THE LENGTH-2 MONAD FOR INSTANTONS OVER ℙ^{3} WITH c_{2} = 3.

ALEXANDER KUZNETSOV

1. Introduction

A mathematical instanton vector bundle (or simply instanton) over ℙ^{3} is a stable 2-dimensional vector bundle F with c_{1}(F) = 0 which is orthogonal to O_{ℙ^{3}}(−2) (that is H^{•}(ℙ^{3}, F ⊗ O_{ℙ^{3}}(−2)) = 0). There is a well-known description of instantons over ℙ^{3}. Namely, one can show (see e.g. [9], II.3.2. Ex. 5) that any instanton F is a cohomology of a monad

O_{ℙ^{3}}(−1)^{⊕n} ⟶ O_{ℙ^{3}}^{⊕(2n+2)} ⟶ O_{ℙ^{3}}(1)^{⊕n},

where n = c_{2}(F). This description allows to construct a moduli space MI_{n}(ℙ^{3}) of instantons with c_{2} = n (see [9], II.4.1.6).

However, with this description the investigation of such problems as rationality and smoothness is very complicated. A very little is known about it. Namely, this problems have been solved only for the values n = 1 (see [1]), n = 2 (see [7]) and n = 3 (see [6]).

In this paper we show that the generic instanton vector bundle over ℙ^{3} with c_{2} = 3 is the cohomology bundle of the following monad of the length 2

R ⟶ Ω_{ℙ^{3}}(3)^{⊕3},  (1)

where R is a certain exceptional vector bundle over ℙ^{3} of rank 11 (see 3.1). Then we get a very nice description of such instantons in terms of nets of quadrics in the dual projective space ℙ^{3}^{√} (see Theorem 2). This immediately implies rationality of MI_{3}(ℙ^{3}) since the space of nets of quadrics in ℙ^{3}^{√} is just the Grassmannian Gr(3, 10). It would be extremely good to get a similar description (i.e. as the cohomology bundle of a monad of the length 2) of generic instanton with arbitrary value of c_{2}.

The paper is organized as follows. In Section 2 we recollect necessary facts about nets of quadrics and rigid plane curves (that is a plane curve with a non-degenerate theta-characteristic). In 2.1we reproduce a theory relating nets of quadrics in ℙ^{3}^{√} to rigid plane quartics (an explicit presentation can be found in [10] or [11]). In 2.2–2.3 we somehow develop this theory. In particular, we define in 2.2.8 the skew web of the rigid quartic and in 2.3.1 we deduce the relation between the net of quadrics and the skew web of the rigid quartic.

In Section 3 we show that there is a one-one correspondence between the spaces of non-Lüroth (i.e. corresponding to non-Lüroth quartics) nets of quadrics in ℙ^{3}^{√} and the space MI_{3}^{T} (ℙ^{3}) of instantons F with c_{2} = 3 satisfying the additional orthogonality condition Hom^{•}(F, T ) = 0, where T stands for the tangent bundle for ℙ^{3}. As a byproduct we prove that any instanton F ∈ MI_{3}^{T} (ℙ^{3}) is the cohomology bundle of a monad (1) and that MI_{3}^{T} (ℙ^{3}) is an open non-empty subspace of MI_{3}(ℙ^{3}). In 3.1 we explain how given a net of quadrics one can construct an instanton in MI_{3}^{T} (ℙ^{3}). In 3.2 we explain how given an instanton F ∈ MI_{3}^{T} (ℙ^{3}) one can reconstruct the net of quadrics A_{F}. In 3.3 we show how the quartic corresponding to the net A_{F} arises in the geometry of non-stable hyperplanes and multiple jumping lines of F.

The author was partially supported by CRDF grant RM1-265.

The author is very grateful to A. Bondal and A.N. Tyurin for the attention and helpful discussions and to the Max-Planck-Institut für Mathematic, where this work was accomplished, for hospitality.

2. Plane quartics and nets of quadrics.

2.1. Recall that a net of quadrics in the projective space ℙ(V) is a 3-dimensional subspace in S^{2}V^{∗} (see [10], [11]). Let W, N be fixed 4-dimensional and 3-dimensional vector spaces and let A : N ⟶ S^{2}W be a net of quadrics in the dual projective space ℙ(W^{∗}).

Given such a net of quadrics one can consider its Hesse curve

H_{A} = {n ∈ ℙ(N) | A(n) is a degenerate quadric} ⊂ ℙ(N)

The curve H_{A} is a plane quartic (because dim W = 4). We denote by α : H_{A} → ℙ(N) its embedding in the plane.

The map β : H_{A} → ℙ(W^{∗}) defined as follows

β : n ∈ H_{A} ↦ Ker(A(n)) ∈ ℙ(W^{∗})

is called the Steiner map and its image S_{A} = β(H_{A}) ⊂ ℙ(W^{∗}) is called the Steiner curve of the net A. Let θ_{A} be a coherent sheaf of O_{H_{A}}-modules defined as follows

θ_{A} = β^{∗}O_{ℙ(W^{∗})}(1) ⊗ α^{∗}O_{ℙ(N)}(−1).  (2)

The sheaf θ_{A} is a so-called generalized non-degenerate theta-characteristic. This means that we have

Hom_{O_{H_{A}}} (θ_{A}, ω_{H_{A}}) ≅ θ_{A}  and  H^{•}(H_{A}, θ_{A}) = 0.  (3)

where ω_{H_{A}} stands for the dualizing sheaf of H_{A}. Note that since H_{A} is a plane quartic we have

ω_{H_{A}} = α^{∗}O_{ℙ(N)}(1).  (4)

The pair (H_{A}, θ_{A}) is called the absolute invariant of the net A. It is well known (see e.g. [2], [8], [10], [11]) that a net A can be uniquely reconstructed from its absolute invariant. In particular, we have the canonical identifications

N = H^{0}(C, ω_{H_{A}})^{∗}, W = H^{0}(C, θ_{A} ⊗ ω_{H_{A}}).

2.1.1. The net A : N ⟶ S^{2}W is called regular if it contains no quadrics of corank 2 and it is called strictly regular if any common point P of the net is a regular point of all quadrics in the net.

Let Nets denote the subspace of all regular nets of quadrics in ℙ(W^{∗}) and Qu denote the space of pairs (C, θ) where C is a plane quartic and θ is a theta-characteristics on C which are the absolute invariants of regular nets of quadrics. An element (C, θ) of Qu we will call the rigid quartic. We denote by α : C → ℙ(N) and β : C → ℙ(W^{∗}) the Hesse and Steiner embeddings of C. The regular net of quadrics, corresponding to the rigid quartic (C, θ) will be denoted by A_{C}.

Repeating once more:

the maps (C, θ) ↦ A_{C}, A ↦ (H_{A}, θ_{A}) are mutually inverse isomorphisms Qu ⇆ Nets.  (5)

2.1.2. There is a natural question, what kind of singularities can have the Hesse curve H_{A} of the regular net A ∈ Nets. The answer is as follows:

the smooth case: H_{A} is smooth;

the singular case: H_{A} has singularities of the second order (and H_{A} may be reducible); the non-reduced case: H_{A} is a double smooth conic.

Example. Let S ⊂ ℙ(W^{∗}) be a rational twisted cubic curve. Then we have dim H^{0}(ℙ(W^{∗}), J_{S}(2)) = 3, where J_{S} stands for the sheaf of ideals of the curve S. Identifying A : N →^{∼} H^{0}(ℙ(W^{∗}), J_{S}(2)) ⊂ H^{0}(ℙ(W^{∗}), O_{ℙ(W^{∗})}(2)) = S^{2}W we get a net of quadrics. One can easily show that its Hesse curve is a double conic and its Steiner curve is a curve S with additional nilpotents in all points x ∈ S in the direction of the touching plane to S at x. One can prove that this case is the only case when the Hesse curve of the net of quadrics is not reduced in all points.

Remark 1. Let n ∈ H_{A} and φ = β(n) ∈ S_{A} ⊂ ℙ(W^{∗}). Then it is easy to see that

T_{n}H_{A} = {n′ ∈ N | A(n′, φ, φ) = 0}/⟨n⟩,

and

T_{φ}S_{A} = {ψ ∈ W^{∗} | A(n, ψ) = A(n′, φ) for some n′ ∈ N}/⟨φ⟩

(see loc. cit.). This means that the isomorphisms of (5) identify the subspace Qu^{s} ⊂ Qu of smooth quartics with the subspace Nets^{s} ⊂ Nets of strictly regular nets.

2.1.3. Given a net A : N → S^{2}W let Ã : W^{∗} ⊗ N^{∗} ⟶ N ⊗ W denote the induced map.

Lemma. The regular nets A with degenerate Ã correspond under the isomorphism of (5) to the subspace Qu^{L} ⊂ Qu of Lüroth quartics.

Proof. See [2, p. 84] (the condition of degenerating Ã is equivalent to the Barth’s (α3) condition).

Definition. Accordingly to the Lemma we will call the nets A with degenerate Ã the Lüroth nets.

The isomorphism (5) identify the subspace Qu^{L} ⊂ Qu of Lüroth quartics with subspace Nets^{L} ⊂ Nets of Lüroth nets.

2.2. The net of quadrics A_{C} ∈ N^{∗} ⊗ S^{2}W corresponding to the rigid quartic C induces a certain section s_{C} ∈ H^{0}(ℙ(N) × ℙ(W^{∗}), W ⊗ O_{ℙ(N)×ℙ(W^{∗})}(1, 1)) = W ⊗ N^{∗} ⊗ W. Consider the zero locus Z_{s_{C}} of s_{C}.

Lemma. Z_{s_{C}} = (α × β)(C) ⊂ ℙ(N) × ℙ(W^{∗})

Proof. First we will check the equality set-theoretically and then we will prove that the differential of α × β induces an isomorphism of tangent spaces of C and Z_{s_{C}}.

Assume that the point (n, φ) ∈ ℙ(N) × ℙ(W^{∗}) is in the zero locus. This means that A_{C}(n, φ) = 0. But this means that the quadric A_{C}(n) is degenerate and φ = Ker(A_{C}(n)) is its kernel, hence

(n, φ) = (α × β)(x), x ∈ C. By the same reasons (α × β)(C) ⊂ Z_{s_{C}}.

Now consider a point (n, φ) = (α × β)(x), x ∈ C. We have

T_{(n,φ)}(ℙ(N) × ℙ(W^{∗})) = T_{n}ℙ(N) ⊕ T_{φ}ℙ(W^{∗}) ≅ N/n ⊕ W^{∗}/φ,

where T stands for the tangent space.

On the other hand the tangent space of the zero locus T_{(n,φ)}Z_{s_{C}} is nothing but the kernel of the

differential d_{s_{c}} : T_{(n,φ)}(ℙ(N)×ℙ(W^{∗})) → W. Its restrictions on summands N/n and W^{∗}/φ are following:

d_{s_{C}} _{|T_{n}ℙ(N)} = f_{φ} := A_{C}(·, φ) : N/n → W, d_{s_{C}} _{|T_{φ}ℙ(W^{∗})} = g_{n} := A_{C}(n, ·) : W^{∗}/φ → W.

Since A_{C}(n) is a quadric of corank 1 the second map is an isomorphism W^{∗}/φ →^{∼} Ann φ ⊂ W. This means that

T_{(n,φ)}Z_{s_{C}} = Ker d_{s_{C}} ≅ f_{φ}^{−1}(Ann φ) = {n′ ∈ N | A(n′, φ, φ) = 0}/⟨n⟩

and Lemma follows from the Remark 1.

2.2.1. The Lemma 2.2 implies that we have the following (Koszul) resolvent of the sheaf (α × β)_{∗}O_{C} on ℙ(N) × ℙ(W^{∗}):

0 ⟶ Λ^{4}W^{∗}(−4, −4) ⟶^{∧s_{C}} Λ^{3}W^{∗}(−3, −3) ⟶^{∧s_{C}} Λ^{2}W^{∗}(−2, −2) ⟶^{∧s_{C}}

⟶^{∧s_{C}} W^{∗}(−1, −1) ⟶^{s_{C}} O_{ℙ(N)×ℙ(W^{∗})} ⟶ (α × β)_{∗}O_{C} ⟶ 0. (6)

Taking the push-forward of this resolvent onto factors of ℙ(N)×ℙ(W^{∗}) we get the following resolvents of α_{∗}O_{C} and β_{∗}O_{C}:

\begin{matrix}0 ⟶ O_{ℙ(N)}(−4) ⟶ O_{ℙ(N)} ⟶ α_{∗}O_{C} ⟶ 0, \\ 0 ⟶ N ⊗ O_{ℙ(W^{∗})}(−4) ⟶ W ⊗ O_{ℙ(W^{∗})}(−3) ⟶ O_{ℙ(W^{∗})} ⟶ β_{∗}O_{C} ⟶ 0. \end{matrix}  (7)

Here we have applied an isomorphisms Λ^{3}N ≅ ℂ and Λ^{3}W^{∗} ≅ W. The first of these resolvents is the usual resolvent of the structure sheaf of the plane quartic.

2.2.2. The equations (2) and (3) imply that we have β^{∗}O_{ℙ(W^{∗})}(2) ≅ α^{∗}O_{ℙ(N)}(3). The resolvents (7) imply that

\begin{aligned}S^{3}N^{∗} & = & H^{0}(ℙ(N), O_{ℙ(N)}(3)) & ≅ & H^{0}(C, α^{∗}O_{ℙ(N)}(3)), \\ S^{2}W & = & H^{0}(ℙ(W^{∗}), O_{ℙ(W^{∗})}(2)) & ≅ & H^{0}(C, β^{∗}O_{ℙ(W^{∗})}(2)). \end{aligned}

Finally we get the natural isomorphism S^{3}N^{∗} ≅ S^{2}W.

2.2.3. Now we will construct a certain 2-dimensional vector bundle on C.

Consider the pull-back via β of the second sequence of (7) twisted by O_{ℙ(W^{∗})}(3). It gives the following exact sequence on C:

0 ⟶ Tor^{2}(β_{∗}O_{C}, β_{∗}O_{C}) ⊗ β^{∗}O_{ℙ(W^{∗})}(3) ⟶ N ⊗ β^{∗}O_{ℙ(W^{∗})}(−1) ⟶

⟶ W ⊗ O_{C} ⟶ Tor^{1}(β_{∗}O_{C}, β_{∗}O_{C})) ⊗ β^{∗}O_{ℙ(W^{∗})}(3) ⟶ 0.

The natural isomorphism

Tor^{2}(β_{∗}O_{C}, β_{∗}O_{C}) ⊗ β^{∗}O_{ℙ(W^{∗})}(3) ≅ Λ^{2}N_{β(C)/ℙ(W^{∗})}^{∗} ⊗ β^{∗}O_{ℙ(W^{∗})}(3) ≅

≅ β^{∗}O_{ℙ(W^{∗})}(−4) ⊗ ω_{C}^{∗} ⊗ β^{∗}O_{ℙ(W^{∗})}(3) ≅ β^{∗}O_{ℙ(W^{∗})}(−1) ⊗ ω_{C}^{∗}

then implies that the quotient sheaf \frac{N ⊗ β^{∗}O_{ℙ(W^{∗})}(−1)}{β^{∗}O_{ℙ(W^{∗})}(−1) ⊗ ω_{C}^{∗}} is naturally embedded in the trivial vector

bundle W ⊗ O_{C}. This gives rise to the Grassmannian embedding γ : C → Gr(2, W) such that

0 ⟶ β^{∗}O_{ℙ(W^{∗})}(−1) ⊗ ω_{C}^{∗} ⟶ N ⊗ β^{∗}O_{ℙ(W^{∗})}(−1) ⟶ γ^{∗}U ⟶ 0,  (8)

where U stands for the tautological subbundle over Gr(2, W).

Taking the determinant we get

γ^{∗}Λ^{2}U ≅ Λ^{3}N ⊗ β^{∗}O_{ℙ(W^{∗})}(−3) ⊗ β^{∗}O_{ℙ(W^{∗})}(1) ⊗ ω_{C} ≅

≅ β^{∗}O_{ℙ(W^{∗})}(−2) ⊗ ω_{C} ≅ α^{∗}O_{ℙ(N)}(−3) ⊗ ω_{C} ≅ ω_{C}^{∗}^{2}. (9)

2.2.4. Now we will introduce a geometrical description of the embedding γ. To this end recall, that the map N ⊗ O_{ℙ(W^{∗})}(−4) → W ⊗ O_{ℙ(W^{∗})}(−3) in (7) over the point φ ∈ ℙ(W^{∗}) is nothing but the map f_{φ} = A_{C}(·, φ) : N → W. Hence

γ(x) = Im f_{β(x)} ⊂ W.

Example. In the non-reduced case (see 2.1.2) one easily sees that the curve γ(C)_{red} is but the curve of the tangent lines to S.

2.2.5. Below we will need the following technical Lemma.

Lemma. Let L = γ(x) ⊂ ℙ(W) be a line. If A_{C}(N, φ) ⊂ L, then φ = β(x).

Proof. Since dim L = 2 < 3 = dim N we have φ = β(y) for some y ∈ C. Assume that y ≠ x. Then β(y) ≠ β(x) and for any ψ ∈ ⟨β(x), β(y)⟩ ⊂ W^{∗} we evidently have A_{C}(N, ψ) ⊂ γ(x). Hence the line M := ⟨β(x), β(y)⟩ is contained in the Steiner curve β(C). We have three possibilities, either M ∩ L^{⊥} = 0, or M ∩ L^{⊥} = ⟨ψ⟩, or M = L^{⊥}.

If M ∩ L^{⊥} = 0 then we have W^{∗} = M ⊕ L^{⊥} and the net A_{C} : N ⟶ S^{2}W is the direct sum A_{C} = A′_{C} ⊕ A′′_{C}, where A′ : N ⟶ S^{2}M^{∗} and A′′ : N ⟶ S^{2}(L^{⊥})^{∗}. If n ∈ ℙ(N) is a common point of the Hesse curve of A′ and of the Hesse curve of A′′ then rank A_{C}(n) ≤ 2 which contradicts the regularity of the net A_{C}.

If M ∩ L^{⊥} = ⟨ψ⟩ then we have A_{C}(N, ψ, L^{⊥}) = 0 (because ψ ∈ M), A_{C}(N, ψ, M) = A_{C}(N, M, ψ) = 0 (because ψ ∈ L^{⊥}) hence A_{C}(N, ψ, M + L^{⊥}) = 0. But dim(M + L^{⊥}) = 3, hence the net A_{C} contains a pencil of quadrics with common kernel ψ. This means that there are three quadrics of corank 2 in this pencil, which again contradicts the regularity of the net A_{C}.

Finally, let M = L^{⊥}. Then we have W^{∗} = M ⊕ L^{∗}, W = M^{∗} ⊕ L and the net A_{C} induces maps A′ : N → Hom(M, L) and A′′ : N → S^{2}L such that A_{C}(n, ψ) = A′(n)(m) + (A′(n))^{∗}(λ) + A′′(n)(λ), where ψ = m + λ, m ∈ M, λ ∈ L^{∗} is the decomposition of ψ, and A′(n)^{∗} : L^{∗} → M^{∗} is the dual operator of A′(n) : M → L.

Let H = {n ∈ ℙ(N) | A′(n) is degenerate} ⊂ ℙ(N), σ : n ∈ H ↦ Ker A′(n) ∈ ℙ(M), τ : n ∈ H ↦ Im A′(n) ∈ ℙ(L), σ′ : n ∈ H ↦ Ker(A′(n)^{∗}) ∈ ℙ(L^{∗}) and τ′ : n ∈ H ↦ A′′(n)(σ′(n)) ∈ ℙ(L). Let n ∈ H be a point such that τ(n) = τ′(n) (that is a point of intersection of the curve (τ × τ′)(H) ⊂ ℙ(L) × ℙ(L) with diagonal). Then we have A_{C}(n, ⟨σ(n), σ′(n)⟩) = 0 that is rank A_{C}(n) ≤ 2, which again contradicts the regularity of the net A_{C}. The Lemma is proved.

2.2.6. The net of quadrics A_{C} corresponding to the rigid quartic C induces a certain section t_{C} ∈ H^{0}(ℙ(W^{∗}) × Gr(2, W)), N^{∗} ⊗ O_{ℙ(W^{∗})}(1) ⊠ W/U) = N^{∗} ⊗ W ⊗ W. Consider the zero locus Z_{t_{C}} of t_{C}.

Lemma. Z_{t_{C}} = (β × γ)(C) ⊂ ℙ(W^{∗}) × Gr(2, W).

Proof. Assume that the point (φ, L) ∈ ℙ(W^{∗})×Gr(2, W) is in the zero locus. This means that Im f_{φ} ⊂ L, hence φ = β(x) and L = γ(x) (see Lemma 2.2.5). The set-theoretical equality follows, hence it is remained to show that the induced map of tangent spaces is an isomorphism.

To this end consider a point (φ, L) = (β × γ)(x), x ∈ C. Then we have

T_{(φ,L)}(ℙ(W^{∗}) × Gr(2, W)) = T_{φ}ℙ(W^{∗}) ⊕ T_{L}Gr(2, W) ≅ W^{∗}/φ ⊕ Hom(L, W/L).

On the other hand, the tangent space of the zero locus T_{(φ,L)}Z_{t_{C}} is nothing but the kernel of the differential

d_{t_{C}} : T_{(φ,L)}(ℙ(W^{∗}) × Gr(2, W)) → N^{∗} ⊗ W/L = Hom(N, W/L). Its restrictions on the first summand W^{∗}/φ is following:

d_{t_{C}} _{|T_{φ}ℙ(W^{∗})} = h_{L} := π_{L} · A_{C} : W^{∗}/φ → Hom(N, W/L),

where π_{L} : W → W/L is the natural projection, A_{C} : W^{∗} → Hom(N, W) is the polarization of A_{C}. Evidently that A_{C} maps φ ∈ W^{∗} to Hom(N, L) ⊂ Hom(N, W) hence π_{L} · A_{C} maps φ to zero and the above composition is well-defined.

The restriction of d_{t_{C}} on the second summand Hom(L, W/L) is following:

d_{t_{C}} _{|T_{L}Gr(2,W)} = k_{φ} := (·) · f_{φ} : Hom(L, W/L) → Hom(N, W/L).

Here we have used the fact that the image of f_{φ} : N → W is L (the map f_{φ} is defined in the proof of Lemma 2.2) hence we can take its composition with any map L → W/L.

Let n = α(x) ∈ N. We have the following exact sequence:

0 ⟶ Hom(N/n, W/L) ⟶ Hom(N, W/L) ⟶^{ev_{n}} W/L ⟶ 0.

The third map here is evaluation at the point n ∈ N.

Since f_{φ} induces an isomorphism N/n → L the map k_{φ} is also an isomorphism Hom(L, W/L) →^{∼} Hom(N/n, W/L), therefore

T_{(φ,L)}Z_{t_{C}} = Ker d_{t_{C}} ≅ h_{L}^{−1}(Hom(N/n, W/L)) = {ψ ∈ W^{∗} | A_{C}(n, ψ) ∈ L}/⟨φ⟩

and the Lemma follows from the Remark 1

2.2.7. The Lemma 2.2.6 implies that we have the following (Koszul) resolvent of the sheaf (β × γ)_{∗}O_{C}:

… ⟶ Λ^{2}(N ⊗ O_{ℙ(W^{∗})}(−1) ⊠ U^{⊥}) ⟶ N ⊗ O_{ℙ(W^{∗})}(−1) ⊠ U^{⊥} ⟶^{t_{C}} O_{ℙ(W^{∗})×Gr(2,W)} ⟶ (β × γ)_{∗}O_{C} ⟶ 0, (10)

where U^{⊥} := (W/U)^{∗}.

2.2.8. The cohomology spectral sequence of (10) multiplied by Λ^{2}U^{∗} shows that the restriction map H^{0}(Gr(2, W), Λ^{2}U^{∗}) → H^{0}(C, γ^{∗}Λ^{2}U^{∗}) is an isomorphism. Therefore we have

Λ^{2}W^{∗} = H^{0}(Gr(2, W), Λ^{2}U^{∗}) ⟶^{∼} H^{0}(C, γ^{∗}Λ^{2}U^{∗}) =

= H^{0}(C, ω_{C}^{2}) = H^{0}(C, α^{∗}O_{ℙ(N)}(2)) = H^{0}(ℙ(N), O_{ℙ(N)}(2)) = S^{2}N^{∗}.

Here we used the equality γ^{∗}Λ^{2}U^{∗} = ω_{C}^{2} which follows from (9).

The dual map of the above composition of isomorphisms

B_{C} : S^{2}N →^{∼} Λ^{2}W

we will call the skew web of the rigid plane quartic C.

Remark 2. The skew web B_{C} : S^{2}N ⟶ Λ^{2}W can be described on the abstract tensor language as the following composition:

S^{2}N ⟶ N^{∗} ⊗ S^{3}N ⟶^{∼} Λ^{2}N ⊗ S^{2}W^{∗} ⟶^{Λ^{2}A_{C}} Λ^{2}W.

Here the first map is dual to the natural pairing N ⊗S^{3}N^{∗} ⟶ S^{2}N^{∗}, the second map is the tensor product of isomorphisms N^{∗} ≅ Λ^{2}N, S^{3}N ≅ S^{2}W^{∗} (see 2.2.2) and the third map is the exterior square of the map A_{C} : N ⊗ W^{∗} ⟶ W. However, in this definition it is absolutely not clear that it is an isomorphism.

2.2.9. Lemma. The quartic C is non-Lüroth iff H^{1}(C, γ^{∗}W/U) = 0.

Proof. The cohomology spectral sequence of (10) multiplied by W/U shows that the space H^{1}(C, γ^{∗}W/U) is isomorphic to the cokernel of a map Ã_{C} : N^{∗} ⊗ W^{∗} → N ⊗ W.

2.2.10. Corollary. The quartic C is non-Lüroth iff Hom(γ^{∗}W/U, ω_{C}) = 0.

Proof. The Serre duality implies that

Hom(γ^{∗}W/U, ω_{C}) ≅ H^{0}(C, (γ^{∗}W/U)^{∗} ⊗ ω_{C})) ≅ H^{1}(C, γ^{∗}W/U))^{∗}.

and it is remained only to apply the above Lemma.

2.3. The relation between tensors. Since the skew web B_{C} : S^{2}N ⟶ Λ^{2}W is an isomorphism we can consider its inverse map

B_{C}^{−1} : Λ^{2}W ⟶ S^{2}N.

Let B̂_{C} : N^{∗} ⊗ Λ^{2}W ⟶ N denote the map induced by B_{C}^{−1}.

2.3.1. Consider also the map

Â_{C} : W^{∗} ⊗ W ⟶^{A_{C}⊗id} N^{∗} ⊗ W ⊗ W ⟶^{id ⊗ Alt} N^{∗} ⊗ Λ^{2}W,

where Alt stands for the alternation.

Proposition. The composition B̂_{C} · Â_{C} : W^{∗} ⊗ W ⟶ N vanishes.

Proof. Let φ = β(x) for some x ∈ C and γ(x) = ⟨w, w′⟩. Chose a basis (n_{1}, n_{2}, n_{3}) in N such that n_{1} = α(x) (i.e. A(n_{1}, φ) = 0) and A(n_{2}, φ) = w, A(n_{3}, φ) = w′. Let (n^{1}, n^{2}, n^{3}) be a dual basis in N^{∗}. By the definition of Â_{C} we have

Â_{C}(φ ⊗ w) = n^{1} ⊗ A(n_{1}, φ) ∧ w + n^{2} ⊗ A(n_{2}, φ) ∧ w + n^{3} ⊗ A(n_{3}, φ) ∧ w = n^{3} ⊗ w′ ∧ w.

Since image of γ(x) in ℙ(Λ^{2}W) is but w′ ∧ w (see 2.2.4) we have

B_{C}^{−1}(w′ ∧ w) = n_{1} · n_{1} ∈ S^{2}N

(up to multiplcative constant). But this means that

B̂_{C}(Â_{C}(φ ⊗ w)) = B̂_{C}(n^{3} ⊗ w′ ∧ w) = ⟨n^{3}, n_{1} · n_{1}⟩ = 2⟨n^{3}, n_{1}⟩n_{1} = 0,

where ⟨ · , · ⟩ stands for the natural pairing. Now the Proposition follows from the following Lemma.

2.3.2. Lemma. The tensors of the type φ ⊗ w, where φ = β(x), w ∈ γ(x) and x runs over C generate all space W^{∗} ⊗ W.

Proof. Assume that they do not. Then there exists a linear map M : W^{∗} ⟶ W^{∗} such that for all x ∈ C, φ = β(x), w ∈ γ(x) we have M(φ)(x) = 0. But this means that the composition

O_{ℙ(W^{∗})}(−1) ⊠ O_{Gr(2,W)} ⟶ W^{∗} ⊗ O_{ℙ(W^{∗})×Gr(2,W)} ⟶^{M} W^{∗} ⊗ O_{ℙ(W^{∗})×Gr(2,W)} ⟶ O_{ℙ(W^{∗})} ⊠ U^{∗}

vanishes over (β × γ)(C), that is the restriction map

W ⊗ W^{∗} = H^{0}(ℙ(W^{∗}) × Gr(2, W), O_{ℙ(W^{∗})}(1) ⊠ U^{∗}) ⟶ H^{0}(C, (β × γ)^{∗}(O_{ℙ(W^{∗})}(1) ⊠ U^{∗}))

sends M to zero. But the cohomology spectral sequence of (10) multiplied by O_{ℙ(W^{∗})}(1) ⊠ U^{∗} shows that this restriction map is an isomorphism and we arrived at the contradiction.

2.3.3. Lemma. We have Ker Â_{C} = ℂ = ⟨id⟩ ⊂ Hom(W^{∗} ⊗ W^{∗}) = W^{∗} ⊗ W is the subspace of all homotheties.

Proof. Consider action of the group GL(W^{∗}) on the space W ⊗ W defined by the formula

gΦ(ϕ, ψ) := Φ(gϕ, ψ), ϕ, ψ ∈ W^{∗}

where Φ ∈ W ⊗ W is considered as a bilinear form on W^{∗}.

Remark 3. If both Φ and gΦ are symmetic forms then it is evident that the kernel of Φ is g-invariant.

Given a subspace N ⊂ S^{2}W (i.e. a net of quadrics in ℙ(W^{∗})) one can consider the subgroup G ⊂ GL(W^{∗}) of all g such that gN ⊂ S^{2}W (in other words Alt(gN) = 0). Considering the net A_{C} we can interpret the kernel of the map Â_{C} as the tangent Lie algebra of G. But the above Remark implies that the curve β(C) is the G fixed-point set, hence G can contain only dilations and the Lemma follows.

2.3.4. The Lemma 2.3.3 and the Proposition 2.3.1 imply that we have the following left-exact complex

0 ⟶ 𝔰𝔩(W) ⟶^{Â_{C}} N^{∗} ⊗ Λ^{2}W ⟶^{B̂_{C}} N ⟶ 0,  (11)

where 𝔰𝔩(W) stands for the (W^{∗} ⊗ W)/ℂ.

Lemma. The complex (11) is exact.

Proof. Since dim N^{∗} ⊗ Λ^{2}W = 18 = 15 + 3 = dim 𝔰𝔩(W) + dim N it suffices to prove the surjectivity of the map B̂_{C}. But this easy follows since B_{C}^{−1} is isomorphism.

3. Instanton bundles

3.1. Let R be an exceptional vector bundle on ℙ(W) obtained by the right mutation of O_{ℙ(W)}(1) through T_{ℙ(W)} (for the definition of exceptional bundles and their mutations consult [4]), that is a cokernel of the natural map

0 ⟶ O_{ℙ(W)}(1) ⟶ W^{∗} ⊗ T_{ℙ(W)} ⟶ R ⟶ 0.  (12)

This sequence implies that rank(R) = 11, c_{1}(R) = 15, c_{2}(R) = 105 and c_{3}(R) = 455.

3.1.1. One can easily prove that Hom(R, Ω_{ℙ(W)}(3)) is naturally isomorphic to S^{2}W. Actually it evidently follows from the fact that both R and Ω_{ℙ(W)}(3) are the members of the certain exceptional collection over ℙ(W) (see 3.2.4).

Given a rigid quartic (C, θ) its net of quadrics A_{C} : N → S^{2}W then gives rise to the natural homomorphism μ_{C} : R → N^{∗} ⊗ Ω_{ℙ(W)}(3).

Proposition. The homomorphism μ_{C} is surjective iff C is non-Lüroth.

Proof. The definition of R implies that it has the following resolvent

0 ⟶ W^{∗} ⊗ O_{ℙ(W)} ⟶ 𝔰𝔩(W) ⊗ O_{ℙ(W)}(1) ⟶ R ⟶ 0.

On the other hand, N^{∗} ⊗ Ω_{ℙ(W)}(3) has the following resolvent

0 ⟶ N^{∗} ⊗ O_{ℙ(W)}(−1) ⟶ N^{∗} ⊗ W ⊗ O_{ℙ(W)} ⟶ N^{∗} ⊗ Λ^{2}W ⊗ O_{ℙ(W)}(1) ⟶ N^{∗} ⊗ Ω_{ℙ(W)}(3) ⟶ 0.

The map μ_{C} can be extended to the map of resolvents and we have the following commutative diagram

0  ⟶ W^{∗} ⊗ O_{ℙ(W)} ⟶ 𝔰𝔩(W) ⊗ O_{ℙ(W)}(1) ⟶  R  ⟶ 0
↓  ↓  ↓^{μ_{C}}
0 ⟶ N^{∗} ⊗ O_{ℙ(W)}(−1) ⟶ N^{∗} ⊗ W ⊗ O_{ℙ(W)} ⟶ N^{∗} ⊗ Λ^{2}W ⊗ O_{ℙ(W)}(1) ⟶ N^{∗} ⊗ Ω_{ℙ(W)}(3) ⟶ 0

The map 𝔰𝔩(W) ⊗ O_{ℙ(W)}(1) ⟶ N^{∗} ⊗ Λ^{2}W^{∗} ⊗ O_{ℙ(W)}(1) here is induced from the map Â_{C}, hence its cokernel is but N ⊗ O_{ℙ(W)}(1). Therefore, it suffices to prove that the arising map

N^{∗} ⊗ W ⊗ O_{ℙ(W)} ⟶ N ⊗ O_{ℙ(W)}(1)

is surjective. Assume that it is not surjective. Then there exists a point w ∈ W and a point ξ ∈ N^{∗} such that the map

N^{∗} ⊗ W ⟶^{id ⊗(∧w)} N^{∗} ⊗ Λ^{2}W ⟶^{id ⊗B_{C}^{−1}} N^{∗} ⊗ S^{2}N ⟶ N ⟶^{ξ} ℂ

vanishes. This means that B_{C}^{−1} maps the subspace (W/w)∧w ⊂ Λ^{2}W on the subspace S^{2} Ann ξ ⊂ S^{2}N. But since B_{C}^{−1} is an isomorphism (because B_{C} is) and the dimension of both subspaces is 3, it follows that the map induces an isomorphism of these subspaces, i.e. we have a commutative diagram

0 ⟶ (W/w) ∧ w ⟶ Λ^{2}W ⟶ Λ^{2}(W/w) ⟶ 0
↓^{B_{C}^{−1}}  ↓^{B_{C}^{−1}}  ↓^{B_{C}^{−1}}
0 ⟶ S^{2} Ann ξ ⟶ S^{2}N ⟶ N ⊗ N/Ann ξ ⟶ 0

and all vertical arrows here are isomorphisms. Consider the inverse dual diagram, that is

0 ⟶ Λ^{2} Ann w ⟶ Λ^{2}W^{∗} ⟶^{w} Ann w ⟶ 0
↓^{B_{C}^{∗}}  ↓^{B_{C}^{∗}}  ↓^{B_{C}^{∗}}
0 ⟶ N^{∗} ⊗ ξ ⟶ S^{2}N^{∗} ⟶ S^{2}(N^{∗}/ξ) ⟶ 0

Here all vertical arrows are also isomorphisms. It has the following geometrical meaning. Consider all hyperplanes in ℙ(Λ^{2}W) which correspond to the subspace Λ^{2} Ann w ⊂ Λ^{2}W^{∗} = H^{0}(O_{ℙ(Λ^{2}W)}(1)). Then all divisors in the linear system |2K_{C}| intersected on the curve γ(C) ⊂ Gr(2, W) ⊂ ℙ(Λ^{2}W) by these hyperplanes form the subsystem ξ_{|C} + |K_{C}| corresponding to the subspace N^{∗} ⊗ ξ ⊂ S^{2}N^{∗} = H^{0}(C, ω_{C}^{2}). In other words, the map

γ^{∗}W/U ⟶^{∧w} γ^{∗}Λ^{2}(W/U) = γ^{∗}O_{Gr(2,W)}(1) ≅ ω_{C}^{2}

factors as

γ^{∗}W/U ⟶ ω_{C} ⟶^{ξ} ω_{C}^{2}

hence Hom(γ^{∗}W/U, ω_{C}) ≠ 0 which by the Corollary 2.2.10 means that C is Lüroth.

The same arguments show that if C is Lüroth then μ_{C} is not surjective.

Remark 4. The fact that for generic N ⊂ S^{2}W the corresponding map R ⟶ N^{∗} ⊗ Ω_{ℙ(W)}(3) is surjective is rather surprising, because counting a parameters it seems that this map should degenerate in codimension 3.

3.1.2. Recall that an instanton vector bundle over ℙ(W) is by definition a 2-dimensional stable vector bundle F with c_{1}(F) = 0 and H^{•}(ℙ(W), F ⊗ O_{ℙ(W)}(−2)) = 0. Let MI_{3}(ℙ(W)) denote the space of instanton vector bundles over ℙ(W) with c_{2} = 3.

Definition. Let C be a non-Lüroth rigid quartic. Denote by R_{C} the kernel of the surjective map μ_{C} : R → N^{∗} ⊗ Ω_{ℙ(W)}(3).

Theorem 1. R_{C} is an instanton bundle over ℙ(W) with c_{2} = 3.

The proof of the Theorem consists of the following three Lemmas.

3.1.3. Lemma. R_{C} is a 2-dimensional vector bundle on ℙ(W) with c_{1}(R_{C}) = 0, c_{3}(R_{C}) = 3.

Proof. By Definition 3.1.2 and Proposition 3.1.1 we have the following exact sequence

0 ⟶ R_{C} ⟶ R ⟶^{μ_{C}} N^{∗} ⊗ Ω_{ℙ(W)}(3) ⟶ 0  (13)

which implies the Lemma.

3.1.4. Lemma. The bundle R_{C} is stable.

Proof. We need to show that R_{C} does not contain an invertible subsheaves of zero degree. This is equivalent to H^{0}(ℙ(W), R_{C}) = 0.

By the Serre duality it suffices to check that H^{3}(ℙ(W), R_{C}^{∗} ⊗ O_{ℙ(W)}(−4)) = 0. Since c_{1}(R_{C}) = 0 the bundle R_{C} is self-dual, hence we need to compute H^{3}(ℙ(W), R_{C} ⊗ O_{ℙ(W)}(−4)). This can be done applying (13). Namely, it suffices to note that

H^{3}(ℙ(W), R(−4)) = H^{2}(ℙ(W), Ω_{ℙ(W)}(−1)) = H^{3}(ℙ(W), Ω_{ℙ(W)}(−1)) = 0.

3.1.5. Lemma. We have H^{•}(ℙ(W), R_{C}(−2)) = 0.

Proof. Follows from H^{•}(ℙ(W), R(−2)) = 0 and H^{•}(ℙ(W), Ω_{ℙ(W)}(1)) = 0.

3.1.6. Here we will state an additional property of R_{C} which as we will see below describes the subset in MI_{3}(ℙ(W)) consisting of instantons F = R_{C}, C ∈ Qu \ Qu^{L}.

Lemma. We have Hom^{•}(R_{C}, T_{ℙ(W)}) = 0.

Proof. Follows from Hom^{•}(R, T_{ℙ(W)}) = 0 and Hom(Ω_{ℙ(W)}(3), T_{ℙ(W)}) = 0.

3.2. Now we will study all instantons which satisfy the orthogonality property of Lemma 3.1.6. Let MI_{3}^{T} (ℙ(W)) ⊂ MI_{3}(ℙ(W)) be subspace parameterizing all instantons F with

Hom^{•}(F, T_{ℙ(W)}) = 0.  (14)

Remark 5. Evidently (14) is an open condition, hence MI_{3}^{T} (ℙ(W)) ⊂ MI_{3}(ℙ(W)) is an open subspace.

Remark 6. Note that c_{1}(F) = 0 implies the self-duality of F

F^{∗} ≅ F

Here isomorphism is unique up to dilations, because F is stable.

3.2.1. Let F ∈ MI_{3}^{T} (ℙ(W)).

Lemma. The following table contains the dimensions h^{i}(F (q)) := dim H^{i}(ℙ(W), F ⊗ O_{ℙ(W)}(q)) of the cohomology spaces of twisted bundles F (q) := F ⊗ O_{ℙ(W)}(q).

\begin{matrix}q & −5 & −4 & −3 & −2 & −1 & 0 & 1 \\ h^{3}(F (q)) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ h^{2}(F (q)) & 1 & 4 & 3 & 0 & 0 & 0 & 0 \\ h^{1}(F (q)) & 0 & 0 & 0 & 0 & 3 & 4 & 1 \\ h^{0}(F (q)) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

Proof. Actually, we only need to show that H^{0}(ℙ(W), F (1)) = 0. Then the proposition follows by the usual arguments (see [3]) and Riemann-Roch Theorem.

To this end consider the Euler exact sequence on ℙ(W) multiplied by F

0 ⟶ F ⟶ W ⊗ F (1) ⟶ F ⊗ T_{ℙ(W)} ⟶ 0.

Since H^{•}(ℙ(W), F ⊗ T_{ℙ(W)}) ≅ H^{•}(ℙ(W), F^{∗} ⊗ T_{ℙ(W)}) ≅ Hom^{•}(F, T_{ℙ(W)}) = 0 we have canonical isomorphisms

H^{•}(ℙ(W), F) ≅ W ⊗ H^{•}(ℙ(W), F (1)).

In particular, H^{0}(ℙ(W), F (1)) = 0 since H^{0}(ℙ(W), F) = 0 (because F is stable).

3.2.2. The proof of the above Lemma gives us natural identification

H^{1}(ℙ(W), F) ≅ W ⊗ H^{1}(ℙ(W), F (1)) = W ⊗ ℂ = W.

Let N := H^{1}(F (−1)). Then the Serre duality gives us the following identification of the cohomology spaces of F (q).

\begin{matrix}q & −5 & −4 & −3 & −2 & −1 & 0 & 1 \\ H^{3}(ℙ(W), F (q)) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ H^{2}(ℙ(W), F (q)) & ℂ & W^{∗} & N^{∗} & 0 & 0 & 0 & 0 \\ H^{1}(ℙ(W), F (q)) & 0 & 0 & 0 & 0 & N & W & ℂ \\ H^{0}(ℙ(W), F (q)) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

3.2.3. Consider the following collection of vector bundles on ℙ(W)

𝔈 = (𝔈_{1}, 𝔈_{2}, 𝔈_{3}, 𝔈_{4}) := (Ω_{ℙ(W)}(−4), O_{ℙ(W)}(−5), O_{ℙ(W)}(−3), O_{ℙ(W)}(−2)).

This is full exceptional collection (see [4]). The dual exceptional collection 𝔈̂ of 𝔈 is following

𝔈̂ = (𝔈̂_{1}, 𝔈̂_{2}, 𝔈̂_{3}, 𝔈̂_{4}) := (T_{ℙ(W)}, R, Ω_{ℙ(W)}(3), O_{ℙ(W)}(2)).

Proposition. There is an exact sequence of vector bundles on ℙ(W)

0 ⟶ F ⟶ R ⟶^{μ_{F}} N^{∗} ⊗ Ω_{ℙ(W)}(3) ⟶ 0.  (15)

Proof. Consider the Beilinson spectral sequence of F with respect to the pair (𝔈, 𝔈̂) of dual exceptional collections

E_{1}^{pq} = H^{q}(F ⊗ 𝔈_{p}) ⊗ 𝔈̂_{p} ⇒ F.

Since

H^{q}(ℙ(W), F ⊗ 𝔈_{1}) = H^{q}(ℙ(W), F ⊗ Ω_{ℙ(W)}(−4)) ≅ H^{3−q}(F^{∗} ⊗ T_{ℙ(W)})^{∗} ≅ Hom^{3−q}(F, T_{ℙ(W)})^{∗} = 0;

H^{q}(ℙ(W), F ⊗ 𝔈_{2}) = H^{q}(ℙ(W), F ⊗ O_{ℙ(W)}(−5)) = \begin{cases}ℂ, & if q = 2 \\ 0, & in other cases; \end{cases}

H^{q}(ℙ(W), F ⊗ 𝔈_{3}) = H^{q}(ℙ(W), F ⊗ O_{ℙ(W)}(−3)) = \begin{cases}N^{∗}, & if q = 2 \\ 0, & in other cases; \end{cases}

H^{q}(ℙ(W), F ⊗ 𝔈_{4}) = H^{q}(ℙ(W), F ⊗ O_{ℙ(W)}(−2)) = 0

the spectral sequence degenerates into (15) and the Proposition follows.

3.2.4. Since 𝔈̂ is dual to 𝔈 we have

Hom(R, Ω_{ℙ(W)}(3)) = Hom(𝔈̂_{2}, 𝔈̂_{3}) ≅ Hom(𝔈_{2}, 𝔈_{3})^{∗} =

= Hom(O_{ℙ(W)}(−5), O_{ℙ(W)}(−3))^{∗} ≅ H^{0}(ℙ(W), O_{ℙ(W)}(2))^{∗} ≅ S^{2}W

Therefore the surjective map μ_{F} : R ⟶ N^{∗} ⊗ Ω_{ℙ(W)}(3) gives rise to the embedding A_{F} : N ⟶ S^{2}W which we can interpret as a net of quadrics in ℙ(W^{∗}).

Lemma. The net A_{F} is a regular non-Lüroth net.

Proof. Assume that for some 0 ≠ n ∈ N the quadric Q := A_{F} (n) is the quadric of corank 2. Let L ⊂ W be the image of the map W^{∗} ⟶ W induced by Q. Consider the composition μ_{Q} : R ⟶^{μ_{F}} N^{∗} ⊗ Ω_{ℙ(W)}(3) ⟶^{ev_{n} ⊗ id} Ω_{ℙ(W)}(3). It should be surjective, because μ_{F} and ev_{n} are.

On the other hand this composition is nothing but the map, induced by the quadric Q ∈ S^{2}W = Hom(R, Ω_{ℙ(W)}(3)). Consider the commutative diagram

0  ⟶ W^{∗} ⊗ O_{ℙ(W)} ⟶ 𝔰𝔩(W) ⊗ O_{ℙ(W)}(1) ⟶  R  ⟶ 0
↓^{Q}  ↓  ↓^{μ_{Q}}
0 ⟶ O_{ℙ(W)}(−1) ⟶ W ⊗ O_{ℙ(W)} ⟶ Λ^{2}W ⊗ O_{ℙ(W)}(1) ⟶ Ω_{ℙ(W)}(3) ⟶ 0

Here the map 𝔰𝔩(W) ⊗ O_{ℙ(W)}(1) ⟶ Λ^{2}W ⊗ O_{ℙ(W)}(1) is induced from the composition

𝔰𝔩(W) ⟶^{id ⊗Q} 𝔰𝔩(W) ⊗ S^{2}W ⟶ W ⊗ W^{∗} ⊗ W ⊗ W ⟶^{id ⊗⟨·,·⟩⊗id} W ⊗ W ⟶^{Alt} Λ^{2}W.

One easily sees that the cokernel of this map is Λ^{2}(W/L). The above commutative diagram implies that the cokernel of the map μ_{Q} is equal to the cokernel of the arising map

W/L ⊗ O_{ℙ(W)} ⟶ Λ^{2}(W/L) ⊗ O_{ℙ(W)}(1)

This map evidently vanishes over ℙ(L) ⊂ ℙ(W). But if the corank of Q is greater then 1 then dim W/L ≥ 2, hence the second term above is not zero, hence the map is not surjective and the regularity follows.

Now since the map μ_{F} is surjective the Proposition 3.1.1 implies that A_{F} is non-Lüroth.

3.2.5. The above results can be summarized as follows.

Theorem 2. The maps

(C, θ) ↦ R_{C},  F ↦ A_{F} ,  A ↦ (H_{A}, θ_{A})

define an isomorphisms

Qu \ Qu^{L} ≅ MI_{3}^{T} (ℙ(W)) ≅ Nets \ Nets^{L} .

3.2.6. Corollary. The space MI_{3}(ℙ(W)) is rational.

Proof. Because Nets \ Nets^{L} is an open subset of the Grassmann variety Gr(3, S^{2}W) and MI_{3}^{T} (ℙ(W)) is an open subset of MI_{3}(ℙ(W)).

Remark 7. The rationality of MI_{3}(ℙ(W)) originally was proved by G. Ellingsrud and S. A. Stromme in [6].

3.3. In this Subsection we will demonstrate how the curve C naturally arises while studying the stability of restriction of F = R_{C} on hyperplanes and lines in ℙ(W).

3.3.1. The hyperplane H ⊂ ℙ(W) is called non-stable for F if F_{|H} is not stable (i.e. if H^{0}(H, F_{|H}) ≠ 0).

Proposition. The set of all non-stable hyperplanes for F = R_{C} is β(C) ⊂ ℙ(W^{∗}).

Proof. By definition F_{|H} is not stable iff H^{0}(H, F_{|H}) ≠ 0. The exact sequence

0 ⟶ F (−1) ⟶^{H} F ⟶ F_{|H} ⟶ 0

and 3.2.2 show that this is equivalent to the non-injectivity of the arising map N ⟶^{H} W. But this map is nothing but A_{C}(·, H) and the Proposition follows.

3.3.2. The line L ⊂ ℙ(W) is called d-jumping line for F if F_{|L} = O_{L}(−d) ⊕ O_{L}(d).

Proposition. If F = R_{C} then F has no 3-jumping lines. The set of all 2-jumping lines for F = R_{C} coincides with γ(C) ⊂ Gr(2, W).

Proof. The line L is d-jumping line for F with d ≥ 2 iff H^{1}(L, F_{|L}) ≠ 0. The exact sequence

0 ⟶ F (−2) ⟶ Ann L ⊗ F (−1) ⟶ F ⟶ F_{|L} ⟶ 0

and 3.2.2 give the following exact sequence

N ⊗ Ann L ⟶ W ⟶ H^{1}(L, F_{|L}) ⟶ 0.

The first map here is just A_{C}(·, ·). Assume that this is not surjective. Then there exists φ ∈ W^{∗} such that A_{C}(N, φ) ⊂ L, hence φ = β(x) for some x ∈ C and L = γ(x) (see Lemma 2.2.5).

To conclude L can not be a 3-jumping line for F = R_{C} because for any L we have R_{|L} = O_{L}(1)^{⊕7} ⊕ O_{L}(2)^{⊕4} and F_{|L} is a subbundle in R_{|L}.

3.3.3. Assume that C is smooth. Then θ is a line bundle.

We denote the divisor class on C corresponding to the line bundle θ by the same letter. We denote by K_{C} the canonical class of C. We have ω_{C} = O_{C}(K_{C}).

Given a point x ∈ C the Riemann-Roch implies that |θ + x| ≠ ∅. The non-degeneracy of θ means that there is a unique effective divisor x_{1} + x_{2} + x_{3} ∈ |θ + x| which we will denote by T_{θ}(x).

Let T_{θ} ⊂ C × C be the following subspace

T_{θ} = {(x, y) | T_{θ}(x) − y ≥ 0} ⊂ C × C.

It is known (see [5]) that T_{θ} is a symmetric (with respect to the permutation of factors) divisor in C × C nonintersecting diagonal (T_{θ} ∩Δ = ∅) and that T_{θ} ∼ Δ + p_{1}^{∗}θ + p_{2}^{∗}θ.

Proposition. A 2-jumping line L = γ(x) is contained in the non-stable hyperplane H = β(y) iff (x, y) ∈ T_{θ} ⊂ C × C.

Proof. Let T_{θ}(x) = y_{1} + y_{2} + y_{3} and let H_{i} = β(y_{i}). The Riemann-Roch says that

H^{0}(C, β^{∗}O_{ℙ(W^{∗})}(1)⊗O_{C}(−y_{1}−y_{2}−y_{3})) = H^{0}(C, O_{C}(K_{C} +θ−y_{1}−y_{2}−y_{3})) = H^{0}(C, O_{C}(K_{C} −x)) = ℂ^{2},

hence the triple of points β(y_{1}), β(y_{2}), β(y_{3}) ∈ ℙ(W^{∗}) lies on a line, that is H_{1} ∩ H_{2} ∩ H_{3} = L′ ⊂ ℙ(W) and L′ is a line. We claim that actually L′ = L.

Indeed, chose a point P ∈ L′ and let D_{P} = z_{1} + z_{2} + z_{3} ∈ |K_{C} + θ − y_{1} − y_{2} − y_{3}| ∈ |K_{C} − x| be the corresponding divisor and l_{P} ⊂ ℙ(N) be the line in ℙ(N) such that D_{P} = α^{−1}(l_{P} ∩ α(C)). Then we have

A_{C}(α(x), β(x), β(z_{j})) = 0, (j = 1, 2, 3).

On the other hand since x, z_{i}, z_{j} ∈ l_{P} we can choose a_{ij}, a′_{ij} ∈ ℂ such that α(z_{i}) = a_{ij}α(x) + a′_{ij}α(z_{j}), hence

A_{C}(α(z_{i}), β(x), β(z_{j})) = A_{C}(a_{ij}α(x) + a′_{ij}α(z_{j}), β(x), β(z_{j})) = 0, (i, j = 1, 2, 3).

This means that A_{C}(l_{P} , β(x)) ∈ ℙ(W) is a common point of hyperplanes β(z_{1}), β(z_{2}), β(z_{3}), that is A_{C}(l_{P} , β(x)) = P. Therefore A_{C}(N, β(x)) = L′, that is L′ = L and the Claim is proven.

Now we see that {y | β(y) ⊃ α(x)} = T_{θ}(x). Note that

β(y) ⊃ α(x) ⇔ A_{C}(N, β(x), β(y)) = 0 ⇔ A_{C}(N, β(y), β(x)) = 0 ⇔ α(y) ⊂ β(x),

and the Proposition follows.

Remark 8. In the non-reduced case (see 2.1.2, 2.2.4) it is evident that a 2-jumping line L = γ(x) is contained in a non-stable hyperplane H = β(y) (x, y ∈ C_{red}) iff (x, y) ∈ Δ ⊂ C_{red} × C_{red}.

References

[1] W. Barth. Some properties of stable rank-2 vector bundles on ℙ_{n}, Math. Ann. 226 (1977), pp. 125–150.

[2] W. Barth. Moduli of vector bundles on the projective plane, Invent. math., 42 (1977), pp. 63–91.

[3] W. Barth, G. Elencwajg. Concernant la cohomologie des fibrés algébriques stables sur ℙ_{n}(ℂ), In Variétés analytiques compactes, Nice 1977, Lecture Notes in Math. 683, pp. 1–24. Springer 1978.

[4] A. Bondal. Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR, Ser. Mat., 53 (1989), pp. 25–44; English transl. in Math. USSR Izvestiya, 34 (1990).

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