so we have a matrix factorization of q. This link between Cℓ-modules and matrix factorizations was first studied by Buchweitz, Eisenbud, and Herzog [5] and has been rediscovered more than once [8, §7.4] [4]. It is interesting to note the resemblance to the Thom class in K-theory [11, App. C].
Finally, let S = coker ϕ and T = coker ψ. These are supported on Q, for ϕ and ψ are isomorphisms where q = 0. Since ϕ • ψ and ψ • ϕ are injective, ϕ and ψ are injective, so S and T have resolutions on PV where N = dim I even = dim I odd = 2 codim W −1 . From these it is easy to compute their cohomology. We ask how far S and T are from being vector bundles. Let K ⊂ V be the kernel of q and recall that the singular locus of Q is PK. Proposition 2.1. The restriction of S to PK ∩PW is trivial of rank 2 codim W −1 . If codim W > 1 then elsewhere on Q, S is locally free of rank 2 codim W −2 . The same is true of T .
Proof. For each v ∈ V , we want to know the rank of the linear map If v ∈ W ∩ K the the map is zero: any v ∈ K commutes with elements of Cℓ even and anti-commutes with elements of Cℓ odd , and any v ∈ W annihilates the generator w 1 · · · w m of I, so any v ∈ W ∩ K annihiliates I.
If v / ∈ W , choose a basis of V starting with v and ending with w 1 , . . . , w m . Then any element of I can be written uniquely as (vξ + η)w 1 · · · w m where v and w 1 , . . . , w m do not appear in ξ and η. Now v · (vξ + η)w 1 · · · w m = vηw 1 · · · w m , which is zero if and only if η = 0. Thus the rank of the map is 2 codim W −2 , so S| Q\PW is a vector bundle of rank 2 codim W −2 . But in §4 we will see that Ext >0 Q (S, O Q ) = 0, which implies that S is a vector bundle on the whole smooth locus of Q.
If codim W = 1 then Q = PW ∪ PW ′ , where W ′ is the other maximal isotropic subspace, and it is easy to check that S = O PW and T = O PW ′ when dim V is odd and vice versa when dim V is even.

Dependence on the Isotropic Subspace
In this section we first summarize how S and T vary with W and how they are related for W s of various dimensions, then prove the corresponding statements about graded Cℓ-modules, and finally show that the functor from modules to sheaves is fully faithful.
If q is non-degenerate then S and T are rigid, so varying W continuously leaves them unchanged. If dim W < 1 2 dim V , so W belongs to a connected family, then S ∼ = T . If dim W = 1 2 dim V , so W belongs to one of two families, then S ∼ = T , and switching W to the other family interchanges S and T . If W is maximal then when dim V is odd, S ∼ = T is the classical spinor bundle, and when dim V is even, S and T are the two spinor bundles.
In short then, on smooth quadrics, maximal isotropic subspaces give the classical spinor bundles, and non-maximal ones give direct sums of them.
If q is degenerate then S and T are not rigid in general, since by Proposition 2.1 we can recover W ∩K from them; but varying W continuously while keeping W ∩ K fixed leaves them unchanged. Let π : V → V /K be the projection, and recall that q descends to a non-degenerate form on and switching π(W ) to the other family (still keeping W ∩ K fixed) interchanges them. For example, consider a line PW on a rank 2 quadric surface: . If PW is not the cone line, it lies on one plane or the other and meets the cone line in a point, and these data determine the isomorphism classes of S and T : varying the line while keeping the point fixed leaves S and T unchanged, switching to the other plane interchanges them, and varying the point deforms them.
Our earlier comments on direct sums are generalized as follows: if W ′ is codimension 1 in W then there are exact sequences That is, if W ∩ K shrinks we get interesting extensions, but if π(W ) shrinks we just get direct sums. So while on smooth quadrics only maximal W s were interesting, on singular quadrics non-maximal W s may be interesting, but only if π(W ) is maximal.
To prove all this, we introduce the following group action. Let G be the subgroup of the group of units Cℓ × generated by the unit vectors, that is, by those u ∈ V with q(u) = 1, and let G act on V by reflections: This preserves q, for The spinor sheaves S and T are equivariant for the action of G even := G∩Cℓ even on Q, as we see from the commutative diagram ?
If q is non-degenerate then G is Pin(V, q), the central extension of the orthogonal group O(V, q) by Z/2, and G even is Spin(V, q) [11, §I.2]. When m < 1 2 dim V , G even acts transitively on the variety of m-dimensional isotropic subspaces, and when m = 1 2 dim V , G even acts transitively on each of its connected components and G odd interchanges them.
If q is degenerate, the natural map G → O(V, q) is not surjective, since O(V, q) acts transitively on K while G even acts as the identity on K and G odd acts as −1. If U ⊂ V is a subspace complementary to K then q| U is nondegenerate, and G contains Pin(U, q| U ). From this it is not hard to see that G can take W to W ′ if W ∩ K = W ′ ∩ K, and that G even can if in addition π(W ) and π(W ′ ) lie in the same family. Now if g ∈ G takes an isotropic subspace W ⊂ V to another one W ′ = gW g −1 , then right multiplication by g −1 takes I to I ′ : so if g ∈ G even then I ∼ = I ′ , and if g ∈ G odd then I ∼ = I ′ [1], where I ′ [1] means I ′ with the odd and even pieces interchanged. Thus we have proved If π(W ) and π(W ′ ) lie in the same family then I ∼ = I ′ . If they lie in opposite families then Inversely, Proof. For the first statement, let dim V /K = 2k. Then there is a basis v 1 , . . . , v n of V in which Observe that ξ := v 1 · · · v k annihilates every element of the associated basis of I except v k+1 · · · v 2k v 2k+1 · · · v 2k+l . Thus if dim W is even then ξ annihilates I odd but not I even , and vice versa if dim W is odd. For the second statement, we saw in the proof of Proposition 2.1 that W ∩ K = V ∩ Ann I, where the latter intersection takes place in Cℓ.
Proof. To see that the sequence is exact, choose a basis w 1 , . . . , w m of W ′ , and extend this to a basis of V ending with w, w 1 , . . . , w m ; then just as we argued in the proof of Proposition 2.1, an element of I ′ can be written as (ξ+ηw)w 1 · · · w m where w and w 1 , . . . , w m do not appear in ξ and η, and if this times w equals zero then ξ = 0. : Inversely, if W ∩ K = W ′ ∩ K then I ′ and I ⊕ I [1] have different annihilators, hence are not isomorphic.
To see that what we have proved about modules implies what we have claimed about spinor sheaves, we study the functor that sends a graded Cℓmodule I to a sheaf S on Q. It is indeed a functor, for a homogeneous map f : and hence a map on cokernels. The functor is exact. Kuznetsov [9,Proposition 4.9], working more generally with quadric fibrations, showed that it is fully faithful; we give a different proof: Proposition 3.4. The natural map Hom Cℓ (I, I ′ ) → Hom Q (S, S ′ ) is an isomorphism.
Proof. The inverse is essentially the map Hom Q (S, S ′ ) → Hom C (I odd , I ′ odd ), where the second object is vector space homomorphisms, given by taking global sections. This is injective because S is generated by global sections. The composition Hom Cℓ (I, I ′ ) → Hom C (I odd , I ′ odd ) sends f to f odd . This too is injective: a map f of graded Cℓ-modules is determined by f odd , for if m ∈ I even and v ∈ V has q(v) = 1 then f (m) = v 2 f (m) = vf (vm).
It remains to check that a linear map I odd → I ′ odd induced by a sheaf map S → S ′ is induced by a module map Taking global sections of (1) and its twist by O(1), we can augment this to 0 I * even ⊗ I ′
where the bottom row and the right column are exact. Thus Hom Q (S, S ′ ) is the set of A ∈ Hom C (I odd , I ′ odd ) for which there is a B ∈ Hom C (I even , I ′ even ) with Aϕ = ϕ ′ B; here we are thinking of A and B as matrices of complex numbers and ϕ and ϕ ′ as matrices of linear forms. Since ϕ ′ * is injective, such a B is unique. Multiplying Aϕ = ϕ ′ B by ψ on the right and ψ ′ on the left, we have is a matrix of linear forms, and plugging in any v ∈ V we get the map I → I given by left multiplication by v. The vs generate Cℓ, so ( B 0 0 A ) is in fact a homomorphism of Cℓ-modules, not just of vector spaces. Since the matrix is block diagonal, it respects the grading.

The Dual Sheaf
To understand the dual sheaf S * := Hom(S, O Q ), we work with some resolutions of S on Q. Restricting (1) to Q, we get → · · · on Q is exact because ϕψ is a matrix factorization of q. We can break it into exact sequences The former is a resolution of S by Hom Q (−, O Q )-acyclics, and applying Hom Q (−, O Q ) we get where ϕ * and ψ * are the transposes of ϕ and ψ. This is exact because ψ * ϕ * is also a matrix factorization of q. Thus S * is the cokernel of and Ext >0 Q (S, O Q ) = 0. That is, S * is not just the dual of S, but the derived dual. Also, we observe that S * * = S, that is, S is reflexive.
From all this we suspect that S * (1) is a spinor sheaf. In fact it is: If codim W is odd then S * ∼ = S(−1) as G even -equivariant sheaves. If codim W is even then S * ∼ = T (−1).
Proof. Let ⊤ be the anti-automorphism of Cℓ determined by (v 1 · · · v k ) ⊤ = v k · · · v 1 . Then I ⊤ is the right ideal w 1 · · · w m · Cℓ. The dual vector space I * is a right Cℓ-module via the action (f · ξ)(−) = f (ξ−). We will show that these two right modules are isomorphic up to a shift. The natural filtration of the tensor algebra descends to Cℓ and the associated graded pieces are F i /F i−1 = i V . In particular Cℓ/F dim V −1 is 1-dimensional, so by choosing a generator we get a linear form tr : Cℓ → C. The pairing Cℓ ⊗ Cℓ → C given by ξ ⊗ η → tr(ξη) is non-degenerate. If v ∈ V and ξ ∈ Cℓ then tr(vξ) = ± tr(ξv).
We claim that I * is generated by tr | I and is isomorphic as an ungraded module to I ⊤ . Since dim I * = dim I ⊤ , it suffices to check that w 1 · · · w m and tr | I have the same annihilator. If tr | I · ξ = 0, that is, if tr(ξηw 1 · · · w m ) = 0 for all η ∈ Cℓ, then tr(w 1 · · · w m ξη) = 0 for all η, so w 1 · · · w m ξ = 0; and conversely. Now tr | I has degree dim V (mod 2), and w 1 · · · w m has degree dim W (mod 2), so I * is isomorphic to I ⊤ or I ⊤ [1] according as codim W is even or odd. So if codim W is even then (4) becomes These isomorphisms are equivariant, as follows. In (3), an element g ∈ G even acts on I by left multiplication by g, so it acts on I * by right multiplication by g −1 and on I ⊤ by right multiplication by g ⊤ . But from the definition of G we see that g ⊤ g = 1.

Linear Sections and Cones
Any singular quadric can be described as a linear section of a smooth quadric or as a cone over a smooth quadric, so we would like to know what happens when spinor bundles are restricted to linear sections or pulled back via projection from the vertex of the cone. We will see that the latter gives spinor sheaves corresponding to maximal linear spaces, while the former gives spinor sheaves corresponding to linear spaces that are maximal in the smooth locus of Q. Thus our spinor sheaves corresponding to linear spaces that meet Q sing in interesting ways interpolate between the two obvious ways of extending spinor bundles to singular quadrics.
Proof. Note that transversality is necessary to make rank S ′ = rank S. Since we have some freedom to move W without changing S, it is not a large restriction. Let I ′ be the Cℓ(U )-module corresponding to W ∩ U . Let I be the Cℓ(V )module corresponding to W ; then I is also a Cℓ(U )-module since Cℓ(U ) is a subring of Cℓ(V ). Restriction is right exact, so we have so it suffices to show that I ∼ = I ′ [codim U ] as Cℓ(U )-modules.
Choose a basis u 1 , . . . , u l of W ∩ U and extend it to a basis u 1 , . . . u l , w l+1 , . . . , w m of W . Then I ′ = Cℓ(U ) · u 1 · · · u l I = Cℓ(V ) · u 1 · · · u l · w l+1 · · · w m and the map Together with Theorem 2.11 of [12] this implies that when q is non-degenerate and W is maximal, our S and T are indeed the classical spinor bundles.
It also allows us to prove an analogue of Horrocks' splitting criterion. Horrocks' criterion is a generalization of Grothendieck's theorem that every vector bundle on P 1 is a sum of line bundles. It states that a vector bundle E on P n is a sum of line bundles if and only if it is arithmetically Cohen-Macaulay (ACM), that is, H i (E(t)) = 0 for 0 < i < n and all t; it is proved by induction on n.
Ottaviani [13] showed that a bundle E on a smooth quadric 3-fold 2 is a sum of line bundles if and only if E and E ⊗ S are ACM, where S is the spinor bundle, and observed that the same induction argument carries the result to higherdimensional smooth quadrics. Ballico [2] observed that if a singular quadric Q is a linear section of a larger smooth quadric and S is the restriction of a spinor bundle, then the same induction argument still works. Here we observe that it works with any of our spinor sheaves: Proof. The "only if" statement follows from (1). For the "if" statement, we induct on dim Q. Suppose that S corresponds to an isotropic subspace W . If dim Q = 3 then Q is smooth and S is either the classical spinor bundle (if W is maximal) or the sum of several copies of it (if W is smaller).
If dim Q > 3, let H be a hyperplane transverse to PW and to Q sing . Let Q ′ = Q ∩ H, whose rank is again at least 5. From the exact sequence Applying Hom(F, −) to the sequence above we get Hom(F, E) → Hom(F, E| Q ′ ) → Ext 1 (F, E(−1)).
The third term vanishes because E is ACM and F is a sum of line bundles, so we can extend f to a mapf : F → E. Now the line bundles det F | Q ′ and det E| Q ′ are isomorphic, so det F and det E are isomorphic, so detf is a non-zero section of (det F ) −1 ⊗ det E ∼ = O Q , hence does not vanish, sof is an isomorphism.
We finish the section with cones: Observe that Q is a cone over a quadric Q ′ ⊂ P(V /U ), with vertex PU . Let p : Q \ PU → Q ′ denote projection from the vertex, and S ′ the spinor sheaf on Q ′ corresponding to W/U . Then Proof. Let I ′ be the Cℓ(V /U )-module corresponding to W/U , which is also a Cℓ(V )-module via the map Cℓ(V ) → Cℓ(V /U ). As usual let I the Cℓ(V )module corresponding to W . Since p * is right exact, as in the proof of Propo-

Stability
In this last section we show that S and T are stable if W is maximal and properly semi-stable otherwise. For background on stability we refer to Huybrechts and Lehn [6]. Recall that the slope of a torsion-free sheaf E on polarized variety X is It can be read from the Hilbert polynomial of E: if For example, from (1) we find that the slope of any spinor sheaf is 1.
If F ⊂ E is a proper, saturated subsheaf (that is, E/F is torsion-free) then either We say that E is semi-stable if µ(F ) ≤ µ(E) for all such F and stable if µ(F ) < µ(E). Every sheaf has a unique Harder-Narasimhan filtration where the quotients F i /F i−1 are semi-stable and µ(F i /F i−1 ) < µ(F i+1 /F i ). Every semi-stable sheaf has a Jordan-Hölder filtration where the quotients F i /F i−1 are stable and µ(F i /F i−1 ) = µ(E). This is not unique, but the associated graded object i=1 F i /F i−1 is. Two semi-stable sheaves whose Jordan-Hölder filtrations have the same associated graded object are called S-equivalent. A sheaf is called polystable if it is a direct sum of stable sheaves of the same slope.
If W is maximal then we will show in a moment that S and T are stable. If W ′ is codimension 1 in W then (2) gives a Jordan-Hölder filtration 0 ⊂ S ⊂ S ′ with S/0 = S and S ′ /S = T , so S ′ is properly semi-stable and S-equivalent to the polystable sheaf S ⊕ T , as is T ′ . If W ′′ is codimension 1 in W ′ the Jordan-Hölder filtration is slightly more complicated, but S ′′ and T ′′ are S-equivalent to S ⊕ S ⊕ T ⊕ T . In general the S-equivalence class of a spinor sheaf depends only on the dimension of the isotropic space.
Our proof that S and T are stable when W is maximal will use the fact that they are simple. We begin with a lemma: Lemma 6.1. If W is maximal then I is irreducible.
Proof. If dim V /K = 2k is even, there is a basis v 1 , . . . , v n of V in which q = x 1 x k+1 + · · · + x k x 2k and W = span(v k+1 , . . . , v n ). Let ξ = v k+1 · · · v n be the generator of I. Then any ξ ′ ∈ I different from zero is of the form ξ ′ = αv i1 · · · v i l ξ + terms of the same or shorter length, where α ∈ C is not zero and 1 ≤ i 1 < · · · < i l ≤ k. I claim that v i l +k · · · v i1+k ξ ′ = αξ. To see this, observe that if 1 ≤ i, j ≤ k then v i anti-commutes with v j+k when i = j and that v i+k ξ = 0, so left multiplication by v i l +k · · · v i1+k annihilates any basis vector not containing Thus any non-zero element of I generates I, so I is irreducible.
If dim V /K = 2k + 1 is odd, there is a basis v 0 , . . . , v n of V in which and W = span(v k+1 , . . . , v n ). Let ξ = v k+1 · · · v n be the generator of I. Let J ⊆ I be a graded submodule. By an argument similar to the one given above, for any non-zero ξ ′ ∈ J there are 1 ≤ i 1 < · · · < i l ≤ k such that v i l · · · v i1 ξ ′ = (α + βv 0 )ξ, where α, β ∈ C are not both zero. Since J is graded, it contains both αξ and βv 0 ξ. If α = 0 then ξ ∈ J; if β = 0 then v 0 · βv 0 ξ = βξ, so again ξ ∈ J, so J = I. Proposition 6.2. If W is maximal then S is simple, that is, Hom Q (S, S) = C. If dim V /K is even, π(W ) is maximal in V /K, and W ∩ K is codimension 1 in K, then again S is simple. Otherwise S is not simple.
Proof. The first statement is immediate from the previous lemma, Schur's lemma, and Proposition 3.4. For the second statement, let W ′ = W + K. Let J be a proper submodule of I, and consider the short exact sequence where v ∈ K \W . Since I ′ is irreducible, either J ∩I ′ = 0 or J ⊃ I ′ . If J ∩I ′ = 0 then Jv is isomorphic to J; since I ′ [1] is irreducible, either Jv = 0, so J = 0, or Jv = I ′ [1], so I = I ′ ⊕ J = I ′ ⊕ I ′ [1], which we know is not true. If J ⊃ I ′ then again either Jv = 0, so J = I ′ , or Jv = I ′ [1], so J = I. Thus the only proper submodule of I is I ′ , and the only proper quotient is I ′ [1]. Since these are not isomorphic, any homomorphism I → I is an isomorphism or zero, so again by Schur's lemma Hom Cℓ (I, I) = C.
For the third statement, if π(W ) is not maximal in V /K then S is a direct sum, hence is not simple. If W ∩ K is codimension 2 or more in K, choose W ′′ ⊃ W ′ ⊃ W with π(W ′′ ) = π(W ′ ) = π(W ); then the composition S ։ T ′ ։ S ′′ ֒→ S ′ ֒→ S is neither zero nor an isomorphism. If W ∩K is codimension 1 in K and dim V /K is odd, let W ′ = W + K; then the composition S ։ T ′ ∼ = S ′ ֒→ S is neither zero nor an isomorphism.
Theorem. Suppose that rank Q > 2. If W is maximal then S and T stable.
Proof. It suffices to show this for S. Suppose that a subsheaf F ⊂ S is invariant under the action of G even introduced in §3. If rank Q > 2 then G even acts transitively on the smooth locus of Q, so F is a vector bundle there. Let p ∈ Q be a smooth point and H ⊂ G even be the stabilizer of p; then according to Ottaviani [12], the representation of H on the fiber S| p is irreducible (recall that G even contains Spin(U, q| U ) for any U complementary to K). Thus either rank F = 0, so F = 0 since S is reflexive and hence torsion-free, or rank F = rank S.
Thus S has no invariant proper saturated subsheaves. The Harder-Narasimhan filtration is unique, hence invariant, so S is semi-stable. Consider the socle of S, that is, the maximal polystable subsheaf, which is necessarily saturated. This too is unique, hence invariant, hence is S; that is, S is a direct sum of stable sheaves. But S is simple, hence indecomposable, so S is stable.
If rank Q = 2 then Q is a union of hyperplanes H and H ′ and S is either O H or O H ′ , which are torsion and thus not eligible for slope stability, but they have no proper saturated subsheaves and thus are Gieseker stable. We excluded from the beginning the non-reduced case rank Q = 1.