On the moduli of pairs of stable sheaves related by non-trivial morphisms

B. V. Karpov^{∗}

March 26, 2001

Introduction

Let M(r, c_{1}, δ) denote the moduli space of semi-stable sheaves by Gieseker on ℙ^{2} over ℂ with the rank r, the first Chern class c_{1} and Euler characteristic (alternated sum of dimensions of cohomology spaces) equal to −δ. It is known [12] that M(r, c_{1}, δ) is a projective scheme, it parametrizes classes of semi-stable sheaves that have isomorphic direct sums of quotients of Jordan-Hölder filtration. In this paper we deal with the case when r and c_{1} are coprime. Then M(r, c_{1}, δ) is smooth and carries a universal family [12, Ex.6.8, Th.6.11]. Besides, in this case semi-stability by Gieseker is equivalent to stability by Mumford – Takemoto, so the word “stable” is used in this paper for the latter notion. Since the Jordan-Hölder filtration of a stable sheaf is trivial, we shall often write “X ∈ M(r, c_{1}, δ)” meaning that X is a stable sheaf with r(X) = r, c_{1}(X) = c_{1} and −χ(X) = δ. Also, we use short notations M_{v} or M(v) for M(r, c_{1}, δ), where v = (r, c_{1}, δ) is a triple of integers with r > 0.

For a sheaf X without torsion denote

Q_{X} = X^{∗∗}/X ,

this is zero-dimensional sheaf i.e. having r(Q_{X}) = 0 and c_{1}(Q_{X}) = 0. For r > 1 the moduli space M(r, c_{1}, δ) has the following stratification:

M(r, c_{1}, δ) = ⋃_{p⩾0}M^{(p)}(r, c_{1}, δ) ,

where M^{(p)}(r, c_{1}, δ) parametrizes the sheaves X such that h^{0}(Q_{X}) = p. We will use also notation M_{v}^{(p)}. Note that the number of strata is finite, it is equal to δ − δ_{min}, where δ_{min} is the lowest integer such that M(r, c_{1}, δ_{min}) is non-empty (see details in 3). Clearly, distinct M^{(p)}(r, c_{1}, δ) are disjoint and M^{(0)}(r, c_{1}, δ) is an open subset of M(r, c_{1}, δ) parametrizing locally free sheaves. In other words, M_{v}^{(0)} is the moduli space of vector bundles.

Let v = (r, c_{1}, δ) and w = (r′, c′_{1}, δ′) be two triples of integers such that the moduli spaces M_{v} and M_{w} are non-empty and

| \begin{matrix}r & c_{1} \\ r′ & c′_{1} \end{matrix} | = 1 .  (1)

∗email: karpov@karpov.mccme.ru

In this paper we study the moduli space M̃_{v→w}, which parameterizes the triples

{(X, Y, [φ]) | X ∈ M_{v}, Y ∈ M_{w}, [φ] ∈ P (Hom(X, Y))} .

We consider only the case r(X) < r(Y), the case r(X) > r(Y) being dual in some sense. In particular, we show that the image of natural map

Φ : M̃_{v→w} ⟶ M_{v} , (X, Y, [φ]) ↦ X

is the union of strata M_{v}^{(p)}, where the lower and the upper bounds of range of values of p depend on the triples v and w (Theorem 3.6).

Besides, Φ is surjective iff M_{v}^{(0)} ⊂ im Φ, in this case the preimage Φ^{−1}(M_{v}^{(0)}) of moduli space of bundles is isomorphic to the projectivization of a vector bundle over the product M_{v}^{(0)} × M_{w−v} (Theorem 4.2). Note that all moduli spaces under consideration are smooth and carry universal families because (1) implies coprimeness of ranks and c_{1}’s for the sheaves.

Preimages of the rest strata are more complicated. In 4.4 we describe Φ^{−1}(X) as a union of parts numerated by subsheaves of Q_{X}, where X ∈ M_{v}^{(p)} and p > 0.

This paper is motivated by the following problem, which can be regarded as mathematical testing of S -duality conjecture ([8],[9],[10]). Consider the sequence {r_{n}}_{n∈ℤ} determined by recurrent relations and initial values

r_{n+1} + r_{n−1} = 3 r_{n} ,  r_{0} = r_{−1} = 1 .  (2)

Several members of this sequence near r_{0} are the following:

\begin{matrix}… & r_{−5} & r_{−4} & r_{−3} & r_{−2} & r_{−1} & r_{0} & r_{1} & r_{2} & r_{3} & r_{4} & r_{5} & … \\ … & 34 & 13 & 5 & 2 & 1 & 1 & 2 & 5 & 13 & 34 & 89 & … \end{matrix}

It is easily seen that r_{n} = r_{−n−1} i.e. the sequence is symmetric w.r.to the “middle” between r_{−1} and r_{0}.

Consider the moduli spaces

M_{n}(δ) := M(r_{n}, −r_{n−1}, δ) .

Note that r_{n} and r_{n−1} are coprime because of the formula

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

which follows from (2) (see Section 5). Hence, as it was pointed above, each M_{n}(δ) is smooth and carries a universal family E_{n,δ} → M_{n}(δ) × ℙ^{2}, which is defined up to tensoring by the pull-back of an invertible sheaf on M_{n}(δ). Suppose that universal families are chosen by a reasonable way, e.g. via construction that describes E_{n,δ} simultaneously, see for example [5]. Let π : M_{n}(δ) × ℙ^{2} → M_{n}(δ) be the projection and put H_{n,δ} = R^{1}π_{∗}E_{n,δ}. This is a locally free sheaf of rank δ over M_{n}(δ), herewith the dimension of M_{n}(δ) is equal to 2r_{n}δ. Mathematical testing of S -duality in this case is examining the functions

Z_{b,n}(τ) = ∑_{k⩾0}c_{top}(H_{n,k})^{2r_{n}}q^{k} ,  where q = e^{2πiτ} , Im τ > 0 ,  (4)

for modular properties. Note that each of the moduli spaces M_{n}(0) is zero-dimensional and parametrizes a unique sheaf, which is a sheaf of sections of an exceptional bundle E_{n} (see [1], or [4], [6] on exceptional bundles). Herewith, E_{−1} = O_{ℙ^{2}}(−2), E_{0} = O_{ℙ^{2}}(−1) and the other bundles E_{n} are obtained from the canonical exact triples

0 ⟶ E_{n−1} ⟶^{ev^{∗}} E_{n} ⊗ Hom(E_{n}, E_{n+1}) ⟶^{ev} E_{n+1} ⟶ 0 ,  (5)

which in view of canonical isomorphism Hom(E_{n}, E_{n+1}) = Hom(E_{n−1}, E_{n})^{∗} give maps of the

form (E_{n}, E_{n+1}) ↦^{L} (E_{n−1}, E_{n}) and (E_{n−1}, E_{n}) ↦^{R} (E_{n}, E_{n+1}) called mutations.

Hypothetically, there should be transformations of moduli spaces M_{n}(δ) concordant with the mutations of exceptional bundles E_{n} = M_{n}(0). These transformations should clarify relations between the functions Z_{b,n}(τ). It seems quite natural to seek such a transformations starting with looking for possible constructions with the sheaves. This paper provides a first move in this direction.

In Section 5 we reformulate description of im Φ for the case when v = (r_{n}, −r_{n−1}, δ), w = (r_{n+1}, −r_{n}, δ′) and n ⩾ 0 (Theorem 5.4) and prove that the map

Ψ : M̃_{v→w} ⟶ M_{w} , (X, Y, [φ]) ⟼ Y

isn’t surjective for almost all values of δ′ (Proposition 5.5).

The author is grateful to S. A. Kuleshov and A. S. Tikhomirov for useful discussions and to Max-Planck-Institut für Mathematik in Bonn for hospitality.

1 Preliminaries

We start with some notation. For a coherent sheaf X over ℙ^{2} put

h^{i}(X) = dim H^{i}(X) ,  χ(X) = ∑_{i}(−1)^{i}h^{i}(X) ,  μ(X) = \frac{c_{1}(X)}{r(X)} ,

the last value is called slope. We represent the image of X in K_{0}(ℙ^{2}) by the vector

v_{X} = (r(X), c_{1}(X), −χ(X)) ,  (6)

which is additive by X. The last component of v_{X} is related with c_{2}(X) by Riemann – Roch theorem

−χ(X) = c_{2}(X) − c_{1}(X)\frac{c_{1}(X) + 3}{2} − r(X) .  (7)

Let Y be also a coherent sheaf over ℙ^{2}. Put

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

Euler characteristic χ(X, Y) is expressed via χ(X) and χ(Y) by Riemann – Roch formula

χ(X, Y) = −r(X)r(Y) − 3r(Y)c_{1}(X) − c_{1}(X)c_{1}(Y) + r(X)χ(Y) + r(Y)χ(X) . (8)

This expression defines non-symmetric bilinear form on ℤ ⊕ ℤ ⊕ ℤ ≅ K_{0}(ℙ^{2}), where the identification is given by (6).

1.1 Remark 1. If X and Y are semistable sheaves such that 0 < μ(Y) − μ(X) < 3, then h^{0}(Y, X) = h^{2}(Y, X) = 0 and h^{1}(Y, X) = −χ(Y, X). In particular, these equalities take place if r(X)c_{1}(Y) − r(Y)c_{1}(X) = 1.

2. For any semi-stable sheaf X with −3 < μ(X) < 0 one has: h^{0}(X) = h^{2}(X) = 0 and −χ(X) = h^{1}(X).

Indeed, the first statement follows immediately from stability properties and Serre duality. Note that the relation r(X)c_{1}(Y) − r(Y)c_{1}(X) = 1 implies the inequality 0 < μ(Y) − μ(X) < 1. The second statement is a particular case of the first one for Y = O_{ℙ^{2}}.

Let v = (r, c_{1}, δ) be a triple of integers such that r > 0, g.c.d.(r, c_{1}) = 1 and the moduli space M_{v} = M(r, c_{1}, δ) is non-empty. Then the dimension of the moduli space is given by the formula

dim M_{v} = 1 − χ(v, v) .  (9)

Indeed, let X be stable sheaf with v_{X} = v, then h^{2}(X, X) = h^{0}(X, X(−3)) = 0 and h^{0}(X, X) = 1, thus h^{1}(X, X) = dim M_{v} = 1 − χ(X, X).

Applying formula (9) to v = (r_{n}, −r_{n−1}, δ) we have:

dim M_{n}(δ) = 2r_{n}δ .

In particular, dim M_{n}(0) = 0 and M_{n}(0) consists of one point E_{n}, which is exceptional vector bundle. Remind that there are two equivalent definitions of exceptional sheaves on ℙ^{2}: a sheaf E is exceptional if

(i) h^{0}(E, E) = 1 and h^{j}(E, E) = 0 for j > 0 ([4], [6]); or

(ii) E is stable and χ(E, E) = 1 ([1]).

Any exceptional sheaf on ℙ^{2} is locally free. The simplest argument is the triviality of Ext^{1}(E, E) and free group action on ℙ^{2}. Besides, this follows from 3.3.

The exceptional bundles E_{n} are known to satisfy the following cohomological conditions: h^{j}(E_{n+1}, E_{n}) = 0 for all j, h^{0}(E_{n}, E_{n+1}) = 3 and h^{j}(E_{n}, E_{n+1}) = 0 for j > 0. On the language of theory of exceptional sheaves, each pair (E_{n}, E_{n+1}) is exceptional and the sequence {E_{n}}_{n∈ℤ} is generated by the pair (E_{−1}, E_{0}) = (O_{ℙ^{2}}(−2), O_{ℙ^{2}}(−1)).

1.2 Lemma. ([9]) Suppose there is an exact triple of sheaves

0 ⟶ X ⟶ X̃ ⟶ Q ⟶ 0 ,

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

a) if X̃ is (semi-)stable, then X is (semi-)stable;

b) if X̃ is torsion-free and X is (semi-)stable, then X̃ is (semi-)stable.

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

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

0 ↓

F ⟶^{id} F
↓  ↓ψ
0 ⟶ X ⟶ X̃ ⟶ Q ⟶ 0
id↓  ↓  ↓
0 ⟶ ker ψ ⟶ X ⟶ G ⟶ Q̃ ⟶ 0
↓  ↓
0  0

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

Suppose r > 1 and X ∈ M^{(p)}(r, c_{1}, δ). From the lemma it follows that taking X to the reflexive span X^{∗∗} defines a map f : M^{(p)}(r, c_{1}, δ) ⟶ M^{(0)}(r, c_{1}, δ − p). The preimage f^{−1}(E) of a point E ∈ M^{(0)}(r, c_{1}, δ −p) is Grothendieck’s quotient scheme Quot(E, p), which parametrizes epimorphisms E ↠ Q modulo automorphisms of Q, where r(Q) = 0, c_{1}(Q) = 0 and h^{0}(Q) = p. It is known that Quot(E, p) for locally free sheaf E is an irreducible [3] projective [7, Th. 2.2.4] scheme of dimension p(r + 1). We have:

dim M^{(p)}(r, c_{1}, δ) = dim M^{(0)}(r, c_{1}, δ − p) + p(r + 1) = 1 − χ(E, E) + p(r + 1) .

From (8) it follows that χ(E, E) − χ(X, X) = 2rp. Therefore, we obtain the following statement.

1.3 Proposition. codim_{M_{v}} M_{v}^{(p)} = p(r − 1) for r > 1. □

Now let us define the moduli space M̃_{v→w}. We are assuming that the vectors v = (r, c_{1}, δ) and w = (r′, c′_{1}, δ′) satisfy (1) and the moduli spaces M_{v} and M_{w} are non-empty. In particular, r and c_{1} are coprime as well as r′ and c′_{1}, so the moduli spaces M_{v} and M_{w} are smooth, projective and carry universal families X and Y, these are coherent sheaves over M_{v} × ℙ^{2} and M_{w} × ℙ^{2} respectively. Denote by π_{v}, π_{w} and π the projections of the product M_{v} × M_{w} × ℙ^{2} onto M_{v} × ℙ^{2}, M_{w} × ℙ^{2} and M_{v} × M_{w}, and put: X̃ = π_{v}^{∗}X, Ỹ = π_{w}^{∗}Y. Suppose z = (X, Y) is a closed point in M_{v} × M_{w} and consider the base change homomorphism

τ^{i}(z) : Ext_{π}^{i}(Ỹ, X̃ ⊗ ω_{π}) ⊗_{M_{v}×M_{w}} ℂ(z) ⟶ Ext_{ℙ^{2}}^{i}(Y, X ⊗ O_{ℙ^{2}}(−3))

for relative Ext-sheaves. Clearly, for i = 3 the last space is trivial, hence, by [13, Th. 1.4], τ^{3}(z) is an isomorphism for all z. It follows that the sheaf Ext_{π}^{3}(Ỹ, X̃ ⊗ ω_{π}) has trivial fibers at all closed points, whence we have: Ext_{π}^{3}(Ỹ, X̃ ⊗ ω_{π}) = 0. Applying the quoted statement again, we conclude that τ^{2}(z) is an isomorphism for all z. By Serre duality we have: Ext_{ℙ^{2}}^{2}(Y, X ⊗ O_{ℙ^{2}}(−3)) = Hom_{ℙ^{2}}(X, Y)^{∗}, therefore the scheme

M̃_{v→w} := ℙ(Ext_{π}^{2}(Ỹ, X̃ ⊗ ω_{π}))

admits the interpretation given in the introduction, i.e. parametrizes the triples (X, Y, [φ]), where φ ∈ P Hom(X, Y). The morphisms Φ : M̃_{v→w} → M_{v} and Ψ : M̃_{v→w} → M_{w} are defined as compositions of the natural morphism ℙ(Ext_{π}^{2}(Ỹ, X̃ ⊗ ω_{π})) ⟶ M_{v} × M_{w} with the projections of the last product onto M_{v} and M_{w} respectively.

2 Properties of pairs of stable sheaves

In this section we give some properties of stable sheaves X, Y on ℙ^{2}, invariants of which satisfy the conditions

| \begin{matrix}r(X) & c_{1}(X) \\ r(Y) & c_{1}(Y) \end{matrix} | = 1 and r(X) < r(Y) .  (10)

Note that all statements of this section hold in a more general situation, namely, for stable sheaves w.r.to an ample divisor H on a smooth projective surface, herewith one should consider c_{1} · H instead of c_{1}.

In particular, all the results can be applied for X ∈ M_{n}(δ) and Y ∈ M_{n+1}(δ′), where n ⩾ 0 (see Section 5).

So, let X and Y be stable sheaves satisfying (10).

2.1 Proposition. Any non-trivial morphism φ : X ⟶ Y is an embedding and the sheaf F = coker φ either is stable or has a torsion.

Proof. Since im φ is a quotient-sheaf of X and a subsheaf of Y , we have 0 < r(im φ) ⩽ r(X) < r(Y), hence μ(X) ⩽ μ(im φ) < μ(Y). Consider the plane with coordinates (r, c_{1}), a sheaf E we will represent by point (r(E), c_{1}(E)) denoted also by E. Herewith, the slope μ(E) = \frac{c_{1}(E)}{r(E)} is equal to the tangent of the angle between r-axis and the line OE.

[figure: r; c_{1}; O; X; B; Z; F; A; Y; r(X); r(F)] Fig. 1.

From (10) it follows that inside the parallelogram XOY Z there is no point with integer coordinates, but the point im φ must belong to the triangle OXA (see Fig. 1). Therefore, there is only possibility (r(im φ), c_{1}(im φ)) = (r(X), c_{1}(X)). But then r(ker φ) = 0, this implies that ker φ = 0 because X has no torsion.

Let us show that if F = coker φ is torsion free then F is stable. Since (r(F), c_{1}(F)) = (r(Y), c_{1}(Y)) − (r(X), c_{1}(X)), the vectors \overrightarrow{OF} and \overrightarrow{OY} form a basis in the integral lattice, hence in the triangle OF B there is no integral points except O and F. Thus, for any quotient sheaf G of F such that 0 < r(G) < r(F) and μ(G) ⩽ μ(F) one has μ(G) < μ(Y) this contradicts stability of Y . Therefore, F is stable. □

It follows that any non-trivial map φ ∈ Hom(X, Y) defines an exact triple

0 ⟶ X ⟶^{φ} Y ⟶ F ⟶ 0 ,  (11)

in which the sheaf F is stable whenever it has no torsion. Let T = T (F) be the torsion subsheaf of F.

2.2 Proposition. The sheaf T has zero-dimensional support and is a subsheaf of Q_{X}. In particular, if X is locally free then T = 0 and F is stable.

Proof. Consider the following commutative diagram, in which F′ is torsion free

\begin{aligned}& & & & & & 0 \\ & & & & & & ↓ \\ & & 0 & & & & T \\ & & ↓ & & & & ↓ \\ 0 & ⟶ & X & ⟶ & Y & ⟶ & F & ⟶ & 0 \\ & & ψ ↓ & & || id & & ↓ \\ 0 & ⟶ & X_{T} & ⟶^{φ′} & Y & ⟶ & F′ & ⟶ & 0 \\ & & ↓ & & & & ↓ \\ & & T & & & & 0 \\ & & ↓ \\ & & 0 \end{aligned}  (12)

We have: r(X) = r(X_{T}) because r(T) = 0. Suppose c_{1}(T) = c_{1}(X_{T}) − c_{1}(X) > 0, then μ(X_{T}) > μ(X) and μ(X_{T}) > μ(Y) since in the triangle OXA (see Fig. 1) there are no integral points except O and X. This contradicts stability of Y . Thus, c_{1}(T) = 0, this means that dim(supp T) = 0.

Now let us show that T ⊂ Q_{X}. We have: r(X) = r(X_{T}), c_{1}(X_{T}) = c_{1}(X), and the embedding ψ from diagram (12) induces isomorphism of reflexive spans X^{∗∗} ≅ X_{T}^{∗∗}. Therefore, we have the diagram

\begin{aligned}& & & & & & 0 \\ & & & & & & ↓ \\ & & 0 & & & & T \\ & & ↓ & & & & ↓ i \\ 0 & ⟶ & X & ⟶ & X^{∗∗} & ⟶ & Q_{X} & ⟶ & 0 \\ & & ψ ↓ & & || & & ↓ \\ 0 & ⟶ & X_{T} & ⟶ & X_{T}^{∗∗} & ⟶ & Q_{X_{T}} & ⟶ & 0 \\ & & ↓ & & & & ↓ \\ & & T & & & & 0 \\ & & ↓ \\ & & 0 \end{aligned}  (13)

from which the required statement follows. □

2.3 Remark 1) The extension

0 ⟶ X ⟶^{ψ} X_{T} ⟶ T ⟶ 0 ,  (14)

in the left column of diagram (13) is defined by the image of i ∈ Hom(T, Q_{X}) from the right column under the map

∂_{T} : Hom(T, Q_{X}) ⟶ Ext^{1}(T, X)

obtained by applying the functor Hom(T, −) to the exact triple

0 ⟶ X ⟶ X^{∗∗} ⟶ Q_{X} ⟶ 0 .  (15)

2) The sheaf X_{T} is stable by Lemma 1.2 since it is torsion free as a subsheaf of X^{∗∗}. Besides, Proposition 2.1 applied to the morphism φ′ : X_{T} ⟶ Y from diagram (12) implies that F′ is stable.

3) It is readily seen that ∂_{T} is an isomorphism, so there is one-to-one correspondence between the pairs (T, i), where T ↪^{i} Q_{X}, modulo automorphisms of T and pairs (X_{T} , ψ) modulo automorphisms of X_{T}. Roughly speaking, subsheaves of Q_{X} are in one-to-one correspondence with stable sheaves X_{T} such that X ⊂ X_{T} ⊂ X^{∗∗}.

2.4 Proposition. Suppose X and F are stable sheaves, | \begin{matrix}r(X) & c_{1}(X) \\ r(F) & c_{1}(F) \end{matrix} | = 1 and the ex-

tension (11) is defined by cocycle e ∈ Ext^{1}(F, X). Then the following statements hold: 1) if X is locally free then Y is stable;

2) if X is not locally free then Y is stable iff e ∉ im ∂, where ∂ : Hom(F, Q_{X}) ⟶ Ext^{1}(F, X) obtained by applying the functor Hom(F, −) to the exact triple (15).

The proof is in [11, Lemma 3.2]. (See similar statement in [9, 4.2].) □

Suppose X and F are as above. Let us describe some necessary and sufficient conditions for existence of stable Y in the middle of (11). Applying the functor Hom(F, −) to the exact triple (15), we have the exact sequence

0 ⟶ Hom(F, Q_{X}) ⟶^{∂} Ext^{1}(F, X) ⟶ Ext^{1}(F, X^{∗∗}) ⟶ Ext^{1}(F, Q_{X}) ⟶ 0 ,

where the zeroes stand for Hom(F, X^{∗∗}) and Ext^{2}(F, X) because of 1.1. We see that ∂ is an embedding, so we will identify Hom(F, Q_{X}) with the subspace im ∂ ⊂ Ext^{1}(F, X) in similar cases. We have:

h^{1}(F, X) − h^{0}(F, Q_{X}) = h^{1}(F, X^{∗∗}) − h^{1}(F, Q_{X}) .

Applying the functor Hom(−, Q_{X}) to the exact triple

0 ⟶ F ⟶ F^{∗∗} ⟶ Q_{F} ⟶ 0

and taking into account the triviality of Ext^{1}(F^{∗∗}, Q_{X}), we get the exact sequence

0 ⟶ Ext^{1}(F, Q_{X}) ⟶ Ext^{2}(Q_{F} , Q_{X}) ⟶ Ext^{2}(F, Q_{X}) ⟶ · · · .

Here Ext^{2}(F, Q_{X}) ≅ Hom(Q_{X}, F ⊗ O_{ℙ^{2}}(−3))^{∗} = 0 and Ext^{2}(Q_{F} , Q_{X}) ≅ Hom(Q_{X}, Q_{F})^{∗}, hence h^{1}(F, Q_{X}) = h^{0}(Q_{X}, Q_{F}). Therefore,

h^{1}(F, X) − h^{0}(F, Q_{X}) = −χ(F, X^{∗∗}) − h^{0}(Q_{X}, Q_{F})

and we obtain the first assertion of the following

2.5 Proposition. Suppose X, F are as in the condition of 2.4; then the following statements are true:

1. There is a stable sheaf Y included in the triple (11) iff h^{0}(Q_{X}, Q_{F}) < −χ(F, X^{∗∗}).

2. There is F_{0} ∈ M(v_{F}) and exact tripe 0 → X → Y_{0} → F_{0} → 0 with stable Y_{0} if and only if −χ(F, X^{∗∗}) > 0.

Proof. To prove 2 note that one can choose F_{0} to be locally free or such that supp Q_{F_{0}} ∩ supp Q_{X} = ∅, then h^{0}(Q_{X}, Q_{F_{0}}) = 0. □

In the rest of this section we examine the question when Y is locally free, still assuming that the sheaves X and Y are stable, satisfy (10) and there is an exact triple of the form

0 ⟶ X ⟶^{φ} Y ⟶ F ⟶ 0 .  (11)

2.6 Proposition. Suppose that F in (11) has no torsion, then the following statements hold:

1. The sheaf Q_{X} is a subsheaf of Q_{Y} . In particular, if Y is locally free than X is locally free too.

2. Let X be locally free, then Y is locally free iff the cocycle e ∈ Ext^{1}(F, X) corresponding to the extension (11) doesn’t belong to the image of the map

Ext^{1}(F′, X) ⟶ Ext^{1}(F, X)  (16)

for any sheaf F′ such that F ⫋ F′ ⊂ F^{∗∗} .

Proof. The embeddings of X and Y into their reflexive spans define the following commutative diagram

\begin{aligned}& & & & & & & & 0 \\ & & & & & & & & ↓ \\ & & & & 0 & & 0 & & T \\ & & & & ↓ & & ↓ & & ↓ \\ & & 0 & ⟶ & X & ⟶ & Y & ⟶ & F & ⟶ & 0 \\ & & & & ↓ & & ↓ & & ↓ \\ & & 0 & ⟶ & X^{∗∗} & ⟶ & Y^{∗∗} & ⟶ & F̃ & ⟶ & 0 \\ & & & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & T & ⟶ & Q_{X} & ⟶ & Q_{Y} & ⟶ & Q & ⟶ & 0 \\ & & & & ↓ & & ↓ & & ↓ \\ & & & & 0 & & 0 & & 0 \end{aligned}  (17)

It follows that T is a torsion sheaf as a subsheaf of Q_{X}. On the other hand, T is a subsheaf of F, whence T = 0, so, we obtain the first statement.

Note that T = T (F) is the torsion subsheaf of F as far as F/T has no torsion being a subsheaf of F̃, which is torsion free by 2.2.

Now let us prove 2. Suppose e is in the image of the map (16), then the triple (11) is included into the following diagram

\begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ 0 & ⟶ & X & ⟶ & Y & ⟶ & F & ⟶ & 0 \\ & & id ↓ & & ↓ & & ↓ \\ 0 & ⟶ & X & ⟶ & Y′ & ⟶ & F′ & ⟶ & 0 \\ & & & & ↓ & & ↓ \\ & & & & Q & ⟶^{id} & Q \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0 \end{aligned}  (18)

where Q is zero-dimensional sheaf. From the middle row of the diagram it follows that Y′ has no torsion as an extension of torsion free sheaves. Hence, the extension in middle column is also non-trivial, thus Y is not locally free.

Now let Y be non-locally-free, than the embeddings of X and Y into reflexive spans induce the diagram of the form (18), in which Y′ = Y^{∗∗}, Q = Q_{Y} and X = X^{∗∗}. It follows that e belongs to the image of the map (16), herewith, F′ has no torsion by 2.2, whence F ⫋ F′ ⊂ F^{∗∗}. □

Note that if we have two sheaves F′ and F′′ such that F ⫋ F′ ⊂ F′′ ⊂ F^{∗∗}, then there are maps

Ext^{1}(F′′, X) ⟶ Ext^{1}(F′, X) ⟶ Ext^{1}(F, X) ,

hence to determine whether is Y locally free or not, it is sufficient to consider minimal (by inclusion) sheaves F′ i.e. non-trivial extensions of the form

0 ⟶ F ⟶ F′ ⟶ O_{x} ⟶ 0 .

Applying to this triple the functor Hom(−, X), one can see that Ext^{1}(F′, X) is a subspace of Ext^{1}(F, X) of codimension r(X). Varying the point x ∈ ℙ^{2}, we obtain two-dimensional family of such subspaces.

2.7 Corollary. Suppose F in (11) has no torsion, then for r(X) > 2 there is an open subset in Ext^{1}(F, X), elements of which correspond to the triples (11) with locally free Y ’s. □

Now assume that we have arbitrary F i.e. F can have a torsion. According to 2.2, the torsion subsheaf T = T (F) ⊂ F is also a subsheaf of Q_{X}.

2.8 Proposition. The sheaf Y in (11) can be locally free only if T (F) = Q_{X}.

Proof. If Y is locally free, then in (17) one has: Y = Y^{∗∗} and Q_{Y} = 0. □

Therefore, if F has a torsion, then to study when a locally free Y exists in the middle of (11) one can “get rid of torsion” by considering the diagram (12) with T = Q_{X}, X_{T} = X^{∗∗}, and apply Proposition 2.6 to the triple 0 → X^{∗∗} → Y → F′ → 0, where F′ = F/T (F).

3 The image of Φ

First we recall the construction of bound for invariants of stable sheaves over ℙ^{2} [1]. For α ∈ ℚ let r_{α} be the smallest r > 0 such that rα ∈ ℤ and put

Δ_{α} = \frac{1}{2} (1 − \frac{1}{r_{α}} ) .

For a pair (α, β) of rational numbers such that 3 + α − β ≠ 0 define a new rational number

α ▪ β = \frac{α + β}{2} + \frac{Δ_{β} − Δ_{α}}{3 + α − β} .

Consider the following map ε : \{\frac{p}{2^{q}} |p, q ∈ ℤ, q ⩾ 0\} → ℚ defined inductively:

ε(n) = n for n ∈ ℤ; ε (\frac{2p + 1}{2^{q+1}}) = ε (\frac{p}{2^{q}} ) ▪ ε (\frac{p + 1}{2^{q}}) .

It is proved in [1] that 𝔊 = im ε ⊂ ℚ is exactly the set of slopes of exceptional bundles and each exceptional bundle is uniquely determined by its slope.

In [1, Theorem B] necessary and sufficient conditions are proved for existence of stable sheaves on ℙ^{2} with given topological invariants. Here we reformulate this result in terms of our coordinates (r, c_{1}, −χ).

3.1 Theorem. ([1]) Let v = (r, c_{1}, δ) be triple of integers with r > 1. Suppose α ∈ 𝔊 is the largest element such that r_{α} ⩽ \frac{r}{2} and α ⩽ \frac{c_{1}}{r}, and β ∈ 𝔊 is the smallest element such that r_{β} ⩽ \frac{r}{2} and \frac{c_{1}}{r} ⩽ β. Then M(r, c_{1}, δ) ≠ ∅ iff

−χ(v, E_{α}) ⩾ 0 and − χ(E_{β}, v) ⩾ 0 ,  (19)

where E_{α} and E_{β} are the exceptional bundles with the slopes α and β respectively. □

Applying formula (8), we see that

−χ(v, E_{α}) ⩾ 0 ⟺ δ ⩾ −r − (3 + α)c_{1} + \frac{r}{r_{α}} χ(E_{α})

and

−χ(E_{β}, v) ⩾ 0 ⟺ δ ⩾ −r − β(3r + c_{1}) + \frac{r}{r_{β}} χ(E_{β}) .

Therefore, there is a unique integer δ_{min}(v) such that the system of conditions (19) is equivalent to the inequality δ ⩾ δ_{min}(v). Clearly, δ_{min}(v) depends only on r and c_{1}.

3.2 Definition. We call vector v = (r, c_{1}, δ) minimal if δ = δ_{min}(v). In other words, v is minimal iff there is a stable sheaf X with v_{X} = v, but there is no stable sheaf X′ with the same rank and c_{1} and −χ(X′) < δ. Also, we call a sheaf X minimal if X is stable and v_{X} is minimal. Put δ_{min}(X) := δ_{min}(v_{X}).

Remark 1.1 implies that if we pose the restriction −3r < c_{1} < 0, then δ_{min}(v) ⩾ 0. In fact, in this case for any stable sheaf X with v_{X} = v we have: δ = h^{1}(X). Other simple properties of minimal sheaves are collected in the following statement.

3.3 Proposition. 1. Any minimal sheaf is locally free. 2. Exceptional bundles are minimal.

3. For r > 1 the maximal number p of the stratum M_{v}^{(p)} ⊂ M_{v} is N = δ − δ_{min}(X).

Proof. If X is stable but non-locally-free, then X^{∗∗} is stable by 1.2 and −χ(X^{∗∗}) = −χ(X)− h^{0}(Q_{X}) < −χ(X), whence X isn’t minimal. This proves 1.

The assertion 2 is proved in [1], but we give a simple argument in our notation. Exceptional bundles are stable [4]. Let E be exceptional bundle with v_{E} = (r, c_{1}, δ), F be stable sheaf with v_{F} = (r, c_{1}, δ′), then dim M_{E} = 0 and

dim M_{F} = dim M_{F} − dim M_{E} = χ(E, E) − χ(F, F) = 2r(δ′ − δ) ⩾ 0 .

Thus, δ′ ⩾ δ.

Now let us prove 3. Suppose X is a stable sheaf with v_{X} = v; then using (15), we have: −χ(X^{∗∗}) = δ − h^{0}(Q_{X}) ⩾ δ_{min}(v), thus p = h^{0}(Q_{X}) ⩽ N. On the other hand, the maximal value p = N is achieved: one can take a stable bundle E with v_{E} = (r, c_{1}, δ_{min}(v)); then the kernel of any epimorphism E ↠ Q, where r(Q) = c_{1}(Q) = 0 and h^{0}(Q) = N, is a stable sheaf X′ ∈ M_{v}^{(N)}. □

3.4 Lemma. Let X and F be stable sheaves such that at least one of them isn’t minimal and 0 < μ(F) − μ(X) < 3. Then Ext^{1}(F, X) ≠ 0.

Proof. Suppose X isn’t minimal, then there is a stable sheaf X′ with the same rank and c_{1} as X and −χ(X′) < −χ(X). We have: μ(X) = μ(X′), this implies that Hom(F, X′) = Ext^{2}(F, X′) = 0, whence −χ(F, X′) = h^{1}(F, X′) ⩾ 0. Comparing the expressions for χ(F, X) and χ(F, X′) by (8), we obtain:

χ(F, X) − χ(F, X′) = r(F)(χ(X) − χ(X′)) < 0 .

Thus, −χ(F, X) = h^{1}(F, X) > 0. The case of non-minimal F is quite similar. □

Observe that for minimal sheaves X and F having invariants satisfying the inequality in the assumption of lemma, Ext^{1}(X, F) can be either trivial or not. Let us give some examples.

3.5 Example 1) X = E_{n}, F = E_{n+1}. We have: Ext^{1}(F, X) = 0.

2) X = E_{n}, n ⩾ 0, F is the cokernel of a non-zero map E_{n} → E_{n+1} (remind that h^{0}(E_{n}, E_{n+1}) = 3). From the proof of 5.3 it follows that F is minimal. Clearly, we have non-trivial extension

0 ⟶ X ⟶ E_{n+1} ⟶ F ⟶ 0 ,

hence h^{1}(F, X) ≠ 0. It is easily shown that Ext^{1}(F, X) is one-dimensional in this example.

In the examples 1) and 2) at least one of moduli spaces M_{v} and M_{u} is zero-dimensional. The next example is less trivial.

3) Let X and F be defined from the exact triples:

0 ⟶ E_{n} ⟶ E_{n+1} ⟶ X ⟶ 0 , 0 ⟶ E_{n−1} ⟶ E_{n} ⟶ F ⟶ 0 ,

where n ⩾ 2. Using the equality χ(E_{i+1}, E_{i}) = 0, it is easy to calculate that −χ(F, X) = h^{1}(F, X) = 3 (χ(E_{n−1}, E_{n+1}) = 8 follows from (5)).

Note that in all the examples above one has | \begin{matrix}r(X) & c_{1}(X) \\ r(F) & c_{1}(F) \end{matrix} | = 1.

The following result describes the image of Φ and necessary and sufficient conditions for emptiness of M̃_{v→w}.

3.6 Theorem. Suppose that the triples of integers v = (r, c_{1}, δ) and w = (r′, c′_{1}, δ′) satisfy (1), 0 < r < r′ and the moduli spaces M_{v} and M_{w} are non-empty. Put N = δ − δ_{min}(v), p_{0} = max{0, δ − δ′ + δ_{min}(w − v)}, v_{0} = (r, c_{1}, δ − p_{0}) and u_{0} = w − v_{0}. Then the image of the map Φ : M̃_{v→w} ⟶ M_{v} is described as follows:

1) If p_{0} < N, then im Φ is the union of strata M_{v}^{(p)}, where p_{0} ⩽ p ⩽ N.

2) If p_{0} = N and χ(u_{0}, v_{0}) ≠ 0, then im Φ = M_{v}^{(p_{0})}.

3) If p_{0} = N and χ(u_{0}, v_{0}) = 0, or p_{0} > N, then the moduli space M̃_{v→w} is empty.

Proof. Suppose X ∈ M_{v}^{(p)}, this means exactly that h^{0}(Q_{X}) = p.

First let us show that if X ∈ im Φ then p_{0} ⩽ p ⩽ N. According to 2.1 and 2.2, any nonzero map φ : X ⟶ Y induces a commutative diagram (12). We have: p ⩾ h^{0}(T) = χ(T). Since

χ(T) = χ(Y) − χ(X) − χ(F′) = −δ′ + δ − χ(F′) ,

we get: p ⩾ δ − δ′ − χ(F′). The sheaf F′ is stable and has invariants v_{F′} = w − v − v_{T}, hence

−χ(F′) ⩾ δ_{min}(w − v − v_{T}) = δ_{min}(w − v)

(remind that v_{T} = (0, 0, −h^{0}(T)) and δ_{min} depends on the two first coordinates only). Therefore, p ⩾ δ − δ′ + δ_{min}(w − v). By the definition, p ⩾ 0, so, p ⩾ p_{0}. From assertion 3 of Proposition 3.3 it follows that p ⩽ N.

In particular, we have proved that for p_{0} > N the moduli space M̃_{v→w} is empty.

Now assume that p_{0} ⩽ p ⩽ N. Consider a subsheaf T ⊂ Q_{X} such that p_{0} ⩽ h^{0}(T). We have:

δ_{min}(w − v) ⩽ δ′ − δ + h^{0}(T) .

Thus, the moduli space M(w−v−v_{T}) is non-empty. Take a stable sheaf F′ ∈ M(w−v−v_{T}). We have: −χ(F′) = δ′ − δ + h^{0}(T), and F′ is minimal iff −χ(F′) = δ_{min}(w − v). This is equivalent to the equality h^{0}(T) = δ − δ′ + δ_{min}(w − v). Therefore, F′ is minimal if and only if h^{0}(T) = p_{0} since h^{0}(T) is non-negative.

Let X_{T} be the stable sheaf constructed via T ↪ Q_{X} as described in 2.3. We have: X_{T}^{∗∗} ≅ X^{∗∗}, and X^{∗∗} is minimal whenever p = N because

−χ(X^{∗∗}) = δ − p ⩾ δ − N = δ_{min}(v) .

Consider the following cases.

1) If p_{0} < N, then for p < N the sheaf X^{∗∗} isn’t minimal and for p = N choose T ⊂ Q_{X} such that h^{0}(T) > p_{0} i.e. F′ isn’t minimal. By 3.4, we have: h^{1}(F′, X^{∗∗}) > 0, so from 2.5 it follows that there is a stable sheaf Y being represented by an extension

0 ⟶ X_{T} ⟶^{φ_{T}} Y ⟶ F′ ⟶ 0 .

Clearly the composition map X ↪ X_{T} ⟶^{φ_{T}} Y is non-zero, so X ∈ im Φ.

2) If p_{0} = p = N, then the both sheaves, X^{∗∗} and F′, are minimal. For the case χ(u_{0}, v_{0}) = χ(F′, X^{∗∗}) < 0 we can apply the above reasoning with T = Q_{X} and X_{T} = X^{∗∗}. In the case χ(u_{0}, v_{0}) = χ(F′, X^{∗∗}) = 0 we have: Ext^{1}(F′, X^{∗∗}) = 0. We claim that Hom(X, Y) = 0 for all Y ∈ M_{Y} , otherwise a non-trivial morphism φ : X → Y induces a diagram of the form (12), in which F′ is minimal, T = Q_{X}, and X_{T} = X^{∗∗} is also minimal. But then from the diagram it follows that Y = F′ ⊕ X^{∗∗}, this contradicts the stability. □

3.7 Corollary. The following statements are equivalent: (i) M_{v}^{(0)} is in the image of Φ; (ii) Φ is surjective; (iii) the moduli space M(w − v) is non-empty.

Clearly, (i) and (ii) are equivalent to the equality p_{0} = 0, i.e. to the inequality δ′ − δ ⩾ δ_{min}(w − v), which is equivalent to (iii). □

3.8 Proposition. Assume that X ∈ M_{v}^{(N)}, Y ∈ M_{w} and w − v is minimal. If for v′ = (r, c_{1}, δ_{min}(v)) one has χ(w − v, v′) = 0, then the cokernel F of any non-trivial morphism φ : X ⟶ Y has a torsion.

Proof. Suppose F has no torsion, then F is stable by 2.1. We have: v_{F} = w − v, so F is minimal and therefore locally free. The condition X ∈ M_{v}^{(N)} means exactly that X^{∗∗} is minimal, in particular, v′ = v_{X^{∗∗}}. We have: h^{1}(F, X^{∗∗}) = −χ(w − v, v′) = 0, whence it follows that the map Hom(F, Q_{X}) ⟶ Ext^{1}(F, X) is an isomorphism. Therefore, Y isn’t stable according to Proposition 2.4. This contradiction completes the proof. □

To conclude this section, we derive from the proved statements some property of the bound of invariants of stable sheaves on ℙ^{2}, which can be regarded as a kind of non-strict local convexity. Of course, the better understanding of this bound can be obtained from remarkable work [1], where the description of the bound is given in terms of another coordinates (slope and discriminant).

3.9 Proposition. Let v = (r, c_{1}, δ) and u = (r′′, c′′_{1}, δ′′) be two minimal vectors such that

| \begin{matrix}r & c_{1} \\ r′′ & c′′_{1} \end{matrix} | = 1 and −χ(u, v) > 0. Then the following inequality holds:

δ_{min}(v + u) ⩽ δ_{min}(v) + δ_{min}(u) .  (20)

Proof. The moduli spaces M(v) and M(u) are non-empty by the definition, so let us consider stable sheaves X and F such that v_{X} = v and v_{F} = u. These are locally free sheaves because of minimality. Using 1.1, we have: h^{1}(F, X) = −χ(u, v) > 0, hence, there is a non-trivial extension of the form (11), where Y has the invariants v_{Y} = v + u and is stable according to 2.4. Therefore,

δ_{min}(v + u) ⩽ −χ(Y) = −χ(X) − χ(F) = δ_{min}(v) + δ_{min}(u)

□

It is easy to show that if v and u don’t correspond to exceptional bundles, then the both values −χ(v + u, v) and −χ(u, v + u) are positive, so one can apply the proposition to the pairs of vectors (v, w) and (w, u), where w = (r + r′′, c_{1} + c′′_{1}, δ_{min}(v + u)).

4 Fibers of the morphism Φ

In this section we apply technical statements of Section 2 to studying the preimages of points in M_{v} under the map Φ : M̃_{v→w} ⟶ M_{v}, where v = (r, c_{1}, δ) and w = (r′, c′_{1}, δ′) satisfy (1) and r < r′. Recall that M̃_{v→w} parametrizes the triples (X, Y, [φ]), where X ∈ M_{v}, Y ∈ M_{w} and [φ] ∈ P (Hom(X, Y)). The map Φ takes a triple (X, Y, [φ]) to X.

Put u = w − v. Fix a sheaf X ∈ M_{v} and consider two cases.

I. X is locally free.

According to 2.1 and 2.2, any non-trivial morphism φ : X ⟶ Y defines exact triple

0 ⟶ X ⟶^{φ} Y ⟶ F ⟶ 0 ,  (11)

where F is stable. Hence, the pair

(Y, φ) , where Y ∈ M_{w} , 0 ≠ φ ∈ Hom(X, Y)  (21)

defines a pair

(F, e) , where F ∈ M_{u} and 0 ≠ e ∈ Ext^{1}(F, X)  (22)

Conversely, any such a pair (F, e) determines an exact triple of the form (11), in which Y is stable by 2.4, i.e. the pair (F, e) gives a pair (Y, φ) of the form (21).

Consider natural equivalence relations on the sets of pairs (21) and (22): (Y, φ) ∼ (Y′, φ′) whenever there is an isomorphism α : Y → Y′ such that φ′ = α ∘ φ, and (F, e) ∼ (F′, e′) iff there is an isomorphism β : F ⟶ F′ such that β̃(e′) = e, where β̃ : Ext^{1}(F′, X) ⟶ Ext^{1}(F, X) is induced by β.

It is easily shown by standard arguments of homological algebra that the correspondence between the sets of pairs (21) and (22) preserves the equivalence relations. Since a stable sheaf has endomorphisms only of the form λ · id ([14]), the pairs (Y, φ) and (Y, φ′) are equivalent if and only if φ′ = λφ, λ ≠ 0. Also, the pairs (F, e) and (F, e′) are equivalent iff e′ = μe, μ ≠ 0. Therefore, the set of pairs (Y, [φ]), where Y ∈ M_{w} and [φ] ∈ P (Hom(X, Y)) is identified with projectivization of “bundle over M_{u} with the fiber Ext^{1}(F, X)”, which is constructed by usual way via universal family.

From (1) it follows that r(F) = r′ − r and c_{1}(F) = c′_{1} − c_{1} are coprime, hence, as it was pointed in the introduction, M_{u} is smooth and admits a universal family F ⟶ M_{u} × ℙ^{2}. Let π : M_{u} × ℙ^{2} → ℙ^{2} be the projection. Consider the sheaf

E_{X} = (R^{1}π_{∗}F ⊗ π^{∗}X^{∗}(−3))^{∗} .

Note that h^{1}(F, X) = −χ(F, X) by 1.1, hence E_{X} is locally free. Using the isomorphisms H^{1}(X^{∗}(−3) ⊗ F)^{∗} = Ext^{1}(X(3), F)^{∗} = Ext^{1}(F, X), one can interpret E_{X} as vector bundle on M_{F} with the fiber Ext^{1}(F, X) at a point F. Therefore, we obtain the following

4.1 Proposition. The preimage Φ^{−1}(X) of a locally free sheaf X coincides with projectivization of the bundle E_{X} on M_{u}. □

Note that the preimage Φ^{−1}(M_{v}^{(0)}) of the moduli space of bundles can be described globally. Denote by π_{3}, π_{2} and π_{1} the projections of M_{v}^{(0)} × M_{u} × ℙ^{2} on M_{v}^{(0)} × M_{u}, M_{v}^{(0)} × ℙ^{2} and M_{u} × ℙ^{2} respectively. Let X ⟶ M_{v}^{(0)} × ℙ^{2} be restriction of the universal family, then according to the construction described above we obtain the following result.

4.2 Theorem. Suppose that v = (r, c_{1}, δ) and w = (r′, c′_{1}, δ′) satisfy (1), r < r′ and the moduli spaces M_{v}, M_{w} and M_{u} = M_{w−v} are non-empty. Then Φ^{−1}(M_{v}^{(0)}) = P (E), where E = R^{1}π_{3∗}(π_{2}^{∗}X^{∗} ⊗ π_{1}^{∗}F ⊗ ω_{π_{3}}) is a vector bundle over M_{v}^{(0)} × M_{u} with the fiber Ext^{1}(F, X).

4.3 Corollary. dim Φ^{−1}(M_{v}^{(0)}) = 1 − χ(w, w) + χ(v, u).

Proof. From the theorem it follows that the dimension of Φ^{−1}(M_{v}^{(0)}) is equal to

dim M_{v}^{(0)} + dim M_{u} − χ(F, X) − 1 = 1 − χ(v, v) − χ(u, u) − χ(u, v) .

Besides, w = v + u, this implies the required formula. □

II. X is non-locally-free.

We can assume that X ∈ M_{v}^{(p)}, where p > 0. Proposition 2.1 implies that any non-trivial morphism φ : X ⟶ Y induces the exact triple (11), but unlike in the previous case the sheaf F can have a torsion. Herewith, by 2.2, the torsion subsheaf T = T (F) ⊂ F is a subsheaf of Q_{X}. Hence, we have the diagram (12), in which F′ is stable. Note that the sheaf X_{T} is stable also by Lemma 1.2 because it has no torsion being a subsheaf of a stable sheaf Y . So, any pair of the form (21) defines the following data

• a subsheaf T ↪^{i} Q_{X}, which determines stable sheaf X_{T} as described in 2.3

• a stable sheaf F′ ∈ M(u − v_{T}) =: M′

• a cocycle e ∈ Ext^{1}(F′, X_{T}) \ Hom(F′, Q_{X_{T}})

Here we identify the space Hom(F′, Q_{X_{T}}) with its image under the inclusion Hom(F′, Q_{X_{T}}) ↪ Ext^{1}(F′, X_{T}) induced by the exact triple

0 ⟶ X_{T} ⟶ X_{T}^{∗∗} ⟶ Q_{X_{T}} ⟶ 0 .

Note that v_{T} = (0, 0, −h^{0}(T)), hence subtracting this vector adds h^{0}(T) to the third component leaving the rank and c_{1} unchanged.

Conversely, this data defines two left columns and the row containing F′ of (12), hence, the whole diagram is restored. This diagram gives the pair (Y, φ) of type (21), where φ = φ′ ∘ ψ.

We leave for the reader checking that the correspondence between the pairs (Y, φ) of type (21) and the data described above preserves natural equivalence relations. Thus, we obtain the following statement.

4.4 Proposition. The preimage of non-locally-free sheaf X admits the following description

Φ^{−1}(X) = ⋃_{T ⊂Q_{X}} ( ⋃_{F′∈M(u−v_{T})} P (Ext^{1}(F′, X_{T})) \ P (Hom(F′, Q_{X_{T}}))) .

5 The case of moduli spaces M_{n}(δ)

Throughout this section we assume that v = (r_{n}, −r_{n−1}, δ) and w = (r_{n+1}, −r_{n}, δ′), where n ⩾ 0 and the sequence {r_{n}}_{n∈ℤ} is defined by relations (2). In other words, M_{v} = M_{n}(δ) and M_{w} = M_{n+1}(δ′). We reformulate here the description of the image of Φ : M̃_{v→w} ⟶ M_{v} given in 3.6 and prove that for almost all values of δ′ the map Ψ : M̃_{v→w} ⟶ M_{w} is non-surjective.

First of all we prove formula (3). Denote the determinant in the left hand side by Δ_{n}. We have: r_{1} = 2 and Δ_{0} = 1. Besides,

| \begin{matrix}r_{n−1} & r_{n} \\ r_{n} & r_{n+1} \end{matrix} | = | \begin{matrix}3r_{n} − r_{n+1} & r_{n} \\ 3r_{n+1} − r_{n+2} & r_{n+1} \end{matrix} | = | \begin{matrix}−r_{n+1} & r_{n} \\ −r_{n+2} & r_{n+1} \end{matrix} | = | \begin{matrix}r_{n} & r_{n+1} \\ r_{n+1} & r_{n+2} \end{matrix} | .

Thus, Δ_{n} = Δ_{n+1} for all n and (3) is proved.

5.1 Lemma. The sequence {r_{n}} is increasing for n ⩾ 0 and decreasing for n ⩽ −1. In particular, all r_{n} > 0.

Proof. Clearly, the sequence is symmetric w.r. to the “middle between r_{−1} and r_{0}”, i.e. r_{n} = r_{−n−1}, so, it is enough to show that r_{n+1} > r_{n} > 0 for n ⩾ 0. If n = 0, then this inequality certainly holds: r_{1} = 2, r_{0} = 1. Using the induction hypothesis, we have: r_{n+2} = 3r_{n+1} − r_{n} > 2r_{n+1} > r_{n+1}, n > 0. □

5.2 Corollary. 1) 0 < r_{n−1} < 3r_{n} for all n; 2) r_{n+1} > 2r_{n} for n > 0. □

The first inequality implies that −3 < \frac{−r_{n−1}}{r_{n}} < 0, hence, for any X ∈ M_{n}(δ) by 1.1 we have: −χ(X) = h^{1}(X).

5.3 Lemma. 1. δ_{min}(v) = δ_{min}(w) = 0. 2. δ_{min}(w − v) = \{ \begin{matrix}0 & for n < 0 \\ −1 & for n = 0 \end{matrix} .

Proof. Suppose X ∈ M_{v} = M_{n}(δ), then δ = h^{1}(X) ⩾ 0. On the other hand, the exceptional bundle E_{n} is stable and has −χ(E_{n}) = 0. This proves 1.

Consider any non-trivial morphism φ : E_{n} ⟶ E_{n+1} (remind that h^{0}(E_{n}, E_{n+1}) = 3). From 2.1 and 2.2 it follows that φ is an embedding and F_{0} = coker φ is a stable sheaf with the vector of invariants v_{F_{0}} = (r_{n+1} − r_{n}, −r_{n} + r_{n−1}, 0). We have: μ(E_{n+1}) < μ(F_{0}) = \frac{−r_{n}+r_{n−1}}{r_{n+1}−r_{n}} < 0 for n > 0 and μ(F_{0}) = 0 for n = 0. From 1.1 we obtain in the former case that for any semi-stable sheaf F with μ(F) = μ(F_{0}) one has: −χ(F) = h^{1}(F) ⩾ 0, and the assertion follows from the equality h^{1}(F_{0}) = 0. For n = 0 we have: v_{F_{0}} = (1, 0, 0). The trivial rank 1 bundle is exceptional, therefore, minimal, and one has v_{O_{ℙ^{2}}} = (1, 0, −1). This completes the proof. □

Note that any torsion free sheaf F_{0} such that v_{F_{0}} = (1, 0, 0) is nothing else but the ideal sheaf of a point x ∈ ℙ^{2}. Indeed, the reflexive span F_{0}^{∗∗} = O_{ℙ^{2}} and c_{2}(F_{0}) = h^{0}(Q_{F_{0}}) = 1. Therefore, M(1, 0, 0) = ℙ^{2}, this is a partial case of the following statement proved in [2]:

M(r_{n+1} − r_{n}, −r_{n} + r_{n−1}, 0) ≅ ℙ^{2} .

Lemma 5.3 permits to give a simpler formulation of Theorem 3.6 in our case. We have: p_{0} = max{0; δ − δ′} and N = δ, hence always p_{0} ⩽ N. Thus, we have two cases:

1) p_{0} = 0, δ ⩽ δ′, v_{0} = v = (r_{n}, −r_{n−1}, δ), u_{0} = w − v = (r_{n+1} − r_{n}, −r_{n} + r_{n−1}, δ′ − δ);

2) p_{0} = δ − δ′ > 0, v_{0} = (r_{n}, −r_{n−1}, δ′), u_{0} = (r_{n+1} − r_{n}, −r_{n} + r_{n−1}, 0).

The equality p_{0} = N is possible only in the second case for δ′ = 0, herewith, −χ(u_{0}, v_{0}) = 1 + (r_{n+1} − r_{n})δ′ is always positive. Therefore, Theorem 3.6 has the following form.

5.4 Theorem. Suppose v = (r_{n}, −r_{n−1}, δ), w = (r_{n+1}, −r_{n}, δ′), and n ⩾ 0; then the map Φ : M̃_{v→w} → M_{v} is surjective for δ ⩽ δ′ and im Φ = ⋃_{p=δ−δ′}^{δ} M_{v}^{(p)} for δ > δ′. In particular, if δ = δ′, then im Φ = M_{v}^{(δ)}. □

5.5 Proposition. Suppose v = (r_{n}, −r_{n−1}, δ) and w = (r_{n+1}, −r_{n}, δ′); then for n = 0, δ′ ⩾ 3, for n = 1, δ′ ⩾ 2 and for n ⩾ 2, δ′ ⩾ 1 the map Ψ : M̃_{v→w} ⟶ M_{w} isn’t surjective.

Proof. The idea is as follows: we represent im Ψ ∩ M_{w}^{(0)} as a union of some schemes and calculate dimensions of their preimages under Ψ. The largest dimension turns to be strictly less than dim M_{w}^{(0)} for the specified values of n and δ′.

Let z = (X, Y, [φ]) be closed point in M̃_{v→w} such that Ψ(z) ∈ M_{w}^{(0)}. This means that Y is locally free and φ : X ⟶ Y is a non-trivial morphism. Put as usual F = coker φ. By 2.8 we have: Q_{X} = T (F) =: T and there is diagram (12), where X_{T} = X^{∗∗}. It follows that φ is a composition of the canonical embedding X ↪ X^{∗∗} and a non-trivial map φ̂ : X^{∗∗} ⟶ Y . Thus, we have:

im Ψ ∩ M_{w}^{(0)} = ⋃_{k=0}^{δ} Λ_{k} , Λ_{k} = Ψ_{k}(Φ_{k}^{−1}(M_{v_{k}}^{(0)})) ,

where v_{k} = (r_{n}, −r_{n+1}, k) and Φ_{k}, Ψ_{k} are the maps from M̃_{v_{k}→w} similar to Φ and Ψ:

\begin{matrix}M̃_{v_{k}→w} & ⟶^{Ψ_{k}} & M_{w} \\ Φ_{k} & ↓ \\ M_{v_{k}} \end{matrix}  (X̂, Y, [φ̂]) ⟼^{Ψ_{k}} Y ; Φ_{k} ↓ ; X̂

By 4.3, we have:

dim Φ_{k}^{−1}(M_{v_{k}}^{(0)}) = 1 − χ(w, w) + χ(v_{k}, w − v_{k}) = 2r_{n+1}δ′ + 2 − r_{n}(δ′ − k) − (r_{n+1} − r_{n})k .

This is linear function on k. For n ⩾ 1 we have:

max_{0⩽k⩽δ} dim Φ_{k}^{−1}(M_{v_{k}}^{(0)}) = dim Φ_{0}^{−1}(M_{v_{0}}^{(0)}) = 2r_{n+1}δ′ + 2 − r_{n}δ′ .

For n = 1, δ′ ⩾ 2 and for n ⩾ 2, δ′ ⩾ 1 this maximum is less than dim M_{w} = 2r_{n+1}δ′, hence im Ψ ∩ M_{w}^{(0)} doesn’t coincide with M_{w}^{(0)} and Ψ isn’t surjective.

For n = 0 we have: dim Φ_{k}^{−1}(M_{v_{k}}^{(0)}) = 3δ′ + 2, this value is independent of k and is less than dim M_{w} = 4δ′ whenever δ′ ⩾ 3, in this case Ψ is also non-surjective. □

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