COMPATIBLE POISSON BRACKETS ASSOCIATED WITH ELLIPTIC CURVES IN G (2 , 5)

. We prove that a pair of Feigin-Odesskii Poisson brackets on P 4 associated with elliptic curves given as linear sections of the Grassmannian G (2 , 5) are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of G (2 , 5).


Introduction
We work over an algebraically closed field k of characteristic 0.
In this paper we continue to study compatible pairs among the Poisson brackets on projective spaces introduced by Feigin-Odesskii (see [1,10]).Their construction associates with every stable vector bundle V of degree n > 0 and rank k on an elliptic curve E, a Poisson bracket on the projective space PH 0 (E, V) * .We refer to such Poisson brackets as FO brackets of type q n,k .
Two Poisson brackets are called compatible if the corresponding bivectors satisfy [Π 1 , Π 2 ] = 0 (equivalently, any linear combination of these brackets is again Poisson).In [9], Odesskii and Wolf discovered 9-dimensional spaces of compatible FO brackets of type q n,1 on P n−1 for each n ≥ 3. Their construction was interpreted and extended in [3], where the authors showed that one gets compatible FO brackets if the elliptic curves are anticanonical divisors on a surface S and the stable bundles on them are restrictions of a single exceptional bundle on S that forms an exceptional pair with O S (see [3,Theorem 4.4]).One can ask whether any two compatible FO brackets of type q n,k on P n−1 appear in this way.In [7] we have shown that this is the case for k = 1 (for some specific rational surfaces containing normal elliptic curves in projective spaces).In the present work, we consider the case of FO brackets of type q 5,2 on P 4 .Note that the question of finding bi-Hamiltonian structures with brackets of type q 5,2 was raised by Rubtsov in [11].
Let V be a 5-dimensional vector space.Consider the Plücker embedding It is well known that for a generic 5-dimensional subspace W ⊂ 2 V the corresponding linear section is an elliptic curve.Furthermore, if U ⊂ V ⊗ O is the universal subbundle on G(2, V ), then one can check that the restriction V W := U ∨ | E W is a stable bundle of rank 2 and degree 5 on E W (see Lemma 2.2.1 below).Thus, we have the corresponding Feigin-Odesskii bracket of type q 5,2 on PH 0 (E W , V W ) * .Furthermore, one can check that the restriction map is an isomorphism (see Lemma 2.2.1).Thus, we get a Poisson bracket Π W on PV (defined up to a rescaling).
On the other hand, we have a natural GL(V )-invariant map π 5,2 : constructed as follows.
Note that we have a natural isomorphism V ≃ H 0 (PV, T (−1)), hence we get a natural map V ⊗ O(1) → T , and hence, the composed map on PV .Taking the 5th exterior power of this map, we get a map where we used the identification det 2 T ≃ det 3 (T ).Note that we have a nondegenerate pairing given by the exterior product, Theorem A. For every 5-dimensional subspace W ⊂ 2 V , such that E W := G(2, V ) ∩ PW is an elliptic curve, one has an equality for some trivializations λ W ∈ 5 W and δ ∈ det 2 (V ).
Theorem A is deduced from the existence of a formula for Π W , depending linearly on the Plücker coordinates of W (which follows from the results of [3]), combined with a representationtheoretic argument employing the fact that the construction of Π W is GL(V )-equivariant.

Theorem B.
(i) For 5-dimensional subspaces W, W ′ ⊂ 2 V such that E W and E W ′ are elliptic curves, the Poisson brackets Π W and Π W ′ are compatible if and only if dim W ∩ W ′ ≥ 4.
(ii) For any collection (W i ) of 5-dimensional subspaces in 2 V , the brackets (Π W i ) are pairwise compatible if and only if either there exists a 6-dimensional subspace U ⊂ 2 V such that each W i is contained in U , or there exists a 4-dimensional subspace K ⊂ 2 V such that each W i contains K.
The idea of proof is to analyze the vanishing [Π W 1 , Π W 2 ] = 0 near a sufficiently generic point where Π W 1 vanishes.An important ingredient of the proof is a 2-dimensional distribution on G(2, V ) associated with W ⊂ 2 V : it corresponds to the rational map from G(2, V ) to P 4 obtained as the composition of the Plucker embedding with the linear projection to P( 2 V /W ) (see Section 3.3).The analysis of the vanishing of the Schouten bracket is used to prove that the elliptic curve E W 1 is everywhere tangent to the distribution associated with W 2 , which implies the result.
Corollary C. The maximal dimension of a linear subspace of Poisson brackets on P(V ), where dim V = 5, spanned by some FO brackets Π W of type q 5,2 , is 6.
Theorems A and B suggest the following Conjecture D. Let W ⊂ 2 V be a 5-dimensional subspace such that E W is an elliptic curve.Consider the subspace V (the quotient of the latter subspace by 5 W is exactly the image of the tangent space to the Grassmannian G 5, 2 V under Plücker embedding).Then the subspace of ξ ∈ 5 2 V satisfying [π 5,2 (ξ), Π W ] = 0 coincides with T W + ker(π 5,2 ).
Note that we know the inclusion one way: the subspace T W is spanned by 5 (W ′ ) such that dim(W ′ ∩W ) ≥ 4 and E W ′ is an elliptic curve, and by Theorems A and B, π 5,2 5

Feigin-Odesskii Poisson brackets of type q n,k
Let E be an elliptic curve, with a fixed trivialization η : O E → ω E , V a stable bundle on E of rank k and degree n > 0. We consider the corresponding Feigin-Odesskii Poisson bracket Π = Π E,V of type q n,k on the projective space PH 1 E, V ∨ defined as in [10].
We will need the following definition of Π in terms of triple Massey products.For nonzero ϕ ∈ H 1 E, V ∨ , we denote by ⟨ϕ⟩ the corresponding line, and we use the identification of the cotangent space to ⟨ϕ⟩ with ⟨ϕ⟩ ⊥ ⊂ H 0 (E, V) where we use the Serre duality where M P denotes the triple Massey product for the arrows

Formula for a family of complete intersections
Let X be a smooth projective variety of dimension n, C ⊂ X a connected curve given as the zero locus of a regular section F of a vector bundle N of rank n − 1, such that det(N ) −1 ≃ ω X .Then the normal bundle to C is isomorphic to N | C , so by the adjunction formula, ω C is trivial.Thus, if C is smooth, it is an elliptic curve.Assume that P is a vector bundle on X, such that the following cohomology vanishing holds: We have the following Koszul resolution for O C : Here the differential δ i (F ) is given by the contraction with F ∈ H 0 (X, N ), so it depends linearly on F .
(i) The natural restriction map H 0 (X, P ) → H 0 (C, P | C ) and the map are isomorphisms.These maps are dual via the Serre duality isomorphisms (ii) Assume in addition that End(P ) = k and we have the following vanishing: Then the bundle P | C is stable.
Proof .(i) This is obtained from the Koszul resolution of O C .For example, the space H 0 (P ⊗ O C ) is computed by tensoring this resolution with P and using the spectral sequence Proof .The computation is completely analogous to that of [8, Proposition 3.1], so we will only sketch it.First, one shows that our Massey product can be computed as the triple product m 3 for the arrows given by s 2 , ϕ and s 1 .Then we use resolutions • N ∨ → O C and • N ∨ ⊗ P → P | C .Thus, we have to calculate the following triple product in the category of twisted complexes: where we view ϕ as a morphism of degree 1 from P to the twisted complex i N ∨ [i].Now, the result follows from the formula for m 3 on twisted complexes (see [5,Section 7.6]).■

Conormal Lie algebra
Let V be a stable bundle of positive degree on an elliptic curve E, with a fixed trivialization of ω E , and consider the corresponding FO bracket Π on the projective space X = PH 0 (V) * = P Ext 1 (V, O).Recall that for every point x of a smooth Poisson variety (X, Π) there is a natural Lie algebra structure on where we consider Π x as a map T * x X → T x X.We call g x the conormal Lie algebra.In the case when Π vanishes on x, we have g x = T * x .Let us consider a nontrivial extension . By Serre duality, we have the corresponding hyperplane ⟨ϕ⟩ ⊥ ⊂ H 0 (V), and we have an identification 3) The following result was proved in [2].
Theorem 2.3.1.The above map induces an isomorphism of Lie algebras from End V /⟨id⟩ to the conormal Lie algebra of Π at the point ϕ.
Note that in particular, the subspace (im Π x ) ⊥ ⊂ ⟨ϕ⟩ ⊥ is equal to the image of the map (2.3).

Proof of Theorem
Proof .Let us denote by F the variety of flags L ⊂ W ⊂ 2 V , where dim(L) = 3, dim(W ) = 5, such that PL ∩ G(2, V ) ̸ = ∅.We claim that F is irreducible of dimension ≤ 30.Note that we have a proper closed subset Z ⊂ F consisting of (L, W ) such that dim(PW ∩ G(2, V )) ≥ 2 (as an example of a point in F \ Z, we can take W such that E W = PW ∩ G(2, V ) is an elliptic curve and pick PL ⊂ PW intersecting E W ). Since Z fibers over Z with fibers G(3, 5), our claim would imply that dim Z = dim Z + 6 < 30, i.e., dim Z < 24, as required.
To estimate the dimension of F , we observe that we have a fibration Namely, as a bundle P on X we take U ∨ , the dual of the universal subbundle.We can view the embedding where . It is easy to see that we have a GL(V )-invariant identification Thus, by adjunction we get an isomorphism 4 ⊗ det(W * ), we can rewrite this as The vanishings (2.1) and (2.2) in this case follow from the well known vanishings (see [4]).Thus, Proposition 2.2.2 gives a formula for Π W .This shows that the association W → Π W gives a regular morphism Furthermore, we claim that Indeed, we have a family of Gorenstein curves π : where Z was defined in Lemma 3.1.1,such that Indeed, this is implied by the argument leading to (3.1), which works for any curve (not necessarily smooth) cut out by PW in G(2, V ).This family of curves is equipped with a family of vector bundles V (the pull-back of we can view the corresponding constant family of projective spaces PV × B as the coarse moduli space of a substack in the relative moduli of complexes on C. Now [3, Proposition 4.1] implies that the relation To show that this map coincides with π 5,2 , up to a constant factor, it remains to show that the space Hom GL(V ) Using the Littlewood-Richardson rule, we deduce where Σ λ denotes the Schur functor associated with a partition λ.It follows that On the other hand, the decomposition of the plethysm e 5 • e 2 (see [6, Section I.8, Example 6, p. 138]) shows that Σ 3,3,2,2 (V ) appears with multiplicity 1 in the GL(V )-representation 5 2 V .This implies the claimed assertion about GL(V )-maps.■ 3.2 Rank stratification for a bracket of type q 5,2 Let E be an elliptic curve, V be a stable vector bundle of rank 2 and degree 5. We consider the FO bracket Π on the projective space P Ext 1 (V, O) ≃ PH 0 (V) * .We want to describe the corresponding rank stratification of PH 0 (V) * = P 4 .More precisely, Π is generically nondegenerate, and we are going to determine the degeneration locus D E ⊂ P 4 (where rk Π ≤ 2) and the zero locus S E of Π.
For every point p ∈ E, we consider the subspace Λ p := V| * p ⊂ H 0 (V) * and the corresponding projective line PΛ p ⊂ PH 0 (V) * .Recall that the rank of Π at a point corresponding to an extension V is equal to 5 − dim End V (see [3,Proposition 2.3]).(i) The bracket Π vanishes at the point of P Ext 1 (V, O) corresponding to an extension if and only if this extension splits under O → O(p) for some point p ∈ E, which happens if and only if V ≃ O(p) ⊕ V ′ , where V ′ is semistable of rank 2 and degree 4. Furthermore, in this case dim End(V ′ ) = 2, so V ′ is either indecomposable, or V ′ ≃ L 1 ⊕ L 2 , where L 1 and L 2 are nonisomorphic line bundles of degree 2.
(ii) The bracket Π has rank ≤ 2 if and only the corresponding extension V is unstable, or equivalently, there exists a line bundle L 2 of degree 2 such that the extension splits over the unique embedding L 2 → V.In other words, the extension class comes from a subspace of the form where we use the unique embedding L 2 → V and consider the induced embedding H 0 (L 2 ) → H 0 (V).
(iii) Each plane PW L 2 ⊂ PV is a Poisson subvariety, and there is an embedding of the curve E into PW L 2 by a degree 3 linear system, so that PW L 2 \ E is a symplectic leaf.
Proof .(i) Suppose a nontrivial extension It follows that dim End(V ′ ) ≥ 2, and so Hence, Π E vanishes on the points of the line PΛ p ⊂ PV , and we have dim End(V ′ ) = 2, which means that either V ′ is indecomposable or Conversely, assume Π vanishes at the point corresponding to V, so dim End V = 5.Then HN-components of V cannot be three line bundles (since they would have to have different positive degrees that add up to 5), so V = L ⊕ V ′ where L is a line bundle and The case deg(L) = 1 leads to the locus discussed above.If deg(L) = 2 and deg(V ′ ) = 3 then dim Hom(V ′ , L) = 1, so we get dim End(V ′ ) = 3 which is impossible.If deg(L) ≥ 3, then deg(V ′ ) ≤ 2 and dim Hom(V ′ , L) ≥ 4, so dim End(V) > 5, a contradiction.
(ii) The rank of Π is ≤ 2 at V if and only if dim End V ≥ 3. Clearly, such V has to be unstable.Conversely, any unstable V would have form L ⊕ V ′ with either Hom(L, V ′ ) ̸ = 0 or Hom(V ′ , L) ̸ = 0, hence dim End V ≥ 3.
Note that µ V = 5/3.Hence, if the extension splits over some L 2 ⊂ V, then V is unstable.Conversely, if V is unstable then either it has a line subbundle of degree 2, or a semistable subbundle V ′ of rank 2 and degree ≥ 4. But any such V ′ has a line subbundle of degree ≥ 2.
(iii) We can identify It is easy to see that the intersection of PW L 2 with the zero locus of Π is exactly the image of E under the map given by |L 3 |.
Given an extension V → V, split over L 2 ⊂ V, the splitting L 2 → V is unique, and the quotient V/L 2 is an extension of L 3 = V/L 2 by O.It is well known that for points of PW L 2 \ E the latter extension is stable, so Now we can calculate the image of the map (2.3).The space End V /⟨id⟩ has a basis ⟨id L 2 , e⟩, where e is a generator of Hom(V L 3 , L 2 ).Their images under (2.3) both factor through L 2 → E, hence the image of (2.3) (which is 2dimensional) is H 0 (L 2 ) ⊂ H 0 (V).But this is exactly the conormal subspace to the projective plane PW L 2 .This shows that PW L 2 \ E (and hence PW L 2 ) is a Poisson subvariety.Since the rank of Π on PW L 2 \ E is equal to 2 and Π| E = 0, we deduce that PW L 2 \ E is a symplectic leaf.
■ By Lemma 3.2.1 (i), the vanishing locus of Π corresponds to extensions V by O, which split over O(p).This is the union S E of the lines PΛ p , where Λ p = V| * p ⊂ PH 0 (V) * , over p ∈ E. The surface S E is the image of the natural map P V ∨ → P(V ), associated with the embedding of bundles V ∨ → V ⊗ O E .We will prove (see Lemma 3.2.3below) that in fact this map induces an isomorphism of the projective bundle P V ∨ with S E .Lemma 3.2.2.Let E be a vector bundle over a smooth curve C and let W → H 0 (C, E) be a linear map from a vector space W , such that for any x ∈ C the composition p x : W → H 0 (C, E) → E| x is surjective, so that we have a morphism f : P E ∨ → P(W * ).Assume that we have a closed subset Z ⊂ P E ∨ with the following properties.
For every x, y ∈ C, x ̸ = y, consider p x (ker(p y )) ⊂ E| x .Then any ℓ ∈ P E ∨ | x , which is orthogonal to p x (ker(p y )), is contained in Z.
For every x ∈ C, consider the map W → H 0 (E| 2x ) and the induced map Then the map P E ∨ \ Z → P(W * ) is a locally closed embedding.
Proof .Assume that for x ̸ = y, we have two nonzero functionals ϕ x : By assumption, this can happen only when ϕ x is in Z.Thus, the map from P E ∨ \ Z is set-theoretically one-to-one.
Next, we need to check that our map is injective on tangent spaces.The tangent space to P E ∨ at a point corresponding to ℓ ⊂ E ∨ | x can be described as follows.Consider the canonical extension Now the quotient ℓ −1 ⊗ H ℓ /k, where we use the natural embedding The restriction of the map which is exactly the tangent map to f .It is injective if and only if the map H ℓ → W * is injective.Equivalently, the dual map W → H * ℓ should be surjective.The latter map is compatible with (surjective) projections to E| x , so this is equivalent to surjectivity of the map

The latter map factors as a composition
By assumption, this never happens for points of P E ∨ \ Z. ■ Lemma 3.2.3.The map P V ∨ → S E is an isomorphism.
Proof .We will check the conditions of Lemma 3.2.2.It suffices to check surjectivity of the maps H 0 (V) → V| x ⊕ V| y for x ̸ = y and of H 0 (V) → H 0 (V| 2x ).But this follows from the exact sequence for any effective divisor D of degree 2 and from the vanishing of H 1 (V(−D)) by stability of V. ■ By Lemma 3.2.1 (ii), the degeneracy locus D E of our Poisson bracket (which is a quintic hypersurface) is the union of planes PW L 2 ⊂ PV over L 2 ∈ Pic 2 (E) (see (3.2)).Let us consider the vector bundle W over E := Pic 2 (E), such that the fiber of W over L 2 is W L 2 .Note that we have a natural identification , where we use a surjection V → L 3 .To define the vector bundle W precisely, we consider the universal line bundle L 3 of degree 3 over E× E ≃ E×Pic 3 (E), normalized so that the line bundle p 2 * Hom(p * 1 V, L 3 ) is trivial.We set W := p 2 * (L 3 ) ∨ .Note that applying p 2 * to the natural surjection p * 1 V → L 3 we get a surjection H 0 (V) ⊗ O → p 2 * (L 3 ).Passing to the dual, we get a morphism P(W) → PV , whose image is D E .Lemma 3.2.4.The morphism P(W) → D E is an isomorphism over D E \ S E .
Proof .We need to check two conditions of Lemma 3.2.2 for the morphism H 0 (V) ⊗ O → W ∨ over E, with Z ⊂ P(W) being the preimage of S E .Note that the intersection of Z with each plane PH 0 (L 3 ) * ⊂ H 0 (V) * is the elliptic curve E embedded by the linear system |L 3 |.
To check the first condition, we use the exact sequence where is nonzero, hence, it identifies L 2 with the subsheaf L ′ 3 (−p) for some point p ∈ E. Hence, the image of H 0 (L 2 ) is precisely the plane H 0 (L ′ 3 (−p)) ⊂ H 0 (L ′ 3 ).Hence, the only point of PH 0 (L ′ 3 ) * orthogonal to this plane is the point p ∈ E ⊂ PH 0 (L ′ 3 ) * , which lies in Z.To check the second condition, we need to understand the map H 0 (V) → H 0 (W ∨ | 2x ) for x ∈ E ≃ Pic 3 (E).For this we observe that this map is equal to the composition ), which is the map induced on H 0 by the morphism of sheaves on E, Note that for x = L 3 , the bundle ) on E is an extension of L 3 by T * x E⊗L 3 , which gives the Kodaira-Spencer map for the family L 3 , so this extension is nontrivial.The composition is the canonical surjection with the kernel L 2 ⊂ V. Hence, α fits into a morphism of exact sequences Note that the map α| L 2 is nonzero, since otherwise we would get a splitting of the extension F x → L 3 .Now the kernel of the map H 0 (V) → W ∨ | x = H 0 (L 3 ) is identified with H 0 (L 2 ), and the induced map H 0 (L 2 ) → T * x E ⊗ H 0 (L 3 ) is given by a nonzero map Hence, its image is the subspace of the form H 0 (L 3 (−p)), and we again deduce that any point of PH 0 (L 3 ) * orthogonal to it lies in Z. ■ Corollary 3.2.5.
(i) There is a regular map D E \ S E → E such that the fiber over L 2 is the symplectic leaf PW L 2 \ E.
(ii) Any line contained in D E is either contained in S E (and so has form PΛ p for some p ∈ E) or in some plane PW L 2 , where L 2 ∈ Pic 2 (E).
Proof .For (ii) we observe that given a line L ⊂ D E not contained in S E , the restriction of the map Similarly, we have a fibration S E → E with fibers PΛ p , so any line contained in S E is one of the fibers.■

Two-dimensional distribution on G(2, 5) associated with the elliptic curve
Let E = E W ⊂ G(2, V ) be the elliptic curve obtained as the intersection with a linear subspace PW ⊂ P 2 V in the Plücker embedding, where dim W = 5.Equivalently, E is cut out by the linear subspace of sections ).As before, we denote by V the restriction of U ∨ , the dual of the universal bundle.Then 2 (V) is the restriction of O(1), and we have an exact sequence In other words, we can identify the dual map to the embedding W → 2 V with the natural map We have a regular map f : G(2, V ) \ E → P 4 given by the linear system ) \ E, we define the subspace as the kernel of the tangent map to f at Λ.
Note that for generic Λ, one has dim D Λ = 2.We have the following characterization of D Λ .
where the second intersection is taken in 2 V / 2 Λ.
(ii) For each v ∈ Λ, let us denote by π v : T Λ G(2, V ) → V /Λ the natural projection.Assume that Π E,v has rank 4, for some nonzero v ∈ Λ.Then D Λ is 2-dimensional, and π v (D Λ ) is the 2-dimensional subspace of V /Λ given as follows: Proof .(i) The map f is the composition of the Plücker embedding G(2, V ) → P 2 V with the linear projection Thus, the tangent map to f at Λ ⊂ W is the composition Equivalently, the map α is the natural map given by l ⊗ (v mod Λ) → l ∧ v mod 2 Λ.Now the assertion follows from the identification (ii) Our identification of Π W from Theorem A implies the following property of the bivector Π W,v ∈ 2 (V /v).Consider the natural map ϕ v : W → 2 (V /v).Recall that S = S E ⊂ PV denotes the surface, obtained as the union of lines corresponding to E ⊂ G(2, V ).We claim that the map ϕ v is injective if and only if ⟨v⟩ is not in S. Indeed, an element in the kernel of ϕ v is an element v ∧ v ′ contained in W , so the plane ⟨v, v ′ ⟩ corresponds to a point of E. Hence, this is true when Π W,v is nonzero.Now assume the rank of Π W,v is 4. We have a nondegenerate symmetric pairing on 2 (V /v) with values in det(V /v), given by the exterior product.Now our description of Π W implies that for ⟨v⟩ ̸ ∈ S, Π W,v is nonzero and Hence, the subspace (Λ/⟨v⟩) ⊗ (V /Λ) cannot be contained in ⟨Π W,v ⟩ ⊥ (this would mean that Λ/⟨v⟩ lies in the kernel of (•, •)).Hence, the intersection is contained in I, we deduce that its dimension is ≤ 2, and so dim D Λ ≤ 2. But we also know that dim D Λ ≥ 2, hence in fact, we have dim The last assertion follows from the fact that under trivialization of Λ/⟨v⟩, the subspace The dual map can be identified with the composition which also factors as the composition We need to check that this map has corank 2, or equivalently the first arrow is an isomorphism.
Set V ′ = V(−p).This is a stable bundle of rank 2 and degree 3. We need to check that the map and it is easy to see that the restriction of the above map to H 0 (O(p ′ )) ∧ H 0 (V ′ ) surjects onto the subspace H 0 ((det V ′ )(−p ′ )) ⊂ H 0 (det V ′ ).Varying the point p ′ ∈ E, we get the needed surjectivity.■ Thus, by Lemma 3.3.2(i), Σ E is exactly the set of points Λ ∈ G(2, V ) \ E where dim D Λ ≥ 3. We have the following geometric description of Σ E .Recall that we have a collection of 3dimensional subspaces W q ⊂ V , associated with points of E = Pic 2 (E) (see (3.2)).Proposition 3.3.5.For Λ ∈ G(2, V ), we have Λ ∈ Σ E if and only if the corresponding line PΛ is contained in some plane PW q , where q ∈ E. In other words, Proof .Assume first that Λ ∈ Σ E .As we have seen above, this means that Λ ∈ G(2, V ) \ E and dim D Λ ≥ 3.By Lemma 3.3.2(ii), this implies that the rank of the Poisson bracket Π W on points of PΛ is ≤ 2. Hence, by Lemma 3.2.1 (ii), PΛ is contained in the quintic D E .By Corollary 3.2.5, this implies that PΛ is contained in some plane PW q .
By Lemma 3.3.2, the space D Λ is isomorphic to the kernel of the composed map Hence, dim(D Λ ) is equal to the corank of the dual map Let B denote the divisor of zeroes of s.We claim that the image of (3.3) is contained in the subspace H 0 2 V(−B) ⊂ H 0 2 V .Indeed, we have an exact sequence where N is a line bundle of degree 2. It is easy to see that the composed map coincides with the natural multiplication map The exact sequence 0 → H 0 (N ) → P → ⟨s⟩ → 0 shows that 2 P ⊂ H 0 (N )∧H 0 (V) and its image in H 0 (N )⊗H 0 (M ) is contained in H 0 (N )⊗⟨s⟩.This proves our claim about the image of the map (3.3).It follows that the corank of this map is ≥ 3, so Λ ∈ Σ E .■ Corollary 3.3.6.The locus of lines in P 4 contained in the degeneration locus Proof .Combine Proposition 3.3.5 with Corollary 3.2.5 (ii).The union is disjoint by Lemma 3.3.4.
(ii) Assume that for generic v ∈ Λ, the rank of Π E,v is 4. Then the map D Λ ⊗O → V /Λ⊗O(1) over the projective line PΛ is an embedding of a rank 2 subbundle.
Proof .(i) Since all elements of 2 M are decomposable, the intersection Q := W ∩ 2 M is a linear subspace consisting of decomposable elements.But all decomposable elements of W are of the form 2 Λ p for some point p ∈ E. Hence, we would get an embedding (ii) From part (i) and from Lemma 3.3.2we get that for any 3-dimensional subspace M ⊂ V containing Λ, one has D Λ ∩ Λ ⊗ M/Λ = 0. Let us set P = V /Λ, and let us consider the exact sequence over PΛ, We want to prove that the rank 1 sheaf Q on P Hence, D Λ has a nontrivial intersection with H 0 (O(1)) ⊗ ⟨v⟩ = Λ ⊗ M/Λ ⊂ Λ ⊗ V /Λ, for some 3-dimensional M ⊂ V , containing Λ.This is a contradiction, as we proved that there could be no such M .
In the case Q ≃ T ⊕ O, we get that D Λ ⊗ O(−1) is contained in the kernel of a surjection P ⊗ O → O, i.e., D Λ ⊗O(−1) is contained in O 2 ⊂ P ⊗O.But any embedding O(−1) 2 → O 2 factors through some O(−1) ⊕ O → O 2 (occurring as kernel of the surjection O 2 → O x , for some point x in the support of the quotient).Hence, we can finish again as in the previous case.■ Remark 3.3.8.The rational map f from G(2, V ) to P 4 has the following interpretation, which can be proved using projective duality.Start with a generic line ℓ ⊂ P(V ).Then the intersection ℓ ∩ D E with the degeneration quintic of Π E consists of 5 points.Taking the images of these points under the projection D E \ S E → E (see Corollary 3.2.5)we get a divisor D ℓ of degree 5 on E. All these divisors will belong to a certain linear system P 4 of degree 5, and the map ℓ → D ℓ is exactly our map f .(i) Let E ⊂ G(2, V ) be the elliptic curve defined by W ⊂ 2 V .Then for each point p ∈ E, the bivector Π E vanishes on the projective line PΛ p ⊂ PV , where Λ p ⊂ V is the 2-dimensional subspace corresponding to p.For a generic point v of Λ p , the Lie algebra g = T * v PV has a basis (h 1 , h 2 , e 1 , e 2 ) such that Equivalently, the linearization of Π E takes form Furthermore, the conormal subspace N ∨ PΛp,v ⊂ g * is spanned by e 1 , e 2 , h 1 + h 2 (dually the tangent space to T PΛp is spanned by (ii) We have an identification Under this identification, the line T p E ⊂ T p G(2, V ) has the property that the corresponding global section of N PΛp evaluated at generic v ∈ PΛ p spans the line Equivalently, the tangent space at v to the surface (iii) Let Π ′ be a Poisson bracket compatible with Π E .Then for p ∈ E and a generic v ∈ Λ p , one has Proof .(i) Extensions V of V by O, corresponding to the line PΛ p , are exactly nontrivial extensions that split under O → O(p).We claim that for a generic point of PΛ p we have where L 1 and L 2 are nonisomorphic line bundles of degree 2. Indeed, by Lemma 3.2.1 (ii), the only other possibility is V ≃ O(p) ⊕ V ′ , where V ′ is a nontrivial extension of M by M , where M 2 ≃ det(V)(−p).Since the corresponding extension splits over the unique embedding M → V, this gives one point on the line PΛ p for each of the four possible line bundles M .
We can compute the Lie algebra g for the point corresponding to V ≃ O(p) ⊕ L 1 ⊕ L 2 using the isomorphism of Theorem 2.3.1,End V /⟨id⟩ ∼ -g ⊂ H 0 (V). (3.5) We consider the following basis in End V /⟨id⟩: Then it is easy to check the claimed commutator relations between these elements.The conormal subspace to PΛ p is identified with Λ ⊥ p = H 0 (V(−p)).The image of the subspace Hom(O(p), L 1 ⊕ L 2 ) under the map (3.5) will consist of compositions which vanish at p, so they are contained in H 0 (V(−p)).We have (ii) To identify the direction corresponding to T p E, we first recall that the map The dual of the natural map V * → H 0 (V| 2p ) fits into a morphism of exact sequences and the map β corresponds to a map Now, given a functional v : V| p → k, the image of T p E under π v : Λ * p ⊗ V /Λ p → V /Λ p corresponds to the composition of (3.6) with v.In other words, it is given by the composition (here we use a trivialization of T p E). Let V → V be the extension corresponding to v.As we have seen in (i), for a generic v, we have V ≃ O(p) ⊕ L 1 ⊕ L 2 , where L i are as above.As we have seen in (i), under the isomorphism (3.5), Λ ⊥ p = H 0 (V(−p)) is the image of the subspace ⟨h 1 + h 2 , e 1 , e 2 ⟩.
Hence, it remains to check that the composition is zero (where the first arrow is induced by (3.5)).Let us consider the element e 1 (the case of e 2 is similar).It maps to the element of H 0 (V(−p)) given by the embedding where we use the composed map L 1 → V → V. Thus, it is enough to check that the composition To this end we use the fact that the extension V is the pull-back of the standard extension O(p) → O p via v. Hence, we have a commutative diagram with exact rows and columns, (iii) This is obtained by a straightforward computation using the vanishing of [Π E , Π ′ ] and the formula for Π lin E from part (i).■ Lemma 3.4.2.Let E, E ′ ⊂ G(2, V ) be a pair of elliptic curves obtained as linear sections, such that Proof .Assume E ⊂ Σ E ′ .Then, by the description of Σ E ′ in Proposition 3.3.5,for every p ∈ E there exists a line bundle L 2 of degree 2 on E ′ such that the image of H where ⟨w 1 , w 2 ⟩ is the tangent plane to the leaf of Π E ′ (i.e., to the projective plane PH 0 (E ′ , L 2 ) ⊥ ).Furthermore, the plane ⟨w 1 , w 2 ⟩ contains the tangent line to PΛ p at v. In the notation of Lemma 3.4.1 (i), the latter tangent line is spanned by ∧ w for some tangent vector w.But we also know by Lemma 3.4.1 (iii) that Π E ′ | v is a linear combination of (2∂ h 1 − ∂ h 2 ) ∧ ∂ e 1 , (2∂ h 2 − ∂ h 1 ) ∧ ∂ e 2 and ∂ h 1 ∧ ∂ h 2 .This is possible only when w ∈ ⟨∂ h 1 , ∂ h 2 ⟩, which is the tangent plane to the surface S E (see Lemma 3.4.1 (ii)).
This implies that S E is tangent to the corresponding projective plane PH 0 (E ′ , L 2 ) ⊥ ⊂ D E ′ .Assume first that S E ̸ ⊂ S E ′ .Then we get that the regular morphism (see Corollary 3.2.5)has zero tangent map at every point.Hence, S E is contained in a projective plane, which is a contradiction (since the map P V ∨ → PH 0 (V) * = PV induces an isomorphism on sections of O(1)). Finally By Lemma 3.3.2(ii), the subspace π v (D E ′ ,p ) ⊂ V /Λ p consists of x such that x ∧ Π norm E ′ ,v = 0. Thus, we deduce the inclusion In other words, the section s generating has the property that for generic point v ∈ PΛ p the evaluation s(v) belongs to the image of the evaluation at v of the embedding D E ′ ,p ⊗ O → V /Λ p ⊗ O(1).Since by Lemma 3.3.7 the latter is an embedding of a subbundle, this implies that in fact s ∈ D E ′ ,p as claimed.This proves the inclusion (3.7) for a generic p ∈ E. But this implies that the composed map has zero derivative everywhere, so it is constant.Hence, E is contained in a linear section of PU ∩ G(2, V ), for some 6-dimensional subspace U ⊂ 2 V containing W ′ .Hence, dim(W + W ′ ) ≤ 6.Conversely, assume W and W ′ are such that U = W + W ′ is 6-dimensional.Then we claim that [Π W , Π W ′ ] = 0. Indeed, since the space of such pairs (W, W ′ ) is irreducible, it is enough to consider the case when the surface S = PU ∩ G(2, V ) is smooth.Then E W and E W ′ are anticanonical divisors on S, and we can apply [3,Theorem 4.4] to the bundle V S := U ∨ | S on S. The fact that (O S , V S ) is an exceptional pair is easily checked using Koszul resolutions, as in Section 2.2.
(ii) It is well known that if a collection of k-dimensional subspaces in a vector space has the property that any two subspaces intersect in a (k − 1)-dimensional space, then either all of them are contained in a fixed (k + 1)-dimensional subspace, or they contain a fixed (k − 1)dimensional subspace.The statement immediately follows from (i) using this fact for k = 5 and the collection (W i ).■ Proof of Corollary C. By Theorem B (ii), the brackets (Π W i ) are pairwise compatible when either there exists a 6-dimensional subspace U ⊂ 2 V , containing all W i , or there is a 4dimensional subspace K ⊂ 2 V , contained in all W i .In the former case the corresponding tensors 5 W i are all contained in the 6-dimensional subspace V .

1
has no torsion.Since deg(Q) = 2 and Q is generated by global sections, we only have to exclude the possibilities Q ≃ O x ⊕O(1) and Q ≃ T ⊕ O, where T is a torsion sheaf of length 2. Assume first that Q ≃ O x ⊕ O(1).Consider the composed surjection f : P ⊗ O → Q → O(1).It is induced by a surjection P → H 0 (O(1)), which has 1-dimensional kernel ⟨v⟩.It follows that the inclusion of D Λ ⊗ O(−1) into P ⊗ O factors as D Λ ⊗ O(−1) → ⟨v⟩ ⊗ O ⊕ O(−1) → P ⊗ O.

3. 4
Calculation of the Schouten bracket and proof of Theorem B Lemma 3.4.1.
and the element id O(p) is mapped under (3.5) to the compositionO → O(p) → V,which also vanishes at p.This proves our claim about the conormal subspace.
For nonzero ϕ ∈ Ext n (P, ω X ) ≃ Ext 1 C (P | C , O C ), and s 1 , s 2 ∈ ⟨ϕ⟩ ⊥ ⊂ H 0 (X, P ), one has Proof of Theorem B. (i) We can assume that E ̸ = E ′ .We will check that for a generic point p ∈ E, one hasT p E ⊂ D E ′ ,p ⊂ T p G(2, V ).(3.7)By Lemma 3.4.2,forageneric p ∈ E, we have p ̸ ∈ Σ E ′ , hence, by Corollary 3.3.6,thelinePΛ p is not contained in the degeneracy locus D E ′ of Π E ′ .Let us pick a generic point v of Λ p , so that the rank of Π E ′ ,v is 4. We want to study the normal projectionΠ norm E ′ ,v ∈ ∧ 2 (T v PV /T v PΛ p ) ≃ ∧ 2 (V /Λ p ) (see Lemma 3.3.2).Recall that in the notation of Lemma 3.4.1, the tangent space to PΛ p at v is spanned by ∂ h 1 − ∂ h 2 .Hence, the inclusion (3.4) implies that Π norm E ′ ,v is proportional to a bivector of the form ∂ h 1 ∧ ξ .By Lemma 3.4.1 (ii), we can reformulate this as , if S E ⊂ S E ′ then E = E ′ ⊂ G(2, V) and, we get a contradiction by Lemma 3.3.4.■