Irreducibility of Lagrangian Quot schemes over an algebraic curve

Let $C$ be a complex projective smooth curve and $W$ a symplectic vector bundle of rank $2n$ over $C$. The Lagrangian Quot scheme $LQ_{-e}(W)$ parameterizes subsheaves of rank $n$ and degree $-e$ which are isotropic with respect to the symplectic form. We prove that $LQ_{-e}(W)$ is irreducible and generically smooth of the expected dimension for all large $e$, and that a generic element is saturated and stable. The proof relies on the geometry of symplectic extensions.


Introduction
Let C be a smooth algebraic curve of genus g ≥ 0 over C. A vector bundle W over C is called symplectic if there exists a nondegenerate skew-symmetric bilinear form ω : W ⊗ W → L for some line bundle L. Such an ω is called an L-valued symplectic form. A subsheaf E of W is called isotropic if ω| E⊗E = 0. By linear algebra, a symplectic bundle has even rank 2n and any isotropic subsheaf has rank at most n. An isotropic subbundle (resp., subsheaf) of rank n is called a Lagrangian subbundle (resp., Lagrangian subsheaf ). For information on semistability and moduli of symplectic bundles, see [1].
For vector bundles, Popa and Roth proved the following result on the irreducibility of Quot schemes. As a corollary, they showed that for sufficiently large d, the Quot scheme Q k,d (V ) is generically smooth of the expected dimension, and a general point of Q k,d (V ) corresponds to an extension 0 → E → V → V /E → 0 where E and V /E are stable vector bundles. A significant feature of this theorem is that it holds for an arbitrary bundle V , with no assumption of generality or semistability.
The main goal of this paper is to show the analogous result for Lagrangian Quot schemes of symplectic bundles (Theorem 4.1). However, the method of [12] does not appear to adapt in an obvious way: Given a symplectic bundle V of rank 2n and for a fixed vector bundle E of rank n, the space parameterizing Lagrangian subsheaves E ⊂ V is a locally closed subset of PH 0 (C, Hom(E, V )), whose irreducibility seems difficult to decide. This is discussed further at the beginning of §4.
We take instead a different approach: We exploit the geometry of symplectic extensions, together with deformation arguments, as developed in [2] and [6]. In particular, Proposition 4.5 gives a geometric interpretation for the statement that a nonsaturated Lagrangian subsheaf can be deformed to a subbundle. The connection between extensions and geometry is via principal parts, as developed in §3. This provides an alternative language toČech cohomology for bundle extensions over curves, and makes transparent the link between the geometric and cohomological properties of the extensions.
We remark that the same argument applies to the vector bundle case, and we expect that similar results can be obtained by these methods for other principal bundles.
We expect that the main result in this paper can be applied to solve the problem on counting maximal Lagrangian subbundles of symplectic bundles, as Holla [7] used the irreducibility of Quot schemes to count maximal subbundles of vector bundles. Also we expect that an effective version of the irreducibility result for semistable bundles would yield an effective base freeness (or very ampleness) result on the generalized theta divisors on the moduli of symplectic bundles, as in [12, §8] for vector bundles. We note that Theorem 4.1 does not give an effective bound on e but only the existence of a bound, mainly due to the existence statement in Lemma 4.3. It would be nice to have an effective and reasonably small uniform bound for semistable symplectic bundles.
Acknowledgements. The first and second authors were supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1A6A3A11930321 and NRF-2017R1D1A1B03034277 respectively). The third author sincerely thanks Konkuk University and the Korea Institute for Advanced Study for financial support and hospitality.
Notation. Throughout, C denotes a complex projective smooth curve of genus g ≥ 0. If W is a vector bundle over C and E ⊂ W a locally free subsheaf, we denote by E the saturation, which is a vector subbundle of W . The Quot scheme Q 0,t (F ) parameterizes all subsheaves E ⊂ F whose quotient F/E is a torsion sheaf of degree t. Since such quotients are called elementary transformations, we write Elm t (F ) := Q 0,t (F ).

Lagrangian Quot schemes
In this section, we define the Lagrangian Quot scheme of a symplectic bundle and study its tangent spaces.
Given a vector bundle V over C, the Quot scheme Q k,d (V ) parameterizes quotient sheaves of V of rank k and degree d; alternatively, subsheaves of V of rank rk V − k and degree deg V − d. Let W be a bundle of rank 2n which carries an L-valued symplectic form, where deg L = ℓ. Then from the induced isomorphism W ∼ = W * ⊗ L, we have deg W = nℓ. We denote by LQ −e (W ) the sublocus in Q n,e+nℓ (W ) consisting of Lagrangian subsheaves of degree −e and call it a Lagrangian Quot scheme.
In view of this, we shall abuse notation and write simply LQ −e (W ).
We recall some other important notions: For each integer e and each x ∈ C we have the evaluation map ev e x : Q n,e+nℓ (W ) Gr(n, W | x ) which sends a subsheaf E to the fiber E| x , when this is defined. Also, let LG(W ) be the Lagrangian Grassmannian bundle of W , that is, the subfibration of Gr(n, W ) whose fiber at x ∈ C is the Lagrangian Grassmannian LG(W | x ). Lemma 2.2. Let W be an L-valued symplectic bundle of rank 2n as above. If g ≥ 2 and e ≥ n(g−1−ℓ) 2 , then the locus LQ −e (W ) is a nonempty closed subset of Q n,e+nℓ (W ).

2
. For e > e 0 , we can take an elementary transformation of the Lagrangian subbundle of degree −e 0 to get a Lagrangian subsheaf of degree −e. This proves the nonemptiness.
For the closedness: Write Indet(ev e x ) for the indeterminacy locus of ev e x : which is a closed subset of Q n,e+nℓ (W ). It is easy to see that .
Remark 2.3. The genus assumption g ≥ 2 is imposed to get the sharp bound e ≥ n(g−1−ℓ) 2 for non-emptyness of LQ −e (W ). This bound is proven in [2] for g ≥ 2, but for the case g = 0 or 1, we still have an existence of a bound to guarantee the non-emptyness of LQ −e (W ).
We denote by LQ −e (W ) 0 the open sublocus of LQ −e (W ) corresponding to vector bundle quotients. The following is a generalization of [3,Lemma 4.3].
(a) Every irreducible component of LQ −e (W ) 0 has dimension at least Since a general member of Y corresponds to a section of π, there is a rational map Y LQ −e (W ) 0 defined on a nonempty open subset. As [σ(C)] is mapped to j, the image of Y lies inside Z. Clearly the map Y Z is generically injective, so we see that dim Z ≥ χ(C, L ⊗ Sym 2 E * ).
(b) Let α : E → W/E ∼ = E * ⊗ L represent a tangent vector to the Quot scheme Q n,e+nℓ (W ) at [j : E → W ]. For each x ∈ C, the section α defines an element α(x) ∈ T j(E|x) Gr(n, W | x ), and the deformation preserves isotropy of E if and only if α(x) is tangent to the Lagrangian Grassmannian LG(W | x ) ⊂ Gr(n, W | x ) for all x.
The result follows from the following description of the tangent space of the Lagrangian Grassmannian: Thus we have equality and LQ −e (W ) 0 is smooth at j.

Symplectic extensions
In this section, we recall or prove some facts on symplectic extensions which we will need later.
If F is a Lagrangian subbundle of a symplectic bundle W , then we have an induced extension 0 → F → W → F * ⊗ L → 0. An extension induced by a symplectic structure in this way will be called a symplectic extension.
Recall that any locally free sheaf V on C has a flasque resolution is the sheaf of sections of V with finitely many poles, and Prin (V ) = Rat (V )/V is the sheaf of principal parts with values in V .
Taking global sections, we have a sequence of Abelian groups A principal part p is represented by a collection (p x : x ∈ C) where p x ∈ Rat (V ) x and p x is regular for all but finitely many x. We have (p ′ x ) = (p x ) if and only if p ′ x − p x is regular for each x. For β ∈ Rat (V ), we denote by β the principal part β mod H 0 (C, V ). If p ∈ Prin(V ), we write [p] for the associated class in H 1 (C, V ).
3.1. Symmetric principal parts and symplectic extensions. Let F be any bundle of rank n. For V = L −1 ⊗ F ⊗ F and a principal part the transpose t p is defined by t p = ( t p x : x ∈ C). Then p is symmetric if t p = p, or equivalently p ∈ Prin(L −1 ⊗ Sym 2 F ). Note that this is stronger than the condition Now any p ∈ Prin(L −1 ⊗ F ⊗ F ) defines naturally an O C -module map F * ⊗ L → Prin (F ), which we also denote p. Suppose p is a symmetric principal part in Prin(L −1 ⊗ Sym 2 F ). Following [10, Chapter 6], we define It is not hard to see that this is an extension of F * ⊗ L by F . Now there is a canonical pairing , : Rat (F ) ⊕ Rat (F * ⊗ L) → Rat (L). By an easy computation (see the proof of [6, Criterion 2.1] for a more general case), the standard symplectic form on Rat (F ) ⊕ Rat (F * ⊗ L) restricts to a regular symplectic form on W p with respect to which the subsheaf F is Lagrangian. This shows that for each symmetric principal part p ∈ Prin(L −1 ⊗ Sym 2 F ) there is a naturally associated symplectic extension of F * ⊗ L by F . We now give a refinement of [ (a) there is an isomorphism of symplectic bundles ι : Proof. (a) As much of this proof is computational, we outline the main steps and leave the details to the interested reader.
Using the facts that F is isotropic and the form is antisymmetric and nondegenerate, one shows that there exist A ∈ Aut (F ) and B ∈ Rat (L −1 ⊗ ∧ 2 F ) such that the given symplectic form ω ′ on the sheaf W p ′ is Using in addition that the restriction of ω ′ to W p ′ is regular, one shows that Hence p := Ap ′ + 1 2 B is a symmetric principal part. Let now W p be defined as in (3.2). As mentioned above, the form ω in (3.3) restricts to a regular symplectic form on W p . A tedious but elementary calculation shows that Part (b) is proven exactly as for extensions of line bundles in [10, Lemma 6.6].

3.2.
Lagrangian subbundles in reference to a fixed symplectic extension. From (3.2), we obtain a splitting Rat (W ) = Rat (F ) ⊕ Rat (F * ⊗ L). This is a vector space of dimension rk (W ) over the field K(C) of rational functions on C.
If β ∈ Rat (Hom(F * ⊗ L, F )), we write Γ β for the graph of the induced map of K(C)-vector spaces Rat (F * ⊗ L) → Rat (F ). Abusing notation, we also denote by Γ β the associated sub-O C -module of Rat (F ) ⊕ Rat (F * ⊗ L).
Proposition 3.2. Let p ∈ Prin(L −1 ⊗ Sym 2 F ) be any symmetric principal part. Let W p be as in (3.2).
(a) There is a bijection between the Proof. Parts (a) and (b) follow from [6, Theorem 3.3 (i) and (iii)]. Note that as the symplectic form on W is given by (3.3), the α referred to in [6] is zero.
Moreover, under the bijection in (a) the set of liftings Γ β ′ ∩ W p with β ′ = β is in canonical bijection with the set of β ′ such that β ′ = β. By (3.1), this is a torsor over H 0 (C, L −1 ⊗ Sym 2 F ).

Remark 3.3.
In part (c) above, we characterize different liftings of Ker(q) for a fixed q ∈ Prin(L −1 ⊗ Sym 2 F ) with δ(W ) = [q]. In general, there can exist also distinct q, q ′ with [q] = δ(W ) = [q ′ ] and Ker(q) = Ker(q ′ ) as subsheaves of F * ⊗L. Such q and q ′ correspond to distinct β and β ′ , and hence different inclusions E ֒→ W . We shall study this phenomenon from a cohomological viewpoint in Lemma 3.6 and discuss its geometric meaning in Remark 3.11. It will be significant in the proof of Proposition 4.5.
We give a slight refinement of Lemma 3.1, essentially allowing us to choose convenient coordinates on W .
Proof. From Lemma 3.1 and Proposition 3.2, we may assume that W is an extension If ω and ω 0 are the standard symplectic forms (3.3) on W p and W p0 respectively, then an easy computation using the symmetry of β shows that ι * ω 0 = ω.
where z is a uniformizer at x on a neighborhood U and η 1 is some regular section of F | U which is nonzero at x.
Complete η 1 to a frame {η i } for F on U and let {φ i } be the dual frame for F * . Then the rincipal part p 0 (φ 1 ) ∈ Prin (F ) is represented by Hence in view of (3.2), a frame for W p on U is given by Now a frame over U for the subsheaf 0 ⊕ Ker(p 0 ) of W p0 is given by Writing (0, z · φ 1 ) in terms of the frame (3.6), we have From this we see that the images of (3.7) in W p | x are independent. Hence 0 ⊕ Ker(p 0 ) ֒→ W p0 is a vector bundle inclusion at x. This computation also shows that the intersection of the subbundles Γ 0 ∩ W p0 and F at x is the line spanned by 3.3. Isotropic liftings of an elementary transformation. Let W be a sym- The following result, generalizing Proposition 3.2 (c), provides the main idea to "linearize" the space of Lagrangian subsheaves of W which respects the fixed symplectic extension and elementary transformation.
Before starting the proof, let us indicate how the intersection of Hom(E, F ) and Proof. Suppose that j 1 : E → W and j 2 : E → W are two liftings of γ to Lagrangian subsheaves. Then each j i (E) is a Lagrangian subbundle. By Proposition 3.2 (a), there exist uniquely defined β 1 , Then we calculate Note that the definition of S γ depends only on γ, and does not make reference to an extension 0 → F → W → F * ⊗ L → 0.
(a) There is a short exact sequence where τ 1 is a torsion sheaf. In particular, S γ is locally free of rank 1 2 n(n+1). (b) There is a short exact sequence where τ 2 is a torsion sheaf. (c) If τ has reduced support, then τ 1 is isomorphic to τ . In particular, in this case deg(S γ ) = deg(L −1 ⊗ Sym 2 F ) + deg(τ ).

Proof. (a) From the sheaf inclusion
As moreover we have rk (S γ ) = rk (L −1 ⊗ Sym 2 F ). The statement follows.
(c) Since the support of τ is reduced, so is that of the torsion sheaf Therefore, at each x ∈ Supp(τ ), the sheaf E * is locally spanned by where {η 1 , . . . , η n } is a suitable local basis of F and λ a local generator of L −1 , and z is a uniformizer at x. Then a local basis of E * ⊗ F is given by Thus a local basis of S γ is given by Therefore, in this case τ 1 is a sum of torsion sheaves of degree 1, each supported at one of the points x ∈ Supp(τ ). The statement follows.

3.4.
A geometric criterion for lifting. Throughout this subsection, we assume that h 1 (C, L −1 ⊗ Sym 2 F ) = 0. Let F → C be a bundle of rank n, and consider the scroll π : PF → C. By Serre duality and the projection formula, there is an isomorphism Thus we obtain a natural map ψ : PF PH 1 (C, L −1 ⊗ Sym 2 F ) with nondegenerate image.
We shall use an explicit description of ψ, given in [2, §2]. For each x ∈ C, there is a sheaf sequence Taking global sections, the associated long exact sequence is a subsequence of (3.1) for V = L −1 ⊗ Sym 2 F . The following is easy to check by explicit computation:  The following is a partial generalization of [11, Proposition 1.1], and was used extensively in [2] and [3]. Proof. For 1 ≤ i ≤ t, let z i be a uniformizer at x i . By Proposition 3.2, the subsheaf E ⊂ F * ⊗ L lifts to a Lagrangian subsheaf of W if and only if δ(W ) can be defined by a symmetric principal part q ∈ Prin(L −1 ⊗ Sym 2 F ) such that E ⊆ Ker (q : F * ⊗ L → Prin (F )). In view of (3.10), such a q must satisfy where µ 1 , . . . , µ t are scalars and λ i is a generator of L −1 near x i , and by abuse of notation, we write η i for a local section of F which spans the line η i ∈ PF | xi . By Lemma 3.8, the projectivization of the set of cohomology classes defined by such q is precisely the linear span of the ψ(η i ) in PH 1 (L −1 ⊗ Sym 2 F ). The statement follows.
Remark 3.12. The lifting of a fixed γ : E → F * ⊗ L corresponding to the principal part (3.11) is a vector bundle inclusion if and only all the µ i are nonzero. If, say, µ 1 = 0 then, by the criterion, a strictly larger subsheaf E 1 lifts to W , fitting into the diagram This illustrates the link between the secant stratification of PH 1 (C, L −1 ⊗ Sym 2 F ) and the Segre invariants of the extensions, which was investigated in [2] and [3]. This phenomenon will appear in Proposition 4.5 when we wish to deform nonsaturated subsheaves to saturated ones. Proof. The proof of Lemma 3.7 (c) shows that S γ is an elementary transformation where the η k and λ k are as in Criterion 3.10. In view of Lemma 3.8, the lemma follows from the associated long exact sequence is Remark 3.14. Suppose that h 0 (C, L −1 ⊗ Sym 2 F ) = 0 and consider again the situation of Remark 3.11. Then by exactness of the above sequence, we see that H 0 (C, S γ ) is the module of syzygies of the points ψ(η k ) in PH 1 (C, L −1 ⊗ Sym 2 F ).
In [2], a principal part q ∈ Prin(Sym 2 F ) of degree t was said to be general if Im (q) has reduced support on C. This definition can be extended in an obvious way to Prin(L −1 ⊗ Sym 2 F ). Clearly, q is general in this sense if and only if it is of the form (3.11) for some collection η 1 , . . . , η t . An elementary transformation 0 → E → F * ⊗ L → τ → 0 is defined to be general if E ∼ = Ker(q) for some general principal part q; equivalently, if τ has reduced support on C.
When h 1 (C, L −1 ⊗ Sym 2 F ) > 0, we shall also use a stronger notion of "general" principal part and elementary transformation: We shall often require in addition that the images of the points η i by some map Ψ : PF P N are in general position, meaning that for each k ≤ N + 1, any k points among the η i span a P k−1 .

Irreducibility of Lagrangian Quot schemes
Let W be an an L-valued symplectic bundle of rank 2n, where deg L = ℓ. In general, the Lagrangian Quot schemes LQ −e (W ) can be reducible, and also there may be irreducible components whose points all correspond to non-saturated subsheaves. In this section, we shall prove the following theorem, showing that for sufficiently large e, these phenomena disappear. However, when W is a symplectic bundle, isotropic subsheaves [j : E → W ] form a locally closed subset of H 0 (C, Hom(E, W )). This seems to be a nonlinear subvariety, whose irreducibility does not follow as easily as in the vector bundle case.
To overcome this difficulty, we use auxiliary Lagrangian subbundles F of W of degree −f ≫ −e. It turns out that the Lagrangian subsheaves E can be parameterized in a linear way if one also records how they are related to a fixed F . We note that other compactifications of LQ −e (W ) 0 have also been studied; more generally, generalizations of Quot schemes to principal G-bundles: Hilbert schemes of sections of LG(W ) as in [9] and moduli of stable maps to LG(W ) as in [8] and [12]. One attractive feature of LQ −e (W ) is that it is naturally contained in the usual Quot scheme, so inherits a universal family of sheaves. This will be used in the proof of our main theorem.
We begin with two lemmas.

Lemma 4.3. Let W be a symplectic bundle. There exists an integer
Proof. Fix y = x ∈ C. By the proof of Ramanathan [ Choosing a suitable frame s 1 , . . . , s 2n for U × C 2n , we may assume that the symplectic form on W is taken into the standard symplectic form on C 2n at each point. Since C has dimension one, there exists an integer k such that each s i : Now each Lagrangian subspace Λ ∈ LG(C 2n ) determines a Lagrangian subbundle of W | U . As C has dimension 1, this extends uniquely to a Lagrangian subbundle of W , which has degree at least −nk. In this way we obtain an injective morphism LG(C 2n ) ֒→ LQ −nk (W ). Pulling back the universal subsheaf over LQ −nk (W ) × C to LG(C 2n ) × C, we obtain an exact sequence 0 → E → π * C W → Q → 0 of coherent sheaves flat over LG(C 2n ). By flatness, the degree of the torsion subsheaf of Q Λ is semicontinuous in Λ. We take f (W ) ≤ nk to be the generic value such that the saturation of a generic E Λ is a Lagrangian subbundle of degree −f (W ).
The easy proof of the next lemma is left to the reader. Given an element [j : E → W ] of Q F , by composing with π : W → F * ⊗ L we get an elementary transformation j = π • j : E → F * ⊗ L. The association j → π • j = j defines a morphism π * : Q F → Elm e+f +nℓ (F * ⊗ L).
To ease notation, we set t := e + f + nℓ.
Next, let Q • F be the open subset of Q F of subsheaves [j : E → W ] such that (i) E is saturated in W ; that is, j is a vector bundle injection; (ii) (F * ⊗ L)/ j(E) ∈ Elm t (F * ⊗ L) has reduced support; and (iii) h 1 (C, S j ) = 0.
If h 1 (C, L −1 ⊗ Sym 2 F ) = 0 then (iii) is immediate from (3.12). Otherwise, by Lemma 3.13, property (iii) is equivalent to the points η 1 , . . . , η t ∈ PF corresponding to the elementary transformation E ⊂ F * ⊗ L being in general position in Note also that the conditions (ii) and (iii) depend only on the map E → F * ⊗ L, and not a priori on W . The following key result guarantees the nonemptiness of Q • F for sufficiently large e in the strongest sense.
As proof of this proposition is rather involved, let us indicate the strategy before starting into the details. By Lemma 3.4, we can assume that W = W p0 for some symmetric principal part p 0 ∈ Prin(L −1 ⊗ Sym 2 F ) and E = Γ 0 ∩ W p0 ∼ = Ker(p 0 ) ⊆ F * ⊗ L.
• Step 1: We construct an explicit one-parameter deformation {p s } of p 0 over a disk ∆ ⊂ C such that [p s ] = δ(W ) for all s ∈ ∆, but for s = 0, the principal part p s has degree t and is general in the sense of §3.5. • Step 2: We show that Ker (p s : F * ⊗ L → Prin (F )) defines a family E of elements of Elm t (F * ⊗ L) with properties (ii) and (iii) for s = 0. • Step 3: We construct a lifting of E to a family of degree −e Lagrangian subsheaves of W with E 0 = E and E s saturated for s = 0.
Step 1 uses the geometric interpretation of H 1 (C, L −1 ⊗ Sym 2 F ) set up in §3. This will be further explained in Remark 4.6. Steps 2 and 3 are more technical. If E is not saturated, then the degree of the variable principal part p s jumps at s = 0, and the issue of flatness requires care.
Proof. In the proof, we simplify the notation by putting L ∼ = O C , since L does not seriously affect the argument. Writing deg F = −f (W ) = −f as above, set Now suppose E ∈ Q F \Q • F . Then the saturation E is a Lagrangian subbundle of W , of degree −ē ≥ −e. By Lemma 3.4, we may assume that W = W p0 as defined in (3.2), and E = Γ 0 ∩ W p0 ∼ = Ker(p 0 : F * → Prin (F )).
where the z i (s) are local coordinates at distinct points x i (s) of C, and η i (s) are local sections of F near x i (s). If E is non-saturated then, after deforming E inside the closed irreducible sublocus Elm e−ē (E) of Q F \Q • F if necessary, we may assume that E/E is supported at distinct points y 1 , . . . , y e−ē disjoint from Supp(p 0 ). Then via the inclusion E → F * , the elementary transformation E ⊂ E is defined by a uniquely determined choice of e −ē points ζ 1 , . . . , ζ e−ē of PF . For 1 ≤ j ≤ e −ē, let w j be a local coordinate at y j near w j . Abusing notation as before, for each j we consider the principal part ζj ⊗ζj wj .
If h 1 (C, Sym 2 F ) > 0, then the class ζj ⊗ζj wj lies over the image of ζ j in PH 1 (C, Sym 2 F ). In view of Lemma 3.8 and since ψ(PF ) is nondegenerate, perturbing the η i (s) and deforming E inside Elm e−ē (E) again if necessary, we may assume that for each s = 0, the e + f points are in general position in H 1 (C, Sym 2 F ). If h 1 (C, Sym 2 F ) = 0 then this perturbation is not necessary; it suffices that the x i (s) and y j be distinct. Now denote by k the largest order of pole in s of the η i (s) at s = 0. Let µ = (µ 1 , . . . , µē +f ) be coordinates on Cē +f . If E is non-saturated, let ν = (ν 1 , . . . , ν e−ē ) be coordinates on C e−ē . We define a family of principal parts by Using the map Prin(Sym 2 F ) → H 1 (C, Sym 2 F ), we obtain a linear map of affine bundles . This concludes Step 1.
Next, the family of principal parts {p s } gives rise to a family of elementary transformations of F * as follows. Write ∆ * := ∆\{0} and consider the family E of sheaves over ∆ * × C given by This is flat over ∆ * , because for s = 0, the Hilbert polynomial of Ker(p s ) is constant with respect to s. We claim that the flat limit E 0 of E at s = 0 is E.
which is exactly E. Hence we can extend E to a flat family on all of ∆ with E 0 ∼ = E as points of Elm e+f (F * ). (It is important to note that for s = 0 the containment E 0 = E ⊆ Ker(p 0 ) ∼ = E may be strict.) Write now γ : E → π * C (F * ) for the inclusion of sheaves over ∆ × C. For s = 0, by construction the torsion sheaf F * γs(Es) has reduced support on C and the corresponding points of F are in general position. Thus for s = 0, the subsheaf E s ⊂ F * satisfies properties (ii) and (iii) in the definition of Q • F . This completes Step 2.
For the rest: There is a complex of sheaves over ∆: where the second and third terms are quasi-coherent but not coherent. The variable principal part p s is a global section of (π ∆ ) * π * C Prin (Sym 2 F ). As by Step 1 we have [p s ] ≡ δ(W ) = [p 0 ], we may choose a global section β s of (π ∆ ) * π * C Rat (Sym 2 F ) lifting the difference π * C (p 0 ) − p s , hence satisfying (4.4) p s = p 0 − β s for each s ∈ ∆.
For v ∈ E s , we have γ s (v) ∈ Ker(p s ) by definition. By (4.4), then, Thus by the description (3.2) we have J s (E s ) ⊆ π * C W p0 . Hence J is a lifting of γ to π * C W . Clearly in fact J s (E s ) ⊆ Γ βs ∩ W p0 . It remains to show that J s (E s ) is saturated for s = 0. By Proposition 3.2 (a), it will suffice to show that we have equality J s (E s ) = Γ βs ∩ W p0 for s = 0. One direction has been shown above. Conversely, suppose (β s (v), v) ∈ Γ βs ∩ W p0 . Then by (3.2) we have β s (v) = p 0 (v), so v ∈ Ker(p−β s ), which by (4.4) is exactly Ker(p s ). But since s = 0, we have Ker(p s ) = γ s (E s ) (cf. (4.3 in Step 2), so v = γ s (v ′ ) for some v ′ ∈ E s . Thus Hence we have equality J s (E s ) = Γ βs ∩ W p0 , as required. This concludes Step 3. (Note that if h 0 (C, Sym 2 F ) = 0 then β is not unique, but an alternative choice β ′ also satisfies Γ β ′ s ∩ W p0 saturated, by Proposition 3.2 (c).) In summary, we have exhibited an irreducible family of elements of LQ −e (W ) containing [E → W ] and of which a general element belongs to Q Remark 4.6. The deformation above is most naturally understood from the point of view of secant geometry. For simplicity, assume that ψ : PF PH 1 (C, Sym 2 F ) is generically an embedding and that E ⊂ F * is a general elementary transformation corresponding to e + f > h 1 (C, Sym 2 F ) general points of PF . By Criterion 3.10, if E is non-saturated in W then δ(W ) lies on the secant spanned by (ē + f ) < (e + f ) of these points. Moving inside the family E then corresponds to perturbing the linear combination defining δ(W ) to be nonzero at all e + f points, so as to obtain saturated subsheaves (cf. Remark 3.12).
is a quotient of H 1 (C, S j ). As the latter space is zero by definition of Q • F , the statement follows from Propositions 2.4 (c) and 4.5. Proof. For [j : E → W ] ∈ Q • F , by Lemma 3.6, the fiber π −1 * j has dimension h 0 (C, S j ). Hence the image of π * | (Q • F ) 1 has dimension equal to h 0 (C, L ⊗ Sym 2 E * ) − h 0 (C, S j ).
By Proposition 4.5, for a general j in the image of π * we can assume that the torsion sheaf (F * ⊗ L)/ j(E) has reduced support and h 1 (C, S j ) = 0. Together with the vanishing result in Corollary 4.7, a Riemann-Roch calculation shows that h 0 (C, L ⊗ Sym 2 E * ) − h 0 (C, S j ) = nt = dim Elm t (F * ⊗ L).
Proposition 4.9. For e ≥ e 1 (W ), the locus Q F is irreducible.
Proof. By Proposition 4.5, it suffices to show that Q • F is irreducible. Suppose (Q • F ) 1 and (Q • F ) 2 were distinct irreducible components of Q • F . By Proposition 4.8, the restriction of π * to either component is dominant. By Lemma 3.6, the fiber π −1 * j is an open subset of a torsor over H 0 (C, S j ). In particular, it is irreducible. Therefore, the two components would have to intersect along a dense subset of a generic fiber. But this would contradict the smoothness of Q • F proven in Corollary 4.7. Thus Q • F is irreducible. If e ≥ f , then the intersection Q e F,π ∩ Q e F ′ ,π ′ is nonempty.
Proof. (a) Let E be any Lagrangian subsheaf of W . By Lemma 4.3, for general x ∈ C we can find a Lagrangian subbundle F of degree −f intersecting E| x in zero.
Thus [E → W ] belongs to Q F . (b) We must find a Lagrangian subsheaf E of degree −e intersecting both F and F ′ generically in rank zero. Choose a general point y ∈ C. Then a general Λ ∈ LG(W | y ) intersects both F | y and F ′ | y in zero. Since e ≥ f = f (W ), by Lemma 4.3 we can find a Lagrangian subsheaf E of degree −e whose saturation E has degree −f and satisfies E| y = Λ. Then [E → W ] is a point of Q F ∩ Q F ′ . By Proposition 4.10 (a), the loci Q F = Q e F,π cover LQ −e (W ). By Proposition 4.9, each Q F is dense in exactly one component of LQ −e (W ), which by Proposition 4.10 (b) must be the same component for all F . Therefore, LQ −e (W ) has only one irreducible component.
Regarding the stability of a general element of LQ −e (W ) as a vector subbundle: If t = e + f + nℓ ≥ n 2 (g − 1) + 1, then a general stable bundle E of degree −e occurs as an elementary transformation of F * ⊗ L. By Proposition 4.8, if we assume that e ≥ max{e 1 (W ), n 2 (g − 1) + 1 − f − nℓ} then a general element of Elm t (F * ⊗ L) lifts to W . Hence, since LQ −e (W ) is irreducible, a general E ∈ LQ −e (W ) is a stable vector bundle.
In analogy with [12, Proposition 6.3], Theorem 4.1 implies immediately the following: Corollary 4.12. If g ≥ 2, then every symplectic bundle W of rank 2n ≥ 2 can be fitted into a symplectic extension 0 → E → W → E * ⊗ L → 0 where E is a stable bundle.