Quiver^{1} representations and moduli spaces of sheaves on P^{2}

S. Kuleshov.

1. Introduction.

Representations of the quivers with two vertices are used for geometrical description of moduli spaces of stable sheaves on the projective plane. These representations are called Kronecker moduli ([1],[2]). Unfortunately, application of this fine construction is bounded by the condition: stable sheaves belong to a subcategory generated by an exceptional pair.

Papers [3],[4],[5] are some generalization of the Kronecker moduli method. In these papers moduli spaces of sheaves on ℙ^{2} are realized as a closed subset in relative Grassman’s variety over the Kronecker moduli space. But these works do not describe the general situation.

On the other hand, it was proved in ([6]), that the derived category of coherent sheaves on ℙ^{2} is equivalent to the derived category of representations of 3 vertices quiver with relations. This gives a hope that there exists a universal construction of geometrical description of stable sheaves moduli on ℙ^{2}. This preprint deals with this general construction.

The idea is the following. Let (E_{0}, E_{1}, E_{2}) be a full exceptional collection on ℙ^{2} (for example, (O(−2), O(−1), O)) and X be a sheaf such that

μ(E_{0}) < μ(X) < μ(E_{2}),  (1.1)

then X is the cohomology of the monad

M(X) :  0 ⟶ U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2} ⟶ 0,  (1.2)

where U_{0}, U_{1}, U_{2} are finite dimension vector spaces such that dimensions of U_{i}’s depend on the topological type of X and the bundles E_{j}’s only (see proposition 3.1).

Moreover, the correspondence between the the stable sheaves X’s and monads M(X) is functorial,. i.e., any morphism of stable sheaves φ : X ⟶ X′, where the slopes of X and X′ satisfy condition (1.1), induces linear maps φ_{i} : U_{i} ⟶ U′_{i} such that the following diagram is commutative^{2}.

M(X) :  0 ⟶ U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2} ⟶ 0
↓ φ_{0}⊗id ↓ φ_{1}⊗id ↓ φ_{2}⊗id
M(X′) :  0 ⟶ U′_{0} ⊗ E_{0} ⟶^{α′} U′_{1} ⊗ E_{1} ⟶^{β′} U′_{2} ⊗ E_{2} ⟶ 0

And vise verse, any morphism of monads induces a morphism of their cohomologies.

The morphism α from (1.2) is determined by a tensor a ∈ U_{0}^{∗} ⊗ U_{1} ⊗ H_{01}, and β is determined by b ∈ U_{1}^{∗} ⊗ U_{2} ⊗ H_{12}, where H_{ij} is the vector space Hom(E_{i}, E_{j}). In such a way, an element (a, b) ∈ R = U_{0}^{∗} ⊗ U_{1} ⊗ H_{01} ⨁ U_{1}^{∗} ⊗ U_{2} ⊗ H_{12} corresponds to monad (1.2). We see that elements (a, b) and (a′b′) correspond to isomorphic monads iff they belong to one orbit the group G = GL(U_{0}) × GL(U_{1}) × GL(U_{2}) action.

1This work was started at the Independent University of Moscow and was done during the author‘s stay at the Max Plank Institute für Mathematik in 1998.

2The collection of morphisms φ_{i}’s, inducing the commutative diagram is called a morphism of monads.

Further, a vector (a, b) ∈ R determines a pair of linear maps

U_{0} ⟶^{a} U_{1} ⊗ H_{01}, U_{1} ⟶^{b} U_{2} ⊗ H_{12}.

These maps are a representation of a quiver Q with 3 vertices and without relations. The quiver Q has h_{01} arrows from the zero vertex to the first and h_{12} arrows from the first one to the second.

The moduli spaces of quiver representations was studded by A. King in [7]. By this work, to construct quotient R/G we need to extend the group G action to the trivial bundle R × ℂ with help of a character χ_{θ} of the group. After this operation the moduli space of semistable representations (w.r.t χ_{θ} ) 𝔑(h_{01}, h_{12}; u_{0}, u_{1}, u_{2}) is defined as the projective spectrum of relative invariants ring.

In the preprint we show that a stable quiver representation corresponds to a stable sheaf X on ℙ^{2}. Moreover, the description of moduli space of stable sheaves has the form.

1.1. Theorem. Let (E_{0}, E_{1}, E_{2}) be a full exceptional collection on ℙ^{2} such that −2 ≤ μ(E_{0}) < μ(E_{2}) ≤ 0. Suppose that integers r, c_{1}, c_{2} satisfy the conditions:

μ = \frac{c_{1}}{r} ∈ (μ(E_{0}), μ(E_{2})), (r, c_{1}) = 1.

Then the moduli space 𝔐^{s}(r, c_{1}, c_{2}) of stable sheaves X on ℙ^{2} with rk (X) = r, c_{1}(X) = c_{1}, and c_{2}(X) = c_{2} is isomorphic to closed subveriety ℨ of the moduli space

𝔑 = 𝔑^{s}(h_{01}, h_{12}; u_{0}, u_{1}, u_{2})

of stable representations of the quiver Q with 3 vertices. The subveriety ℨ is defined as the zero set of the section of the universal sheaf U_{0}^{∗} ⊗ U_{2} ⊗ H_{02} on 𝔑.

The author is grateful to A. Gorodentsev for useful discussions. I am thankful to Max-Plank-Institut für Mathematik for hospitality.

2. Notations and preliminaries.

Here we use the following notations. O(n) is a line bundle on ℙ^{2}, in particular, O is the structure sheaf of projective plane; F (n) is the tensor product F ⊗ O(n). For an exceptional collection (E_{0}, E_{1}, E_{2}) on ℙ^{2} r_{i} is rank of bundle E_{i}; d_{i} is the first Chern class of E_{i}; μ_{i} = \frac{d_{i}}{r_{i}} is the slope of E_{i}; H_{ij} is the vector space Hom(E_{i}, E_{j}); h_{ij} = dim H_{ij}. We identify a bundle with the locally free sheaf of its local sections. The stability of sheaves

we understand in the sense of Mumford – Takemoto.

Exceptional bundles.^{3} Recall that a sheaf E on ℙ^{2} is called exceptional if it is simple and superrigid, i.e.,

3The details about the exceptional bundles on ℙ^{2} can be found in [8] and [9].

Hom(E, E) = ℂ and Ext^{q}(E, E) = 0 for q > 0.

It is known that the exceptional sheaves on ℙ^{2} are locally free and stable.

An ordered collection (E_{0}, E_{1}, E_{2}) of exceptional bundles on ℙ^{2} is called exceptional collection or exceptional triple if Ext^{q}(E_{i}, E_{j}) = 0 ∀q whenever i > j. It is known that such the collections on ℙ^{2} are full, in other words, the minimal triangulated subcategory in the derived category of coherent sheaves on ℙ^{2} (D^{b}ℙ^{2}) containing the bundles of the collection coincide with D^{b}ℙ^{2}.

The braid group acts on the set of all exceptional triples on ℙ^{2} by mutations of neighboring exceptional pairs. If (E, F) is an exceptional pair, then its the left mutation is defined as (L_{E}F, E), where L_{E}F is the kernel of the canonical epimorphism

0 ⟶ L_{E}F ⟶ E ⊗ Hom(E, F) ⟶^{lcan} F ⟶ 0.  (2.1)

The canonical morphism lcan corresponds to the tensor can ∈ Hom(E, F)^{∗} ⊗ Hom(E, F) such that the map can ∈ End(Hom(E, F)) is identical. The right mutation of the pair (E, F) is (F, R_{F} E), where R_{F} E the cokernel of the canonical morphism

0 ⟶ E ⟶^{rcan} Hom(E, F)^{∗} ⊗ F ⟶ R_{F} E ⟶ 0.  (2.2)

The morphism rcan corresponds the identical map can ∈ End(Hom(E, F)) too.

If (E_{0}, E_{1}, E_{2}) is a full exceptional collection on ℙ^{2}, then

R_{E_{2}}R_{E_{1}}E_{0} = E_{0}(3) and L_{E_{0}}L_{E_{1}}E_{2} = E_{2}(−3).

Besides ranks r_{i} of the bundles from the exceptional collection satisfy Markov’s equation

r_{0}^{2} + r_{1}^{2} + r_{2}^{2} = 3r_{0}r_{1}r_{2}.  (2.3)

Dimensions of the spaces H_{ij} = Hom(E_{i}, E_{j}) can be calculated by the following formulae

h_{01} = 3r_{2},  h_{12} = 3r_{0},  h_{02} = 3r̄_{1} = 3(3r_{0}r_{2} − r_{1}).  (2.4)

Finally, the most important fact from the theory of exceptional bundles is the generalization of Beilinson spectral sequence to the case of arbitrary exceptional collection. For the projective plane this generalization has the form:

2.1. Theorem. (Gorodentcev) Let (E_{0}, E_{1}, E_{2}) be a full exceptional collection on ℙ^{2}. Let us define its the left dual one as (F_{0}, F_{1}, F_{2}), where

F_{0} = E_{2},  F_{1} = R_{E_{2}}E_{1},  F_{2} = E_{0}(3).

Then for any coherent sheaf X on ℙ^{2} there exists a spectral sequence, converging to X on the principal diagonal with the E_{1}-part in the form

E_{1}^{pq} = Ext^{q}(F_{−p}, X) ⊗ E_{2−p}.

Quiver representations. Hear we formulate some results of paper [7] for the 3 vertices quiver Q with h_{01} arrows from the zero vertex to the first one and h_{12} arrows from the first vertex to the second one.

Definition. A pair of linear maps (a, b)

a : U_{0} ⟶ U_{1} ⊗ H_{01} and b : U_{1} ⟶ U_{2} ⊗ H_{12}

is called the representation of the quiver Q with the dimension vector (u_{0}, u_{1}, u_{2}), where

dim H_{ij} = h_{ij}, u_{i} = dim U_{i}.

We see that the representation is an element of the vector space

R = U_{0}^{∗} ⊗ U_{1} ⊗ H_{01} ⨁ U_{1}^{∗} ⊗ U_{2} ⊗ H_{12}.

The representation (a, b) is isomorphic to another one (a′, b′) if there exists isomorphisms g_{i}’s of the vector spaces U_{i}’s such that the following diagrams are commutative

\begin{matrix}U_{0} & ⟶^{a} & U_{1} ⊗ H_{01} \\ g_{0} ↓ & & ↓ g_{1}⊗id \\ U_{0} & ⟶^{a′} & U_{1} ⊗ H_{01} \end{matrix} \begin{matrix}U_{1} & ⟶^{b} & U_{2} ⊗ H_{12} \\ g_{1} ↓ & & ↓ g_{2}⊗id \\ U_{1} & ⟶^{b′} & U_{2} ⊗ H_{12} \end{matrix}.

Thus the group

G = GL(U_{0}) × GL(U_{1}) × GL(U_{2}).

acts on the vector space R.

To construct moduli space of the quiver representations let us choose a character χ of G (χ : G ⟶ ℂ^{∗}) such that the restriction of χ to the kernel of the action Δ is trivial. Let us extend the action of G to the trivial bundle R × ℂ by the following rule g · (v, z) = (g · v, χ^{−1}(g)z). Recall that a function f ∈ ℂ[R] is called a relative invariant of weight χ if f(g · v) = χ(g)f(v) ∀g ∈ G and v ∈ R. The set of all such invariants is denoted by ℂ[R]^{G,χ}.

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

ii) A vector v ∈ R is χ-stable if there is a relative invariant f ∈ ℂ[R]^{G,χ^{n}} with n ≥ 1 such that f(v) ≠ 0 and, further, dim G · v = dim G/Δ and the G-action on {v ∈ R|fv ≠ 0} is closed.

Hence the moduli space of the quiver representations can be described as

Proj(⨁_{n≥0} ℂ[R]^{G,χ^{n}}).

It is easily shown, that in our case the kernel of G-action Δ consists of homotheties of the spaces U_{i} with a common coefficient. A character of the group G has the form

χ(g_{0}, g_{1}, g_{2}) = (det g_{0})^{θ_{0}}(det g_{1})^{θ_{1}}(det g_{2})^{θ_{2}},

where θ_{i} ∈ ℤ. Therefore a choice of a character χ such that χ|_{Δ} = 1 is equivalent to a choice of on additive function θ on ℤ^{3} satisfying the condition

θ(u_{0}, u_{1}, u_{2}) = ∑_{i=0}^{2} θ_{i}u_{i} = 0,  (2.5)

where u_{i} = dim U_{i}. Such the functions are called polarizations. Furthermore, we assume that the notion χ_{θ}-stable is the same as θ-stable.

The following numerical criterion was proved in [7].

2.2. Theorem. A representation (a, b) of the quiver is θ-semistable, if for any its subrepresentation a′ : U′_{0} ⟶ U′_{1} ⊗ H_{01}, b′ : U′_{1} ⟶ U′_{2} ⊗ H_{12} the inequality ∑ θ_{i}u′_{i} ≥ 0 holds (the strong inequality for θ-stable representation).

Below we will fix the polarization and thus will speak of stable representations w.r.t. this polarization, without specifying it each time. We denote by 𝔑(h_{01}, h_{12}; u_{0}, u_{1}, u_{2}) the moduli space of the quiver representations with the dimension vector (u_{0}, u_{1}, u_{2}).

It was noted in the paper [7] that the quasi-projective subveriety 𝔑^{s}(h_{01}, h_{12}; u_{0}, u_{1}, u_{2}) consisting of only stable representation is fine whenever the dimension vector (u_{0}, u_{1}, u_{2}) is primitive, i.e. the integers u_{0}, u_{1}, u_{2} are coprime.

Moreover, if the system

\{ \begin{matrix}∑ θ_{i}v_{i} = 0 \\ 0 < v_{i} ≤ u_{i} \end{matrix}

has no solutions except (u_{0}, u_{1}, u_{2}), then 𝔑^{s}(h_{01}, h_{12}; u_{0}, u_{1}, u_{2}) = 𝔑(h_{01}, h_{12}; u_{0}, u_{1}, u_{2}).

3. A map from sheaves moduli space to representations moduli space.

3.1. Proposition. Let (E_{0}, E_{1}, E_{2}) be a full exceptional collection on ℙ^{2} and X be a stable sheaf such that μ_{0} = μ(E_{0}) < μ(X) < μ(E_{2}) = μ_{2}. Then X is the cohomology of the monad

M(X) :  0 ⟶ U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2} ⟶ 0,  (3.1)

where U_{0} = Ext^{1}(E_{0}(3), X), U_{1} = Ext^{1}(R_{E_{2}}E_{1}, X), U_{2} = Ext^{1}(E_{2}, X).

Proof. By theorem 2.1, there exists a spectral sequence converging to X on the principal diagonal with E_{1}-part in the form

E_{1}^{−2,q} = Ext^{q}(E_{0}(3), X) ⊗ E_{0}, E_{1}^{−1,q} = Ext^{q}(R_{E_{2}}E_{1}, X) ⊗ E_{1}, E_{1}^{0,q} = Ext^{q}(E_{2}, X) ⊗ E_{2}.

It follows from the stability of exceptional bundles and the condition

μ(F) − 3 < μ(X) < μ(F) for F ∈ {E_{2}, R_{E_{2}}E_{1}, E_{0}(3)}

that Ext^{q}(F, X) = 0 if q ≠ 1. Hence the diagram of the E_{1}-part has the form

0  0  0

Ext^{1}(E_{0}(3), X) ⊗ E_{0} Ext^{1}(R_{E_{2}}E_{1}, X) ⊗ E_{1} Ext^{1}(E_{2}, X) ⊗ E_{2}

0  0  E_{1}^{0,0} = 0

We see that E_{2}^{p,q} = E_{∞}^{p,q}. This completes the proof.

Taking into account the functorial properties of the spectral sequence, we obtain

3.2. Corollary. A morphism of stable sheaves induces a morphism of monads. Moreover, two stable sheaves are isomorphic iff corresponding monads are isomorphic.

Now we note, that any sequence of morphisms α and β from (3.1) determines a representation (a, b) of the quiver Q with dimension vector (u_{0}, u_{1}, u_{2}):

U_{0} ⟶^{a} U_{1} ⊗ H_{01} U_{1} ⟶^{b} U_{2} ⊗ H_{12}.

Whence, for a stable sheaf X satisfying the assumptions of proposition 3.1 we can construct a representation (a_{X}, b_{X}) of the quiver Q.

Consider the following composition

γ : U_{0} ⟶^{a} U_{1} ⊗ H_{01} ⟶^{b⊗id} U_{2} ⊗ H_{12} ⊗ H_{01} ⟶^{id⊗δ} U_{2} ⊗ H_{02},

where δ : H_{12} ⊗ H_{01} ⟶ H_{02} is the composition map, which takes each tensor f ⊗ g to f ∘ g ∈ H_{02} = Hom(E_{0}, E_{2}). We say that a representation (a, b) has zero composition (or (a, b) is representation with zero composition) whenever the composition γ is trivial.

3.3. Lemma. The sequence of sheaves morphisms (3.1) is a complex iff the corresponding representation (a, b) has zero composition.

Proof. We fix basses {e_{i}^{j}}_{i=1,...,u_{j}} in the spaces U_{j} (j = 0, 1, 2). Then the morphisms α and β can be wrote as matrices

A = (g_{kj})_{{k=1,...,u_{1} \atop j=1,...,u_{0}}} , B = (f_{ik})_{{i=1,...,u_{2} \atop k=1,...,u_{1}}} ,

where g_{kj} ∈ H_{01}, f_{ik} ∈ H_{12}. Therefore the morphisms composition β ∘α : U_{0} ⊗E_{0} ⟶ U_{2} ⊗E_{2} has the form

B · A = (∑_{k=1}^{u_{1}} f_{ik} ∘ g_{kj})_{{i=1,...,u_{2} \atop j=1,...,u_{0}}}

Thus sequence (3.1) is a complex iff for each pair of induces i, j the following relation ∑_{k=1}^{u_{1}} f_{ik} ∘

g_{kj} = 0 holds.

On the other hand, the linear map a takes the basis vector e_{j}^{0} of U_{0} to the tensor ∑_{k=1}^{u_{1}} e_{k}^{1} ⊗g_{kj}.

The linear map b takes the basis vector e_{k}^{1} of U_{1} to ∑_{i=1}^{u_{2}} e_{i}^{2} ⊗ f_{ik}. Therefore the action of γ to

the vector e_{j}^{0} has the form

e_{j}^{0} ⟶^{a} ∑_{k=1}^{u_{1}} e_{k}^{1} ⊗ g_{kj} ⟶^{b⊗id} ∑_{k,i} e_{i}^{2} ⊗ f_{ik} ⊗ g_{kj} ⟶^{id⊗δ} ∑_{i=1}^{u_{2}} e_{i}^{2} ⊗ ∑_{k=1}^{u_{1}} f_{ik} ∘ g_{kj}.

In other words, γ = 0 iff for any pair of i, j the condition ∑_{k=1}^{u_{1}} f_{ik} ∘ g_{kj} = 0 holds.

With this moment we fix the topological type of stable sheaf X from proposition 3.1. That is we fix the vector x = (r, c_{1}, c_{1}^{2} − 2c_{2}) in K_{0}(ℙ^{2}) (here r is rank of sheaf, c_{1}, c_{2} are Chern classes of it) and dimensions u_{i} of the spaces U_{i}. Note that

r = u_{1}r_{1} − u_{0}r_{0} − u_{2}r_{2} > 0.

To construct the moduli space of the quiver representations, let us choose a polarization θ, it is let us fix integers θ_{i} (i = 0, 1, 2) satisfying condition (2.5). By definition put

θ_{0} = u_{2}r̄_{1} − u_{1}r_{2} θ_{1} = u_{0}r_{2} − u_{2}r_{0} θ_{2} = u_{1}r_{0} − u_{0}r̄_{1},  (3.2)

where r_{i} = rk E_{i} and r̄_{1} = rk R_{E_{2}}E_{1}. Bellow we understand stability of representations as the stability w.r.t the polarization θ with θ_{i} fixed above.

By given v = (v_{0}, v_{1}, v_{2}) ∈ ℤ^{3} we define an element of K_{0}(ℙ^{2}) by the rule

x(v) = x(v_{0}, v_{1}, v_{2}) = v_{1}e_{1} − v_{0}e_{0} − v_{2}e_{2},

where the vectors of K_{0}(ℙ^{2}) e_{i}’s correspond to the exceptional bundles E_{i}’s. Besides we define

a slope function μ on K_{0}(ℙ^{2}), μ = \frac{r(x)}{c_{1}(x)}. In these notations the following lemma holds

3.4. Lemma. Suppose that v = (v_{0}, v_{1}, v_{2}) ∈ ℤ^{3} has nonzero coordinates and the inequality v_{1}r_{1} − v_{0}r_{0} − v_{2}r_{2} > 0 holds, where r_{i} = rk E_{i}. Then

μ(x(v)) ≥ μ(x(u_{0}, u_{1}, u_{2}))  ⟺  ∑_{i} θ_{i}v_{i} ≤ 0.

Proof. Since the functions r and c_{1} are additive on K_{0}(ℙ^{2}), the inequality μ(x(v)) ≥ μ(x(u_{0}, u_{1}, u_{2})) is equivalent to

\frac{v_{1}d_{1} − v_{0}d_{0} − v_{2}d_{2}}{v_{1}r_{1} − v_{0}r_{0} − v_{2}r_{2}} ≥ \frac{u_{1}d_{1} − u_{0}d_{0} − u_{2}d_{2}}{u_{1}r_{1} − u_{0}r_{0} − u_{2}r_{2}} ,

where d_{i} = c_{1}(E_{i}). Because of the denominators of these fractions are positive, the direct calculations show that this inequality is equivalent to

v_{0}(u_{1}r_{0}r_{1}(μ_{1} − μ_{0}) − u_{2}r_{0}r_{2}(μ_{2} − μ_{0}))+
v_{1}(u_{2}r_{1}r_{2}(μ_{2} − μ_{1}) − u_{0}r_{0}r_{1}(μ_{1} − μ_{0}))+
v_{2}(u_{0}r_{0}r_{2}(μ_{2} − μ_{0}) − u_{1}r_{1}r_{2}(μ_{2} − μ_{1})) ≥ 0

(here μ_{i} = μ(E_{i})). Recall that if i < j, the pair (E_{i}, E_{j}) is exceptional. In particular, χ(E_{j}, E_{i}) = 0 and

h_{ij} = χ(E_{i}, E_{j}) − χ(E_{j}, E_{i}).

We use Riemann – Roch formula for the projective plane,

χ(E, F) = rk E · rk F (1 + \frac{3}{2}(μ(F) − μ(E)) + q(E) + q(F) − \frac{c_{1}(E)c_{1}(F)}{rk Erk F}) ,  (3.3)

where q(E) = \frac{c_{1}^{2} − 2c_{2}}{2rk E} to obtain

χ(E_{i}, E_{j}) − χ(E_{j}, E_{i}) = 3r_{i}r_{j}(μ_{j} − μ_{i}).

On the other hand, dimensions of the spaces Hom(E_{i}, E_{j}) for exceptional pairs satisfy the relations (2.4). Therefore,

3r_{0}r_{1}(μ_{1} − μ_{0}) = h_{01} = 3r_{2}, 3r_{0}r_{2}(μ_{2} − μ_{0}) = h_{02} = r̄_{1}, 3r_{1}r_{2}(μ_{2} − μ_{1}) = h_{12} = 3r_{0}.

By the choice of the polarizations

u_{1}r_{0}r_{1}(μ_{1} − μ_{0}) − u_{2}r_{0}r_{2}(μ_{2} − μ_{0}) = u_{1}r_{2} − u_{2}r̄_{1} = −θ_{0},
u_{2}r_{1}r_{2}(μ_{2} − μ_{1}) − u_{0}r_{0}r_{1}(μ_{1} − μ_{0}) = u_{2}r_{0} − u_{0}r_{2} = −θ_{1},
u_{0}r_{0}r_{2}(μ_{2} − μ_{0}) − u_{1}r_{1}r_{2}(μ_{2} − μ_{1}) = u_{0}r̄_{1} − u_{1}r_{0} = −θ_{2}.

This implies our lemma.

We need the following remark.

3.5. Remark. The integers θ_{i}, from (3.2), can be rewrote in the form

θ_{0} = r_{0}r(μ_{0} − μ), θ_{1} = r_{1}r(μ − μ_{1}), θ_{2} = r_{2}r(μ_{2} − μ),

where r and μ are the values of the rank function and the slope function on the vector x(u_{0}, u_{1}, u_{2}) respectively.

Proof. Using the formulae, obtaining in the previous proof, we get

θ_{0} = u_{2}r_{0}r_{2}(μ_{2} − μ_{0}) − u_{1}r_{0}r_{1}(μ_{1} − μ_{0}).

Consequently,

θ_{0} = u_{2}(d_{2}r_{0} − d_{0}r_{2}) − u_{1}(d_{1}r_{0} − d_{0}r_{1}) = d_{0}(u_{1}r_{1} − u_{0}r_{0} − u_{2}r_{2}) − r_{0}(u_{1}d_{1} − u_{2}d_{2} − u_{0}d_{0}).

But u_{1}r_{1} −u_{0}r_{0} −u_{2}r_{2} = r, and \frac{u_{1}d_{1} − u_{2}d_{2} − u_{0}d_{0}}{u_{1}r_{1} − u_{0}r_{0} − u_{2}r_{2}} = μ. Therefore, θ_{0} = rr_{0}(μ_{0} −μ). Similarly

we can obtain the formulae for θ_{1} and θ_{2}.

3.6. Proposition. Let X be a stable sheaf on ℙ^{2} with slop μ satisfying the condition μ_{0} < μ < μ_{2}. Then the corresponding quiver representation (a_{X}, b_{X}) is stable too.

Proof. Recall that X is the cohomology of monad (3.1) (proposition 3.1). Therefore (a_{X}, b_{X}) is the representation with zero composition (see 3.3), i.e. the maps composition

U_{0} ⟶^{a_{X}} U_{1} ⊗ H_{01} ⟶^{b_{X}⊗id} U_{2} ⊗ H_{12} ⊗ H_{01} ⟶^{id⊗δ} U_{2} ⊗ H_{02}

is trivial. Suppose that the representation (a_{X}, b_{X}) is no stable. Taking into account numerical criterion 2.2, we see that there exists a subrepresentation

a′ : U′_{0} ⟶ U′_{1} ⊗ H_{01} b′ : U′_{1} ⟶ U′_{2} ⊗ H_{12}

with dimension vector (u′_{0}, u′_{1}, u′_{2}) such that

∑_{i} θ_{i}u′_{i} ≤ 0.  (3.4)

Consider the following commutative diagram

\begin{matrix}U_{0} & ⟶^{a_{X}} & U_{1} ⊗ H_{01} & ⟶^{b_{X}⊗id} & U_{2} ⊗ H_{12} ⊗ H_{01} & ⟶^{id⊗δ} & U_{2} ⊗ H_{02} \\ ↑ φ_{0} & & ↑ φ_{1}⊗id & & ↑ φ_{2}⊗id & & ↑ φ_{2}⊗id \\ U′_{0} & ⟶^{a′} & U′_{1} ⊗ H_{01} & ⟶^{b′⊗id} & U′_{2} ⊗ H_{12} ⊗ H_{01} & ⟶^{id⊗δ} & U′_{2} ⊗ H_{02} \\ ↑ & & ↑ & & ↑ & & ↑ \\ 0 & & 0 & & 0 & & 0 \end{matrix}.

The diagram yields that (a′, b′) has zero composition, i.e., the corresponding sequence of sheaves morphisms is a complex. Besides, the following diagram is commutative too.

\begin{aligned}M(X) : & 0 ⟶ & U_{0} ⊗ E_{0} & ⟶^{α} & U_{1} ⊗ E_{1} & ⟶^{β} & U_{2} ⊗ E_{2} & ⟶ 0 \\ & & ↑ φ_{0}⊗id & & ↑ φ_{1}⊗id & & ↑ φ_{2}⊗id \\ & & U′_{0} ⊗ E_{0} & ⟶^{α′} & U′_{1} ⊗ E_{1} & ⟶^{β′} & U′_{2} ⊗ E_{2} \\ & & ↑ & & ↑ & & ↑ \\ & & 0 & & 0 & & 0 \end{aligned} .

Whence we have, α′ is a monomorphism and the restriction of φ_{2} ⊗ id to B = im β′ is a monomorphism too, i.e., we get a monomorphism of monads

\begin{aligned}M(X) : & 0 ⟶ & U_{0} ⊗ E_{0} & ⟶^{α} & U_{1} ⊗ E_{1} & ⟶^{β} & U_{2} ⊗ E_{2} & ⟶ 0 \\ & & ↑ φ_{0}⊗id & & ↑ φ_{1}⊗id & & ↑ φ_{2}⊗id \\ M(X′) : & 0 ⟶ & U′_{0} ⊗ E_{0} & ⟶^{α′} & U′_{1} ⊗ E_{1} & ⟶^{β′} & B & ⟶ 0 \\ & & ↑ & & ↑ & & ↑ \\ & & 0 & & 0 & & 0 \end{aligned} .  (3.5)

Denote by X′ the cohomology of the monad M(X′). At first we show that inequality (3.4) implies the following

μ(X′) ≥ μ(X),  (3.6)

i.e.,

\frac{u′_{1}d_{1} − d(B) − u′_{0}d_{0}}{u′_{1}r_{1} − r(B) − u′_{0}r_{0}} ≥ \frac{d}{r},

where r(B) and d(B) are rank and the first Chern class of the sheaf B, respectively. It is easy to prove that this inequality is equivalent to

u′_{0}rr_{0}(μ_{0} − μ) + u′_{1}rr_{1}(μ − μ_{1}) + rr(B)(μ(B) − μ) ≤ 0.

Taking into account remark 3.5 and inequality (3.4), we see that to prove the last inequality it remains to check that

rr(B)(μ(B) − μ) ≤ u′_{2}rr_{2}(μ_{2} − μ).

Since B is a subsheaf of the semistable bundle U′_{2} ⊗ E_{2}, then μ(B) ≤ μ_{2}, i.e.,

rr(B)(μ(B) − μ) ≤ rr(B)(μ_{2} − μ).

Furthermore, by the assumptions of the lemma, μ_{2} − μ > 0, and r(B) ≤ u′_{2}r_{2}. Hence,

rr(B)(μ_{2} − μ) ≤ u′_{2}rr_{2}(μ_{2} − μ).

Thus inequality (3.6) is proved.

Commutative diagram (3.5) is divided into two diagrams with exact columns and rows

0  0  0
↑  ↑  ↑
0 ⟶ ker β/ ker β′ ⟶ (U_{1}/U′_{1}) ⊗ E_{1} ⟶ (U_{2} ⊗ E_{2})/B ⟶ 0
↑  ↑  ↑
0 ⟶  ker β  ⟶ U_{1} ⊗ E_{1}  ⟶^{β} U_{2} ⊗ E_{2} ⟶ 0
↑ φ′  ↑ φ_{1}⊗id  ↑ φ_{2}⊗id
0 ⟶  ker β′  ⟶ U′_{1} ⊗ E_{1}  ⟶^{β′}  B  ⟶ 0
↑  ↑  ↑
0  0  0

0  0  0 ↑  ↑  ↑ 0 ⟶ (U_{0}/U′_{0}) ⊗ E_{0} ⟶ ker β/ ker β′ ⟶ coker φ ⟶ 0 ↑  ↑  ↑ 0 ⟶ U_{0} ⊗ E_{0}  ⟶^{α}  ker β  ⟶ X ⟶ 0 ↑ φ_{0}⊗id  ↑ φ′  ↑ φ 0 ⟶ U′_{0} ⊗ E_{0}  ⟶^{α′}  ker β′  ⟶ X′ ⟶ 0 ↑  ↑  ↑ 0  0  ker φ .

Besides, the sequence

0 ⟶ ker φ ⟶ (U_{0}/U′_{0}) ⊗ E_{0} ⟶ (ker β/ ker β′) ⟶ coker φ ⟶ 0  (3.7)

is exact too.

Suppose ker φ = 0, then X′ ⊂ X and the inequality (3.6) contradicts the stability of the sheaf X. This contradiction proves the proposition in this case.

If ker φ ≠ 0, it follows from (3.7) and the stability of exceptional bundles that μ(ker φ) ≤ μ_{0}. By the assumption of the proposition, μ_{0} < μ(X). Combining these inequalities with (3.6), we get μ(ker φ) < μ(X′). Hence, μ(coker φ) > μ(X′) ≥ μ(X). On the other hand, coker φ is a subsheaf of the stable sheaf X. This contradiction proves the proposition.

Thus we have the set theoretical inclusion of the set of stable sheaves corresponding to the vector x(u_{0}, u_{1}, u_{2}) ∈ K_{0}(ℙ^{2}) to the moduli space 𝔑 = 𝔑^{s}(h_{01}, h_{12}; u_{0}, u_{1}, u_{2}) of stable representations of the quiver Q. Note that there exists a compact coarse moduli space 𝔐(x) of semistable in Gieseker sense sheaves on ℙ^{2} with the vector x. To construct algebraic morphism 𝔐(x) ⟶ 𝔑 we need to recall the definition of coarse moduli space.

Denote by Σ_{x}(S) the set of equivalence classes of families of semistable by Gieseker sheaves on ℙ^{2} with the vector x and base S, i.e.,

Σ_{x}(S) = {[X] | X is a sheaf on S × ℙ^{2} : X |_{s×ℙ^{2}} is a stable sheaf on ℙ^{2} with the vector x ∈ K_{0}(ℙ^{2})}

In such a way, the contravariant functor

Σ_{x} : Alg ⟶ Ens

from the category of algebraic varieties to the set category is defined. An algebraic variety 𝔐(x) is called the coarse moduli space for Σ_{x}, if the following conditions hold

1) there exists a natural transformation of contravariant functors

Σ_{x} ⟶ Hom(·, 𝔐(x)),

which is bijective for any (reduced) point;

2) for every algebraic variety 𝔑 and every natural transformation Σ_{x} ⟶ Hom(·, 𝔑) there is a unique algebraic mapping f : 𝔐(x) ⟶ 𝔑 such that the diagram

[figure: Σ_{x}; Hom(·, 𝔐(x)); Hom(·, 𝔑); f_{∗}]

is commutative.

3.7. Proposition. Let x be a vector from K_{0}(ℙ^{2}) such that r(x) and c_{1}(x) are coprime, μ(x) ∈ (μ_{0}, μ_{2}), and the moduli space 𝔐(x) is not empty. Then there exists algebraic monomorphism m : 𝔐(x) ⟶ 𝔑.

Proof. It follows from the definition that it is sufficient to prove that there exists a natural transformation of functors Σ_{x} ⟶ Hom(·, 𝔑). Note that the condition (r(x), c_{1}(x)) = 1 implies that any semistable in Gieseker sense sheaf from 𝔐(x) is stable in the sense of Mumford – Takemoto.

Let X be a family of stable sheaves on ℙ^{2} with vector x and base S, i.e. X is a sheaf on the direct product S × ℙ^{2} such that the restriction of it to s × ℙ^{2} is a stable sheaf on ℙ^{2} with topological invariants vector x.

By the relative variant of theorem 2.1, there exists a spectral sequence on S ×ℙ^{2} converging to X with E_{1}-part in the form

E_{1}^{p,q} = π^{∗}R_{π_{∗}}^{q}Hom(g^{∗}F_{−p}, X) ⊗ g^{∗}E_{2−p},

where (F_{0}, F_{1}, F_{2}) is the left dual collection to the collection (E_{0}, E_{1}, E_{2}) and π and g are projections of the direct product S × ℙ^{2} to the first factor and the second one respectively. By the same reasons as in 3.1, we obtain that the sheaf X is the cohomology of the monad

0 ⟶ π^{∗}F_{0} ⊗ g^{∗}E_{0} ⟶^{A} π^{∗}F_{1} ⊗ g^{∗}E_{1} ⟶^{B} π^{∗}F_{2} ⊗ g^{∗}E_{2} ⟶ 0,

where F_{i} = R_{π_{∗}}^{1}Hom(g^{∗}F_{−p}, X). Let us tensoring by g^{∗}E_{0}^{∗} the morphism A of the monad, and the morphism B tensoring by g^{∗}E_{1}^{∗}:

π^{∗}F_{0} ⊗ g^{∗}(E_{0}^{∗} ⊗ E_{0}) ⟶^{A⊗id} π^{∗}F_{1} ⊗ g^{∗}(E_{0}^{∗} ⊗ E_{1})

π^{∗}F_{1} ⊗ g^{∗}(E_{1}^{∗} ⊗ E_{1}) ⟶^{B⊗id} π^{∗}F_{2} ⊗ g^{∗}(E_{1}^{∗} ⊗ E_{2})

Applying the functor R_{π_{∗}} to these sequences, we get morphisms of bundles on S:

F_{0} ⟶^{A} F_{1} ⊗ H_{01}, F_{1} ⟶^{B} F_{2} ⊗ H_{12}.

By construction, the fibers A and B over a point X of the base S determine the representation (a_{X}, b_{X}) of the quiver Q corresponding to the sheaf X. Proposition 3.6 yields that the representation (a_{X}, b_{X}) is stable. Hence the family of stable sheaves with the base S determines the family of stable representations of the quiver with the same base.

Since (r(x), c_{1}(x)) = 1, the dimension vector (u_{0}, u_{1}, u_{2}) is primitive. Therefore the moduli space of stable representations 𝔑 is fine (see the end of section 2). Thus there exists an algebraic map s : S ⟶ 𝔑 such that F_{i} = s^{∗}U_{i}, where U_{i}’s are the universal bundles over 𝔑. In other words, there is a natural transformation of functors. This completes the proof.

4. The image of 𝔐(x) under the map m.

In the previous section we constructed the algebraic map m from the moduli space of sheaves 𝔐(x) to the moduli space of the stable representations 𝔑. Now we describe the image of 𝔐(x) under this map. At first we note that dimension of tangent space of 𝔐(x) in a point X equals dim Ext^{1}(X, X). Since stable sheaves are simple and the anticanonical class on ℙ^{2} is ample, we have, dim Ext^{1}(X, X) = 1 − χ(X, X).

On the other hand, x = u_{1}e_{1}−u_{0}e_{0}−u_{2}e_{2}, where e_{i}’s are vectors from K_{0}(ℙ^{2}) corresponding to the bundles of the exceptional collections. Therefore, χ(X, X) = ∑ u_{i}^{2} +u_{0}u_{2}h_{02} −u_{0}u_{1}h_{01} − u_{1}u_{2}h_{12}, i.e., the variety 𝔐(x) is smooth and

dim 𝔐(x) = u_{0}u_{1}h_{01} + u_{1}u_{2}h_{12} − u_{0}u_{2}h_{02} − ∑ u_{i}^{2} + 1.

Dimension of moduli space 𝔑 can by calculated as the difference of dimension of the vector space R and dimension of the group G/Δ, i.e.,

dim 𝔑 = u_{0}u_{1}h_{01} + u_{1}u_{2}h_{12} − ∑ u_{i}^{2} + 1.

Besides, by lemma 3.3 we get that the image of the variety 𝔐(x) under the map m is a subset of the zero set ℨ of morphism Γ : U_{0} ⟶ U_{2} ⊗ H_{02}. We see that if the map Γ is general, then dimension of ℨ coincide with dimension of 𝔐(x). We will prove that the the image coincides with ℨ too. This will imply theorem 1.1.

We start with a simple remark.

4.1. Remark. Let (a, b) be a stable representation with zero composition such that the corresponding complex of sheaves (3.1) is a monad. Then the cohomology of this monad X is stable sheaf.

Proof. Suppose that X is non-stable. Then there exists a subsheaf X′ such that rk X′ < rk X and μ(X′) ≥ μ(X). Without loss of generality it can be assumed that X′ is stable.

Since μ(X′) ≥ μ(X) > μ(F_{i}(−3)), we have Ext^{2}(F_{i}, X′) = 0 for i = 0, 1, 2 (hear F_{i}’s are the bundles of the left dual collection for (E_{0}, E_{1}, E_{2}) as before).

On the other hand, X′ ⊂ X and Hom(F_{i}, X) = 0. This yields, Hom(F_{i}, X′) = 0. Therefore, the spectral sequence constructed by the collection (E_{0}, E_{1}, E_{2}) and the sheaf X′ is degenerated into the monad

M(X′) 0 ⟶ U′_{0} ⊗ E_{0} ⟶^{α′} U′_{1} ⊗ E_{1} ⟶^{β′} U′_{2} ⊗ E_{2} ⟶ 0

with cohomology X′. The monomorphism of sheaves X′ ⟶ X induces a monomorphism of the monads M(X′) ⟶ M(X) ( 3.2). Hence the representation (a_{X′}, b_{X′}) corresponding to the subsheaf X′ is a subrepresentation of (a_{X}, b_{X}).

Now using lemma 3.4 and numerical criterion 2.2, we get that the subrepresentation (a_{X′}, b_{X′}) contradicts to the stability of (a_{X}, b_{X}). This contradiction proves the remark.

Thus we see that, for theorem 1.1 to be proved it is necessary and sufficient to check that any stable representation with zero composition induces a monad namely. We start with the case of exceptional collection (O(−2), O(−1), O).

4.2. Proposition. Suppose that (E_{0}, E_{1}, E_{2}) = (O(−2), O(−1), O) and

a : U_{0} ⟶ U_{1} ⊗ H_{01}, b : U_{1} ⟶ U_{2} ⊗ H_{12}

is a representation with zero composition. In addition we assume that slope μ of the vector x(u_{0}, u_{1}, u_{2}) satisfy the inequality μ_{0} < μ < μ_{2}. Then the corresponding morphisms of sheaves α : U_{0} ⊗ O(−2) ⟶ U_{1} ⊗ O(−1), β : U_{1} ⊗ O(−1) ⟶ U_{2} ⊗ O are a monomorphism and an epimorphism (respectively).

Proof. Taking into account the duality of these morphisms, it is sufficient to check that α is a monomorphism of bundles. Namely, we show that the fiber of α over any point x ∈ ℙ^{2} α_{x} : U_{0} ⊗ ℂ ⟶ U_{1} ⊗ ℂ is injective.

Recall that if we fixed basses for the spaces U_{i}’s, the linear map a and a morphism α can be wrote as a matrix A = (g_{kj})_{{k=1,...,u_{1} \atop j=1,...,u_{0}}} with g_{kj} ∈ H^{0}(O(1)). Assume that the morphism α fiber over a point x α_{x} has nonzero kernel, i.e., there is a vector u ∈ U_{0} such that α_{x}(u) = 0. Without loss of generality it can be assumed that u is the first basis vector of the space U_{0}. Then the morphisms g_{k1} ∈ H_{01} = H^{0}(O(1)) vanish at the point x when k = 1, ..., u_{1}. Because of the zero set of a morphism g ∈ H^{0}(O(1)) is a line on the projective plane, dimension of the linear span H = ⟨g_{11}, ..., g_{u_{0}1}⟩ less or equal to 2. This implies that the matrix A in suitable basses {e_{i}^{j}} of the spaces U_{j} (j = 0, 1) has the form

A = ( \begin{matrix}g_{11} & ∗ & ⋯ & ∗ \\ g_{21} & ∗ & ⋯ & ∗ \\ 0 & ∗ & ⋯ & ∗ \\ ⋯ & ⋯ & ⋯ & ⋯ \\ 0 & ∗ & ⋯ & ∗ \end{matrix} )

Holding fixed a basis for U_{2} we can write the linear map b as a matrix B = (f_{ik})_{{i=1,...,u_{2} \atop k=1,...,u_{1}}} with

f_{ik} ∈ H^{0}(O(1)). By assumption, the representation (a, b) has zero composition, i.e., B · A = 0. In particular,

f_{i1} ∘ g_{11} + f_{i2} ∘ g_{21} = 0 ∀i = 1, ..., u_{2}.  (4.1)

Let us consider 3 cases.

Case 1. Suppose that dim H = 2, that is the morphisms g_{11} and g_{21} are not proportional. Then the exact sequence takes place.

0 ⟶ O(−2) ⟶^{g_{11}⊕g_{21}} O(−1) ⊕ O(−1) ⟶ J_{x} ⟶ 0,

where J_{x} is the ideal sheaf of the point x. Applying the functor Hom(·, O) we have,

0 ⟶ Hom(J_{x}, O) ⟶ Hom(O(−1), O)^{⊕2} ⟶^{δ} Hom(O(−2), O) ⟶ 0.

Here the morphism δ takes a pair (f_{1}, f_{2}) ∈ Hom(O(−1), O)^{⊕2} to f_{1} ∘ g_{11} + f_{2} ∘ g_{21}. Since dim Hom(J_{x}, O) = 1, then dim⟨(f_{i1}, f_{i2})⟩ ≤ 1 (see (4.1)). Therefore, changing basis of the space U_{2}, the matrix B can be rewrote in the form

B = ( \begin{matrix}f_{11} & f_{12} & ∗ & ⋯ & ∗ \\ 0 & 0 & ∗ & ⋯ & ∗ \\ & ⋯ & ⋯ & ⋯ & ⋯ \\ 0 & 0 & ∗ & ⋯ & ∗ \end{matrix} )

In such a way we constructed a subrepresentation (a′, b′) of the representation (a, b) with dimension vector (1, 2, 1). Let us compute value of the polarization θ at this vector. In the case when exceptional collection is (O(−2), O(−1), O) the polarization has the form

θ_{0} = 2u_{2} − u_{1} = r(μ_{0} − μ), θ_{1} = u_{0} − u_{2} = r(μ − μ_{1}), θ_{2} = u_{1} − 2u_{0} = r(μ_{2} − μ) (4.2)

(see (3.2), 3.5). Hence, θ_{0} + 2θ_{1} + θ_{2} = 0,. This contradicts the numerical criterion 2.2 and the assumption of stability of the representation (a, b).

Case 2. Suppose that dim H = 1. We can assume that g_{12} = 0, i.e., the condition D · A = 0 implies the inequality f_{i1} ∘ g_{11} = 0 ∀i = 1, ..., u_{2}. We see that this is possible in the case f_{i,1} = 0 only. Therefore the representation (a, b) has a subrepresentation with dimension vector (1, 1, 0). We have

θ(1, 1, 0) = u_{2} + u_{0} − u_{1} = \frac{1}{2}(2u_{2} − u_{1} + 2u_{0} − u_{1}).

By the assumption of the proposition, 2u_{2} − u_{1} = r(μ_{0} − μ) < 0, u_{1} − 2u_{0} = r(μ_{2} − μ) > 0, i.e., θ(1, 1, 0) < 0. By the stability of (a, b), it is impossible.

Case 3. Now suppose that dim H = 0. Then the representation (a, b) has a subrepresentation with dimension vector (1, 0, 0). This contradicts stability of (a, b) too.

This completes the proof.

To check that α is a monomorphism and β is an epimorphism in the general case we need a mutations of representations. Further we deal with the mutations.

4.3. Lemma. Let a representation U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12} with zero composition corresponds to a complex of sheaves U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2}, where (E_{0}, E_{1}, E_{2}) is a full exceptional collection. Denote by L_{0}U_{1} the kernel of the map a, i.e.,

0 ⟶ L_{0}U_{1} ⟶^{L_{0}a} U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1},

and by L_{0}H_{12} the linear space Hom(RE_{0}, E_{2}). Then there exits a linear map L_{0}b : U_{0} ⟶ U_{2} ⊗ L_{0}H_{12} such that the representation L_{0}U_{1} ⊗ H_{01} ⟶^{L_{0}a} U_{0} ⟶^{L_{0}b} U_{2} ⊗ L_{0}H_{12} corresponds to

the sheaves complex L_{0}U_{1} ⊗E_{1} ⟶^{L_{0}α} U_{0} ⊗RE_{0} ⟶^{L_{0}β} U_{2} ⊗E_{2}, where, as usually, RE_{0} = R_{E_{1}}E_{0}

is the right mutation of the exceptional pair ^{4} (E_{0}, E_{1}). Moreover it follows from the fact that α is monomorphism that L_{0}α is a monomorphism too. If β and a are epimorphisms, then L_{0}β is epimorphism too.

Proof. At first we check that the diagram is commutative

\begin{matrix}U_{0} ⊗ E_{0} & ⟶^{α} & U_{1} ⊗ E_{1} \\ ↑ id & & ↑ a⊗id \\ U_{0} ⊗ E_{0} & ⟶^{id⊗rcan} & U_{0} ⊗ H_{01}^{∗} ⊗ E_{1} \end{matrix}  (4.3)

The sheaf morphism α and the linear map a are determined by one tensor of U_{0}^{∗} ⊗ U_{1} ⊗ H_{01}, which in a fixed basis f_{1}, ..., f_{n} for H_{01} has the form ∑_{i} φ_{i} ⊗ f_{i}, where φ_{i} ∈ Hom(U_{0}, U_{1}). The

fiber of α over a point x ∈ ℙ^{2} acts by the rule α_{x}(v ⊗ e) = ∑_{i} φ_{i}(v) ⊗ f_{i,x}(e) The linear map a

takes each tensor v ⊗ g ∈ U_{0} ⊗ H_{01}^{∗} to the vector a(v ⊗ g) = ∑_{j} φ_{j}(v)⟨g, f_{j}⟩, where ⟨·, ·⟩ is the

natural pairing on the tensor product H_{01}^{∗} ⊗ H_{01}.

Besides, the canonical map rcan : E_{0} ⟶ H_{01}^{∗} ⊗ E_{1} is determined by ∑_{i} f_{i} ⊗ f^{i}, where

f^{1}, ..., f^{n} is the dual basis for H_{01}^{∗}. Therefore the action of this map on a fiber vector e ∈ E_{0,x} has the form e ⟶ ∑_{i} f_{i,x}(e) ⊗ f^{i}.

Combining these formulae together, we obtain

v⊗e ⟶^{id⊗rcan} ∑_{i} v⊗f_{i,x}(e)⊗f^{i} ⟶^{a⊗id} ∑_{i}(∑_{j} φ_{j}(v)⟨f^{i}, f_{j}⟩)⊗f_{i,x}(e) = ∑_{i} φ_{i}(v)⊗f_{i,x}(e) = α_{x}(v⊗e).

This proves the commutativity of the diagram.

Since diagram (4.3) is commutative, we have one more commutative diagram with exact columns and rows

U_{0} ⊗ E_{0} ⟶^{α}  U_{1} ⊗ E_{1}  ⟶ coker α ⟶ 0 ↑ id  ↑ a⊗id  ↑ ∃β′ 0 ⟶ U_{0} ⊗ E_{0} ⟶^{id⊗rcan} U_{0} ⊗ H_{01}^{∗} ⊗ E_{1} ⟶^{id⊗lcan} U_{0} ⊗ RE_{0} ⟶ 0 ↑  ↑ L_{0}a⊗id  ↑ L_{0}α 0  ⟶  L_{0}U_{1} ⊗ E_{1}  ⟶^{id} L_{0}U_{1} ⊗ E_{1} ⟶ 0 ↑ 0 ,

where L_{0}α = (id ⊗ lcan) ∘ (L_{0}a ⊗ id). As before we can check that the morphism L_{0}α and the linear map a are determined by one tensor from (L_{0}U_{1})^{∗} ⊗ U_{0} ⊗ H_{01}^{∗}. The commutative diagram implies

(i) β′ ∘ L_{0}α = 0,

(ii) if a is an epimorphism, then β′ is an epimorphism too,

(iii) if α is an inclusion of bundles, then L_{0}α is also an inclusion of bundles.

4Recall that Hom(E_{0}, E_{1})^{∗} ≅ Hom(E_{1}, RE_{0}).

By assumption the following commutative diagram takes place

U_{2} ⊗ E_{2} ⟶^{id} U_{2} ⊗ E_{2} ↑ β  ↑ ∃β′′ U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶ coker α ⟶ 0 ↑ id  ↑ U_{0} ⊗ E_{0} ⟶^{α} ker β ↑ 0 .

In particular, we see that the fact that β is an epimorphism gives that β′′ is an epimorphism too. Let us define the left mutation L_{0}β as the composition L_{0}β = β′′ ∘ β′ and let L_{0}b be the linear map U_{0} ⟶ U_{2} ⊗ L_{0}H_{12}, corresponding to the same tensor form U_{0}^{∗} ⊗ U_{2} ⊗ L_{0}H_{12}, as L_{0}β corresponds.

4.4. Remark. We say that the representation (L_{0}a, L_{0}b) constructed in the previous lemma is the left mutation of the representation (a, b). By construction the left mutation (L_{0}a′, L_{0}b′) of a subrepresentation (a′, b′) of (a, b) is a subrepresentation of (L_{0}a, L_{0}b).

4.5. Lemma. Let a representation with zero composition

U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12}

corresponds to the complex of sheaves U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2}, where (E_{0}, E_{1}, E_{2}) is a full exceptional collection. Denote by R_{1}U_{0} the cokernel of the map a, i.e,

U_{0} ⟶^{a} U_{1} ⊗ H_{01} ⟶^{R_{1}a} R_{1}U_{0} ⟶ 0.

Then there exists a linear map R_{1}b : R_{1}U_{0} ⟶ U_{2} ⊗ H_{02} such that the representation U_{1} ⊗ H_{01} ⟶^{R_{1}a} R_{1}U_{0} ⟶^{R_{1}b} U_{2} ⊗ H_{02} corresponds to the complex of sheaves

U_{1} ⊗ LE_{1} ⟶^{R_{1}α} R_{1}U_{0} ⊗ E_{0} ⟶^{R_{1}β} U_{2} ⊗ E_{2},

where, as usually, LE_{1} = L_{E_{0}}E_{1} is the left mutation of the bundle E_{1}. Moreover, if α is a monomorphism of bundles and a is injective, then R_{1}α is a monomorphism of bundles. If β is an epimorphism, then R_{1}β is too.

Proof. Application of the functor Hom(E_{0}, ·) to the sequence

U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶ coker α ⟶ 0

gives

U_{0} ⟶^{a} U_{1} ⊗ H_{01} ⟶^{R_{1}a} Hom(E_{0}, coker α).

By the assumption of the lemma R_{1}U_{0} is the image of the map R_{1}a. Hence there exists the commutative diagram

0  0  0 ↑  ↑  ↑ U_{0} ⊗ E_{0} ⟶^{α}  U_{1} ⊗ E_{1}  ⟶ coker α ⟶ 0 ↑  ↑ id⊗lcan  ↑ β′ U_{0} ⊗ E_{0} ⟶^{a⊗id} U_{1} ⊗ H_{01} ⊗ E_{0} ⟶^{R_{1}a⊗id} R_{1}U_{0} ⊗ E_{0} ⟶ 0 ↑  ↑ id⊗rcan  ↑ R_{1}α 0  ⟶ U_{1} ⊗ LE_{1}  ⟶ U_{1} ⊗ LE_{1} ↑ 0 ,

where R_{1}α = (R_{1}a ⊗ id) ∘ (id ⊗ rcan), and β′ is the restriction of the canonical map lcan : Hom(E_{0}, coker α) ⊗ E_{0} ⟶ coker α to the subspace R_{1}U_{0} ⊂ Hom(E_{0}, coker α). We see that R_{1}α and R_{1}a are determined by one tensor from U_{1}^{∗} ⊗ R_{1}U_{0} ⊗ H_{01}^{∗}. This diagram implies

(i) β′ is epimorphism,

(ii) β′ ∘ R_{1}α = 0,

(iii) if α is an inclusion of bundles, then R_{1}α is inclusion of bundles too.

Further, as before, the following commutative diagram takes place

U_{2} ⊗ E_{2} ⟶^{id} U_{2} ⊗ E_{2} ↑ β  ↑ ∃β′′ U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶ coker α ⟶ 0 ↑ id  ↑ U_{0} ⊗ E_{0} ⟶^{α} ker β ↑ 0 .

Moreover, if β is an epimorphism, we see that β′′ is an epimorphism too. To prove the lemma it is remains to define R_{1}β as the composition β′′ ∘ β′, and we define R_{1}b as the corresponding linear map.

4.6. Remark. The representation (R_{1}a, R_{1}b) constructed in the previous lemma is called the right mutation of the representation (a, b). By construction, the right mutation (R_{1}a′, R_{1}b′) of a subrepresentation (a′, b′) of (a, b) is a subrepresentation of (R_{1}a, R_{1}b).

The next lemma follows from the proves of the lemmas 4.3 and 4.5.

4.7. Lemma (i) If the map a : U_{0} ⊗ H_{01}^{∗} ⟶ U_{1} of a representation (a, b) is an epimorphism,

then (R_{1}L_{0}a, R_{1}L_{0}b) = (a, b).

(ii) If the map a : U_{0} ⟶ U_{1} ⊗ H_{01} of a representation (a, b) is a monomorphism, then (L_{0}R_{1}a, L_{0}R_{1}b) = (a, b).

Below we show when the mutations of stable representations are stable.

4.8. Lemma. Suppose that μ = \frac{u_{1}d_{1} − u_{0}d_{0} − u_{2}d_{2}}{u_{1}r_{1} − u_{0}r_{0} − u_{2}r_{2}} = \frac{d}{r} ∈ (μ_{0}, μ_{2}) and a representation with

zero composition U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12} is stable w.r.t. the polarization θ with θ_{0} = u_{2}r̄_{1} − u_{1}r_{2} = rr_{0}(μ_{0} − μ) < 0, θ_{1} = u_{0}r_{2} − u_{2}r_{0} = rr_{1}(μ − μ_{1}), θ_{2} = u_{1}r_{0} − u_{0}r̄_{1} = rr_{2}(μ_{2} − μ) > 0. Then the right mutation of the representation U_{1} ⊗ H_{01} ⟶^{R_{1}a} R_{1}U_{0} ⟶^{R_{1}b} U_{2} ⊗ H_{02} is stable

w.r.t. the polarization R_{1}θ with R_{1}θ_{0} = u_{2}(3r_{2}r̄_{1} − r_{0}) − (3r_{2}u_{1} − u_{0})r_{2} = rr_{0}(μ̄_{1} − μ) < 0, R_{1}θ_{1} = u_{1}r_{2} − u_{2}r̄_{1} = rr_{0}(μ − μ_{0}) > 0, R_{1}θ_{2} = (3r_{2}u_{1} − u_{0})r̄_{1} − u_{1}(3r_{2}r̄_{1} − r_{0}) = u_{1}r_{0} − u_{0}r̄_{1} = rr_{2}(μ_{2} − μ) > 0,

where μ̄_{1} = μ(L_{E_{0}}E_{1}).

Proof. At first we show that the map a : U_{0} ⟶ U_{1} ⊗ H_{01} is a monomorphism. Assume that it is not true, i.e., a has a nonzero kernel U′_{0} ⊂ U_{0}. Then the pair of trivial maps U′_{0} ⟶ {0} ⊗ H_{01}, {0} ⟶ {0} ⊗ H_{12} determines a subrepresentation of (a, b) with dimension vector (u′_{0}, 0, 0). Since θ_{0} < 0, the values of the polarization θ at this vector is negative. This contradicts the stability of (a, b).

Thus, a is an monomorphism and by the previous lemma we have

(L_{0}R_{1}a, L_{0}R_{1}b) = (a, b).  (4.4)

Besides, dim R_{1}U_{0} = h_{01}u_{1} − u_{0}. Further, since h_{01} = 3r_{2}, then dim R_{1}U_{0} = 3r_{2}u_{1} − u_{0}. Using this fact we can prove that R_{1}θ is a polarization, i.e., R_{1}θ_{0}u_{1} + R_{1}θ_{1} dim R_{1}U_{0} + R_{1}θ_{2}u_{2} = 0.

Now we assume that the representation (R_{1}a, R_{1}b) is no stable. By criterion 2.2, it has a subrepresentation U′_{0} ⊗ H_{01} ⟶^{a′} U′_{1} ⟶^{b′} U′_{2} ⊗ H_{02} with dimension vector (u′_{0}, u′_{1}, u′_{2}) such that

∑ R_{1}θ_{i}u′_{i} ≤ 0.  (4.5)

If a′ is not epimorphism, then taking im a′ in place of U′_{1} we get a subrepresentation (a′′, b′′) of (R_{1}a, R_{1}b) such that a′′ is an epimorphism and the value of the polarization R_{1}θ at dimension vector of (a′′, b′′) is non-positive too (by assumption R_{1}θ_{1} > 0). Therefore we can assume that a′ : U′_{0} ⊗ H_{01} ⟶ U′_{1} is an epimorphism.

Taking into account 4.4 and (4.4) we get that the left mutation (L_{0}a′, L_{0}b′) of the subrepresentation (a′, b′) is a subrepresentation of (a, b) with dimension vector (3r_{2}u′_{0} − u′_{1}, u′_{0}, u′_{2}). We have

θ(3r_{2}u′_{0} − u′_{1}, u′_{0}, u′_{2}) = (u_{2}r̄_{1} − u_{1}r_{2})(3r_{2}u′_{0} − u′_{1}) + (u_{0}r_{2} − u_{2}r_{0})u′_{0} + (u_{1}r_{0} − u_{0}r̄_{1})u′_{2}.

Substituting the values of R_{1}θ_{i} to inequality (4.5), we obtain

u′_{0}(u_{2}(3r_{2}r̄_{1} − r_{0}) − r_{2}(3r_{2}u_{1} − u_{0})) + u′_{1}(u_{1}r_{2} − u_{2}r̄_{1}) + u′_{2}(u_{1}r_{0} − u_{0}r̄_{1}) ≤ 0.

This is equivalent to

θ(3r_{2}u′_{0} − u′_{1}, u′_{0}, u′_{2}) ≤ 0.

We get the contradiction with the stability of the representation (a, b). This concludes the proof.

4.9. Lemma. Suppose that μ = \frac{u_{1}d_{1} − u_{0}d_{0} − u_{2}d_{2}}{u_{1}r_{1} − u_{0}r_{0} − u_{2}r_{2}} = \frac{d}{r} ∈ (μ_{1}, μ_{2}) and a representation with

zero composition U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12} is stable w.r.t. the polarization θ from the previous lemma. Then the left mutation of the representation

L_{0}U_{1} ⊗ H_{01} ⟶^{L_{0}a} U_{0} ⟶^{L_{0}b} U_{2} ⊗ L_{0}H_{12}

is stable w.r.t. the polarization L_{0}θ with L_{0}θ_{0} = u_{2}r_{0} − u_{0}r_{2} = rr_{0}(μ_{1} − μ) < 0, L_{0}θ_{1} = (3r_{2}u_{0} − u_{1})r_{2} − u_{2}r_{1} = r(3r_{2}r_{1} − r_{0})(μ − μ(RE_{0})), L_{0}θ_{2} = u_{0}r_{1} − (3r_{2}u_{0} − u_{1})r_{0} = rr_{2}(μ_{2} − μ) > 0.

Proof. If the image of the map a does not coincide with U_{1}, then the representation (a, b) has a subrepresentation with dimension vector (u_{0}, dim(im a), u_{2}). By assumption μ > μ_{1}, hence θ_{1} = rr_{1}(μ−μ_{1}) > 0 and θ(u_{0}, dim(im a), u_{2}) < 0. This contradicts the stability of (a, b). Therefore the map a is an epimorphism and from lemma 4.7 it follows that (R_{1}L_{0}a, R_{1}L_{0}b) = (a, b). Besides, dim L_{0}U_{1} = 3r_{2}u_{0} − u_{1}. Using this equality, it is easy to show that L_{0}θ is a polarization for representations with dimension vector (3r_{2}u_{0} − u_{1}, u_{0}, u_{2}).

Suppose that the representation (L_{0}a, L_{0}b) has a destabilizing subrepresentation (a′, b′) with dimension vector (u′_{0}, u′_{1}, u′_{2}), i.e.,

L_{0}θ(u′_{0}, u′_{1}, u′_{2}) ≤ 0.  (4.6)

As before, we can assume that a′ : U′_{0} ⟶ U′_{1} ⊗H_{01}^{∗} is an inclusion. In other words, (R_{1}a′, R_{1}b′) is a subrepresentation of (a, b) with dimension vector (u′_{0}, 3r_{2}u′_{1} − u′_{0}, u′_{2}). It can by proved that the subrepresentation (R_{1}a′, R_{1}b′) destabilizes the representation (a, b). This contradiction proves the lemma.

By the dual reasons we can prove the following lemmas.

4.10. Lemma. Let U_{0}⊗H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2}⊗H_{12} — be a representation with zero composition associated with the complex of sheaves U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2}. Denote by R_{2}U_{1} the cokernel of the map b, i.e.,

U_{1} ⟶^{b} U_{2} ⊗ H_{12} ⟶^{R_{2}b} R_{2}U_{1} ⟶ 0.

Then there exists a linear map R_{2}a : U_{0}⊗Hom(E_{0}, LE_{2})^{∗} ⟶ U_{2} such that the representation

(R_{2}a, R_{2}b) corresponds to the complex of sheaves U_{0} ⊗ E_{0} ⟶^{R_{2}α} U_{2} ⊗ LE_{2} ⟶^{R_{2}β} R_{2}U_{1} ⊗ E_{1}.

Moreover, if β is an epimorphism, then R_{2}β is an epimorphism too. If α is a monomorphism of bundles, R_{2}α is a monomorphism of bundles too.

4.11. Lemma. Let U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12} be a representation with zero composition associated with the complex of sheaves U_{0} ⊗ E_{0} ⟶^{α} U_{1} ⊗ E_{1} ⟶^{β} U_{2} ⊗ E_{2}. Denote by L_{1}U_{2} the kernel of the map b, i.e.

0 ⟶ L_{1}U_{2} ⟶^{L_{1}b} U_{1} ⊗ H_{12}^{∗} ⟶^{b} U_{2}.

Then there exists a linear map L_{1}a : U_{0} ⟶ L_{1}U_{2} ⊗ H_{02} such that the representation

(L_{1}a, L_{1}b) corresponds to the complex of sheaves U_{0} ⊗ E_{0} ⟶^{L_{1}α} L_{1}U_{2} ⊗ E_{2} ⟶^{L_{1}β} U_{1} ⊗ RE_{1}.

Moreover, if β and b are epimorphisms, then L_{1}β is epimorphism too. If α is a monomorphism of bundles, L_{1}α is a monomorphism of bundles too.

4.12. Remark. 1. If (a′, b′) is a subrepresentation of (a, b) then (L_{1}a′, L_{1}b′) is subrepresen-

tation of (L_{1}a, L_{1}b) and (R_{2}a′, R_{2}b′) is subrepresentation of (R_{2}a, R_{2}b).

2. If the map b of (a, b) is inclusion, then (L_{1}R_{2}a, L_{1}R_{2}b) = (a, b). If it is an epimorphism, then (R_{2}L_{1}a, R_{2}L_{1}b) = (a, b).

4.13. Lemma. Suppose that a representation U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12} with zero composition is stable w.r.t. the polarization θ with

θ_{0} = u_{2}r̄_{1} − u_{1}r_{2}, θ_{1} = u_{0}r_{2} − u_{2}r_{0}, θ_{2} = u_{1}r_{0} − u_{0}r̄_{1}.

If, an additional, μ = \frac{d}{r} ∈ (μ_{0}, μ_{1}), then the representation (R_{2}a, R_{2}b) is stable w.r.t. R_{2}θ with

R_{2}θ_{0} = (3r_{0}u_{2} − u_{1})r_{2} − u_{2}r_{1},

R_{2}θ_{1} = u_{0}r_{1} − (3r_{0}u_{2} − u_{1})r_{0},

R_{2}θ_{2} = u_{2}r_{0} − u_{0}r_{2}. If μ ∈ (μ_{0}, μ_{2}), then the representation (L_{1}a, L_{1}b) is stable w.r.t. L_{1}θ with

L_{1}θ_{0} = u_{1}(3r̄_{1}r_{0} − r_{2}) − (3r_{0}u_{1} − u_{2})r̄_{1} = u_{2}r̄_{1} − u_{1}r_{2},

L_{1}θ_{1} = u_{0}r̄_{1} − u_{1}r_{0},

L_{1}θ_{2} = (3r_{0}u_{1} − u_{2})r_{0} − u_{0}(3r̄_{1}r_{0} − r_{2}).

Now we prove the following proposition, which proves theorem 1.1.

4.14. Proposition. Let σ = (E_{0}, E_{1}, E_{2}) be a full exceptional collection with −2 ≤ μ_{0} < μ_{2} ≤ 0 and u = (u_{0}, u_{1}, u_{2}) be a collection of nonnegative integer such that^{5} μ(x(u)) ∈ (μ_{0}, μ_{2}). Suppose that a representation with zero composition U_{0} ⊗ H_{01}^{∗} ⟶^{a} U_{1} ⟶^{b} U_{2} ⊗ H_{12} is stable w.r.t. the polarization θ with θ_{i} from (3.2), and dimension vector of (a, b) is equal to u = (u_{0}, u_{1}, u_{2}). Then the corresponding complex of sheaves U_{0}⊗E_{0} ⟶^{α} U_{1}⊗E_{1} ⟶^{β} U_{2}⊗E_{2} is a monad.

5See the definition of x(u) on p. 6

Proof. Real we need to prove that α is a monomorphism and β is an epimorphism. It is known that all exceptional collection on ℙ^{2}, with slopes of bundles satisfying the condition of the proposition can be obtained by the ”interior” mutations from the collection (O(−2), O(−1), O), i.e., by the mutations L_{0} and R_{2}. Because of this we study the properties of the morphisms α and β with help of the mutation induction. The base of the induction (the case of the collection (O(−2), O(−1), O)) was done in proposition 4.2.

Assume that the proposition is true for a collection τ = (F_{0}, F_{1}, F_{2}), and the collection σ was obtained from τ by one of the mutations

L_{1}τ = (E_{0}, LE_{2}, E_{1}), R_{1}τ = (E_{1}, RE_{0}, E_{2}).

We assume that σ = R_{1}τ, i.e., E_{0} = F_{1}, E_{1} = RF_{0}, E_{2} = F_{2}. the second case can be proved by the same method. Thus, τ = L_{0}σ.

The mutations (R_{1}a, R_{1}b) of the representation (a, b) is stable w.r.t. R_{1}θ (4.5) and it is associated with the sheaves complex U_{1} ⊗ F_{0} ⟶^{R_{1}α} R_{1}U_{0} ⊗ F_{1} ⟶^{R_{1}β} U_{2} ⊗ F_{2}. By the inductive hypothesis, R_{1}α is a monomorphism and R_{1}β is an epimorphism. Since the representation (a, b) is stable, the map a : U_{0} ⟶ U_{1}⊗H_{01} is inclusion. Therefore, by remark 4.7, (L_{0}R_{1}a, L_{0}R_{1}b) = (a, b). By 4.3, we get that α = L_{0}R_{1}α is a monomorphism and β = L_{0}R_{1}β is an epimorphism.

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