Moduli spaces of sheaves necessary for testing of S-duality conjecture.

S. Kuleshov^{1}

Abstract

In this text we give a geometrical description for several series of moduli spaces of stable sheaves on ℙ^{2} and ℙ^{1} × ℙ^{1}, which are interesting for mathematical testing of the S-duality conjecture.

Introduction.

It is known that mathematical testing of the S-duality conjecture for SYM QFT in four dimensions is possible only for scaling invariant theories. It is equivalent that the coefficient of the Gell-Mann – Low β–function on the field configuration of the theory vanishes. In this case there appears a constant

τ = \frac{θ}{2π} + \frac{4πi}{g^{2}}

as a parameter of our theory, where θ is the vacuum angle and g is the strong coupling constant. The S-duality conjecture predicates that the partition function

Z_{τ} = ∑_{δ≥0} a_{δ}(e^{2πiτ} )^{δ}

of our theory admits some modular properties under integral linear fractional transformations of τ. The coefficient a_{δ} of the partition function is equal to the solution number of field equations (they are enumerated by δ) for a general metric on an algebraic surface S with K_{S} ≤ 0 considering as a real 4-manifold. The Kählerian metric is not general and there exists a space 𝔐(δ) of gauge orbits of the field equations solutions. In this case the coefficient a_{δ} is calculated as the top Chern class of an obstruction bundle H_{δ} over 𝔐(δ).

From the mathematical viewpoint the field configuration is a collection of representations of Lie algebra for fiber-wise gauge group of our theory. There are known three cases when the β–function coefficient is zero for collections of 𝔲(r)–representations (see [13]).

In the first case the collection consists of one adjoint representation. Testing of Sduality conjecture for this case is suggested in the paper of Vafa and Witten [16]. In the second case, when the coefficient of β-function is zero, the collection consists of 2r canonical representations of 𝔲(r) in ℂ^{r}. QFT and the field equations with this field configuration are described in [6] for r = 2 and in [13] for arbitrary r. The space 𝔐(δ) in these papers are interpreted as moduli spaces 𝔐(r, c_{1}, δ) of stable (in the Mumford – Takemoto sense) bundles X_{δ} on a Del Pezzo surface S with fixed rk (X_{δ}) = r, c_{1}(X_{δ}) = c_{1} and positive −χ(X_{δ}) = δ such that X_{δ}’s and 𝔐(δ) satisfy the conditions

1e-mail: kuleshov@ium.ips.ras.ru

H^{0}(X_{δ}) = H^{2}(X_{δ}) = 0,  (1)
dim 𝔐(δ) = 2rδ.  (2)

Suppose that the moduli space 𝔐(δ) = 𝔐(r, c_{1}, δ) is fine. Then we can consider a universal bundle H_{δ} over 𝔐(δ) such that its fiber over a point X_{δ} ∈ 𝔐(δ) is H^{1}(X_{δ}). In this notation the obstruction bundle is H_{δ}^{⊕2r}. Therefore, the coefficients a_{δ} of Z_{τ} is the top Chern class of H_{δ}^{⊕2r}

a_{δ} = c_{top}(H_{δ}^{⊕2r}).

Condition (1) follows from the stability of the sheaf X_{δ} and the assumption

−μ(O_{S}(K_{S})) < μ(X_{δ}) < 0.  (3)

where = −K_{S} is the anti-canonical divisor of the Del Pezzo surface S (the definition of μ(X) see in the next section). Condition (2) is equivalent to the fact that X_{0} is an exceptional bundle such that

Ext^{i}(O_{S}, X_{0}) = 0 ∀i,  (4)

i.e., X_{0} is orthogonal to O_{S} ([13]). We see that the series of the moduli spaces 𝔐(δ) corresponds to the exceptional bundles X_{0}.

In this preprint, using Ellingsrud’s and Strømme’s method ([3]), we give a geometrical description for several series of moduli spaces on ℙ^{2} and ℙ^{1} × ℙ^{1}, and explicitly construct the bundle H_{δ} on 𝔐(δ).

In the case of ℙ^{2} we deal with moduli spaces series 𝔐_{n}(δ) corresponding to any exceptional bundle E_{n} such that E_{n} satisfies conditions (3) and (4). These bundles can be obtained from E_{0} = O_{ℙ^{2}}(−2) and E_{1} = O_{ℙ^{2}}(−1) by mutations (see proposition 1.7 ). As a result, theorem 5.1 gives a geometrical realization of 𝔐_{n}(δ) and H_{δ}.

In the case of ℙ^{1} × ℙ^{1} we consider the moduli space series 𝔐_{2k}(δ), 𝔐′_{2k+1}(δ), and 𝔐′′_{2k+1}(δ) corresponding to exceptional bundles from three-block collections

(Q_{2k}, Q′_{2k+1}, Q′′_{2k+1}, O_{ℙ^{1}×ℙ^{1}})

(see the definition in (10)). In theorems 6.1 and 6.2 we describe these moduli spaces and the corresponding obstruction sheaves.

I am grateful to B. V. Karpov and D. O. Orlov for many valuable discussions, and I would like to thank the MPI for hospitality.

1 Notations and preliminaries..

In this preprint the surface S means either ℙ^{2} or ℙ^{1} ×ℙ^{1} over complex field ℂ. Here we deal with coherent sheaves on S only. Because of this we omit the word ”coherent”. Besides, there will be no notational distinction made between algebraic bundles and the locally free sheaves of their sections.

We denote by K_{S} the canonical class of S. For a sheaf F we fix the following notation: rk F is rank, c_{1}(F) and c_{2}(F) are the first and the second Chern classes of F. The symbol deg(F) means the degree of F w.r.t the anticanonical class −K_{S} of the surface S, i.e.,

deg(F) = −\frac{(c_{1}(F), K_{S})}{\sqrt{(K_{S}, K_{S})}}, where (·, ·) denotes the cup product of divisors.

The term ”stability” means the stability in sense of Mumford – Takemoto w.r.t. −K_{S}. In another words, a torsion free sheaf F on S is stable iff for any its subsheaf of lesser rank

E the inequality μ(E) < μ(F) holds, where μ(F) = \frac{deg(F)}{rk F} is a slope of F.

The following remark is often used without explanations

1.1 Remark. Let G and X be semistable sheaves on S such that

μ(G(−K_{S})) < μ(X) < μ(G),

then Ext^{i}(G, X) = 0, when i ≠ 1.

To calculate dimensions of the group Ext^{i}(X, Q) for sheaves X and Q on S, we use Riemann-Roch formula in the form

χ(X, Q) = χ(X)rk (Q) + χ(Q)rk (X) − rk (X)rk (Q) − s · deg(X)rk (Q) − (c_{1}(X), c_{1}(Q)),

where s = 3 if S = ℙ^{2} and s = 2 when S = ℙ^{1} × ℙ^{1}. Here

χ(X, Q) = ∑(−1)^{i} dim Ext^{i}(X, Q) = ∑(−1)^{i}h^{i}(X, Q).

Further we give some bases definitions and facts about exceptional sheaves and exceptional collections on the surface S ([7], [15]).

1.2 Definition. 1. A sheaf E on the surface S is called exceptional if

Hom(E, E) = ℂ,  and  Ext^{i}(E, E) = 0, when i > 0.

2. An ordered pair (E, F) of exceptional sheaves on S is called exceptional, if Ext^{i}(F, E) = 0 ∀i.

1.3 Proposition. 1) Any exceptional sheaf on S is locally free and stable;

2) for any exceptional pair (E, F) on ℙ^{2} the groups Ext^{i}(E, F) are trivial, when i > 0 and Hom(E, F) ≠ 0;

3) for any exceptional pair (E, F) on ℙ^{1} × ℙ^{1} Ext^{2}(E, F) are trivial, and only one of the groups ether Ext^{0}(E, F) or Ext^{1}(E, F) does not equal zero.

1.4 Definition. An exceptional pair (E, F) on S is called a hom-pair if Ext^{0}(E, F) ≠ 0. It is called an ext-pair if Ext^{1}(E, F) ≠ 0. Provided Ext^{0}(E, F) = Ext^{1}(E, F) = 0, the pair (E, F) is called a zero-pair.

1.5 Remark. 1. An exceptional pair (E, F) on S is a hom-pair iff μ(E) < μ(F), it is an ext-pair iff μ(E) > μ(F) and it is a zero-pair iff μ(E) = μ(F).

2. All exceptional pair on ℙ^{2} are hom-pairs.

For example, the pairs (O_{ℙ^{2}}(n), O_{ℙ^{2}}(n + 1)) are exceptional hom-pairs and the pair of the line bundles on the quadric (O_{ℙ^{1}×ℙ^{1}}(0, 1), O_{ℙ^{1}×ℙ^{1}}(1, 0)) is an exceptional zero-pair.

We can obtain new exceptional sheaves and pairs with help of mutations.

1.6 Proposition. Let (E, F) be an exceptional hom-pair on S and

lcan : Hom(E, F) ⊗ E ⟶ F

be the canonical map, corresponding to the identity element of Hom(E, F)^{∗} ⊗ Hom(E, F), then lcan is either an epimorphism or a monomorphism. Besides,

1) if the canonical map is an epimorphism, then the sheaf L_{E}F = ker lcan is exceptional and the pair (L_{E}F, E) is an exceptional hom-pair;

2) if the canonical map is a monomorphism, then the sheaf L_{E}F = coker lcan is exceptional and the pair (L_{E}F, E) is an exceptional ext-pair.

1.7 Proposition. Let (E, F) be an exceptional hom-pair on S and

rcan : E ⟶ Hom(E, F)^{∗} ⊗ F

be the canonical map, corresponding to the identity element of Hom(E, F)^{∗} ⊗ Hom(E, F), then rcan is either an epimorphism or a monomorphism. Besides,

1) if the canonical map is a monomorphism, then the sheaf R_{F} E = coker rcan is exceptional and the pair (F, R_{E}F) is an exceptional hom-pair.

2) if the canonical map is an epimorphism, then the sheaf R_{F} E = ker can is exceptional and the pair (F, R_{E}F) is an exceptional ext-pair.

1.8 Proposition. Let (E, F) be an exceptional ext-pair on S and e be the identity element of Ext^{1}(E, F)^{∗} ⊗ Ext^{1}(E, F), then

1) the sheaf L_{E}F from the universal extension

0 ⟶ F ⟶ L_{E}F ⟶ Ext^{1}(E, F) ⊗ E ⟶ 0,

corresponding to e is exceptional and the pair (L_{E}F, E) is exceptional too;

2) the sheaf R_{F} E from the universal extension

0 ⟶ Ext^{1}(E, F)^{∗} ⊗ F ⟶ R_{F} E ⟶ E ⟶ 0,

corresponding to e is exceptional and the pair (F, R_{F} E) is exceptional too.

1.9 Definition. The sheaves L_{E}F (R_{F} E) defining in propositions 1.6 , 1.7 and 1.8 are called the left (right) shifts of F (E); the pairs (L_{E}F, E) are called the left mutations of the pair (E, F) and the pairs (F, R_{F} E) are called the right one.

The left (right) mutation of a pair (E, F) is called regular provided the pair (E, F) is a hom-pair and the map lcan is an epimorphism (rcan is a monomorphism.)

1.10 Remark. If (E, F) is an exceptional pair on ℙ^{2}, then the left and right mutations of it are regular.

1.11 Proposition. let (E, F′, F′′) be an exceptional collection such that μ(E) < μ(F′) = μ(F′′). Consider the canonical map

rcan : E ⟶ \begin{matrix}Hom(E, F′)^{∗} ⊗ F′ \\ ⨁ \\ Hom(E, F′′)^{∗} ⊗ F′′ \end{matrix} ,

corresponding to the element

(id, id) ∈ Hom(E, F′)^{∗} ⊗ Hom(E, F′) ⊕ Hom(E, F′′)^{∗} ⊗ Hom(E, F′′).

Then rcan is a monomorphism and coker rcan = R_{F′′}R_{F′}E. We’ll denote this sheaf by R_{F} E. In the same way the sheaf L_{E}F can be define for an exceptional collection (E′, E′′, F) such that μ(E′) = μ(E′′) < μ(F).

1.12 Definition. An ordered collection (E_{0}, …, E_{n}) on S is called exceptional if each its pair (E_{i}, E_{j}) (i < j) is exceptional. It is called full, if the minimal triangulated subcategory in the derived category D^{b}(S) of coherent sheaves on S, containing the collection’s objects, coincides with D^{b}(S).

1.13 Proposition. Let σ = (E_{0}, …, E_{n}) be an exceptional collection on S, then

1) the collections

L_{i}σ = (E_{0}, …, L_{E_{i}}E_{i+1}, E_{i}, E_{i+2}, …, E_{n})

and

R_{i}σ(E_{0}, …, E_{i−2}, E_{i}, R_{E_{i}}E_{i−1}, …, E_{n})

are exceptional,

2) if σ is full, then L_{i}σ and R_{i}σ are full too,

3) if σ is full, then

L_{0}L_{1} ⋯ L_{n−1}σ = (E_{n}(K_{S}), E_{0}, …, E_{n−1}),
R_{n}R_{n−1} ⋯ R_{1}σ = (E_{1}, …, E_{n}, E_{0}(−K_{S})).

1.14 Definition. Let σ = (E_{0}, …, E_{k}) be an exceptional collection on S. The minimal full triangulated subcategory in the derived category of coherent sheaves on S, containing the collection’s objects, is called the subcategory generated by σ and is denoted by ⟨E_{0}, …, E_{k}⟩_{cat}.

1.15 Lemma (A. Bondal,[1]) Let (E_{0}, …, E_{n}) be a full exceptional collection and X be a sheaf on S, then

X ∈ ⟨E_{0}, …, E_{s}⟩_{cat} ⟺ Ext^{i}(E_{j}, X) = 0 ∀i, j > s;

X ∈ ⟨E_{s}, …, E_{n}⟩_{cat} ⟺ Ext^{i}(X, E_{j}) = 0 ∀i, j < s.

1.16 Definition Let σ = (E_{0}, …, E_{k}) be an exceptional collection on S. The collection (^{∨}E_{k}, …,^{∨}E_{0}) is called the left dual collection, whenever

∨E_{k} = E_{k}, ^{∨}E_{k−1} = R_{E_{k}}E_{k−1}, ^{∨}E_{k−2} = R_{E_{k}}R_{E_{k−1}}E_{k−2}, …

1.17 Theorem. (A. Gorodentsev, [5].) Let (E_{0}, …, E_{k}) be an exceptional collection and X be a sheaf on S such that X ∈ ⟨E_{0}, …, E_{k}⟩_{cat}. There exists a spectral sequence E^{p,q} converging to X with the first part of the form

E^{p,q} = Ext^{q−Δ_{p+k}}(^{∨}E_{k+p}, X) ⊗ E_{k+p},

where Δ_{p+k} is the number of non regular mutations needed for constructing the sheaf ^{∨}E_{k+p}.

In this preprint we are interesting exceptional bundles E’s on S that are orthogonal to O_{S}, i.e.,

Ext^{i}(O_{S}, E) = 0 ∀i.  (5)

The case of ℙ^{2}. It is obvious that (O_{ℙ^{2}}(−2), O_{ℙ^{2}}(−1), O_{ℙ^{2}}) is a full exceptional collection. Therefore, if an exceptional bundle E satisfy condition (5), then E belongs to the subcategory ⟨O_{ℙ^{2}}(−2), O_{ℙ^{2}}(−1)⟩_{cat} (1.15 ). Moreover, E can be obtained from O_{ℙ^{2}}(−2), O_{ℙ^{2}}(−1) by mutations. By definition put,

E_{0} = O_{ℙ^{2}}(−2), E_{1} = O_{ℙ^{2}}(−1), E_{n+1} = R_{E_{n}}E_{n−1}, r_{n} = rk (E_{n}),  (6)

i.e., for each natural n we have the exact triple

0 ⟶ E_{n−1} ⟶^{rcan} Hom(E_{n−1}, E_{n})^{∗} ⊗ E_{n} ⟶ E_{n+1} ⟶ 0.

It is easy to prove that the following relations are true:

deg(E_{n}) = −r_{n−1}, r_{n+1} = 3r_{n} − r_{n−1}, r_{n}^{2} + r_{n−1}^{2} + 1 = 3r_{n}r_{n−1}.  (7)

Using this relations, we can calculate that

deg(E_{n−1})rk (E_{n}) − deg(E_{n})rk (E_{n−1}) = r_{n−1}^{2} − r_{n−2}r_{n} = −1.  (8)

In the case of ℙ^{1} × ℙ^{1} we consider so called exceptional three-block collections ([11]).

1.18 Definition. A full exceptional collection (Q_{0}, Q_{1}, Q_{2}, Q_{3}) on ℙ^{1} ×ℙ^{1} is called threeblock collection if for one of i ∈ {0, 1, 2} the pair (Q_{i}, Q_{i+1}) is a zero-pair, i.e., μ(Q_{i}) = μ(Q_{i+1}). It is known that in this case μ(Q_{j}) ≤ μ(Q_{j+1}) ∀j.

For example,

(O_{ℙ^{1}×ℙ^{1}}(−1, −1), O_{ℙ^{1}×ℙ^{1}}(−1, 0), O_{ℙ^{1}×ℙ^{1}}(0, −1), O_{ℙ^{1}×ℙ^{1}})

is the full exceptional three-block collection.

By definition, put

Q_{0} = O_{ℙ^{1}×ℙ^{1}}(−1, −1), Q′_{1} = O_{ℙ^{1}×ℙ^{1}}(−1, 0), Q′′_{1} = O_{ℙ^{1}×ℙ^{1}}(0, −1),
Q_{2k} = R_{Q_{2k−1}}Q_{2k−2}, Q′_{2k+1} = R_{Q_{2k}}Q′′_{2k−1}, Q′′_{2k+1} = R_{Q_{2k}}Q′_{2k−1};  (9)

(definition of R_{Q_{2k−1}}Q_{2k−2} see in proposition 1.11 )

q_{2k} = rk Q_{2k}, q_{2k+1} = rk Q′_{2k+1} = rk Q′′_{2k+1},

It follows from [11], that for any integer k the collections

(Q_{2k}, Q′_{2k+1}, Q′′_{2k+1}, O_{ℙ^{1}×ℙ^{1}}) and (Q′_{2k−1}, Q′′_{2k−1}, Q_{2k}, O_{ℙ^{1}×ℙ^{1}})  (10)

are full exceptional three-block collections. Since,

h^{0}(Q_{2k}, Q′_{2k+1}) = h^{0}(Q_{2k}, Q′′_{2k+1}) = h^{0}(Q′_{2k−1}, Q_{2k}) = h^{0}(Q′′_{2k−1}, Q_{2k}) = 2,

we obtain,

q_{2k+1} = 2q_{2k} − q_{2k−1},  q_{2k+2} = 4q_{2k+1} − q_{2k}  (11)

It is not hard to prove the following arithmetic relations

c_{1}(Q_{2k}) = −(q_{2k−1}, q_{2k−1}), c_{1}(Q′_{2k+1}) = −\frac{1}{2}(q_{2k}+1, q_{2k}−1), c_{1}(Q′′_{2k+1}) = −\frac{1}{2}(q_{2k}−1, q_{2k}+1);

deg(Q_{n−1})rk Q_{n} − deg(Q_{n})rk Q_{n−1} = −1,  (12)

where Q_{n} is an exceptional bundle of the sequence of bundles Q_{0}, Q′_{1}, Q′′_{2}, Q_{3}, …, moreover, the following analog of Markov equation takes place

q_{2k}^{2} + 2q_{2k±1}^{2} + 1 = 4q_{2k}q_{2k±1}  (13)

2 Some useful statements.

2.1 Lemma. Let

0 ⟶ A ⟶ B ⟶ C ⟶ 0

be an exact sequence of objects from an Abelian category 𝔄 and D be another object from 𝔄. Then for any φ ∈ Hom(D, C) such that e_{D} = δ_{D}(φ) ≠ 0, where δ_{D} : Hom(D, C) ⟶ Ext^{1}(D, A) is the coboundary map, there exists a commutative diagram

\begin{aligned}& & 0 \\ & & ↓ \\ 0 & ⟶ & A & ⟶ & B̃ & ⟶ & D & ⟶ & 0 \\ & & id ↓ & & φ̃ ↓ & & φ ↓ \\ 0 & ⟶ & A & ⟶ & B & ⟶ & C & ⟶ & 0 \\ & & ↓ \\ & & 0 \end{aligned} ,  (14)

where the first row of the diagram is determined by the element δ_{D}(φ) ∈ Ext^{1}(D, A). Proof. Consider the commutative diagram

Hom(C, C) ⟶^{φ_{0}} Hom(D, C) δ_{C} ↓ δ_{D} ↓ Ext^{1}(C, A) ⟶^{φ_{1}} Ext^{1}(D, A) ,

where δ_{C}, δ_{D} are coboundary morphisms and φ_{0}, φ_{1} are induced by φ. The second row of diagram(14) is determined by e_{C} = δ_{C}(id_{C}) ∈ Ext^{1}(C, A). On the other hand, φ_{0}(id_{C}) = φ and e_{D} = δ_{D}(φ) = φ_{1}(e_{C}). Hence there exists a map φ̃ : B̃ ⟶ B such that diagram (14) commutative.

2.2 Lemma. Let X and F be stable sheaves on a smooth projective surface S such that their slopes μ(X) = \frac{a}{r_{1}} , μ(F) = \frac{b}{r_{2}} satisfy the condition

ar_{2} − br_{1} = −1,

then a nontrivial extension

0 ⟶ X ⟶ X′ ⟶ F ⟶ 0  (15)

is stable if and only if a) X is locally free;

b) X is not locally free, but the element e from Ext^{1}(F, X) determining the extension does not belong to the subspace Hom(X^{∗∗}/X, F) ⊂ Ext^{1}(F, X), where X^{∗∗} is the reflexive span of X.

Proof. Let X be locally free and Q is a torsion free quotient of X′ with the minimal slope and the minimal rank. By construction Q is stable. Suppose X′ is not stable, then μ(Q) < μ(X′) < μ(F). This implies that Hom(F, Q) = 0 and there exists a commutative diagram

\begin{aligned}0 & ⟶ & X & ⟶ & X′ & ⟶ & F & ⟶ & 0 \\ & & φ ↓ & & ↓ \\ & & Q & ⟶^{id} & Q \end{aligned} .

Since the sheaves X and Q are stable, we have μ(X) ≤ μ(Q).

Consider the following picture

M̃_{X}M̃_{X′} M_{X′} M_{F} M_{X}  2r_{1} r_{1} r_{2}  r c_{1} b 2a a O

By the assumption of the proposition, area of parallelogram OM_{X}M_{X′}M_{F} equals 1. Hence, area of parallelogram M_{X} M̃_{X} M̃_{X′}M_{X′} equals 1 too. This implies that there are not points with integer coordinates interior parallelogram OM̃_{X} M̃_{X′}M_{F}. On the other hand, rk Q < rk X′ and μ(X) ≤ μ(Q) < μ(X′). Therefore, μ(Q) = μ(X) and morphism φ is isomorphism in a general point, i.e. we have the exact triple:

0 ⟶ X ⟶ Q ⟶ Q̃ ⟶ 0,

where dim Supp Q̃ = 0.

Using the fact that the group Ext^{1}(Q̃, X) = 0 for any locally free sheaf X and torsion sheaf Q̃ with zero-dimensional support on a surface ([8]), we obtain that Q = X ⊕ Q̃. This contradicts the stability of Q. Hence Q ≅ X.

Recall that the stability of X implies the fact that X is simple (Hom(X, X) ≅ ℂ) and obtained nontrivial map X′ ⟶ X shows that sequence (15) splits. This contradiction proves the first statement of proposition.

Now suppose that X is not locally free and consider the commutative diagram

\begin{aligned}& & 0 & & 0 \\ & & ↓ & & ↓ \\ 0 & ⟶ & X & ⟶ & X′ & ⟶ & F & ⟶ & 0 \\ & & can ↓ & & ↓ & & ↓ id \\ 0 & ⟶ & X^{∗∗} & ⟶ & X̃ & ⟶ & F & ⟶ & 0 \\ & & ↓ & & ↓ \\ & & X^{∗∗}/X & & X^{∗∗}/X \\ & & ↓ & & ↓ \\ & & 0 & & 0 \end{aligned} ,

where the first line is exact triple (1.8 ) determined by an element e ∈ Ext^{1}(F, X) and the second one is the extension corresponding to the element can(e) ∈ Ext^{1}(F, X^{∗∗}).

Since X^{∗∗} is locally free, in the case can(e) ≠ 0 X̃ is stable (by the first statement of the proposition). But μ(X′) = μ(X̃) and any subsheaf of X′ is a subsheaf of X̃ too. Hence X′ is stable also.

Suppose can(e) = 0, i.e. e ∈ Hom(F, X^{∗∗}/X) ⊂ Ext^{1}(F, X). Then the following commutative diagram takes place (see lemma 2.1 ).

0 ⟶ X ⟶ X′ ⟶ F  ⟶ 0 id ↓  ψ ↓  e ↓ 0 ⟶ X ⟶ X^{∗∗} ⟶ X^{∗∗}/X ⟶ 0 ↓  ↓  ↓ 0 ⟶ coker ψ ⟶ coker e ⟶ 0 ↓  ↓ 0  0 .

Since dim Supp coker ψ = 0, slope of im ψ is equal to slope of X^{∗∗}, i.e. μ(im ψ) < μ(X′), and X′ is not stable. This concludes the proof of the proposition.

2.3 Lemma. Let

V ⊗ X ⟶ X′ ⟶ F ⟶ 0

be an exact sequence of sheaves on a smooth projective complex surface S such that the sheaf X is locally free, stable, the space Hom(X′, X) = 0, and F has no torsion. Suppose that the following conditions is true

deg(X)rk (F) − deg(F)rk (X) = −1,

then the stability of F is equivalent to the stability of X′.

Proof. First we note that the assumptions of the lemma implies

μ(X) < μ(X′) < μ(F).

Suppose that the sheaf F is not stable, i.e., it has a quotient sheaf F ⟶ G ⟶ 0 with μ(F) ≥ μ(G), rk (G) < rk (F). By the same method as in the previous proof we can show that μ(X) ≥ μ(G). But μ(X′) > μ(X) and G is the quotient of X′. Therefore, X′ is not stable either.

Now suppose that the sheaf F is stable and X′ has a quotient sheaf Q such that

μ(Q) ≤ μ(X′) rk (Q) < rk (X′).

Without loss of generality we can assume that Q is stable. Hence, Hom(F, Q) = 0 and, consequently, Hom(X, Q) ≠ 0. This yields the inequality μ(X) ≥ μ(Q).

By the same arguments as before, we obtain, μ(X) = μ(Q). Let us recall that Q and X are stable sheaves and X is locally free. Therefore, any nontrivial map f : X ⟶ Q is an isomorphism. Thus we have a some nonzero map X′ ⟶ X. This contradiction proves the lemma.

2.4 Remark. The previous lemma holds in the case when X = X_{1}⊕X_{2} with stable bundles X_{i} such that μ(X) = μ(X_{1}) = μ(X_{2}).

2.5 Lemma. Let X and F be a stable sheaves on a smooth projective complex surface S such that the sheaf X is locally free and

deg(X)rk (F) − deg(F)rk (X) = −1.

Suppose φ ∈ Hom(U ⊗ X, F) is nontrivial, then

1) either rk (im φ) = rk F, or im φ is a semistable sheaf with a filtration

im φ = Gr(X, …, X),

2) provided X is exceptional, ker φ = U′ ⊗ X for some subspace U′ ⊂ U.

Proof. Assume that rk (im φ) < rk (F). Taking into account the stability of X and F, we have

μ(X) ≤ μ(im φ) < μ(F).

This inequality and the assumptions of the lemma imply that

μ(X) = μ(im φ)  (16)

and the semistability of im φ follows from the stability of X.

Let us show that im φ = Gr(X, …, X) by induction on rk (im φ). We have an epimorphism φ : U ⊗ X ⟶ im φ ⟶ 0. Therefore, there exists a nonzero map φ_{1} : X ⟶ im φ. Since X is a stable bundle and im φ is semistable, taking into account (16), we obtain that φ_{1} is a monomorphism.

Suppose that coker φ_{1} has a torsion T. It is obvious that deg(coker φ) = 0, consequently, its support is a finite set of points, hence Ext^{1}(T, X) = 0. Consider the following commutative diagram

0 ↓ 0  0  T ↓  ↓  ↓ 0 ⟶ X ⟶^{φ_{1}} im φ ⟶ coker φ_{1} ⟶ 0 ↓  ↓  ↓ 0 ⟶ X̄ ⟶ im φ ⟶ (coker φ_{1})/T ⟶ 0 ↓  ↓  ↓ T  0  0 ↓ 0 .

We see that X̄ = X ⊕ T and X ⊕ T ⊂ im φ. But im φ has no torsion (as a subsheaf of a torsion free sheaf). Therefore, T = 0 and coker φ_{1} is semistable. Now, application of the inductive hypothesis to coker φ_{1} proves the first statement of the lemma.

Further, in additional, suppose that X is exceptional, i.e. Ext^{1}(X, X) = 0. Hence im φ = U′′ ⊗ X and we have an exact triple

0 ⟶ ker φ ⟶ U ⊗ X ⟶^{φ} U′′ ⊗ X ⟶ 0.

Since X is simple, the morphism φ is induces by a linear map of the vector spaces φ′ : U ⟶ U′′. Therefore, ker φ = X ⊗ ker φ′. This completes the proof.

3 Monads.

Our method moduli space description is a generalization of the method of G.Ellingsrud and S.A.Strøme ([3]) and uses monads. Here we construct the monads on ℙ^{2} and ℙ^{1} × ℙ^{1}.

3.1 Proposition. Let (E, F, G) be a full exceptional collection on ℙ^{2}. Then

1) any semistable sheaf X on ℙ^{2} satisfying the slope conditions

μ(G) − 3 < μ(E) < μ(X) < μ(G)

is the cohomology of the monad M(X)

0 ⟶ U_{0}(X) ⊗ E ⟶ U_{1}(X) ⊗ F ⟶ U_{2}(X) ⊗ G ⟶ 0

with U_{0}(X) = Ext^{1}(E(3), X), U_{1}(X) = Ext^{1}(L_{E}F (3), X), U_{2}(X) = Ext^{1}(G, X);

2) two semistable sheaves X and X′ on ℙ^{2} are isomorphic iff their monads M(X) and M(X′) are.

Proof. By theorem 1.17 we have the spectral sequence E^{pq} converging to X with the first part in the form

E_{1}^{pq} = Ext^{q−Δ_{2+p}}(^{∨}G_{2+p}, X) ⊗ G_{2+p}.

where G_{0} = E, G_{1} = F, G_{2} = G, ^{∨}G_{2} = G, ^{∨}G_{1} = R_{G}F, ^{∨}G_{2} = R_{G}R_{F} E. Taking into account proposition 1.13 and the fact that O_{ℙ^{2}}(K_{ℙ^{2}}) = O_{ℙ^{2}}(−3), we obtain,

R_{G}F = L_{E}F (3) and R_{G}R_{F} E = E(3).

Besides, all mutations on ℙ^{2} are regular (see 1.10 ), i.e., Δ_{2+p} = 0 ∀p.

On the other hand, by the slope condition of the proposition and the stability of exceptional bundles (1.3 ) and X, we get E_{1}^{pq} = 0, whenever q ≠ 1. This means that the spectral sequence degenerates into the monad of the proposition.

The second point of the proposition is proved in the same way as lemma 4.1.3. of [9] with help of the spectral sequence functoriality.

3.2 Proposition Let (E, F′, F′′, G) be a full exceptional three-block collection on ℙ^{1} × ℙ^{1} (μ(F′) = μ(F′′)). Then

1) any semistable sheaf X on ℙ^{1} × ℙ^{1} satisfying the slope conditions

μ(G) − 4 < μ(E) < μ(X) < μ(G)

is the cohomology of the monad M_{1}(X)

0 ⟶ U_{0}(X) ⊗ E ⟶ \begin{matrix}U_{1}(X) ⊗ F′ \\ ⊕ \\ U_{2}(X) ⊗ F′′ \end{matrix} ⟶ U_{3}(X) ⊗ G ⟶ 0

with U_{0}(X) = Ext^{1}(E(2, 2), X), U_{1}(X) = Ext^{1}(L_{E}F′(2, 2), X), U_{2}(X) = Ext^{1}(L_{E}F′′(2, 2), X), U_{3}(X) = Ext^{1}(G, X);

2) two semistable sheaves X and X′ on ℙ^{1} ×ℙ^{1} are isomorphic iff their monads M_{1}(X) and M_{1}(X′) are.

Proof. This proposition can be proved by the same method as the previous one with help of the Beilinson–Gorodentsev spectral sequence. But here we give another method, using a projections to subcategory generated by sub collection.

At the beginning we note that the stability and our assumptions imply

Ext^{i}(G, X) = Ext^{i}(X, E) = 0, when i ≠ 1.  (17)

Consider the universal extension

0 ⟶ Ext^{1}(X, E)^{∗} ⊗ E ⟶ X_{1} ⟶ X ⟶ 0.  (18)

It follows from the long cohomology sequence, corresponding to (18) that Ext^{i}(X_{1}, E) = 0 ∀i. Therefore (1.15 ),

X_{1} ∈ ⟨F′, F′′, G⟩_{cat}.

Besides,

Ext^{i}(G, X_{1}) ≅ Ext^{i}(G, X) ∀i.

Further, taking the next universal extension

0 ⟶ X_{1} ⟶ X_{2} ⟶ Ext^{1}(G, X_{1}) ⊗ G ⟶ 0,  (19)

we obtain,

X_{2} ∈ ⟨F′, F′′⟩_{cat}.

But (F′, F′′) is a zero-pair and any sheaf Y ∈ ⟨F′, F′′⟩_{cat} has the form Y = U_{1}(Y) ⊗ F′ ⊕ U_{2}(Y)⊗F′′ for some vector spaces U_{1}(Y), U_{2}(Y). This fact can be easily proved with help of the Beilinson–Gorodentsev spectral sequence, associated with the exceptional pair (F′, F′′).

Note that the Serre duality implies the equality Ext^{1}(X, E)^{∗} ≅ Ext^{1}(E(2, 2), X). Therefore, combining (18) and (19) together, we get the monad from the formulation of the proposition. The fact that U_{1}(X) = Ext^{1}(L_{E}F′(2, 2), X), U_{2}(X) = Ext^{1}(L_{E}F′′(2, 2), X) is proved by the standard method, using long cohomology sequences. This concludes the proof of the first point of the proposition.

The second point is proved by the same arguments as lemma 4.1.3 of [9] and here we limit the following remark. Any morphism of sheaves f : X ⟶ X′ induces a morphism of the universal extensions (18) and (19).

This completes the proposition proof.

3.3 Proposition. Let (E′, E′′, F, G) be a full exceptional three-block collection on ℙ^{1} ×ℙ^{1} (μ(E′) = μ(E′′)). Then 1) any semistable sheaf X on ℙ^{1} × ℙ^{1} satisfying the slope conditions

μ(G) − 4 < μ(E′) < μ(X) < μ(G)

is the cohomology of the monad M_{2}(X)

0 ⟶ \begin{matrix}U_{0}(X) ⊗ E′ \\ ⊕ \\ U_{1}(X) ⊗ E′′ \end{matrix} ⟶ U_{2}(X) ⊗ F ⟶ U_{3}(X) ⊗ G ⟶ 0

with U_{0}(X) = Ext^{1}(E′(2, 2), X), U_{1}(X) = Ext^{1}(E′′(2, 2), X), U_{2}(X) = Ext^{1}(L_{E}F (2, 2), X), U_{3}(X) = Ext^{1}(G, X);

2) two semistable sheaves X and X′ on ℙ^{1} ×ℙ^{1} are isomorphic iff their monads M_{2}(X) and M_{2}(X′) are.

Proof. The proof of this proposition differs from the previous proof at the first step only. Namely, we have to project the sheaf X to the subcategory ⟨F, G⟩_{cat}. Consider the universal extension

0 ⟶ \begin{matrix}Ext^{1}(X, E′)^{∗} ⊗ E′ \\ ⊕ \\ Ext^{1}(X, E′′)^{∗} ⊗ E′′ \end{matrix} ⟶ X_{1} ⟶ X ⟶ 0,

determining by the element

(id × id) ∈ (Ext^{1}(X, E′)^{∗} ⊗ Ext^{1}(X, E′)) ⊕ (Ext^{1}(X, E′′)^{∗} ⊗ Ext^{1}(X, E′′)).

Then Ext^{i}(X_{1}, E′) = Ext^{i}(X_{1}, E′′) = 0 ∀i, i.e., X_{1} ∈ ⟨F, G⟩_{cat}. Further, taking the universal extension

0 ⟶ X_{1} ⟶ X_{2} ⟶ Ext^{1}(G, X_{1}) ⊗ G ⟶ 0,

we get X_{2} ∈ ⟨F ⟩_{cat}. Therefore, X_{2} = U_{2}(X) ⊗ F and so on.

3.4 Remark. In addition to the assumptions of the previous proposition, suppose that E′′ = i_{∗}E′ and X = i^{∗}X, where i : ℙ^{1} ×ℙ^{1} ⟶ ℙ^{1} ×ℙ^{1} is the natural involution, transposing the factors. Then the monad M_{2}(X) can be rewritten in the form

0 ⟶ U_{1}(X) ⊗ ( \begin{matrix}E′ \\ ⊕ \\ E′′ \end{matrix} ) ⟶ U_{2}(X) ⊗ F ⟶ U_{3}(X) ⊗ G ⟶ 0,

where U_{1}(X) = Ext^{1}(X, E′)^{∗} ≅ Ext^{1}(X, E′′)^{∗}.

4 Kronecker moduli and representations of quivers.

To describe required moduli spaces we need Kronecker moduli and moduli space of some quiver representations. We start with Kronecker moduli by [2].

4.1 Definition. Let U_{0}, U_{1}, V be finite dimensional vector spaces with u_{0} = dim U_{0}, u_{1} = dim U_{1}, v = dim V. A linear map t : U_{0} ⟶ U_{1} ⊗V up to the natural action of the reductive group G = GL(U_{0}) × GL(U_{1})/ℂ · {id × id} is called a Kronecker module.

4.2 Remark. There exists a compact Kronecker moduli space 𝔑(v; u_{0}, u_{1}), i.e. the space of the semistable orbits of the G-action in the vector space U_{0}^{∗} ⊗ U_{1} ⊗ V. Moreover, in the case, when u_{0}, u_{1} are coprime, the moduli space 𝔑(v; u_{0}, u_{1}) is fine, i.e., there exists a universal Kronecker module U_{0}^{∗} ⊗ U_{1} ⊗ V over 𝔑(v; u_{0}, u_{1}).

J.-M. Drezet in [2] shows that we can describe some moduli spaces of bundles on ℙ^{2} as Kronecker moduli space. Namely, any morphism of sheaves α : U_{0} ⊗ F ⟶ U_{1} ⊗ G corresponds to a Kronecker module t_{α} : U_{0} ⟶ U_{1} ⊗ Hom(F, G). In this notation the following statements are true.

4.3 Proposition. Let (E, F, G) be a full exceptional collection on ℙ^{2} and U_{0}, U_{1} be vector spaces with u_{0} = dim U_{0}, u_{1} = dim U_{1} such that

1) u_{0}rk (F) > u_{1}rk (G), 2) μ(E) < \frac{u_{0} deg F − u_{1} deg G}{u_{0}rk (F) − u_{1}rk (G)},

then t_{α} is stable iff α is an epimorphism and ker α is a stable bundle.

4.4 Theorem In additional to the assumption of the previous proposition suppose that u_{0}, u_{1} are coprime, then the moduli space of stable sheaves Y ’s on ℙ^{2} with topological invariants, satisfying the conditions

rk Y = u_{0}rk F − u_{1}rk G, c_{1}(Y) = u_{0}c_{1}(F) − u_{1}c_{1}(G), χ(Y, E) = 0

is isomorphic to the smooth complied fine Kronecker moduli space 𝔑(h, u_{0}, u_{1}), where h = dim Hom(F, G). Moreover, all the sheaves Y are locally free.

These Drezet’s statements were generalized by V. Karpov to the case of quadric ℙ^{1} × ℙ^{1} [12]

4.5 Proposition. Let (D, E, F, G) be a full exceptional collection on ℙ^{1} × ℙ^{1} such that μ(E) < μ(F) < μ(G) and the pair (F, G) is obtained from a line bundle pair by regular mutations. Suppose that dimensions of vector spaces u_{0} = dim U_{0}, u_{1} = dim U_{1} satisfy the conditions

1) u_{0}rk (F) > u_{1}rk (G), 2) μ(E) < \frac{u_{0} deg F − u_{1} deg G}{u_{0}rk (F) − u_{1}rk (G)},

then t_{α} is stable iff α is an epimorphism and ker α is a stable bundle.

4.6 Theorem. In additional to the assumption of the previous proposition suppose that u_{0}, u_{1} are coprime, then the moduli space of stable sheaves Y ’s on ℙ^{1} × ℙ^{1} with topological invariants, satisfying the conditions

rk Y = u_{0}rk F − u_{1}rk G, c_{1}(Y) = u_{0}c_{1}(F) − u_{1}c_{1}(G), χ(Y, E) = χ(Y, D) = 0

is isomorphic to the smooth projective fine Kronecker moduli space 𝔑(h, u_{0}, u_{1}), where h = dim Hom(F, G). Moreover, all the sheaves Y are locally free.

To describe moduli spaces of bundles on the quadric we need a some more generalization of Drezet’s theorem. At first we formulate results of [14] concerning the quiver Q with three vertecis p_{0}, p_{1}, p_{2}, h_{01} arrows from p_{0} to p_{1}, and h_{02} arrows from p_{0} to p_{2}.

4.7 Definition. A linear map of vector spaces

t : U_{0} ⟶ \begin{matrix}U_{1} ⊗ V_{01} \\ ⊕ \\ U_{2} ⊗ V_{02} \end{matrix}

with dim V_{0i} = h_{0i} is called a representation of the quiver Q.

The group G = GL(U_{0}) × GL(U_{1}) × GL(U_{2}) acts on the vector space R = U_{0}^{∗} ⊗ (U_{1} ⊗

V_{01} ⊕U_{2} ⊗V_{02}) in the standard way. Obviously, the action orbits coincide with isomorphism classes of the representations.

Let us choose a character of G

χ_{θ}(g) = (det g_{0})^{θ_{0}} · (det g_{1})^{θ_{1}} · (det g_{2})^{θ_{2}},

where (g_{0}, g_{1}, g_{2}) ∈ G, and let us lift the G-action to the direct product R × ℂ as follows g(v, z) = (g(v), χ_{θ}^{−1}(g) · z).

4.8 Definition. 1. A point v ∈ R is χ_{θ}-semistable if there is a relative invariant f ∈ ℂ[R]^{G,χ_{θ}^{n}} with n ≥ 1, such that f(v) ≠ 0.

2. A point v ∈ R is χ_{θ}-stable if it is semistable and dim G · v = dim G/G_{0}, where G_{0} is the kernel of the G-action, and the G-action on {v ∈ R | f(v) ≠ 0} is closed.

Let us fix a dimension vector u = (u_{0}, u_{1}, u_{2}), where u_{i} = dim U_{i} and let us choose a character of G θ = (θ_{0}, θ_{1}, θ_{2}) such that

∑ θ_{i}u_{i} = 0.  (20)

This means that the restriction of the character χ_{θ} to the G-action kernel equals 1.

4.9 Definition. A line map t′ : U′_{0} ⟶ (U′_{1} ⊗V_{01} ⊕U′_{2} ⊗V_{02}) is called a subrepresentation of the representation t if U′_{i} ⊂ U_{i} and the following commutative diagram takes place:

\begin{matrix}U′_{0} & ⟶^{t′} & (U′_{1} ⊗ V_{01} ⊕ U′_{2} ⊗ V_{02}) \\ ↓ & & ↓ \\ U_{0} & ⟶^{t} & (U_{1} ⊗ V_{01} ⊕ U_{2} ⊗ V_{02}) \end{matrix} .

4.10 Proposition. A representation t ∈ R is χ_{θ}-(semi)stable if and only if, for every its subrepresentation t′ with a dimension vector u′ the condition ∑ θ_{i}u′_{i} > 0 (≥ 0 ) holds.

4.11 Theorem. Let u = (u_{0}, u_{1}, u_{2}) be a dimension vector such that gcd(u_{0}, u_{1} + u_{2}) = 1, then the variety

R_{ij}//G = Proj(⨁_{n≥0} ℂ[R]^{G}, χ_{θ}^{n}) = 𝔑^{θ}(h_{01}, h_{02}, u)

is a fine moduli space for families of χ_{θ}-stable representations. In other words, there exists the universal bundle U_{0}^{∗} ⊗ (U_{1} ⊗ V_{01} ⊕ U_{2} ⊗ V_{02}) on 𝔑^{θ}(h_{01}, h_{02}, u) such that the global tautological section of it can be interpreted as the bundle morphism

τ : U_{0} ⟶ (U_{1} ⊗ V_{01} ⊕ U_{2} ⊗ V_{01})

and the fiber of τ over each point t coincides with the representation t.

Further we fixed θ. Namely, by definition, put θ_{0} = −(u_{1} + u_{2}), θ_{1} = u_{0}, θ_{2} = u_{0}. Then the inequality of theorem 4.10 has the form

\frac{u′_{0}}{u′_{1} + u′_{2}} < \frac{u_{0}}{u_{1} + u_{2}} (≤ in semistable case).

Therefore, the fraction \frac{u_{0}}{u_{1} + u_{2}} we shall call the slope of a representation t and shall denote it by ν(t). In these terms, a representation of the quiver Q is χ_{θ}-stable if and only if any its subrepresentation has a lesser slope.

Later in this text the term ”stable representation of the quiver Q” means χ_{θ}-stable representation with respect to the character χ_{θ} with θ = (−(u_{1} + u_{2}), u_{0}, u_{0}).

Furthermore in this section we construct a moduli space of stable bundles Y ’s on ℙ^{1} ×ℙ^{1} such that the monad from proposition 3.2 for Y has the form

0 ⟶ Y ⟶ \begin{matrix}U_{1}(Y) ⊗ Q′_{2k+1} \\ ⊕ \\ U_{2}(Y) ⊗ Q′′_{2k+1} \end{matrix} ⟶^{α} U_{0}(Y) ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ 0.  (21)

In another words, the slope of Y satisfies the condition μ(Q_{2k}) < μ(Y) < μ(O_{ℙ^{1}×ℙ^{1}}), and Ext^{i}(Y, Q_{2k}) = 0 ∀i, where (Q_{2k}, Q′_{2k+1}, Q′′_{2k+1}, O_{ℙ^{1}×ℙ^{1}}) is the full exceptional three-block collection on ℙ^{1} × ℙ^{1} (see (10)). Namely, we prove the following theorem.

4.12 Theorem. Let u_{0}, u_{1}, u_{2} be natural numbers such that

1) gcd(u_{0}, u_{1} + u_{2}) = 1, 2)(u_{1} + u_{2})q_{2k+1} > u_{0}, 3) \frac{u_{0}}{u_{1} + u_{2}} < \frac{1}{2q_{2k−1}} ,

then the moduli space of stable sheaves Y ’s from the exact sequence (21) is isomorphic to the fine moduli space of the quiver representations 𝔑(2q_{2k}, 2q_{2k}; u_{o}, u_{1}, u_{2}), where 2q_{2k} = h^{0}(Q′_{2k+1}) = h^{0}(Q′′_{2k+1})

We start with

4.13 Lemma. Let

α : ℂ^{u_{0}} ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ \begin{matrix}ℂ^{u_{1}} ⊗ Q_{2k+1}^{′∗} \\ ⊕ \\ ℂ^{u_{2}} ⊗ Q_{2k+1}^{′′∗} \end{matrix}

be a morphism of bundles and

t_{α} : ℂ^{u_{0}} ⟶ \begin{matrix}ℂ^{u_{1}} ⊗ V_{01} \\ ⊕ \\ ℂ^{u_{2}} ⊗ V_{02} \end{matrix}

be the corresponding representation of the quiver (V_{01} = H^{0}(Q_{2k+1}^{′∗}), V_{02} = H^{0}(Q_{2k+1}^{′′∗})). Suppose that t_{α} is stable and ν(t_{α}) < \frac{1}{2q_{2k−1}} , then α is an embedding of bundles (here q_{2k−1} = rk (Q′_{2k−1})).

Proof. The representation t_{α} is determined by (u_{0} ×(u_{1} +u_{2}))-matrix F with elements f_{ij} ∈ V_{01} ∪ V_{02}. Consider the restriction α_{x} of the morphism α to the fiber at a point x.

The linear map α_{x} is determined by the matrix F_{x}, which can be obtained from the matrix F by taking the fibers of F_{ij} at the point x.

Assume that α_{x} has a nonzero kernel. Without loss of generality it can be assumed that α_{x}(e_{1}) = 0, where e_{1} is the first basis vector of the space ℂ^{u_{0}}. It means that f_{i1}(x) = 0 ∀i, i.e., f_{i1} ∈ H^{0}(Q_{2k+1}^{′∗} ⊗ J_{x}) ∪ H^{0}(Q_{2k+1}^{′′∗} ⊗ J_{x}), where J_{x} is the ideal sheaf of the point x. Therefore, the representation t_{α} has the subrepresentation t′ : ⟨e_{1}⟩ ⟶ U_{1} ⊗ V_{01} ⊕ U_{2} ⊗ V_{02}, where

dim U_{1} = dim U_{2} = h^{0}(Q_{2k+1}^{′∗} ⊗ J_{x}) = h^{0}(Q_{2k+1}^{′∗}) − rk Q_{2k+1}^{′∗}.

It is easily check that h^{0}(Q_{2k+1}^{′∗}) = 2 deg(Q_{2k+1}^{′∗}) = 2q_{2k}. Thus,

ν(t′) = \frac{1}{2(2q_{2k} − q_{2k+1})} =^{11} \frac{1}{2q_{2k−1}} > ν(t_{α}).

This contradiction concludes the proof.

Because of the following proposition is proved in the same way as the analogous propositions 4.3 and 4.5 from [2] and [12] respectively, we limit its formulation only.

4.14 Proposition. Let (O_{ℙ^{1}×ℙ^{1}}, Q_{2k+1}^{′∗}, Q_{2k+1}^{′′∗}, Q_{2k}^{′∗}) is the full exceptional three-block collection on ℙ^{1} × ℙ^{1} (10) and u_{0}, u_{1}, u_{2} be natural numbers such that

1) (u_{1} + u_{2})q_{2k+1} > u_{0}, 2)\frac{u_{0}}{u_{1} + u_{2}} < \frac{1}{2q_{2k−1}} ,

then a representation of the quiver

t_{α} : ℂ^{u_{0}} ⟶ \begin{matrix}ℂ^{u_{1}} ⊗ V_{01} \\ ⊕ \\ ℂ^{u_{2}} ⊗ V_{02} \end{matrix}

is stable iff the corresponding morphism of sheaves

α : ℂ^{u_{0}} ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ \begin{matrix}ℂ^{u_{1}} ⊗ Q_{2k+1}^{′∗} \\ ⊕ \\ ℂ^{u_{2}} ⊗ Q_{2k+1}^{′′∗} \end{matrix}

is a monomorphism and coker α is a stable bundle.

As a corollary of this proposition we obtain the following theorem

4.15 Theorem. Let u_{0}, u_{1}, u_{2} be natural numbers such that

1) gcd(u_{0}, u_{1} + u_{2}) = 1 2) (u_{1} + u_{2})q_{2k+1} > u_{0}, 3) \frac{u_{0}}{u_{1} + u_{2}} < \frac{1}{2q_{2k−1}} ,

then the moduli space of stable sheaves X’s from the exact sequence

0 ⟶ ℂ^{u_{0}} ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ \begin{matrix}ℂ^{u_{1}} ⊗ Q_{2k+1}^{′∗} \\ ⊕ \\ ℂ^{u_{2}} ⊗ Q_{2k+1}^{′′∗} \end{matrix} ⟶ X ⟶ 0,

is isomorphic to the smooth compact fine moduli space of the quiver representations

𝔑(2q_{2k}, 2q_{2k}; u_{0}, u_{1}, u_{2}),

where 2q_{2k} = h^{0}(Q′_{2k+1}) = h^{0}(Q′′_{2k+1}). Moreover, each of the sheaves X’s is locally free.

Since all the X’s from this theorem are locally free, we obtain theorem 4.12 as the dual case.

5 Moduli spaces on the projective plane.

In this section we describe a moduli space 𝔐_{n}(δ) = 𝔐(r_{n}, −r_{n−1}, δ) of stable sheaves X_{n} on ℙ^{2} such that their rank and the first Chern class coincide with rank and the first Chern class of the exceptional bundle E_{n} (see 6) and −χ(X_{n}) = h^{1}(X_{n}) = δ. Namely, let 𝔑_{n} = 𝔑(3r_{n−1}, δr_{n−1} + 1, δ) be the Kronecker moduli space (see remark 4.2 ) with the universal bundle U_{1}^{∗} ⊗ U_{2} ⊗ V_{02} and π_{𝔑} : 𝔊_{n} = Grass_{δr_{n−1}}(U_{1} ⊗ V_{01}) ⟶ 𝔑_{n} be the Grassman bundle of (δr_{n−1})−planes in the bundle U_{1} ⊗ V_{01} with the universal bundle S (here V_{01} = Hom(E_{n−1}, O_{ℙ^{2}}), V_{02} = Hom(E_{n−1}, E_{n})). Then the following theorem takes place.

5.1 Theorem. ^{2} 1. The smooth compact fine moduli space 𝔐_{n}(δ) of stable sheaves on ℙ^{2} is isomorphic to the sub-variety ℨ_{n} ⊂ 𝔊_{n} defined as the zero scheme of a section of the

bundle S^{∗} ⊗ π𝔑^{∗}U_{2} ⊗ V_{02}.

2. The universal bundle H_{δ} over 𝔐_{n}(δ), i.e., the bundle with H^{1}(X) as the fiber over a point X ∈ 𝔐_{n}(δ), is isomorphic to the restriction of π𝔑^{∗}U_{2} to the scheme ℨ_{n}.

Proof. Let X_{n} be the sheaf as above. It is clear, that μ(E_{n−1}) < μ(X_{n}) < μ(O_{ℙ^{2}}). Therefore, the sheaf X_{n} is the cohomology of the monad

0 ⟶ U_{0}(X_{n}) ⊗ E_{n−1} ⟶^{β} U_{1}(X_{n}) ⊗ E_{n} ⟶^{α} U_{2}(X_{n}) ⊗ O_{ℙ^{2}}⟶0  (22)

(see proposition 3.1 ), where

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

By Riemann-Roch formula for ℙ^{2} and relations (7), we have,

dim U_{0}(X) = δr_{n−1}, dim U_{1}(X) = δr_{n−2} + 1, dim U_{2}(X) = δ.

Construction of the moduli space base. 1. A map to the Kronecker moduli space. The monad splits into two exact triples

0 ⟶ Y ⟶ U_{1}(X_{n}) ⊗ E_{n} ⟶^{α} U_{2}(X_{n}) ⊗ O_{ℙ^{2}}⟶0,

0 ⟶ U_{0}(X_{n}) ⊗ E_{n−1} ⟶^{β} Y ⟶ X_{n} ⟶ 0.

Since

deg(E_{n−1})rk (X_{n}) − deg(X_{n})rk (E_{n−1}) = deg(E_{n−1})rk (E_{n}) − deg(E_{n})rk (E_{n−1}) = −1

(see (8)) and E_{n−1} is a stable bundle ( 1.3 ), it follows from lemma 2.3 that Y is stable iff X_{n} is. Therefore, we have some map from the set of all isomorphism classes of stable sheaves X_{n} ∈ 𝔐_{n}(δ) to the set of all isomorphism classes of stable sheaves Y. By theorem 4.4 , there exists a smooth compact fine moduli space of Y ’s, which isomorphic to the Kronecker moduli space 𝔑_{n} = 𝔑_{n}(3r_{n−1}; δr_{n−1} + 1, δ) (h^{0}(E_{n}, O_{ℙ^{2}}) = 3r_{n−1}). On 𝔑_{n} there exists a universal bundle U_{1}^{∗} ⊗ U_{2} ⊗ V_{12}, where V_{12} = Hom(E_{n}, O_{ℙ^{2}}). Its tautological section can be interpreted as the canonical epimorphism of bundles

τ : U_{1} ⊗ V_{12}^{∗} ⟶ U_{2} ⟶ 0

2This theorem was proved with A. Gorodentsev.

such that the restriction of τ to a point t_{α} ∈ 𝔑_{n} gives the stable Kronecker module t_{α} : U_{1} ⊗ V_{12}^{∗} ⟶ U_{2}, corresponding to the exact triple

0 ⟶ ker (α) ⟶ U_{1} ⊗ E_{m} ⟶^{α} O_{ℙ^{2}} ⟶ 0.

2. A lift the map to the Grassman bundle. By definition, put

V_{01} = Hom(E_{n−1}, E_{n}), V_{02} = Hom(E_{n−1}, O_{ℙ^{2}}).

Let γ ∈ V_{01}^{∗}⊗V_{12}^{∗}⊗V_{02} be the tensor, corresponding to the composition γ : V_{01}⊗V_{12} ⟶ V_{02}. This tensor can be interpreted as the map γ : V_{01} ⟶ V_{12}^{∗} ⊗ V_{02}. We denote by ψ the composition of the maps

U_{1} ⊗ V_{01} ⟶^{id⊗γ} U_{1} ⊗ V_{12}^{∗} ⊗ V_{02} ⟶^{τ⊗id} U_{2} ⊗ V_{02}.

Let 𝔊_{n} = Grass_{δr_{n−1}}(U_{1} ⊗ V_{01}) be the Grassman bundle of (δr_{n−1})-planes in the bundle U_{1} ⊗ V_{01}. Let π_{𝔑} : 𝔊_{n} ⟶ 𝔑_{n} be the projection of 𝔊_{n} to the base and Ũ_{i} = π𝔑^{∗}U_{i}. In this notation we can write the sequence of bundles on 𝔊_{n}.

S ⟶^{η} Ũ_{1} ⊗ V_{01} \underrightarrow{_{ψ}} Ũ_{2} ⊗ V_{02},  (23)

where S is the universal subbundle on 𝔊_{n}. By ℨ_{n} we denote the zero scheme of the composition ψ̃ ∘ η.

2. (1–1)-correspondence between 𝔐_{n}(δ) and the zero scheme ℨ_{n}. To show that there exists this one-to-one correspondence it is sufficient to check that each class of isomorphic monads (22) corresponds to a unique point of ℨ_{n} and vice versa.

Monad (22) is determined by two morphisms β and α. These morphisms correspond to the linear map

t_{β} : U_{0}(X_{n}) ⟶ U_{1}(X_{n}) ⊗ V_{01}, t_{α} : U_{1}(X_{n}) ⊗ V_{12}^{∗} ⟶ U_{2}(X_{n}).

Since β is a monomorphism, t_{β} is a monomorphism too. Therefore, monad (22) corresponding to a stable sheaf X_{n} defines a point z_{X_{n}} = (t_{β}(U_{0}(X_{n})), t_{α}) of 𝔊_{n} such that α ∘ β = 0 (we’ll denote it by (t_{β}, t_{α})).

Let us show that z_{X_{n}} ∈ ℨ_{n}. Because of η|_{z_{Xn}} = t_{β} and ψ̃|_{z_{Xn}} = ψ|_{t_{α}}, it is sufficient to check that the composition

U_{0}(X_{n}) ⟶^{t_{β}} U_{1}(X_{n}) ⊗ V_{01} ⟶^{id⊗γ} U_{1}(X_{n}) ⊗ V_{12}^{∗} ⊗ V_{02} ⟶^{t_{α}⊗id} U_{2}(X_{n}) ⊗ V_{02}

is trivial. Let us fix bases {a_{i}} in the space U_{0}(X_{n}), {b_{j}} in U_{1}(X_{n}) and {c_{s}} in U_{2}(X_{n}). In these bases the morphism β of the monad is setting by the matrix F = (f_{ji}), where f_{ji} ∈ V_{01}. The morphism α is setting by the matrix G = (g_{sj}), where g_{sj} ∈ V_{12}. Therefore, the composition α ∘ β has the form G · F. Thus, the condition α ∘ β = 0 is equivalent to the equality

∑_{j} g_{sj}f_{ji} = 0.  (24)

At the same time, the linear maps t_{β} and t_{α} in the fixed bases are setting by the same matrices. Hence,

t_{β}(∑_{i} x_{i}a_{i}) = ∑_{i} x_{i} ∑_{j} b_{j} ⊗ f_{ji} = ∑_{j} b_{j} ⊗ ∑_{i} x_{i}f_{ji}.

If we fix a basis {h_{k}} in the space V_{02}, we can write the element (id ⊗ γ) ∘ t_{β}(∑_{i} x_{i}a_{i}) as

∑_{j} b_{j} ⊗ ∑_{i} x_{i}γ(f_{ji}) = ∑_{j} b_{j} ⊗ ∑_{i} x_{i} ∑_{k} λ_{ji}^{k} ⊗ h_{k},  (25)

where λ_{ji}^{k} ∈ V_{12}^{∗} and ∑_{k} λ_{ji}^{k} ⊗ h_{k} = γ(f_{ji}) is the morphism of V_{12} to V_{02} such that for all

g ∈ V_{12} we have

γ(f_{ji}) : g ↦ ∑_{k}⟨λ_{ji}^{k}, g⟩ ⊗ h_{k} = g ∘ f_{ji},  (26)

where ⟨λ_{ji}^{k}, g⟩ is the natural pairing V_{12}^{∗} ⊗ V_{12} ⟶ ℂ.

To apply the map t_{α} ⊗ id to (25) let us recall that

t_{α} ∑_{j} b_{j} ⊗ λ_{j} = ∑_{s} c_{s} ∑_{j}⟨g_{sj}, λ_{j}⟩.

Now we have,

t_{α} ⊗ id ∑_{k}(∑_{j} b_{j} ⊗ ∑_{i} x_{i}λ_{ji}^{k}) ⊗ h_{k} = ∑_{k}(∑_{s} c_{s} ∑_{j}⟨g_{sj}, ∑_{i} x_{i}λ_{ji}^{k}⟩ ⊗ h_{k}) =

= ∑_{s} c_{s} ∑_{i} x_{i} ∑_{j}(∑_{k}⟨g_{sj}, λ_{ji}^{k}⟩ ⊗ h_{k}) =^{(26)} ∑_{s} c_{s} ∑_{i} x_{i} ∑_{j} g_{sj} ∘ f_{ji}.

Combining this and (24), we obtain that the point z_{X_{n}} = (t_{β}, t_{α}) corresponding to the stable sheaf X_{n} belongs to ℨ_{n}.

On the other hand, any point (U_{0}, t_{α}) ∈ 𝔊_{n} gives a monomorphism t_{β} and an epimorphism t_{α}

t_{β} : U_{0} ⟶ U_{1} ⊗ V_{01}, t_{α} : U_{1} ⊗ V_{12}^{∗} ⟶ U_{2},

i.e., two morphisms of bundles

β : U_{0} ⊗ E_{n−1} ⟶ U_{1} ⊗ E_{n}, α : U_{1} ⊗ E_{n} ⟶ U_{2} ⊗ O_{ℙ^{2}}.

Moreover, the previous arguments tell us that if the point (U_{0}, t_{α}) belongs to ℨ_{n}, then the composition α ∘ β is trivial. Thus we proved the following statement: point of the closed subscheme ℨ_{n} ⊂ 𝔊_{n} one-to-one correspond to classes of isomorphic complexes (22) such that the morphism α is an epimorphism (α corresponds to a stable Kronecker module).

Let us check that the morphism β is a monomorphism. We know that the composition α ∘ β = 0. Therefore, we can assume that β : U_{0} ⊗ E_{n−1} ⟶ ker α. It is easily to check that the sheaves E_{n−1} and ker α satisfy the assumptions of lemma 2.5 , consequently, ker β = U ⊗ E_{n−1} for some subspace U ⊂ U_{0}. This contradicts the fact that t_{β} is a monomorphism. Thus we proved that point of the closed subscheme ℨ_{n} ⊂ 𝔊_{n} one-to-one correspond to points of 𝔐_{n}(δ). In another words we constructed a base of the moduli space 𝔐_{n}(δ).

Construction of the universal sheaf. Denote by Ū_{i} and S̄ the restriction of the bundles U_{i} and S to ℨ_{n} respectively. In this notation the restriction of sequence (23) to ℨ_{n} is the monad on ℨ_{n}

0 ⟶ S̄ ⟶ Ū_{1} ⊗ V_{01} ⟶ Ū_{2} ⊗ V_{02} ⟶ 0.  (27)

Denote by p_{1} and p_{2} the projections of the direct product ℙ^{2} ×ℨ_{n} to the first and the second factors respectively and by definition put,

U′_{i} = p_{2}^{∗}Ū_{i} (i = 1, 2), U′_{0} = p_{2}^{∗}S̄.

It can be proved that sequence (27) induces the monad on the direct product ℙ^{2} × ℨ_{n}

0 ⟶ U′_{0} ⊗ p_{1}^{∗}E_{n−1} ⟶ U′_{1} ⊗ p_{1}^{∗}E_{n} ⟶ U′_{2} ⊗ p_{1}^{∗}O_{ℙ^{2}} ⟶ 0  (28)

with a cohomology X_{n}. Note that the restriction of this monad to ℙ^{2} × z_{X_{n}} is the monad with the cohomology X_{n}.

Now we’ll prove that the family X_{n} over the base ℨ_{n} is universal, i.e., we’ll show that for any family F of stable sheaves (with the topological invariants from the theorem) over a base 𝔅 there exists a map φ : 𝔅 ⟶ ℨ_{n} such that the sheaf F on ℙ^{2} × 𝔅 is isomorphic to (id × φ)^{∗}X_{n}.

On the direct product ℙ^{2} × 𝔅 there exists a Beilinson-Gorodentsev relative spectral sequence E^{p,q} converging to the sheaf F on the diagonal ([10]) with

E_{1}^{p,q} = π𝔅^{∗}Ext_{π_{𝔅}}^{q}(π^{∗}(^{∨}G_{2+p}), F) ⊗ π^{∗}G_{2+p}, q = 0, 1, 2; p = −2, −1, 0,

where π_{𝔅} and π are the projections of ℙ^{2} × 𝔅 to the factors and

(G_{0}, G_{1}, G_{2}) = (E_{n−1}, E_{n}, O_{ℙ^{2}}), (^{∨}G_{2},^{∨}G_{1},^{∨}G_{0}) = (O_{ℙ^{2}}, E_{n−2}(3), E_{n−1}(3)).

It is not hard to prove, that this spectral sequence is degenerated into a monad

0 ⟶ π𝔅^{∗}F_{0} ⊗ π^{∗}E_{n−1} ⟶^{β} π𝔅^{∗}F_{1} ⊗ π^{∗}E_{n} ⟶^{α} π𝔅^{∗}F_{2} ⊗ π^{∗}O_{ℙ^{2}}⟶0,  (29)

with the cohomology F, where

F_{i} = Ext_{π_{𝔅}}^{1}(π^{∗}(^{∨}G_{2+p}), F).

The restriction of this monad to ℙ^{2}×[F] is the standard monad (22) on ℙ^{2} constructed by the sheaf F. We take each point [F] ∈ 𝔅 to the Kronecker module t_{α}(F) corresponding to the restriction of α to ℙ^{2}×[F]. Denote by f : 𝔅 ⟶ 𝔑_{n} this map. Note that f^{∗}U_{i} = F_{i}, i = 1, 2, when U_{i}’s are the universal bundles on 𝔑.

If we tensoring monad (29) by π^{∗}E_{n−1}^{∗} and apply the functor R^{0}π_{𝔅}_{∗}, we obtain an inclusion of bundles

F_{0} ↪ F_{1} ⊗ V_{01} = f^{∗}(U_{1} ⊗ V_{01}).

Therefore, the map f can be extended to a map f′ : 𝔅 ⟶ 𝔊_{n} (see [4]) such that F_{0} = f^{′∗}S.

It is clear, that f′(𝔅) ⊂ ℨ_{n} and the inverse image of monad (28) under the map id × f′ coincides with (29). Thus (id × f′)^{∗}X_{n} = F. This concludes the proof.

6 Moduli spaces on the quadric.

In this section we describe a moduli spaces

𝔐_{2k}(δ) = 𝔐(q_{2k}, −(q_{2k−1}, q_{2k−1}), δ),

𝔐′_{2k+1}(δ) = 𝔐(q_{2k+1}, −\frac{1}{2}(q_{2k}+1, q_{2k}−1), δ), 𝔐′′_{2k+1}(δ) = 𝔐(q_{2k+1}, −\frac{1}{2}(q_{2k}−1, q_{2k}+1), δ)

of stable sheaves X’s on the quadric ℙ^{1} × ℙ^{1} such that −χ(X) = δ, and their rank and the first Chern class coincide with rank and the first Chern class of the exceptional bundles Q_{2k}, Q′_{2k+1}, Q′′_{2k+1} respectively (see (9)). Note that the natural involution i on the quadric ℙ^{1} × ℙ^{1} induces the isomorphism of the moduli spaces i^{∗} : 𝔐′_{2k+1}(δ) ⟶ 𝔐′′_{2k+1}(δ). Therefore, it is sufficient to construct one of them only. Because of the construction of these moduli spaces is the same as in the section 5, we omit details.

Let us start with the moduli space 𝔐_{2k}(δ). Each sheaf X ∈ 𝔐_{2k}(δ) satisfies the inequality

μ(Q′′_{2k−1}) = μ(Q′_{2k−1}) < μ(X) < μ(O_{ℙ^{1}×ℙ^{1}}).

Therefore, it is the cohomology of the monad

0 ⟶ U_{0}(X) ⊗ ( \begin{matrix}Q′_{2k−1} \\ ⊕ \\ Q′′_{2k−1} \end{matrix} ) ⟶^{β} U_{1}(X) ⊗ Q_{2k} ⟶^{α} U_{2}(X) ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ 0  (30)

(see remark 3.4 ), where

U_{0}(X) = Ext^{1}(X, Q′_{2k−1})^{∗} = Ext^{1}(X, Q′′_{2k−1})^{∗},
U_{1}(X) = Ext^{1}(X, L_{Q_{2k−1}}Q_{2k})^{∗} = Ext^{1}(X, Q_{2k−2})^{∗}, U_{2}(X) = H^{1}(X).

By Riemann-Roch formula for ℙ^{1} × ℙ^{1} and relations (11), (12), (13), we have

dim U_{0}(X) = δq_{2k−1}, dim U_{1}(X) = δq_{2k−2} + 1, dim U_{2}(X) = δ.

Monad (30) splits into two exact sequences

0 ⟶ Y ⟶ U_{1}(X) ⊗ Q_{2k} ⟶^{α} U_{2}(X) ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ 0,

0 ⟶ U_{0}(X) ⊗ ( \begin{matrix}Q′_{2k−1} \\ ⊕ \\ Q′′_{2k−1} \end{matrix} ) ⟶^{β} Y ⟶ X ⟶ 0.  (31)

It follows from the definition of the sheaf X and (12) that exact triple (31) satisfies the assumptions of lemma 2.3 and remark 2.4 . Hence the sheaf Y is stable iff X is. Whence we obtain a map of 𝔐_{2k}(δ) to the Kronecker moduli space 𝔑_{2k} = 𝔑(4q_{2k−1}, δq_{2k−2} + 1, δ) with the universal Kronecker module τ : U_{1} ⊗ V_{12}^{∗} ⟶ U_{2}, where 4q_{2k−1} = h^{0}(Q_{2k}, O_{ℙ^{1}×ℙ^{1}}).

By definition, put

V_{01} = Hom(Q′_{2k−1} ⊕ Q′′_{2k−1}, Q_{2k}), V_{12} = Hom(Q_{2k}, O_{ℙ^{1}×ℙ^{1}}),
V_{02} = Hom(Q′_{2k−1} ⊕ Q′′_{2k−1}, O_{ℙ^{1}×ℙ^{1}}).

Let γ : V_{01} ⟶ V_{12}^{∗}⊗V_{02} be the map corresponding to the composition map V_{01}⊗V_{12} ⟶ V_{02}. Denote by ψ the composition

U_{1} ⊗ V_{01} ⟶^{id⊗γ} U_{1} ⊗ V_{12}^{∗} ⊗ V_{02} ⟶^{τ⊗id} U_{2} ⊗ V_{02}.

Further, as in the previous section we lift the map 𝔐_{2k}(δ) ⟶ 𝔑_{2k} to a map 𝔐_{2k}(δ) ⟶ 𝔊_{2k} = Grass_{δq_{2k−1}}(U_{1} ⊗ V_{01}) and prove that points of 𝔐_{2k}(δ) (1–1)-correspond to points of the zero scheme ℨ_{2k} of the composition

S ⟶ π𝔑^{∗}U_{1} ⊗ V_{01} ⟶^{ψ̃} π𝔑^{∗}U_{2} ⊗ V_{02},

where S is the universal subbundle over 𝔊_{2k} ⟶^{π_{𝔑}} 𝔑_{2k}. Repeating the reasons of the previous section, the following theorem can be proved.

6.1 Theorem. 1. The smooth compact fine moduli space 𝔐_{2k}(δ) is isomorphic to the zero scheme ℨ_{2k} of a section of the bundle S^{∗} ⊗ π𝔑^{∗}U_{2} ⊗ V_{02} on the Grassman bundle

Grass_{δq_{2k−1}}(U_{1} ⊗ V_{01}) ⟶^{π_{𝔑}} 𝔑(4q_{2k−1}, δq_{2k−2} + 1, δ)

over the Kronecker moduli space.

2. The universal bundle H over 𝔐_{2k}(δ), i.e., the bundle with H^{1}(X) as the fiber over a point X ∈ 𝔐_{2k}(δ), is isomorphic to the restriction of π𝔑^{∗}U_{2} to the zero scheme ℨ_{2k}.

Now, using the moduli space of quiver representation (see theorem 4.11 ), we construct the moduli space 𝔐′_{2k+1}(δ). It is clear that each sheaf X ∈ 𝔐′_{2k+1}(δ) satisfies the assumptions of proposition 3.2 with the full exceptional collection (Q_{2k}, Q′_{2k+1}, Q′′_{2k+1}, O_{ℙ^{1}×ℙ^{1}}). Hence the sheaf X is the cohomology of the monad

0 ⟶ U_{0}(X) ⊗ Q_{2k} ⟶^{β} \begin{matrix}U_{1}(X) ⊗ Q′_{2k+1} \\ ⊕ \\ U_{2}(X) ⊗ Q′′_{2k+1} \end{matrix} ⟶^{α} U_{3}(X) ⊗ O_{ℙ^{1}×ℙ^{1}} ⟶ 0,

where dimensions of the vector spaces U_{i}(X) can be calculated by Riemann-Roch formula for ℙ^{1} × ℙ^{1}:

dim U_{0}(X) = δq_{2k}, dim U_{1}(X) = δq_{2k−1} + 1, dim U_{2}(X) = δq_{2k−1}, dim U_{3}(X) = δ.

It can easily be checked that the sheaves Q_{2k} and X satisfy the assumptions of lemma 2.3 . Therefore, taking each stable sheaf X ∈ 𝔐′_{2k+1}(δ) to the quiver representation

\begin{matrix}U_{1}(X) ⊗ Hom(Q′_{2k+1}, O_{ℙ^{1}×ℙ^{1}})^{∗} \\ ⊕ \\ U_{2}(X) ⊗ Hom(Q′′_{2k+1}, O_{ℙ^{1}×ℙ^{1}})^{∗} \end{matrix} ⟶^{t_{α}} U_{3}(X),

we obtain a map of 𝔐′_{2k+1}(δ) to the fine smooth compact quiver representation moduli space

𝔑′_{2k+1} = 𝔑(2q_{2k}, 2q_{2k}; δ, δq_{2k−1}, δq_{2k−1} + 1)

(see 4.11 ) with the universal representation

τ : \begin{matrix}U_{1}(X) ⊗ V_{13}^{∗} \\ ⊕ \\ U_{2}(X) ⊗ V_{23}^{∗} \end{matrix} ⟶U_{3}(X),

where

V_{13} = Hom(Q′_{2k+1}, O_{ℙ^{1}×ℙ^{1}}), V_{23} = Hom(Q′′_{2k+1}, O_{ℙ^{1}×ℙ^{1}}).

By definition, put

V_{01} = Hom(Q_{2k}, Q′_{2k+1}), V_{02} = Hom(Q_{2k}, Q′′_{2k+1}), V_{03} = Hom(Q_{2k}, O_{ℙ^{1}×ℙ^{1}})

and denote by γ_{i} the linear map γ_{i} : V_{0i} ⊗ V_{i3} ⟶ V_{03} of composition. It can be interpreted as a map

γ_{i} : V_{0i} ⟶ V_{i3}^{∗} ⊗ V_{03}.

Denote by ψ the composition of the maps on 𝔑′_{2k+1}:

\begin{matrix}U_{1}(X) ⊗ V_{01} \\ ⊕ \\ U_{2}(X) ⊗ V_{02} \end{matrix} ⟶^{γ} \begin{matrix}U_{1}(X) ⊗ V_{13}^{∗} ⊗ V_{03} \\ ⊕ \\ U_{2}(X) ⊗ V_{23}^{∗} ⊗ V_{03} \end{matrix} ⟶^{τ⊗id} U_{3} ⊗ V_{03},

where γ = (id ⊗ γ_{1}) ⊕ (id ⊗ γ_{2}).

Let 𝔊′_{2k+1} = Grass_{δq_{2k}}(U_{1} ⊗ V_{01} ⊕ U_{2} ⊗ V_{02}) be the Grassman bundle with the projection π_{𝔑} to the base and S as the universal subbundle. Denote by ℨ′_{2k+1} the zero scheme of the composition

S ⟶ π𝔑^{∗} ( \begin{matrix}U_{1}(X) ⊗ V_{01} \\ ⊕ \\ U_{2}(X) ⊗ V_{02} \end{matrix} ) ⟶^{ψ̃} π𝔑^{∗}U_{3} ⊗ V_{03}.

It can be shown in the same way as before that the following theorem takes place.

6.2 Theorem. 1.The smooth compact fine moduli space 𝔐′_{2k+1}(δ) is isomorphic to the zero scheme ℨ′_{2k+1} of a section of the bundle S^{∗} ⊗ π𝔑^{∗}U_{3} ⊗ V_{03} on the Grassman bundle

Grass_{δq_{2k}}(U_{1} ⊗ V_{01} ⊕ U_{2} ⊗ V_{02}) ⟶^{π_{𝔑}} 𝔑(2q_{2k}, 2q_{2k}; δ, δq_{2k−1}, δq_{2k−1} + 1)

over the quiver representations moduli space.

2. The universal bundle H_{δ} over 𝔐′_{2k+1}(δ), i.e., the bundle with H^{1}(X) as the fiber over a point X ∈ 𝔐′_{2k+1}(δ), is isomorphic to the restriction of π𝔑^{∗}U_{3} to the zero scheme ℨ′_{2k+1}.

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