On correlation between testing S -duality conjecture and exceptional bundles.

B.V. Karpov

July 27, 1999

Introduction

In this paper algebro-geometrical problems are considered, which are possibly interesting for physicists because of relation to testing S -duality conjecture for super-symmetric Yang-Mills quantum field theories in four dimensions. This conjecture asserts that such a theory has a symmetry exchanging strong and week coupling and exchanging also electric and magnetic fields.

In the work [14], with which S -duality conjecture was originated, Montonen and Olive proposed existence of ℤ_{2} -symmetry involving only the real coupling constant g. In further development this constant was combined with one more parameter, so-called θ angle^{1}, in a complex parameter

τ = \frac{θ}{2π} + \frac{4πi}{g^{2}} ,

see [VW] and further references there. For physical reasons it is natural to extend the ℤ_{2} symmetry to SL(2, ℤ) action on τ

τ ↦ \frac{aτ + b}{cτ + d} , where ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) .

The original Montonen-Olive symmetry takes place when θ = 0 for the action by ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ).

One of the ways to test the conjecture deals with some quantities that can be interpreted as correlation functions in the theories. These are power series in q = e^{2πiτ} . One has to examine these quantities for SL(2, ℤ) -invariance or more generally for some modular properties under the SL(2, ℤ) -action.

In this paper such power series are considered not for QFT itself but for its semiclassical limit i.e. for classical gauge theory, from which the given QFT is obtained by quantization.

It is important to have zero quantum corrections betweem quantum and semiclassical partition functions. For this it is sufficient for the value of β -function on field configuration of the theory to be zero. Then the theory is scaling-invariant, τ is really a constant, and quantum correlation functions can be calculated at semi-classical limit.

We describe all cases of vanishing of beta-function in 1.4. There are six ”regular” field configurations labelled as a), b), c), d), e), f), which have zero β -function for arbitrary rank of the gauge group, and some special cases. We present here the power series corresponding to the regular cases only.

1or vacuum angle

Let S be del Pezzo surface over ℂ, K be the canonical class of S. Denote by M(r, c_{1}, k) the moduli space of Mumford-Takemoto stable vector bundles over S w.r. to −K with the rank r, the Chern classes c_{1} and c_{2} , where

k = c_{2} − \frac{1}{2}c_{1} · (c_{1} − K) − r .

The Giesecker-Marujama compactification \overline{M}(r, c_{1}, k) of this space is a coarse moduli space of semi-stable sheaves by Giesecker with the same invariants. It is known [13] that \overline{M}(r, c_{1}, k) is a projective scheme. Besides, if r and c_{1} · K are coprime, then \overline{M}(r, c_{1}, k) carries a universal family [13, 6.11] and is smooth [13, 6.7] as far as any semi-stable sheaf with these invariants is stable.

Consider the following power series.

a) Put with fixed r and c_{1} :

Z_{a}(τ) = ∑_{k⩾0}χ(\overline{M}(r, c_{1}, k))q^{k} .  (1)

Here χ denotes topological Euler characteristic.

b) Suppose that the following condition holds:

−K^{2} < \frac{c_{1} · (−K)}{r} < 0  (2)

Let F be semistable sheaf with the invariants (r, c_{1}, k) ; then from (2), Serre duality and properties of stability it follows that H^{0}(F) = H^{2}(F) = 0 , whence we have:

dim H^{1}(F) = −χ(F) = c_{2} − \frac{1}{2}c_{1} · (c_{1} − K) − r = k .  (3)

Observe that this dimension depends only on topological invariants of the sheaf F . Therefore, a sheaf H_{k} over \overline{M}(r, c_{1}, k) with fiber (H_{k})_{F} = H^{1}(F) is locally free.

Restrict the invariants by the condition

dim M(r, c_{1}, k) = 2r dim H^{1}(E)  (4)

and consider the following power series (with r and c_{1} fixed):

Z_{b}(τ) = ∑_{k⩾0}c_{top}(H_{k})^{2 r}q^{k} .  (5)

The equality (4) is necessary for the coefficients of Z_{b}(τ) to be a numbers.

For the rest power series from c) to f) we replace (2) by a stronger condition

−\frac{1}{2}K^{2} < \frac{c_{1} · (−K)}{r} < 0 .  (6)

It is shown in Section 3 that for a semi-stable sheaf F with topological invariants satisfying this condition, the dimensions of H^{1}(S^{2}F) and H^{1}(Λ^{2}F) depend only on the the invariants of F . Let S_{k} and L_{k} be bundles over M(r, c_{1}, k) with fibers respectively H^{1}(S^{2}F) and H^{1}(Λ^{2}F) .

c) Under the equality of dimensions

dim M(r, c_{1}, k) = dim H^{1}(S^{2}F) + dim H^{1}(Λ^{2}F)  (7)

consider the power series

Z_{c}(τ) = ∑_{k} c_{top}(S_{k} ⊕ L_{k}) q^{k} ,  (8)

where r and c_{1} are fixed; the same is assumed for the rest power series.

d) Under the assumption

dim M(r, c_{1}, k) = (r + 2) dim H^{1}(F) + dim H^{1}(Λ^{2}F)  (9)

consider the power series

Z_{d}(τ) = ∑_{k⩾0}c_{top}(H_{k})^{r+2}c_{top}(L_{k})q^{k} .  (10)

e) Under the assumption

dim M(r, c_{1}, k) = (r − 2) dim H^{1}(F) + dim H^{1}(S^{2}F)  (11)

consider the power series

Z_{e}(τ) = ∑_{k⩾0}c_{top}(H_{k})^{r−2}c_{top}(S_{k})q^{k} .  (12)

f) Under the assumption

dim M(r, c_{1}, k) = 4 dim H^{1}(F) + 2 dim H^{1}(Λ^{2}F)  (13)

consider the power series

Z_{f}(τ) = ∑_{k⩾0}c_{top}(H_{k})^{4}c_{top}(L_{k})^{2}q^{k} .  (14)

For the power series (1) and (5) there are descriptions of gauge theories such that Z_{a}(τ) and Z_{b}(τ) can be interpreted as quantum correlation functions calculated in semi-classical limit (see [16] for Z_{a}(τ) , where r = 2 , and [8] for Z_{b}(τ) ). The rest power series are composed in a similar way from mathematical viewpoint, but I believe that they have analogous physical meaning.

Observe that for all cases except a) one can construct correlation functions in different ways. In particular, for b) there are 2r + 1 sub-cases that correspond to replacing k of 2r copies of H^{1}(F) with H^{1}(F^{∗}) in the right hand side of (4). However, it is difficult to calculate dim H^{1}(F^{∗}) , so, we consider only sub-case described above. For the cases c), d), e), f) the situation is similar.

The case c) was considered in [7]; there is a unique solution to the system of (6) and (7): S is the plane with 8 blowed points, r = 3 and c_{1} = K.

In the case b) the restrictions (2) and (4) distinguish a subset in the set of (r, c_{1}, k) , the most interesting elements of which are related to exceptional bundles. It is shown in [8] that any exceptional bundle that satisfies (2) and has zero cohomology groups produces a correlation function for testing S -duality. Namely, each such a bundle E is initial (zerodimensional) term of the sequence of moduli spaces \overline{M}(r(E), c_{1}(E), k) , k ⩾ 0 , that gives a power series of the form (5). We describe this relation in Section 2.

The set of exceptional sheaves has an interesting combinatorial structure: there are some transformations, so called mutations, which permit to obtain an infinite set of exceptional sheaves from a finite exceptional collection. Moreover, it is often possible to choose the initial exceptional collection such that the sheaves in it are locally free, and the mutations give new exceptional bundles. Herewith, if all cohomology groups of exceptional bundles in the initial collection are trivial, then the same is true for the new exceptional bundles obtained by mutations. Thus, there arises the following question: do the mutations induce any transformations of correlation functions? (See Example 2.3.) For the moment there is no any answer. The first step in attempt to answer this question is certainly related to looking for transformations of the above-mentioned moduli spaces \overline{M}(r(E), c_{1}(E), k) that generalize the mutations. We discuss an approach to possible constructions of such transformations for moduli spaces over ℙ^{2} in Section 4.

In Section 3 we examine the exceptional bundles that give correlation functions for d),

e) and f) cases. There are ”less” such bundles than in the case b) because the degree (i. e. c_{1} · (−K) ) of such a bundle is a function of its rank (see 3.2). However, this condition can be compatible with mutations as it is shown in Example 3.4.

In Section 6 we give a description of a part of a moduli space \overline{M}(r(E), c_{1}(E), k) over ℙ^{2} consisting of non-locally-free sheaves, hoping that this can help to construct transformations of moduli spaces.

I am grateful to Max-Planck Institute für Mathematik in Bonn for hospitality, to A. N. Tyurin for attention to this work, and to A. Levin for useful discussions, which helped me to understand the relation between Dynkin index and second Chern class.

I dedicate this work to memory of my grandmother.

1 Beta-function

This is a function of QFT under consideration. For a classical gauge theory one can calculate beta-function as well. It depends on representations of fiber-wise gauge group that arise from the action on fields. Such a collection of representations we call field configuration of the given theory. So, we start with representations.

Let γ be an irreducible finite dimensional complex representation of su(r) (the Lie algebra for the fiber-wise gauge group), can be the canonical representation of su(r) in ℂ^{r}. Consider the invariant form

B_{γ}(X; Y) = T r(γ(X) γ(Y)) .

It is well-known that all such forms are proportional to each other. Denote by c_{2}(γ) the coefficient between B_{γ} and the trace form

(X; Y) := T r(X Y) = B_{can}(X; Y) .  (15)

We have:

B_{γ}(X; Y) = c_{2}(γ)(X; Y) .

The number c_{2}(γ) is positive integer, it is called Dynkin index of the representation γ.

One can define c_{2} for an irreducible u(r) -representation γ̃ as c_{2}(γ) , where γ is the restriction of γ̃ on the subalgebra su(r) ⊂ u(r) . All the results of this section hold also for u(r) -representations.

For the dual representation we have: B_{γ^{∗}}(X; Y) = B_{γ}(X; Y) . Therefore, we obtain:

c_{2}(γ^{∗}) = c_{2}(γ) .  (16)

Any complex representation of su(r) can be extended to its complexification sl(r, ℂ) without changing c_{2} , so we consider γ as an irreducible finite dimensional representation of sl(r, ℂ) . There is a well-known formula that expresses c_{2}(γ) in terms of the higher weight.

1.1 Proposition. Let R be the root system for sl(r, ℂ) , λ be the highest weight of γ with respect to a base of R, ρ be the half-sum of the positive roots. Then

c_{2}(γ) = \frac{(λ; λ + 2ρ)}{r^{2} − 1} dim Γ = \frac{(λ; λ + 2ρ)}{r^{2} − 1} ∏_{α∈R_{+}} \frac{(λ + ρ; α)}{(ρ; α)} .  (17)

Here Γ is the representation space for γ and ( ; ) is the dual to the trace form (15).

1.2 Corollary. ([8])

c_{2}(Λ^{k}(can)) = \binom{r − 2}{k − 1} , c_{2}(S^{k}(can)) = \binom{r + k}{k − 1} and c_{2}(ad) = 2 r .  (18)

Consider a finite collection of representations {γ_{1}, …, γ_{N_{a}}}, among which there are some isomorphic to each other^{2}. The beta-function takes the collection {γ_{1}, …, γ_{N_{a}}} to the value

β({γ_{i}}) = −c_{2}(ad) + ∑_{i} c_{2}(γ_{i}) ,  (19)

(see [1]). We describe here all cases when β({γ_{i}}) = 0 . The positiveness of c_{2}(γ) imposes an essential restriction on representations included in field configuration with zero beta-function (19):

c_{2}(γ) ⩽ 2 r = c_{2}(ad) .  (20)

The above-mentioned values of c_{2} for Λ^{k}(can) and S^{k}(can) show that there is ”a small amount” of such representations, since the c_{2} ’s grow as polynomials of r. Here is the exact statement.

1.3 Theorem. ([9]) For 3 ⩽ r ⩽ 5 or r ⩾ 9 the list of all irreducible representations γ of su(r) such that c_{2}(γ) ⩽ c_{2}(ad) , is as follows:

ad , can , can^{∗} ≅ Λ^{r−1}(can) , S^{2}(can) , S^{2}(can^{∗}) , Λ^{2}(can) and
Λ^{2}(can^{∗}) ≅ Λ^{r−2}(can) .

For r = 6, 7, 8 there are additional representations

Λ^{3}(can) and Λ^{3}(can^{∗}) ≅ Λ^{r−3}(can) .

2From physical viewpoint this collection describes configuration of fields for QFT under consideration. A subcollection consisting of isomorphic to each other representations corresponds to a particle, the number of representations in the subcollection is the number N_{f} of flavours of the particle, and the dimension r of the canonical representation of the gauge group is the number of colors. So, N_{a} is the joint number for the flavours of particles in our theory.

Now it is clear which form can have a collection of representations with zero beta-function. By the definition, c_{2}(can) = 1 , so any collection {γ_{i}} with β({γ_{i}}) < 0 can be completed by an appropriate number of can ’s or can^{∗} ’s to a collection with β = 0 . All such collections are described in the following statement.

1.4 Proposition. ([9]) Let {γ_{1}, …, γ_{N_{a}}} be a collection of irreducible (possibly equivalent) representations of su(r) such that β({γ_{i}}) = 0 . Then {γ_{i}} is one of the following collections.

1. For r = 2 the cases a) and b) below; for r = 3 — a),b),c); for r = 4, 5 and r ⩾ 9 — the cases a) , …, f).

a) N_{a} = 1 and γ_{1} = ad.

b) N_{a} = 2r and each of γ_{i} is either can or can^{∗} ≅ Λ^{r−1}(can) .

c) N_{a} = 2 , one of γ_{i} is either Λ^{2}(can) , or Λ^{2}(can^{∗}) and the other is either S^{2}(can) or S^{2}(can^{∗}) .

d) N_{a} = r + 3 , one of γ_{i} is either Λ^{2}(can) or Λ^{2}(can^{∗}) and the other r + 2 representations are either can or can^{∗}.

e) N_{a} = r − 1 , one of γ_{i} is either S^{2}(can) , or S^{2}(can^{∗}) and the other r − 2 representations are either can or can^{∗}.

f) N_{a} = 6 , two of γ_{i} are either Λ^{2}(can) , or Λ^{2}(can^{∗}) and the rest four are either can or can^{∗}.

2. For r = 6 the following cases add to above-mentioned cases: g_{6} ) N_{a} = 7 , one of γ_{i} is either Λ^{3}(can) or Λ^{3}(can^{∗}) and the rest six are either can or can^{∗}. h_{6} ) N_{a} = 4 , one of γ_{i} is either Λ^{3}(can) or Λ^{3}(can^{∗}) , another one is either Λ^{2}(can) or Λ^{2}(can^{∗}) and the rest two are either can or can^{∗}. i_{6} ) N_{a} = 2 , each representation is either Λ^{3}(can) or Λ^{3}(can^{∗}) .

3. For r = 7 the following case adds to a), …,f): g_{7} ) N_{a} = 5 , one of γ_{i} is either Λ^{3}(can) or Λ^{3}(can^{∗}) and the rest four are either can or can^{∗}.

4. For r = 8 the following case adds to a), …,f): g_{8} ) N_{a} = 2 , one of γ_{i} is either Λ^{3}(can) , or Λ^{3}(can^{∗}) and the other is either can or can^{∗}.

For su(2) and su(3) in some of the cases above the algebro-geometric problem is been studied by method of Vafa and Witten. For instance, for P^{2} and r = 2 the case a) is investigated in [16], the case b) in [5]. Geometric realization of the case b) for del Pezzo surfaces is mystically connected with exceptional bundles ([8], [9]). We briefly describe this connection in the next section.

1.5 Dynkin index and second Chern class.

The relation between Dynkin indices and second Chern classes is, of course, known for experts. We describe it here because this fact explains why the conditions on dimensions (4), (7), (9),

(11) and (13) expressed via topological invariants are independent of the second Chern class.

Let E be a vector bundle with structure group G. We restrict our attention to the cases when G is one of the following groups:

SU(r), SL(r, ℂ), U(r), GL(r, ℂ) .

Suppose the bundle E is given by trivializations {U_{α}, φ_{α} : E|_{U_{α}} ⟶^{≅} U_{α} × ℂ^{r}} and transition functions g_{αβ} : U_{α} ∩ U_{β} → G.

For a complex representation γ : G → GL(Γ) define a new vector bundle γE to have trivializations γE|_{U_{α}} ≅ U_{α} × Γ and transition functions γ ∘ g_{αβ}. The correspondent representation of Lie algebra 𝔤 of the group G we denote also by γ. Then the relation between Dynkin index c_{2}(γ) and second Chern classes c_{2}(E) , c_{2}(γE) is as follows

c_{2}(γE) = c_{2}(γ) c_{2}(E) for G = SU(r), SL(r, ℂ) ;
c_{2}(γE) = c_{2}(γ) c_{2}(E) + N_{γ} c_{1}^{2}(E) for G = U(r), GL(r, ℂ) ,

where N_{γ} is some coefficient. In other words, the coefficient at c_{2}(E) in the expression of c_{2}(γE) in terms of the Chern classes of the bundle E is just the Dynkin index c_{2}(γ) .

To explain these relations we use the definition of Chern classes via differential geometry [6]. Let a be a connection in the bundle E with the curvature Θ . If the curvature is given locally by collection {Θ_{α}} of matrices of two-forms, then over U_{α} ∩ U_{β} we have

Θ_{α} = g_{αβ}Θ_{β}g_{αβ}^{−1}.

Therefore, if P is an invariant polynomial of degree m on 𝔤 , i.e. such that P (A) = P (gAg^{−1}) , g ∈ G; then P (Θ) is a differential 2m -form. Moreover, dP (Θ) = 0 and the cohomology class [P (Θ)] ∈ H^{2m}(M) does not depend on Θ (see [6, 3.3]). The Chern classes of E are defined as

c_{k}(E) = [P_{k}(\frac{i}{2π}Θ)] ∈ H^{2k}(M) ,

where P_{k} are the basic invariant polynomials such that

det(A + tI) = ∑_{k=0}^{r} P_{k}(A)t^{r−k} .

We have: P_{0} ≡ 1 , P_{1}(A) = tr(A) , and P_{r}(A) = det(A) . More generally, P_{k}(A) = tr(∧^{k}A) , where elements of matrix ∧^{k}A are minors of k -th order of A. Observe that the basic invariant polynomials are nothing else as elementary symmetric functions of the eigenvalues.

It is clear now that the coefficients is the expression of c_{2}(γE) via c_{2}(E) and c_{1}^{2}(E) are exactly the same as the ones in expression of the invariant polynomial P_{2}(γ(A)) via P_{2}(A) and P_{1}^{2}(A) . For our purposes it is sufficient to determine the coefficient at P_{2}(A) , so we can consider the restriction to the set of traceless matrices. From the identity

tr(A^{2}) = (tr A)^{2} − 2P_{2}(A)

it follows that

tr(A^{2}) = −2P_{2}(A) whenever tr A = 0 .

From the other hand, by the definition of Dynkin index, we have

tr(γ(A)^{2}) = c_{2}(γ) tr(A^{2}) for traceless matrices A .

This implies our assertions.

2 Exceptional bundles in realization of the case b)

For a coherent sheaves E, F over del Pezzo surface S we use the following notation:

μ(E) := \frac{c_{1}(E) · (−K)}{r(E)} the slope; d(E) := c_{1}(E) · (−K) the degree;

h^{i}(E) := dim H^{i}(E); h^{i}(E, F) := dim Ext^{i}(E, F); χ(E, F) := ∑_{i}(−1)^{i}h^{i}(E, F) .

The image of E in K_{0}(S) we represent by the vector

v(E) = (r(E), c_{1}(E), −χ(E)) .

The last component is related with c_{2}(E) by the Riemann – Roch formula

−χ(E) = c_{2}(E) − \frac{c_{1}(E)^{2}}{2} + \frac{c_{1}(E) · K}{2} − r(E) = − ch_{2}(E) + \frac{c_{1}(E) · K}{2} − r(E) .  (21)

Now it is easy to express χ(E, F) calculated by the Riemann – Roch formula

χ(E, F) = r(E)r(F)+ \frac{r(E)d(F)}{2} − \frac{r(F)d(E)}{2} +r(E) ch_{2}(F)+r(F) ch_{2}(E)−c_{1}(E)·c_{1}(F) (22)

in terms of −χ(E) and −χ(F) . We have:

χ(E, F) = −r(E)r(F) + r(F)c_{1}(E) · K − c_{1}(E) · c_{1}(F) + r(E)χ(F) + r(F)χ(E) (23)

In this section we interpret M(r, c_{1}, k) as the space of orbits of solutions to fields equations by gauge group action (see details in [8]). We consider correlation function (5) under assumptions (2) and (4) made in the introduction.

For a stable sheaf F with (r(F), c_{1}(F), −χ(F)) = (r, c_{1}, k) satisfying (2) the equality (4) is equivalent to

c_{1}^{2} − r c_{1} · K + r^{2} + 1 = 0 .  (24)

In particular, this condition does not depend of k and hence of c_{2}(F) .

Indeed, we have the formula (3). Besides, dim Hom(F, F) = 1 by the properties of stability. Using Serre duality, we get: dim Ext^{2}(F, F) = dim Hom(F, F ⊗ K) = 0 . It follows that

dim M(r, c_{1}, k) = dim Ext^{1}(F, F) = 1 − χ(F, F) = 2rc_{2} − (r − 1)c_{1}^{2} − r^{2} + 1 (25)

the last equality holds by Riemann – Roch theorem. Substituting this expression and (3) to (4), we obtain (24).

Recall that a sheaf E is exceptional if dim Hom(E, E) = 1 and Ext^{i}(E, E) = 0 for i > 0 . It is proved in [10] that exceptional sheaf over del Pezzo surface is either locally free or torsion sheaf of the form O_{ℓ}(d) , where ℓ is an exceptional curve. Consider the full subcategory

O^{⊥} = {F : Ext^{i}(O, F) = 0}

in the category of coherent sheaves over S.

2.1 Theorem. ([9]) Suppose that the invariants (r, c_{1}) of exceptional bundle E satisfy the equality (2); then the following statements are equivalent:

(i) E ∈ O^{⊥} (ii) For any semistable sheaf F with the invariants (r, c_{1}, k) , k ⩾ 0 the following equality holds:

dim \overline{M}(r, c_{1}, k) = 2r dim H^{1}(F)

(This is exactly (4) for the sheaf F .)

Proof. It is known from [10] that exceptional bundles over S are stable in the sense of Mumford – Takemoto w.r.t. −K_{S}. This implies that H^{0}(E) = H^{2}(E) = 0 . Therefore, E is in O^{⊥} if and only if H^{1}(E) = 0 . This is equivalent to (4) and to (24) for E because dim \overline{M}(r, c_{1}, 0) = 0 by exceptionality of E. Since the equality (24) is independent of k, we obtain the required. Observe that there are no semistable sheaves F with the invariants (r, c_{1}, l) for l < 0 because in this case formula (25) gives negative dimension of Ext^{1}(F, F) . This concludes the proof. □

The theorem means that each exceptional bundle from O^{⊥} satisfying (2) gives a correlation function for testing S -duality for the field configuration (can, …, can) of the type b).

2.2 Remark. Under the conditions of the theorem all the moduli spaces \overline{M}(r, c_{1}, k) , k ⩾ 0 , are smooth, projective, and carry universal families. Indeed, for any exceptional bundle E the integers r(E) and c_{1}(E) · K are coprime [10].

2.3 Example. S = ℙ^{2} . Over the projective plane the exceptional bundles belonging to the subcategory O^{⊥} form a sequence E_{n} such that E_{−1} = O_{ℙ^{2}}(−2) , E_{0} = O_{ℙ^{2}}(−1) and the other E_{n} can be defined from the exact triples

0 ⟶ E_{n−1} ⟶ Hom(E_{n}, E_{n+1}) ⊗ E_{n} ⟶ E_{n+1} ⟶ 0 .  (26)

There is a canonical identification Hom(E_{n}, E_{n+1}) ≅ Hom(E_{n−1}, E_{n})^{∗} and the both morphisms of bundles are canonical. Observe that E_{1} ≅ Ωℙ^{2}(1) ≅ Tℙ^{2}(−2) and for n = 0 the exact triple (26) is the Euler sequence twisted by (−2) .

The equality (24) that is equivalent to (4) has the following interpretation: the pair (r, c_{1}) satisfies(24) iff the triple (r, −c_{1}, 1) is a solution of Markov equation

x^{2} + y^{2} + z^{2} = 3 xyz ,  (27)

as long as c_{1} · K = −3 c_{1} for ℙ^{2} .

The solution set of (27) is well-known [3]. Namely, any solution can be obtained from (1, 1, 1) by a finite number of mutations. A mutation is changing one variable to another root of the quadratic equation while the rest variables are fixed. So, for z = 1 we have a sequence of mutations that gives the set of all solutions (r, c_{1}) to (24) for ℙ^{2} . Here are several first solutions such that r > −c_{1} :

(2, −1) , (5, −2) , (13, −5) , (34, −13) , (89, −34) , (233, −89) , …  (28)

The pair after (r, c_{1}) in this sequence is (3r + c_{1}, −r) .

It is known also that the collection of ranks of exceptional bundles from a helix coil^{3} is a solution to (27). Conversely, for any solution of (27) there is an infinitely many helix coils (because of twisting) having the rank collection just the same as the given solution. The set of solutions of (27) of the form (x, y, 1) corresponds up to twisting to the set of helix coils (E_{n}, E_{n+1}, O_{ℙ^{2}}) . Herewith, if r(E_{n}) = r_{n}, then c_{1}(E_{n}) = −r_{n−1} .

The bundles E_{n} = \overline{M}(r_{n}, −r_{n−1}, 0) are the initial points for the sequences of moduli spaces \overline{M}(r_{n}, −r_{n−1}, k) , where k ⩾ 0 . Therefore, we have the diagram:

⋮  ⋮  ⋮
O_{ℙ^{2}}(−2) Hilb^{1}(ℙ^{2})  Hilb^{2}(ℙ^{2}) … Z_{−1} is a polynomial of degree ⩽ 4

O_{ℙ^{2}}(−1)  {|| \atop ℙ^{2}}  Hilb^{2}(ℙ^{2}) … Z_{0} = 1 + q
Ω_{ℙ^{2}}(1) \overline{M}(2, −1, 1) \overline{M}(2, −1, 2) … Z_{1} = q + 13q^{4} + 729q^{5} + 85025q^{6}+
E_{2}  \overline{M}(5, −2, 1) \overline{M}(5, −2, 2) …  +15650066q^{7} + ⋯
E_{3}  \overline{M}(13, −5, 1) \overline{M}(13, −5, 2) …
⋮  ⋮  ⋮  ...

Each row of this diagram produces a partition function Z_{n}. All what is known about these functions for the moment is written in the right hand side. On the other hand, there are mutations (26) for the terms of the first column.

Question. Are there transformations between the whole rows of the diagram that generalize the mutations (26)?

Conjecture. If such a transformation exists, then one can calculate all the partition functions knowing only two of them.

Observe that the first seven coefficients of Z_{1} are calculated in [5] with the aid of the Bott’s formula. Unfortunately, the authors of this paper don’t receive a general formula or recurrent relation for the coefficients because the combinatorial structure of fixed components of the torus action on \overline{M}(2, −1, k) is very complicated. The calculations have been performed on computer using MAPLE V.3. In the first part of [5] there is given an explicit description of \overline{M}(2, −1, k) . Similar description for all \overline{M}(r_{n}, −r_{n−1}, k) and for some series of moduli spaces over ℙ^{1} × ℙ^{1} , which start with exceptional bundles satisfying the conditions of 2.1, is given in [12].

2.4 Remark. 1. For the case S = P^{2} it can be easily checked that the inequality (2) holds for all bundles E_{n} from O^{⊥}. However, it is not so in general. For example, the line bundles O(−1, m) over ℙ^{1} × ℙ^{1} belong to O^{⊥} and have c_{1} · (−K_{ℙ^{1}×ℙ^{1}}) = 2 (m − 1) .

2. The sequence of bundles E_{n} from Example 2.3 is a particular case of system of bundles generated by exceptional pair. Such a system is defined as follows: the pair (E_{i}, E_{i+1}) is the right mutation of (E_{i−1}, E_{i}) and the left mutation of (E_{i+1}, E_{i+2}) (see [4] or [10] for mutations). Herewith, if an exceptional pair belongs to O^{⊥}, then all the system generated by this pair also belongs to O^{⊥}. Therefore, there appears a diagram of moduli spaces similar to the above-mentioned one (because of the restriction (2) the diagram can be bounded). For such a diagram we put the similar question and the similar conjecture as above.

3the same as full exceptional collection for ℙ^{2}

3 Exceptional bundles in realization of d), e) and f) cases.

In this section we deal with power series (10),(12) and (14) under the conditions (9),(11) and (13) respectively. Studying modular properties of the series Z_{d}(τ) , Z_{e}(τ) and Z_{f}(τ) is a problem related, as we believe, to mathematical testing of S -duality. However, we don’t examine this here. Our interest in this section is in exceptional bundles that generate these power series.

3.1 Proposition. Let F be a semi-stable sheaf over del Pezzo surface with the invariants (r, c_{1}, k) such that r ⩾ 3 and

−\frac{1}{2}K^{2} < −\frac{c_{1} · K}{r} < 0 .  (6)

Then the spaces H^{i}(S^{2}F) and H^{i}(Λ^{2}F) for i ≠ 1 are zero.

Proof. The spaces H^{0}(S^{2}F) and H^{0}(Λ^{2}F) are naturally identified with subspaces of H^{0}(F ⊗ F) , which is contained in H^{0}(F^{∗∗} ⊗ F) because of the canonical embedding F ↪ F^{∗∗}. The last space is isomorphic to Hom(F̃, F) , where F̃ is the dual bundle to F^{∗∗}. The singularity set of F is at least 2-codimensional since F has no torsion, whence c_{1}(F) = c_{1}(F^{∗∗}) = −c_{1}(F̃) . From Lemma 6.1 it follows that F^{∗∗} is a semistable bundle, then by the equality μ(F) < 0 < μ(F̃) we have: Hom(F̃, F) = 0 . Thus, we get: H^{0}(S^{2}F) = H^{0}(Λ^{2}F) = 0 .

The spaces H^{2}(S^{2}F) = Hom(S^{2}F, K)^{∗} and H^{2}(Λ^{2}F) = Hom(Λ^{2}F, K)^{∗} are naturally identified with subspaces in Hom(F ⊗ F, K)^{∗}. Applying the functor Hom(−, K) to the exact sequence

0 ⟶ F ⊗ F ⟶ F^{∗∗} ⊗ F^{∗∗} ⟶ Q ⟶ 0 ,

we have: Hom(F ⊗ F, K) ≅ Hom(F^{∗∗} ⊗ F^{∗∗}, K) because r(Q) = 0 , c_{1}(Q) = 0 , and thus Hom(Q, K) = Ext^{1}(Q, K) = 0 . The sheaf F^{∗∗} is locally free, whence

Hom(F^{∗∗} ⊗ F^{∗∗}, K) = H^{0}(F̃ ⊗ F̃ ⊗ K) = Hom(F^{∗∗}, F̃ ⊗ K) .

Since 0 < μ(F̃) < \frac{1}{2}K^{2} , we have:

μ(F̃ ⊗ K) < −\frac{1}{2}K^{2} < μ(F) = μ(F^{∗∗}) .

It follows that Hom(F^{∗∗}, F̃ ⊗ K) = 0 and thus H^{2}(S^{2}F) = H^{2}(Λ^{2}F) = 0 . The statement is proved. □

The proposition implies that the dimensions of H^{1} of the symmetric and exterior square of F can be expressed in terms of the topological invariants via Riemann – Roch formula:

dim H^{1}(S^{2}F) = −χ(S^{2}F) = (r + 2)k − \frac{r + 3}{2}c_{1}^{2} + \frac{r + 1}{2}c_{1} · K − \frac{r(r + 1)}{2} ,

dim H^{1}(Λ^{2}F) = −χ(Λ^{2}F) = (r − 2)k − \frac{r − 1}{2}c_{1}^{2} + \frac{r − 1}{2}c_{1} · K − \frac{r(r − 1)}{2} .

Therefore, the equalities (9) and (11) can be expressed by the following way:

case d):  (9) ⟺ \frac{3}{2} c_{1}^{2} − (r + \frac{1}{2}) c_{1} · K + \frac{1}{2} (r^{2} + 3 r + 2) = 0 .  (29)

case e):  (11) ⟺ \frac{3}{2} c_{1}^{2} − (r − \frac{1}{2}) c_{1} · K + \frac{1}{2} (r^{2} − 3 r + 2) = 0 .  (30)

As well as (24), these conditions don’t depend on k and, hence, on c_{2} .

Let now E be exceptional bundle. We say that E generates correlation function for the case d), e) or f) if for any semistable sheaf F ∈ \overline{M}(r, c_{1}, k) , where r = r(E) , c_{1} = c_{1}(E) , k ⩾ 0 , the equality (9), (11) or respectively (13) holds. In other words, the correspondent power series (10), (12) or (14) is well-defined.

3.2 Theorem. Suppose exceptional bundle E satisfies −K^{2}/2 < μ(E) < 0 . Then the following statements hold:

1. E generates correlation function for the case d) iff E ∈ O^{⊥} and d(E) = 1 − 2r(E) .

2. E generates correlation function for the case e) iff E ∈ O^{⊥} and d(E) = −1 − 2r(E) .

3. E generates correlation function for the case f) iff E ∈ O^{⊥} and d(E) = 1 − 2r(E) .

Proof. E generates correlation function for the case d) iff the invariants (r, c_{1}) of E satisfy (29) or equivalently E satisfies (9). The moduli space \overline{M}(r, c_{1}, 0) is zero-dimensional, whence we have

dim H^{1}(E) = 0 and  dim H^{1}(Λ^{2}E) = 0 .  (31)

Thus, E generates correlation function for the case d) iff the invariants of E satisfy the both (24) and (29). Eliminating from these equalities the terms with c_{1}^{2} , we have:

\frac{1 − r}{2} c_{1} · K + r^{2} − \frac{3}{2} r + \frac{1}{2} = 0 ,

whence after dividing by \frac{1}{2}(1 − r) we obtain: c_{1} · K = 2 r − 1 .

From the proof of 2.1 it follows that (24) is equivalent to H^{1}(E) = 0 that under (6) means that E ∈ O^{⊥}. This proves the first statement, the other statements are analogous. Observe that an exceptional bundle E generates correlation functions for d) and for f) cases equivalently, since each condition of (9) and (13) for E means that H^{1}(E) = H^{1}(Λ^{2}E) = 0 . □

3.3 Remark. For any exceptional bundle E the equality d(E) = ±1 − 2r(E) is equivalent to μ(E) = ±\frac{1}{r(E)} − 2 as far as d(E) and r(E) are coprime.

It is not hard to check that there is no nontrivial examples of exceptional bundles over ℙ^{2} that generate correlation functions for cases d)–f). Indeed, the exceptional bundles in subcategory O^{⊥} are E_{n}, n ∈ ℤ described in Example 2.3. It is readily seen that the sequence {μ(E_{n})} is increasing. Consider two cases.

1. n ⩾ 1 . The sequence {r(E_{n})} is increasing and the sequence {−2 + 1/r(E_{n})} is decreasing; besides, μ(E_{n}) = −2 + 1/r(E_{n}) for n = 1 , E_{1} = Ωℙ^{2}(1) . It is clear that {μ(E_{n})} ≠ {−2 + 1/r(E_{n})} for n > 1 . The equality μ(E_{n}) = −2 − 1/r(E_{n}) is impossible because μ(E_{n}) > −2 for n ⩾ 1 .

2. n ⩽ 0 . In this case the equality μ(E_{n}) = −2+1/r(E_{n}) is impossible because μ(E_{n}) < −2 . The another equality μ(E_{n}) = −2 − 1/r(E_{n}) holds only for n = 0 , E_{0} = O_{ℙ^{2}}(−1) , since μ(E_{n}) < −3 < −2 − 1/r(E_{n}) for n < 0 .

Obviously, there is no exceptional bundle over ℙ^{1} × ℙ^{1} generating correlation function for the cases d)–f) because c_{1} · K_{ℙ^{1}×ℙ^{1}} is even.

In the rest of this section we give an example of exceptional bundles over the plane with five blowed points that generate correlation functions for d) and f) cases.

Denote by X_{m} the plane with m blowed points in general position, 1 ⩽ m ⩽ 8 . We have:

Pic X_{m} = ℤℓ_{0} ⊕ ℤℓ_{1} ⊕ ℤℓ_{2} ⊕ ⋯ ⊕ ℤℓ_{m} ,

where ℓ_{0} is the preimage of the line in ℙ^{2} and ℓ_{i}, 1 ⩽ i ⩽ m are exceptional curves i.e. preimages of the blowed points. The canonical class of X_{m} is

K = −3ℓ_{0} + ∑_{i=1}^{m} ℓ_{i} ,

the intersection form is given by the relations

ℓ_{0}^{2} = 1 , ℓ_{i}^{2} = −1 , 1 ⩽ i ⩽ m; ℓ_{i} · ℓ_{j} = 0 , i ≠ j .

Put c_{1} = aℓ_{0} + ∑_{i=1}^{m} b_{i}ℓ_{i}, then we have: c_{1} · K = −3a − ∑_{i=1}^{m} b_{i}; c_{1}^{2} = a^{2} − ∑_{i=1}^{m} b_{i}^{2} ; K^{2} = μ(−K) =

9 − m.

Consider a standard full exceptional collection over X_{m}:

O_{ℓ_{1}}(−1), …, O_{ℓ_{m}}(−1), O_{X_{m}}(−ℓ_{0}), O_{X_{m}}, O_{X_{m}}(ℓ_{0}).

Shifting the torsion sheaves O_{ℓ_{i}}(−1) = O_{ℓ_{i}}(−ℓ_{0} + ℓ_{i}) to the right over O_{X_{m}}(−ℓ_{0}) , we have the exceptional collection

O_{X_{m}}(−ℓ_{0}), O_{X_{m}}(−ℓ_{0} + ℓ_{1}), …, O_{X_{m}}(−ℓ_{0} + ℓ_{m}), O_{X_{m}}, O_{X_{m}}(ℓ_{0}).

Shifting O_{X_{m}}(ℓ_{0}) to the left over all the other sheaves, we obtain full exceptional collection

O_{X_{m}}(ℓ_{0} + K), O_{X_{m}}(−ℓ_{0}), O_{X_{m}}(−ℓ_{0} + ℓ_{1}), …, O_{X_{m}}(−ℓ_{0} + ℓ_{m}), O_{X_{m}}.

All the line bundles except O_{X_{m}} in this collection generate the subcategory O_{X_{m}}^{⊥} and all mutations of these bundles don’t lead outside O_{X_{m}}^{⊥} .

3.4 Example. Put m = 5 . We have: μ(O_{X_{5}}(ℓ_{0} + K)) = −1 and μ(O_{X_{5}}(−ℓ_{0})) = −3 , so the exceptional pair (O_{X_{5}}(ℓ_{0} + K), O_{X_{5}}(−ℓ_{0})) is an ext-pair. Put V = Ext^{1}(O_{X_{5}}(ℓ_{0} + K), O_{X_{5}}(−ℓ_{0})) , dim V = 2 .

Consider the system of bundles {E_{n}}_{n∈ℤ} generated by the pair (O_{X_{5}}(ℓ_{0} +K), O_{X_{5}}(−ℓ_{0})) . By the definition, we have : E_{0} = O_{X_{5}}(ℓ_{0} + K) , E_{1} = O_{X_{5}}(−ℓ_{0}) , the bundles E_{2} and E_{−1} are the universal extensions

0 ⟶ O_{X_{5}}(−ℓ_{0}) ⊗ V^{∗} ⟶ E_{2} ⟶ O_{X_{5}}(ℓ_{0} + K) ⟶ 0 ,
0 ⟶ O_{X_{5}}(−ℓ_{0}) ⟶ E_{−1} ⟶ O_{X_{5}}(ℓ_{0} + K) ⊗ V ⟶ 0 ,

the bundles E_{i−1} for i ⩽ −1 are included into exact triples

0 ⟶ E_{i−1} ⟶ E_{i} ⊗ Hom(E_{i}, E_{i+1}) ⟶ E_{i+1} ⟶ 0 ,

and the bundles E_{i+1} , i ⩾ 2 are included into the same exact triples with Hom(E_{i}, E_{i+1}) replaced by canonically isomorphic to it Hom^{∗}(E_{i−1}, E_{i}) ; herewith the both spaces are twodimensional. It is easily checked that the bundles E_{i}, i ⩽ −1 have the invariants

r(E_{i}) = 2 · (−i) + 1 and c_{1}(E_{i}) = 4i − 1 = −2r(E_{i}) + 1 ,

thus, by Theorem 3.2, these bundles generate correlation functions for the cases d) and f). Therefore, we have a diagram similar to the diagram from page 10, for which we put the same question and formulate the same conjecture as in page 10.

4 An approach to transformations of moduli spaces

In this section we look for approaches on possible constructions of transformations between the moduli spaces

\overline{M}_{n}(δ) = \overline{M}(r_{n}, −r_{n−1}, δ)

described in 2.3, which generalize the mutations (26) of the exceptional bundles E_{n} = \overline{M}_{n}(0) .

We consider here only the case n ⩾ 1 , for which r_{n} > r_{n−1} , and use the recurrent relation of minimal degree on ranks

r_{n+1} + r_{n−1} = 3r_{n} .  (32)

Remind that the triple of positive integers (r_{n}, r_{n−1}, 1) satisfies Markov equation (27), whence

| \begin{matrix}r_{n−1} & r_{n} \\ r_{n} & r_{n+1} \end{matrix} | = r_{n−1}(3r_{n} − r_{n−1}) − 3r_{n}^{2} = 1  (33)

Denote by X_{n} semi-stable sheaf with v(X_{n}) = (r_{n}, −r_{n−1}, δ) . Observe that X_{n} is stable as far as r_{n} and r_{n−1} are coprime.

Since dim \overline{M}_{n}(δ) = 2 r_{n} δ, we easily see from (32) that

dim \overline{M}_{n−1}(δ) + dim \overline{M}_{n+1}(δ) = 3 dim \overline{M}_{n}(δ) .  (34)

Then the naive move is to take for the left hand side an extension of arbitrary X_{n+1} via arbitrary X_{n−1} and for the right hand side to look for a filtration of this extension with the factors (X_{n}, X′_{n}, X′′_{n}) , where X′_{n} and X′′_{n} are stable sheaves with the same collection of invariants v as v(X_{n}) . But then there will be no equality of the vectors v because the third component

−χ(X_{n−1}) = δ, −χ(X_{n+1}) = δ; −χ(X_{n}) = −χ(X′_{n}) = −χ(X′′_{n}) = δ .

So one needs to increase the −χ -component in the left hand side. The sheaves are taken according to (34). In order to not destroy this equality one has to take again the same X_{n+1} or the same X_{n−1} ; then the other components of the vectors v are balanced by adding appropriate number of exceptional bundles. Thus, possible construction would be based on (34) and one of the following equalities

v(E_{n−1}) + v(X_{n−1}) + 2v(X_{n+1}) = 3v(E_{n}) + 3v(X_{n}) ,  (35)

or

2v(E) + v(X_{n+1}) + v(E_{n+1}) = 3v(E_{n}) + 3v(X_{n}) .  (36)

We formulate here more concrete assumptions on transformations of moduli spaces.

Conjectures. 1. Suppose X_{n−1} and X_{n+1} are given; then there is a way to construct a sheaf F having two filtrations with

Gr^{(1)}(F) = E_{n−1} ⊕ X_{n−1} ⊕ X_{n+1} ⊕ X_{n+1} and Gr^{(2)}(F) = E_{n} ⊕ E_{n} ⊕ E_{n} ⊕ X_{n} ⊕ X′_{n} ⊕ X′′_{n}

where the last two sheaves have the same invariants as X_{n}.

2. This construction gives a ”good” map (finite or birational isomorphism) between \overline{M}_{n−1}(δ) × \overline{M}_{n+1}(δ) and \overline{M}_{n}(δ) × \overline{M}_{n}(δ) × \overline{M}_{n}(δ) .

Let us analyze our conjectures. The following statement gives some necessary conditions for possible construction to exist.

4.1 Proposition. Suppose there is an exact sequence

0 ⟶ F′ ⟶ F ⟶ F′′ ⟶ 0

of coherent sheaves with μ(F′) = μ(F) = μ(F′′) . Then the following statements hold:

1) F is semistable if and only if F′ and F′′ are semistable.

2) suppose that F is semistable and has a semistable subsheaf E such that μ(E) < μ(F) and the quotient G = F/E is semistable; then Hom(E, F′′) ≠ 0 and Hom(F′, G) ≠ 0 .

Proof. The first statement is well-known (see, for example, similar statement in [13, 1.4]). Let us prove the second one.

Observe that μ(F) < μ(G) . There is commutative diagram with exact rows and columns:

0  0
↓  ↓
0 ⟶ V ⟶ E ⟶^{g} F′′ ⟶ W ⟶ 0
↓ ↓ ||
0 ⟶ F′ ⟶ F ⟶ F′′ ⟶ 0
f↓  ↓
G = G
↓  ↓
W  0

↓ 0

Here V = ker f = ker g and W = coker f = coker g. Let us show that the maps g and f are non-zero. Indeed, if we had g = 0 , then we would have W = F′′ as a quotient-sheaf of G, that contradicts semi-stability of G. Similarly, if we had f = 0 , then we would have F′ = V as a subsheaf of E, which violates semi-stability of E. □

Suppose that a construction conjectured above exists. Then by the first statement of the proposition, F must be semi-stable as far as the second filtration has stable quotients with the slope equal to μ(E_{n}) . Hence, there is only one direction to make extensions, namely, E_{n−1} and X_{n−1} in Gr^{(1)}(F) must be sub-sheaves, but not quotient-sheaves of F , and X_{n+1} ’s must be quotient-sheaves, but not sub-sheaves of F .

The second statement leads to the following problem: to describe the cases when the spaces Hom(X_{n−1}, X_{n}) and Hom(E_{n−1}, X_{n}) are non-trivial. Some moves in this direction are made in the next section.

4.2 Theorem. Suppose X_{n−1} and X_{n+1} are as above but with arbitrary −χ(X_{n−1}) and −χ(X_{n+1}) ; then any non-trivial extension

0 ⟶ X_{n−1} ⟶ F ⟶ X_{n+1} ⟶ 0 .  (37)

has the following properties:

(i) if X_{n−1} is locally free, then F is semistable; (ii) for X_{n−1} non-locally-free, F is semi-stable if and only if the corresponding to (37) cocycle e doesn’t belong to the image of the map Hom(X_{n+1}, X_{n−1}^{∗∗}/X_{n−1}) → Ext^{1}(X_{n+1}, X_{n−1}) induced by the canonical embedding X_{n−1} ↪ X_{n−1}^{∗∗}; (iii) if F is non-stable, then the Harder – Narasimhan filtration of F has only two terms 0 = F_{0} ⊂ F_{1} ⊂ F_{2} = F , where (r(F_{1}), c_{1}(F_{1})) = (r_{n+1}, −r_{n}) and (r(F/F_{1}), c_{1}(F/F_{1})) = (r_{n−1}, −r_{n−2}) .

Proof. Observe that F has no torsion as an extension of torsion-free sheaves. Let E ⊂ F be a subsheaf such that 0 < r(E) < r(F) . Without loss of generality we can assume that the quotient G = F/E is without torsion (see [15, Ch. II, 1.2.2]). Consider the following commutative diagram with exact rows and columns

\begin{aligned}& & & & & & 0 \\ & & & & & & ↓ \\ & & & & & & E & ⟶^{id} & E \\ & & & & & & ↓ & & ↓ψ \\ & & 0 & ⟶ & X_{n−1} & ⟶ & F & ⟶ & X_{n+1} & ⟶ & 0 \\ & & & & id↓ & & ↓ & & ↓ \\ 0 & ⟶ & I & ⟶ & X_{n−1} & ⟶ & G & ⟶ & Q & ⟶ & 0 \\ & & & & & & ↓ & & ↓ \\ & & & & & & 0 & & 0 \end{aligned}  (38)

where I = ker ψ and Q = coker ψ. Observe that r(F) = 3r_{n} and c_{1}(F) = −3r_{n−1} , so μ(F) = μ(E_{n}) .

Suppose μ(E) > μ(F) > μ(G) ; then ψ ≠ 0 , otherwise E is isomorphic to a subsheaf of X_{n−1} and has the slope less than the slope of F .

Consider the real (x, y) -plane. For a coherent sheaf E we denote by the same symbol E the point (r(E), c_{1}(E)) . The equality (33) implies that the vectors \overrightarrow{OE_{n}} and \overrightarrow{OX_{n+1}} form a basis in the integral lattice of the plane (see Fig. 1, where for clearness different scales are chosen on the axes). If we had μ(E) ⩽ μ(X_{n+1}) , then the point E would belong to the triangle OF A, where A = OX_{n+1} ∩ {x = 3r_{n}}. But it is easily seen that the only integral points of this triangle are E_{n}, B = (2r_{n}, −2r_{n−1}) , F and G because of the inequality r_{n+1} + r_{n} > 3r_{n}, which follows from r_{n} > r_{n−1} and r_{n+1} + r_{n−1} = 3r_{n}. Thus, there are the following possibilities: μ(E) > μ(X_{n+1}) or r(E) = r_{n+1} and c_{1}(E) = −r_{n}.

[figure: x; y; O; E_{n}; B; F; X_{n+1}; A; 3r_{n}; (r_{n+1} + r_{n})] Fig. 1.

The case μ(E) > μ(X_{n+1}) is impossible because in this case I ≠ 0 and from the stability of X_{n−1} it follows that μ(I) < μ(X_{n−1}) < μ(E) , hence μ(I) < μ(E) < μ(J) , where J = im ψ is a subsheaf of torsion-free sheaf X_{n+1} . We obtain: μ(J) > μ(X_{n+1}) , that contradicts the stability.

Thus, we conclude that r(E) = r_{n+1} and c_{1}(E) = −r_{n}. This implies immediately that r(G) = r_{n−1} and c_{1}(G) = −r_{n−2} . It is impossible to have I ≠ 0 because in this case μ(I) < μ(X_{n−1}) < μ(E) = μ(X_{n+1}) < μ(J) , that contradicts the stability of X_{n+1} . Hence, I = 0 , r(J) = r_{n+1} and c_{1}(J) = −r_{n}. It follows that r(Q) = 0 and c_{1}(Q) = 0 . Therefore, E is stable.

Let X_{n−1} be locally free, then for Q ≠ 0 the lowest row of (38) is a trivial extension because Ext^{1}(Q, X_{n−1}) = 0 (remind that I = 0 ). This contradicts the choice of G to be torsion-free. Thus, we have: Q = 0 , X_{n−1} ≅ G and the extension (37) is trivial. Therefore, our assumption μ(E) > μ(F) > μ(G) is impossible and F is semistable. This proves (i).

Now suppose that X_{n−1} isn’t locally free. Applying the functor Hom(X_{n+1}, −) to the exact triple

0 ⟶ X_{n−1} ⟶ X_{n−1}^{∗∗} ⟶ X_{n−1}^{∗∗}/X_{n−1} ⟶ 0 ,

and taking into account stability of X_{n−1}^{∗∗}, which follows from 6.1, we have the exact sequence

0 ⟶ Hom(X_{n+1}, X_{n−1}^{∗∗}/X_{n−1}) ⟶ Ext^{1}(X_{n+1}, X_{n−1}) ⟶^{α} Ext^{1}(X_{n+1}, X_{n−1}^{∗∗}) .

Suppose that the extension (37) is given by cocycle e ∈ Ext^{1}(X_{n+1}, X_{n−1}) , then there is the following diagram

\begin{aligned}& & 0 & & 0 \\ & & ↓ & & ↓ \\ 0 & ⟶ & X_{n−1} & ⟶ & F & ⟶ & X_{n+1} & ⟶ & 0 \\ & & ↓ & & ↓ & ↓id \\ 0 & ⟶ & X_{n−1}^{∗∗} & ⟶ & F̃ & ⟶ & X_{n+1} & ⟶ & 0 \\ & & ψ′′↓ & & ↓ψ \\ & & Q & ⟶^{id} & Q \\ & & ↓ & & ↓ \\ & & 0 & & 0 \end{aligned}  (39)

Here Q = X_{n−1}^{∗∗}/X_{n−1} and the middle row is given by α(e) . Clearly, e isn’t in the image of Hom(X_{n+1}, X_{n−1}^{∗∗}/X_{n−1}) iff α(e) ≠ 0 , in this case F̃ is semi-stable by (i), hence F is semi-stable by 6.1.

Suppose α(e) = 0 ; then F̃ = X_{n+1} ⊕ X_{n−1}^{∗∗} and the map ψ from the diagram (39) is given by the pair of maps (ψ′, ψ′′) , where ψ′′ is contained in (39). Denote by Q′ the image of ψ′ and consider the following diagram, which has the middle column the same as in the diagram (39)

0  0  0
↓  ↓  ↓
0 ⟶ E ⟶  F  ⟶ G ⟶ 0
↓  ↓  ↓id
0 ⟶ X_{n+1} ⟶ X_{n+1} ⊕ X_{n−1}^{∗∗} ⟶ X_{n−1}^{∗∗} ⟶ 0
ψ′↓  ↓ψ  ↓
0 ⟶ Q′ ⟶  Q  ⟶ Q′′ ⟶ 0
↓  ↓  ↓
0  0  0

Here E = ker ψ′, (r(E), c_{1}(E)) = (r_{n+1}, −r_{n}) , whence μ(E) > μ(F) and F is non-stable. This proves (ii).

Let F be non-stable, then the first term F_{1} of the Harder-Narasimhan filtration of F is the subsheaf of maximal rank among sub-sheaves of maximal Gieseker slope. This description determines F_{1} uniquely up to isomorphism. In particular, μ(F_{1}) > μ(F) , the equality is impossible, otherwise F is semi-stable. But we have proved that any subsheaf of F with the slope greater than μ(F) has the rank r_{n+1} and the first Chern class −r_{n}. Moreover, there exists the diagram (38) with E = F_{1} , I = 0 and r(Q) = 0 , c_{1}(Q) = 0 . By 6.1, G = F/F_{1} is stable, hence F/F_{1} is Gieseker-stable. This implies that F/F_{1} has maximal slope among sub-sheaves of F/F_{1} , so the second term of the Harder – Narasimhan filtration of F coincides with F . This concludes the proof. □

4.3 Remark. In the conditions of the theorem, the set of cocycles e ∈ Ext^{1}(X_{n+1}, X_{n−1}) such that the corresponding extension gives a semistable sheaf F is open.

Indeed, by standard stability arguments h^{0}(X_{n+1}, X_{n−1}) = h^{2}(X_{n+1}, X_{n−1}) = 0 , whence

h^{1}(X_{n+1}, X_{n−1}) = −χ(X_{n+1}, X_{n−1}) = 1 + h^{1}(X_{n−1})r_{n+1} + h^{1}(X_{n+1})r_{n−1} ,

so the difference

h^{1}(X_{n+1}, X_{n−1}) − h^{0}(X_{n+1}, Q) = 1 + h^{1}(X_{n+1})r_{n−1} + (h^{1}(X_{n−1}) − h^{0}(Q))r_{n+1}

is always positive as long as h^{0}(Q) ⩽ h^{1}(X_{n−1}) .

Thus, one can take two extensions: (37) and some extension of the form 0 → E_{n−1} → F′ → X_{n+1} → 0 and look for an extension of F with F′ (or of F′ with F ) that has a filtration similar to Gr^{(2)} in the conjecture. We shall investigate this in subsequent articles.

5 Morphisms between stable sheaves

Here, as in the previous section, we deal with moduli spaces \overline{M}_{n}(δ) = \overline{M}(r_{n}, −r_{n−1}, δ) over ℙ^{2} , described in 2.3. We have:

dim \overline{M}_{n}(δ) = 2 r_{n} δ .

Let E_{n} be exceptional bundle with rank r_{n} and c_{1}(E_{n}) = −r_{n−1} , X_{n} be stable sheaf with

v(X_{n}) = (r_{n}, −r_{n−1}, δ) .

By the Riemann — Roch theorem we have: χ(X_{n−1}, X_{n}) = 3 − δ(r_{n} + r_{n−1}) , this value is negative for almost all δ and n; so, by the semi-continuity theorem for general (X_{n−1}, X_{n}) one has

Hom(X_{n−1}, X_{n}) = 0 , dim Ext^{1}(X_{n−1}, X_{n}) = −χ(X_{n−1}, X_{n}) .

Here we are interesting in a special cases when Hom(X_{n−1}, X_{n}) ≠ 0 . It is proved in [12] that any sheaf X_{n} can be represented as the cohomology of the monad

M(X_{n}) :  0 ⟶ U_{0} ⊗ E_{n−1} ⟶^{β} U_{1} ⊗ E_{n} ⟶^{α} U_{2} ⊗ O ⟶ 0 ,  (40)

where

U_{0} = Ext^{1}(E_{n−1}(3), X_{n}) , U_{1} = Ext^{1}(E_{n−2}(3), X_{n}) , and U_{2} = H^{1}(X_{n}) .

The dimensions of the spaces are

dim U_{0} = δ r_{n−1} , dim U_{1} = δ r_{n−2} + 1 , and dim U_{2} = δ .

Observe that the monad M(X_{n}) is nothing else than the first term of Gorodentsev – Beilinson spectral sequence constructed via the full exceptional collection (E_{n−1}, E_{n}, O) and the sheaf X_{n} (see [12, 3.3]).

Herewith, the sheaf

Y_{n} = ker α

is locally free and stable ([12]) and the Kronecker’s Hom(E_{n}, O)^{∗} -module

t_{α} : U_{1} ⊗ Hom(E_{n}, O) ⟶ U_{2}

corresponding to α in the monad, is also stable (see [2] on Kronecker’s moduli).

Applying the functor Hom(E_{n−1}, −) to β in (40) and taking into account exceptionality of E_{n−1} , we have an inclusion of vector spaces U_{0} ↪ U_{1} ⊗ Hom(E_{n−1}, E_{n}) . So, we regard U_{0} as a subspace of U_{1} ⊗ Hom(E_{n−1}, E_{n}) .

5.1 Proposition. Suppose δ > 1 and V_{1} ⊂ U_{1} is a hyperplane such that

1) the restriction t′ : V_{1} ⊗ Hom(E_{n}, O)^{∗} ⟶ U_{2} of t_{α} is stable Kronecker’s module;

2) the subspace U_{0} isn’t contained in V_{1} ⊗ Hom(E_{n−1}, E_{n}) . Then there exists a μ -stable sheaf X_{n−1} and a monomorphism X_{n−1} ⟶ X_{n} with the quotient isomorphic to E_{n}/E_{n−1} with respect to some embedding E_{n−1} ↪ E_{n}.

Proof. Let us show that the corresponding to t′ morphism of bundles

α′ : V_{1} ⊗ E_{n} ⟶ U_{2} ⊗ O

is an epimorphism. We apply here Drezet’s result [2, 27] to the dual morphism

α^{′∗} : U_{2}^{∗} ⊗ O ⟶ V_{1}^{∗} ⊗ E_{n}^{∗} .

The triad (E, G, F) and the bundle S used in [2] correspond to the full exceptional collection (O, E_{n}^{∗}, E_{n−1}^{∗}) and E_{n−2}^{∗} in our case. We see that

\frac{dim U_{2}^{∗}}{dim V_{1}^{∗}} = \frac{δ}{δ r_{n−2}} = \frac{1}{r_{n−2}} = \frac{r(O)}{r(E_{n−2})} = \frac{r(E)}{r(S)} .

From the proof of [2, 27(ii)] it follows that if α^{′∗} is not injective at a point x, then there is a submodule t′ : H′_{0} ⊗ Hom(E_{n}, O)^{∗} ⟶ H′_{1} of the module t′ with dim H′_{0} = 1 and dim H′_{1} = r_{n−2} . This contradicts stability of t′ because by our assumptions δ > 1 . Thus, α^{′∗} is injective and α′ is surjective as morphisms of bundles, whence it follows that α′ is sheaf epimorphism.

From the assumption U_{0} ⊄ V_{1} ⊗ Hom(E_{n−1}, E_{n}) it follows that β in (40) gives a non-zero map from U_{0} ⊗ E_{n−1} to (U_{1}/V_{1}) ⊗ E_{n} ≅ E_{n}. Moreover, one can choose a line L ⊂ U_{0} such that the restriction φ of this map to L ⊗ E_{n−1} ≅ E_{n−1} is non-zero. Fixing any projection U_{0} ↠ L with the kernel V_{0} , we obtain the commutative diagram with exact rows

\begin{aligned}0 & ⟶ & V_{0} ⊗ E_{n−1} & ⟶^{i} & U_{0} ⊗ E_{n−1} & ⟶ & E_{n−1} & ⟶ & 0 \\ & & β′↓ & & β↓ & & φ↓ \\ 0 & ⟶ & V_{1} ⊗ E_{n} & ⟶ & U_{1} ⊗ E_{n} & ⟶^{j} & E_{n} & ⟶ & 0 \end{aligned}  (41)

where φ ≠ 0 and the map β′ is induced by β ∘ i as far as j ∘ β ∘ i = 0 .

The morphism β is a sheaf embedding, φ is also sheaf monomorphism by [12, 3.5]. Thus, the restriction of β to V_{0} ⊗ E_{n−1} is also a sheaf embedding. Therefore, we have a morphism of monads

\begin{aligned}0 & ⟶ & V_{0} ⊗ E_{n−1} & ⟶^{β′} & V_{1} ⊗ E_{n} & ⟶^{α′} & U_{2} ⊗ O & ⟶ & 0 \\ & & ↓ & & ↓ & & id↓ \\ 0 & ⟶ & U_{0} ⊗ E_{n−1} & ⟶^{β} & U_{1} ⊗ E_{n} & ⟶^{α} & U_{2} ⊗ O & ⟶ & 0 \end{aligned}  (42)

where all vertical arrows are sheaf monomorphisms. Denote by X_{n−1} the cohomology of the upper monad, then we have a monomorphism X_{n−1} → X_{n}.

It remains to prove that X_{n−1} is stable and X_{n}/X_{n−1} ≅ E_{n}/E_{n−1} . Observe that X_{n−1} is torsion free as a subsheaf of the stable sheaf X_{n}. By [2, 27], the locally free sheaf coker α^{′∗} is stable by Gieseker, hence the bundle Y_{n−1} = (coker α^{′∗})^{∗} = ker α′ is μ -semistable. Dividing the upper monad from (42) into exact triples, we have one of them:

0 ⟶ V_{0} ⊗ E_{n−1} ⟶^{β′} Y_{n−1} ⟶ X_{n−1} ⟶ 0 .  (43)

Calculating the topological invariants of Y_{n−1} , we obtain with the aid of the recurrent relation and Markov equation

c_{1}(Y_{n−1}) = −r_{n−1} dim V_{1} = −δr_{n−2}r_{n−1} ,
r(Y_{n−1}) = r_{n} dim V_{1} − δ = δ(r_{n−2}r_{n} − 1) = δ(3r_{n−1}r_{n} − r_{n}^{2} − 1) = δr_{n−1}^{2} .

Thus, we conclude that μ(Y_{n−1}) = μ(E_{n−1}) = μ(X_{n−1}) . Any quotient of X_{n−1} is also a quotient of Y_{n−1} , so X_{n−1} is semistable. The stability follows from the fact that c_{1}(X_{n−1}) and r(X_{n−1}) are coprime numbers.

Display of a monad gives two ways to divide the monad into exact triples [15, 3.1]. One way is used in (43); to show that X_{n}/X_{n−1} ≅ E_{n}/E_{n−1} we need the other. Put Q′ = coker β′, Q = coker β, and regard the diagram (41) as an exact triple of complexes; then the long cohomology sequence has the form

0 ⟶ Q′ ⟶ Q ⟶ coker φ ⟶ 0

as long as the vertical arrows are embeddings. Therefore, there is the commutative diagram

0  0
↓  ↓
0 ⟶ X_{n−1} ⟶ Q′ ⟶ U_{2} ⊗ O ⟶ 0
↓  ↓  id↓
0 ⟶ X_{n} ⟶ Q ⟶^{α} U_{2} ⊗ O ⟶ 0
↓  ↓
coker φ ⟶^{id} coker φ
↓  ↓
0  0

(44)

This completes the proof. □

Unfortunately, for the moment there is no information on the set of hyper-planes of U_{1} that satisfy the conditions of the proposition. Nevertheless, the following statement shows that for any hyperplane V_{1} ⊂ U_{1} the submodule t′ of t_{α} is semistable.

5.2 Proposition. Let r and δ be positive integers. Suppose t : U_{1} ⊗ L → U_{2} is stable Kronecker’s L -module with dim U_{1} = δr + 1 and dim U_{2} = δ, t′ : U′_{1} ⊗ L → U′_{2} is a submodule of t such that U′_{1} ⊂ U_{1} is a hyperplane. Then U′_{2} = U_{2} and t′ is semi-stable.

Proof. We shall use the following criterion of stability [2, Prop.15]: a non-zero Kronecker’s module t is semi-stable (resp. stable) if and only if for any submodule t′ of t with U′_{1} ≠ 0

\frac{dim U′_{2}}{dim U′_{1}} ⩾ \frac{dim U_{2}}{dim U_{1}} (resp. >) or equivalently | \begin{matrix}dim U′_{1} & dim U_{1} \\ dim U′_{2} & dim U_{2} \end{matrix} | ⩽ 0 (resp. >).

If we had dim U′_{2} < δ, then we would have

| \begin{matrix}dim U′_{1} & dim U_{1} \\ dim U′_{2} & dim U_{2} \end{matrix} | = | \begin{matrix}δr & δr + 1 \\ dim U′_{2} & δ \end{matrix} | =

= δr(δ − dim U′_{2}) − dim U′_{2} > δr − dim U′_{2} > 0 .

Hence, the submodule t′ would violate the semi-stability criterion. Thus, dim U′_{2} = δ and U′_{2} = U_{2} .

Now consider a submodule τ : V_{1} ⊗ L → V_{2} of t′. From the stability criterion applied to τ as a submodule of t we have

| \begin{matrix}dim V_{1} & δr + 1 \\ dim V_{2} & δ \end{matrix} | = δ(dim V_{1} − r dim V_{2}) − dim V_{2} < 0 .

From the inequality dim V_{2} ⩽ δ it follows that

δ(dim V_{1} − r dim V_{2}) = | \begin{matrix}dim V_{1} & δr \\ dim V_{2} & δ \end{matrix} | = | \begin{matrix}dim V_{1} & dim U′_{1} \\ dim V_{2} & dim U′_{2} \end{matrix} | ⩽ 0 .

Therefore, t′ is semistable by the criterion. This completes the proof. □

In the rest of this section we discuss another sufficient condition for non-trivial morphism X_{n−1} → X_{n} to exist.

Fix a sheaf X_{n}, then the monad (40) defines the exact triple

0 ⟶ Y_{n} ⟶ U_{1} ⊗ E_{n} ⟶^{α} U_{2} ⊗ O ⟶ 0 ,  (45)

Our claim is to show that any epimorphism Y_{n} ↠ E_{n} defines a sheaf X_{n−1} such that Hom(X_{n−1}, X_{n}) ≠ 0 . Observe that the data, in terms of which the sheaf X_{n−1} is determined, is the same as in Proposition 5.1 because there is the canonical identification

Hom(Y_{n}, E_{n}) ≅ U_{1}^{∗} .

obtained by applying the functor Hom(−, E_{n}) to the triple (45), so hyper-planes in U_{1} correspond to non-zero elements of Hom(Y_{n}, E_{n}) up to scaling.

5.3 Proposition. Let φ : Y_{n} → E_{n} be nontrivial morphism; then im φ is a stable sheaf with v(im φ) = (r_{n}, −r_{n−1}, δ′) and ker φ is semistable with v(ker φ) = (δr_{n−1}^{2}, −δr_{n−2}r_{n−1}, δ′′) , where δ′ and δ′′ are non-negative and δ′ + δ′′ = δ.

Proof. Consider the real (x, y) -plane. As above, for a coherent sheaf E we denote by the same symbol E the point (r(E), c_{1}(E)) . The inequality μ(E_{n−1}) < μ(Y_{n}) < μ(E_{n}) implies that the ray OY_{n} lies between the rays OE_{n−1} and OE_{n} (see Fig. 2). From stability of Y_{n} and E_{n} it follows that μ(Y_{n}) ⩽ μ(im φ) ⩽ μ(E_{n}) ; besides, r(im φ) ⩽ r_{n}. Thus, the point im φ belongs to the triangle OE_{n}A, where A = OY_{n} ∩ {x = r_{n}}. From the equality

| \begin{matrix}−r_{n−2} & −r_{n−1} \\ r_{n−1} & r_{n} \end{matrix} | = −1 (see (33)) it follows that the parallelogram OE_{n−1}BE_{n} has no points

with integer coordinates except the vertices. The triangle OE_{n}A lies in the strip between the parallel lines OE_{n−1} and E_{n}B, which also doesn’t contain interior integer points. Therefore, the only possibility for (r(im φ), c_{1}(im φ)) is (r_{n}, −r_{n−1}) . Clearly, im φ is stable.

Now let us show that ker φ is semistable. Consider the exact triple

0 ⟶ ker φ ⟶ Y_{n} ⟶ im φ ⟶ 0 .  (46)

Since im φ has the rank and c_{1} to be equal to (r_{n}, −r_{n−1}) , the rank and c_{1} of ker φ coincide with (r(Y_{n−1}), c_{1}(Y_{n−1})) from the proof of 5.1:

r(ker φ) = δr_{n−1}^{2} ; c_{1}(ker φ) = −δr_{n−2}r_{n−1} .

[figure: y; x; O; E_{n−1}; E_{n}; ker φ; B; Y_{n}; A; r_{n}] Fig. 2.

We see that (r(ker φ), c_{1}(ker φ)) = δr_{n−1}(r_{n−1}, −r_{n−2}) ; in other words, the point ker φ lies on the ray OE_{n−1} , hence Y_{n} ∈ E_{n}B.

Suppose Q ⊂ ker φ is a subsheaf such that 0 < r(Q) < r(ker φ) and μ(Q) ⩾ μ(ker φ) ; then μ(Q) < μ(Y_{n}) as far as Q is a subsheaf of Y_{n}. It follows that the point Q belongs to the triangle O ker φY_{n}. By the above arguments, all the points of this triangle with integer coordinates lie on the side O ker φ. Therefore, μ(Q) = μ(ker φ) and ker φ is semistable.

By standard arguments, the H^{0} and H^{2} groups of the sheaves ker φ and im φ are trivial, therefore, dim H^{1} = −χ for these sheaves. This implies that δ′, δ′′ ⩾ 0 . The equality δ′ + δ′′ = δ follows immediately from (46). This concludes the proof. □

5.4 Lemma. Let Y_{n−1} be semistable subsheaf of Y_{n} with the invariants

v(Y_{n−1}) = (δ r_{n−1}^{2}, −δ r_{n−2}r_{n−1}, δ) .

Then Y_{n}/Y_{n−1} ≅ E_{n}.

Proof. Denote the quotient Y_{n}/Y_{n−1} by Q and consider the Gorodentsev – Beilinson spectral sequences (see [12] or [4]) constructed via full exceptional collection (E_{n−1}, E_{n}, O) that converge to Y_{n−1} and Y_{n}. Each sequence gives an exact triple and the embedding Y_{n−1} ↪^{i} Y_{n} induces the commutative diagram

\begin{aligned}0 & ⟶ & Y_{n−1} & ⟶ & V_{1} ⊗ E_{n} & ⟶ & H^{1}(Y_{n−1}) ⊗ O & ⟶ & 0 \\ & & i↓ & & ↓ & & ↓ \\ 0 & ⟶ & Y_{n} & ⟶ & U_{1} ⊗ E_{n} & ⟶ & H^{1}(Y_{n}) ⊗ O & ⟶ & 0 \end{aligned}  (47)

Here

V_{1} = Ext^{1}(E_{n−2}(3), Y_{n−1}) , U_{1} = Ext^{1}(E_{n−2}(3), Y_{n}) ,

dim V_{1} = dim U_{1} − 1 , dim H^{1}(Y_{n−1}) = dim H^{1}(Y_{n}) = δ .

Regarding (47) as an exact triple of complexes, we have the long cohomology sequence

0 ⟶ V ⊗ E_{n} ⟶ W ⊗ O ⟶^{φ} Q ⟶^{ψ} U ⊗ E_{n} ⟶ W′ ⊗ O ⟶ 0 ,

where dim V = dim U − 1 and dim W = dim W′. The sheaf I = im φ = ker ψ is a quotientsheaf of W ⊗ O and at the same time a subsheaf of U ⊗ E_{n}, whence I = 0 . Thus, V ⊗ E_{n} is isomorphic to W ⊗ O, that is possible only for dim V = dim W = 0 . Therefore, we get: Q ≅ U ⊗ E_{n} with U one-dimensional. □

In particular, from two previous statements it follows that epimorphisms Y_{n} ↠ E_{n} are in one-to-one correspondence with semistable sub-sheaves Y_{n−1} of Y_{n} having the invariants as in the previous lemma.

5.5 Proposition. Suppose X_{n} is a stable sheaf with the invariants v(X_{n}) = (r_{n}, −r_{n−1}, δ) , Y_{n} is the kernel of the map α in the monad (40), and φ : Y_{n} ↠ E_{n} is an epimorphism; then there is a hyperplane V_{0} ⊂ U_{0} such that β (see (40)) induces an embedding of V_{0} ⊗ E_{n−1} ↪ ker φ, the quotient X_{n−1} = ker φ/V_{0}⊗E_{n−1} is a stable sheaf with v(X_{n−1}) = (r_{n−1}, −r_{n−2}, δ) , and the embedding ker φ ↪ Y_{n} induces an embedding X_{n−1} ↪ X_{n}.

Proof. According to 5.3, Y_{n−1} = ker φ is stable. Consider the composition map

φ ∘ β : U_{0} ⊗ E_{n−1} ⟶^{β} Y_{n} ⟶^{φ} E_{n} .

Observe that the invariants (r, c_{1}) of the sheaves U_{0} ⊗ E_{n−1} and Y_{n−1} coincide. If we had φ ∘ β = 0 ; then β would induce an embedding U_{0} ⊗ E_{n−1} ↪ Y_{n−1} , cokernel of which would be an Artinian sheaf with Euler characteristics equal to χ(Y_{n−1}) < 0 . But this is impossible, hence, φ∘β ≠ 0 . As in the proof of 5.1, we can choose a line L ⊂ U_{0} such that the restriction ψ of φ ∘ β to L ⊗ E_{n−1} ≅ E_{n−1} is non-zero. Fixing any projection U_{0} ↠ L with the kernel V_{0} , we obtain the commutative diagram with exact rows

0 ⟶ V_{0} ⊗ E_{n−1} ⟶^{i} U_{0} ⊗ E_{n−1} ⟶ E_{n−1} ⟶ 0
β′↓  β↓  ψ↓
0 ⟶ Y_{n−1} ⟶ Y_{n}  ⟶^{φ} E_{n} ⟶ 0

where, as in the proof of 5.1, β′ is induced by β ∘ i as far as φ ∘ β ∘ i = 0 . Clearly, β is a sheaf embedding by construction of the monad, β′ is an embedding as a restriction of β, therefore, ψ is a sheaf embedding^{4}. Denote by X_{n−1} the sheaf coker β, we have:

v(X_{n−1}) = v(Y_{n−1}) − (dim U_{0} − 1)v(E_{n−1}) = (r_{n−1}, −r_{n−2}, δ) .

Regarding the last diagram as an exact triple of complexes, we have the long cohomology sequence

0 ⟶ X_{n−1} ⟶ X_{n} ⟶ coker ψ ⟶ 0 .

It follows that X_{n−1} is torsion-free. Besides, μ(X_{n−1}) = μ(Y_{n−1}) = μ(E_{n−1}) , whence X_{n−1} is semistable by 4.1 and stable because it has the rank and c_{1} to be coprime. This completes the proof. □

4Remark that any non-zero map E_{n−1} → E_{n} is an embedding by [12, 3.5].

6 Non locally free part of M_{n}(δ).

We describe here the subset of M_{n}(δ) parametrizing non locally free sheaves. For such a sheaf X consider the reflexive span X^{∗∗}, which is locally free. The sheaf X has no torsion, so the singularity set of X is at least 2-codimensional and the quotient X^{∗∗}/X is an Artinian sheaf i.e. has zero rank and c_{1} .

The following statement implies that X^{∗∗} is stable.

6.1 Lemma. Suppose there is an exact triple of sheaves

0 ⟶ F ⟶ F̃ ⟶ Q ⟶ 0 ,  (48)

where r(Q) = 0 and c_{1}(Q) = 0 . Then the following statements hold:

a) if F̃ is (semi-)stable, then F is (semi-)stable;

b) if F̃ is torsion-free and F is (semi-)stable, then F̃ is (semi-)stable.

Proof. We have: r(F) = r(F̃) and c_{1}(F) = c_{1}(F̃) . The statement a) is trivial because F has no torsion as a subsheaf of a torsion-free sheaf and any subsheaf of F is also a subsheaf of F̃. Let us prove b).

Suppose E is a subsheaf of F̃ such that 0 < r(E) < r(F̃) , then there is the commutative diagram with exact rows and columns

\begin{aligned}& & & & & 0 \\ & & & & & ↓ \\ & & & & & E & ⟶^{id} & E \\ & & & & & ↓ & & ↓ψ \\ 0 & ⟶ & F & ⟶ & F̃ & ⟶ & Q & ⟶ & 0 \\ & & & id↓ & & ↓ & & ↓ \\ 0 & ⟶ & ker ψ & ⟶ & F & ⟶ & G & ⟶ & Q̃ & ⟶ & 0 \\ & & & & & ↓ & & ↓ \\ & & & & & 0 & & 0 \end{aligned}  (49)

Here Q̃ = coker ψ. We have: r(im ψ) = 0 and c_{1}(im ψ) = 0 because im ψ is a subsheaf of Q; whence r(ker ψ) = r(E) , c_{1}(ker ψ) = c_{1}(E) , and μ(ker ψ) = μ(E) . But ker ψ is a subsheaf of F , so μ(ker ψ) < μ(F) (respectively ⩽) as far as F is stable (resp. semistable). Therefore, μ(E) < μ(F̃) (resp. ⩽) that means stability (semi-stability) of F̃. □

The following statement is very close to [15, 1.2.8], it describes the spaces of homomorphisms between two stable sheaves with the same rank and c_{1} , one of them being locally free.

6.2 Proposition. Suppose X, E are stable sheaves with r(X) = r(E) and c_{1}(X) = c_{1}(E) , and E is locally free. Then the following statements are true:

a) Hom(X, E) can be only zero- or one-dimensional, in the latter case E ≅ X^{∗∗};

b) Hom(E, X) = 0 .

Proof. From the proof of [15, 1.2.8] it follows that any non-zero morphism φ from one of these sheaves to the other is a monomorphism and generically an epimorphism. Therefore, Q = coker φ is an Artinian sheaf.

Suppose 0 ≠ φ ∈ Hom(X, E) ; then applying the functor Hom(−, E) to the exact triple

0 ⟶ X ⟶^{φ} E ⟶^{ψ} Q ⟶ 0 ,  (50)

we obtain Hom(E, E) ≅ Hom(X, E) ≅ ℂ as far as Hom(Q, E) = Ext^{1}(Q, E) = 0 and stable bundle E is simple. The embedding φ induces a nontrivial map X^{∗∗} → E^{∗∗} = E, which is an isomorphism as far as X^{∗∗} and E are stable bundles with the same rank and c_{1} . This proves a). To prove b), let’s apply the functor Hom(E, −) to the triple (50). We obtain an exact sequence 0 → Hom(E, X) → ℂ · id_{E} → Hom(E, Q) , where id_{E} comes to non-zero map ψ. This implies that Hom(E, X) = 0 . □

We are now ready to describe the strata of M_{n}(δ) . Put by definition

M_{n}^{δ′}(δ) = {X ∈ M_{n}(δ)| h^{1}(X^{∗∗}) = δ − δ′}

For each X ∈ M_{n}^{δ′}(δ) we have the exact triple

0 ⟶ X ⟶ X^{∗∗} ⟶ Q ⟶ 0 ,  (51)

where h^{0}(Q) = h^{1}(X) − h^{1}(X^{∗∗}) = δ′. Clearly, we have: 0 ⩽ δ′ ⩽ δ, and M_{n}^{0}(δ) consists of locally free sheaves. Thus, we get:

X ∈ M_{n}^{δ′}(δ) ⟺ X^{∗∗} ∈ M_{n}^{0}(δ′′) , where δ′ + δ′′ = δ .

Denote by A(δ′) the set of Artinian sheaves Q such that h^{0}(Q) = δ′. We have a map

π : M_{n}^{δ′}(δ) ⟶ M_{n}^{0}(δ′′) × A(δ′)

which takes X to the pair (X^{∗∗}, Q) according to (51). Obviously, this map is well-defined. Let us describe the preimage π^{−1}(X^{∗∗}, Q) . By Lemma 6.1, any epimorphism φ : X^{∗∗} ↠ Q gives a stable sheaf X = ker φ. Another such an epimorphism φ′ has ker φ′ ≅ X if and only if φ′ = g ∘ φ for some g ∈ Aut(Q) . Indeed, suppose that the rows are given of the following diagram

0 ⟶ X ⟶^{i} X^{∗∗} ⟶^{φ} Q ⟶ 0
ψ↓  id↓  g↓
0 ⟶ X′ ⟶^{i′} X^{∗∗} ⟶^{φ′} Q ⟶ 0

If there is an isomorphism ψ, take i = i′ ∘ ψ, then the left square is commutative, whence there is an isomorphism g : Q → Q. And vice versa, if φ′ = g ∘ φ, then the right square is commutative, hence there is an isomorphism ψ closing the diagram. Therefore, we conclude that

π^{−1}(X^{∗∗}, Q) = (set of epimorphisms in Hom(X^{∗∗}, Q))/ Aut(Q) .

The set A(δ′) has rather complicated description, we give only idea of this. Let A_{k}(δ′) be the set of Artinian sheaves Q ∈ A(δ′) such that there exists an epimorphism O^{⊕k} ↠ Q, but there is no epimorphism O^{⊕(k−1)} ↠ Q. In other words, any sheaf from A_{k}(δ′) can be generated by k sections, but can’t be generated by k − 1 sections. Obviously, we have: A_{1}(δ′) = Hilb^{δ′}(ℙ^{2}) , and there exists a map

⋃_{j=1}^{δ′−1} Z_{j} ⟶ A_{2}(δ′) ,

where Z_{j} is a cycle in Hilb^{j}(ℙ^{2}) × Hilb^{δ′−j}(ℙ^{2}) consisting of (ξ_{1}, ξ_{2}) such that supp ξ_{1} ∩ supp ξ_{2} ≠ ∅, and the map takes (ξ_{1}, ξ_{2}) to the sheaf O_{ξ_{1}} ⊕ O_{ξ_{2}} . This map is, of course, surjective but non-injective. Observe that each Z_{j} has positive codimension in Hilb^{j}(ℙ^{2}) × Hilb^{δ′−j}(ℙ^{2}) . Clearly, there are similar maps for the other A_{k}(δ′) .

In the rest part of this subsection we assume that Q = O_{ξ} ∈ Hilb^{δ′}(ℙ^{2}) i.e. investigate the map π over the ”biggest part” of A(δ′) . Let us show that the corresponding restriction of π is surjective, this is equivalent to existence of an epimorphism E ↠ O_{ξ} for all E ∈ M_{n}^{0}(δ′′) and ξ ∈ Hilb^{δ′}(ℙ^{2}) .

6.3 Lemma. Suppose E is a vector bundle and the canonical map

E ⊗ Hom(E, O) ⟶ O  (52)

is epimorphic. Then for any finite collection of points {x_{i}}_{i=1}^{δ} there is φ ∈ Hom(E, O) such that the fiber map at each point φ_{x_{i}} : E_{x_{i}} → O_{x_{i}} is surjective. Moreover, the set of such maps is open in Hom(E, O) .

Proof. By the lemma of Nakayama we can work out with fibers as the fibers of vector bundles. We don’t change the notation for simplicity. Then φ_{x} is surjective if and only if φ_{x} ≠ 0 . The proof is by induction on δ.

It is clear that for any point x there is φ ∈ Hom(E, O) having φ_{x} ≠ 0 ; otherwise the canonical map (52) would be non-epimorphic at x. Thus, the statement is true for δ = 1 .

Suppose δ = k; then by induction hypothesis there is ψ : E → O such that all ψ_{x_{i}} , i = 1, …, k are surjective. Consider one more point, x_{k+1} . Suppose ψ_{x_{k+1}} = 0 and take ψ′ ∈ Hom(E, O) such that ψ′_{x_{k+1}} ≠ 0 . If ψ′_{x_{i}} is not proportional to ψ_{x_{i}} , then ψ′_{x_{i}} + ψ_{x_{i}} ≠ 0 . Therefore, choosing λ ∈ ℂ^{∗} \ {λ_{i} | ψ′_{x_{i}} = λ_{i}ψ_{x_{i}}, i = 1, …, k}, we obtain that ψ′ − λψ is epimorphic at all points x_{i}, i = 1, …, k + 1 . The last statement follows easily from the fact that the set of φ ∈ Hom(E, O) such that φ_{x} = 0 at a given point x is closed. The lemma is proved. □

6.4 Proposition. Let E be locally free sheaf, ξ ⊂ ℙ^{2} be zero-dimensional subscheme; then there exists an epimorphism of sheaves E ↠ O_{ξ}. Besides, the set of epimorphisms is open in Hom(E, O_{ξ}) .

Proof. Observe that Hom(E, O_{ξ}) = Hom(E(k), O_{ξ}) for any k ∈ ℤ because the sheaf O_{ξ} is invariant under tensoring by invertible sheaf. So, without loss of generality we can assume that E^{∗} is generated by global sections. Suppose the map of stalks at some point x induced by (52) is non-epimorphic; then by lemma of Nakayama the corresponding map of fibers of bundles E_{x} ⊗ Hom(E, O) → O_{x} is also non-epimorphic i.e. vanishes. Thus, for any e ∈ E_{x} and f ∈ H^{0}(E^{∗}) one has ⟨e, f_{x}⟩ = 0 . This implies that f_{x} = 0 for all f ∈ H^{0}(E^{∗}) . This contradicts our assumption that E^{∗} is generated by global sections. Hence, under this assumption the canonical map (52) is epimorphic.

From the lemma it follows that there exist φ ∈ Hom(E, O) having epimorphic maps of fibers φ_{x} for all x ∈ supp(ξ) . Therefore, one can obtain the required map as composition of φ with the standard epimorphism O ↠ O_{ξ}.

In addition, tensoring E by an invertible sheaf we can assume that H^{1}(E^{∗} ⊗ J_{ξ}) = 0 .

Thus, we have a surjective linear map Hom(E, O) −↠^{j} Hom(E, O_{ξ}) . Denote by U the set of epimorphisms in Hom(E, O_{ξ}) , by V the set of φ ∈ Hom(E, O) such that φ_{x} : E_{x} → O_{x} is epimorphic for all x ∈ supp ξ. It readily seen that V = j^{−1}(U) . Choose a linear subspace H ⊂ Hom(E, O) such that Hom(E, O) = H ⊕ ker j, then the restriction j|_{H} is an isomorphism that identifies the rings of polynomial functions on H and on Hom(E, O_{ξ}) . Besides, j|_{H} identifies U and V ∩ H. Thus, for any polynomial f on Hom(E, O) such that the zero-set (f)_{0} of f contains Hom(E, O) \ V the restriction f|_{H} is identified with a polynomial on Hom(E, O_{ξ}) such that (f|_{H})_{0} ⊃ Hom(E, O_{ξ}) \ U . Therefore, U is open. This concludes the proof. □

Denote by M_{n}^{δ′}(δ)_{0} the set of X ∈ M_{n}^{δ′}(δ) such that X^{∗∗}/X = ⨁_{i=1}^{δ′} O_{p_{i}} , where p_{i} ≠ p_{j} for i ≠ j. Clearly,

Aut (⨁_{i=1}^{δ′} O_{p_{i}}) ≅ \underbrace{ℂ^{∗} × ⋯ × ℂ^{∗}}_{δ′}

whence dim Aut(⨁_{i=1}^{δ′} O_{p_{i}}) = δ′. Therefore, we have:

dim(M_{n}^{δ′}(δ)_{0}) = dim(M_{n}^{0}(δ′′)) + dim Hilb^{δ′}(ℙ^{2}) + h^{0}(X^{∗∗}, O_{ξ}) − δ′

Substituting to this the values

dim(M_{n}^{0}(δ′′)) = 2r_{n}δ′′ , dim Hilb^{δ′}(ℙ^{2}) = 2δ′ , and h^{0}(X^{∗∗}, O_{ξ}) = r_{n}δ′ ,

we obtain the following statement.

6.5 Proposition.

dim(M_{n}^{δ′}(δ)_{0}) = 2r_{n}δ′′ + (r_{n} + 1)δ′; codim(M_{n}^{δ′}(δ)_{0}) = (r_{n} − 1)δ′ .

□

References.

[1] R. Y. Donagi. Seiberg-Witten integrable systems// Proceedings of Symposia in Pure Math. V. 62, 2 (1997). P. 3–43.

[2] J.-M. Drezet. Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur ℙ^{2}(ℂ) // J. Reine Angew. Math. 380 (1987), 14-58.

[3] A. L. Gorodentsev, A. N. Rudakov. Exceptional Vector Bundles on Projective Spaces// Duke Math. J., 54 (1987) 115–130.

[4] A. L. Gorodentsev. Helix theory and non-symmetric bilinear forms// Proceedings of the 8th Alg. Geom. Conf., Yaroslavl’ 1992; Aspects of Math. E-25. P. 47–59.

[5] A.L.Gorodentsev, M.I.Leenson. How to calculate the correlation function in twisted SYM N = 2 , N_{f} = 4 QFT on projective plane// Preprint MPI 96-49

[6] P. Griffits, J. Harris. Principles of Algebraic Geometry// New York, 1978.

[7] B. V. Karpov. On the Algebraic Geometry of S -duality// Math. Notes, Vol. 61, No 1 (1997), 133-145.

[8] B. Karpov. Testing S-duality conjecture and exceptional bundles// Preprint MPI 97-40.

[9] B. Karpov. Testing S-duality conjecture and exceptional bundles//To appear in Russian Acad. Sci. Izv. Math.

[10] S. A. Kuleshov and D. O. Orlov. Exceptional sheaves on del Pezzo surfaces// Russian Acad. Sci. Izv. Math. Vol. 44 (1995) No 1. P. 479–513.

[11] S.A.Kuleshov. Moduli Space of Bundles on a Quadric// Warwick Preprints: 18/1996 [12] S.A.Kuleshov. Moduli Spaces of sheaves necessary for testing S -duality conjecture// Preprint MPI 97-32.

[13] M. Maruyama. Moduli of stable sheaves, II// J. Math. Kyoto Univ. V. 18, 3 (1978). P. 557–614.

[14] C. Montonen, D. Olive, Phys. Lett. 72B (1977), 117-132.

[15] C. Okonek, M. Shneider, H. Spindler. Vector Bundles on Complex Projective Spaces// Birkhäuser Boston, 1980.

[16] G. Vafa, E. Witten. A strong coupling test of S-duality// Nucl. Phys. B 431 (1995). P. 3–77.