GL_{4}-EQUIVARIANT DESINGULARIZATION OF THE UNIVERSAL CURVE OVER THE ELLINGSRUD–PIENE–STRØMME COMPACTIFICATION OF THE SPACE OF TWISTED RATIONAL CUBIC CURVES IN ℙ^{3}.

KUZNETSOV A.G.

1. Introduction

Let W be 4-dimensional complex vector space. Let 𝔚^{∘} be the space of smooth rational twisted cubic curves in the projective space ℙ(W) and denote by 𝔚 its Ellingsrud–Piene–Strømme compactification via

nets of determinantal quadrics (see [2]). Let I^{∘} ⊂ ℙ(W) × 𝔚^{∘} be the universal curve, i.e. the point-curve

incidence subspace I^{∘} = {(x, C) | x ∈ C} and let I be its closure in the space ℙ(W) × 𝔚. It is easy to see that I is singular. This paper introduces a certain desingularization π̃ : M̃ → I of the space I. Moreover, some exact sequences, relating the pull-backs of the universal bundles over 𝔚 and ℙ(W) to M̃ are found.

Now we describe briefly our construction. Both ℙ(W) and 𝔚 are realized as moduli spaces of stable representations of the quiver Q_{small} = • ⟶^{W} • of dimensions (1, 3) and (2, 3) correspondingly. We consider the quiver Q = • ⟶^{ℂ} • ⟶^{W} • and the moduli space of its stable (1, 2, 3)-dimensional representations. The latter moduli spaces, as was pointed by A. King in [3], depend on the choice of the character of the group GL(1) × GL(2) × GL(3). When the character varies these moduli spaces transform. This phenomenon was investigated by M. Thaddeus in [7].

In particular, for certain characters χ_{0} and χ_{1} the corresponding moduli spaces are nothing but ℙ(W) and 𝔚. Now we consider the family of fractional characters χ_{t} = (1 − t)χ_{0} + tχ_{1}, t ∈ ℚ and the corresponding moduli spaces M_{t} of χ_{t}-semistable representations of Q. We describe the transformations of M_{t} that take place while t varies from t = 0 to t = 1 and define the space M̃ as the fiber product of intermediate moduli spaces (see Corollary 4.6).

The space M̃ may be applied to investigation of the derived category of coherent sheaves on the space 𝔚 (see the discussion after the Corollary 5.7).

Also it can be applied to description of the derived category of coherent sheaves on the Fano threefolds V_{22}. The first step in this direction was made in [4] where an exceptional collection of vector bundles on these threefolds was constructed. But it appears to be much more difficult problem to prove that this collection is full, i.e. generates the derived category. However it is possible to do it using the space M̃. This will be done in the forthcoming paper.

The paper is organized as follows. In the Section 2 we recall some basic definitions and facts about representations of quivers and their moduli spaces. In Section 3 we describe the projective space ℙ(W) and the Ellingsrud–Piene–Strømme compactification 𝔚 of the space of twisted cubics in ℙ(W) as a certain moduli spaces of (1, 2, 3)-dimensional representations of the quiver Q and introduce the family M_{t} of moduli spaces such that M_{0} = ℙ(W) and M_{1} = 𝔚. We also describe the set of critical values t_{i} in which transformations of the moduli spaces M_{t} occur, namely we show that the set of critical values consists of t_{0} = 0, t_{1} = 2/3 and t_{2} = 1. In Section 4 we describe the transformations that occur at the points t_{0} and t_{2}.Namely choosing 1/3 and 5/6 as typical representatives in the intervals (t_{0}, t_{1}) and (t_{1}, t_{2}) we show that M_{1/3} is a ℙ^{10}-fibration over ℙ(W) = M_{0} and M_{5/6} is a ℙ^{1}-fibration over 𝔚 = M_{1} (see Propositions 4.1 and 4.2). Afterwards we apply the Thaddeus theory to the transformation which occur at the point t_{1} = 2/3 and get its coarse description. We define here the space M̃ as the fiber product M_{1/3} ×_{M_{2/3}} M_{5/6} and construct the desingularization map π̃ : M̃ → I. In Section 5 we describe the last transformation in details. After that we deal with universal families of representations of Q over M_{1/3} and M_{5/6}. We construct an exact sequence including the pull-backs to M̃ of these universal families. This sequence is important for applications to Fano threefolds V_{22}.

I was partially supported by CRDF grant RM1-265.

The main results of the paper are Theorem 4.7 and Propositions 5.1, 5.2 describing the space M̃.

Notations.

1. Spaces:

W — fixed 4-dimensional vector space; 𝔚^{∘} — space of smooth rational twisted cubic curves in ℙ(W);

𝔚 — compactification of 𝔚^{∘} via nets of determinantal quadrics (see [2]);

I^{∘} ⊂ ℙ(W) × 𝔚^{∘} — the universal curve;

I — closure of I^{∘} in ℙ(W) × 𝔚; ℜ = ℜ(α) = (A_{1}^{∗} ⊗A_{2})⊕(W^{∗} ⊗A_{2}^{∗} ⊗A_{3}) — vector space parameterizing α-dimensional representations

of Q, see 2.1; ℜ_{t}^{ss} ⊂ ℜ — subset of χ_{t}-semistable representations; ℜ^{−} = ℜ_{2/3}^{ss} \ ℜ_{5/6}^{ss}, ℜ^{+} = ℜ_{2/3}^{ss} \ ℜ_{1/3}^{ss}, ℜ^{0} = ℜ^{+} ∩ ℜ^{−}; M_{t} = M_{Q}((1, 2, 3), χ_{t}) = ℜ_{t}^{ss}//GL(α) — moduli space of α-dimensional χ_{t}-semistable representa-

tions of Q; M̃ = M_{1/3} ×_{M_{2/3}} M_{5/6} — see Corollary 4.6; M^{−} = ℜ^{−}//GL(α) = ℙ_{M^{0}}(S^{−}) ⊂ M_{1/3} = ℙ(R), M^{+} = ℜ^{+}//GL(α) = ℙ_{M^{0}}(S^{+}) ⊂ M_{5/6} = ℙ(E); M^{0} = ℜ^{0}//GL(α) = Gr(2, W) × ℙ(W^{∗}) ⊂ M_{2/3}; M̃ = M^{−} ×_{M^{0}} M^{+} ⊂ M̃.

2. Sheaves:

E, K — universal 2-dimensional and 3-dimensional vector bundles on 𝔚; R — 11-dimensional exceptional vector bundle on ℙ(W), see (9); L_{E}, L_{R} — tautological linear subbundles on ℙ(E) and ℙ(R); U — tautological 2-dimensional subbundle on Gr(2, W) and its pull-back on M^{0} = Gr(2, W)×ℙ(W^{∗}); η — tautological linear subbundle on ℙ(W^{∗}) and its pull-back on M^{0}; N^{w} — weight component of N_{G·A/ℜ} with respect to action of Ĝ_{A}; L — linear bundle on ℜ^{0}, defined before Corollary 4.6; S^{±} — vector bundles on M^{0}, defined before Corollary 4.6; L^{±} — tautological linear subbundles on M^{±} = ℙ_{M^{0}}(S^{±}); L′ = p^{−}^{∗}(Λ^{2}U ⊗ η) ⊗ L^{−} — linear subbundle in p^{−}^{∗}U on M^{−}; E^{−} — 2-dimensional exceptional vector bundle on ℙ(R), see Remark 7; E^{+}, K^{+} — 2 and 3-dimensional vector bundles on M^{+}, defined before Proposition 5.3; O_{X}, T_{X}, N_{Y/X} — structure sheaf of X, tangent sheaf of X and normal sheaf of Y ⊂ X.

3. Maps:

π_{0} : ℙ(R) → ℙ(W), π_{1} : ℙ(E) → 𝔚 — natural projections;

π′_{0} : ℙ(W) × 𝔚^{∘} → ℙ(W), π′_{1} : ℙ(W) × 𝔚^{∘} → 𝔚^{∘} — natural projections; π̂_{0} : M̃ → ℙ(R), π̂_{1} : M̃ → ℙ(E) — blow-ups along M^{−} and M^{+} respectively; π̃_{0} = π_{0} · π̂_{0} : M̃ → ℙ(W), π̃_{1} = π_{1} · π̂_{1} : M̃ → 𝔚 — projections; π̃ = π̃_{0} × π̃_{1} : M̃ → I ⊂ ℙ(W) × 𝔚; p^{−} : ℙ(R) = M_{1/3} → M_{2/3}, p^{+} : ℙ(E) = M_{5/6} → M_{2/3}, see 3.6; p = p^{−} · π̂_{0} = p^{+} · π̂_{1} : M̃ → M^{0}.

4. Other:

Q, Q_{small} — quivers, see 2.1, 2.4; A^{0}, A^{1} — canonical subrepresentations in A, see proofs of Propositions 3.1, 3.2; GL(α) = GL(α_{1}) × GL(α_{2}) × GL(α_{3}), see 2.1; Δ ⊂ GL(α), G = GL(α)/Δ — diagonal subgroup and quotient group; G̃_{A} and G_{A} — stabilizers of A ∈ ℜ^{0} in the groups GL(α) and G; Ĝ_{A} — subgroup in GL(α), see Lemma 4.3; μ_{χ}(A) — slope of A with respect to χ, see 2.2; χ_{t} — family of characters of GL(α), see (5); v — generator of Hom(G_{A}, ℂ^{∗}), see Remark 10; A^{−}, A^{+}, — universal families on M_{1/3}, M_{5/6}, see (10) and (11); A′, A′′ — universal families on M((1, 1, 2), χ_{2/3}), M((0, 1, 1), χ_{2/3}), see Proposition 5.3.

The author is very grateful to A. Bondal and D. Orlov for useful discussions and to the Max-Planck-Institut für Mathematik where this paper was prepared for hospitality.

2. Representations of the quiver and the moduli spaces.

2.1. Let W be 4-dimensional vector space. Let Q be the following quiver

• \underrightarrow{ℂ} • \underrightarrow{W} •

where ℂ is considered as one-dimensional vector space.

Recall basic definitions of the theory of representations of quivers. An α = (α_{1}, α_{2}, α_{3})-dimensional representation A of the quiver Q is a triple (A_{1}, A_{2}, A_{3}) of vector spaces such that dim A_{i} = α_{i}, together with homomorphisms

a_{12} : ℂ → Hom(A_{1}, A_{2}), a_{23} : W → Hom(A_{2}, A_{3}), a_{13} : W → Hom(A_{1}, A_{3})

such that

a_{13}(w) = a_{23}(w) · a_{12}(1), (w ∈ W)

Fixing vector spaces A_{i} we see, that the representation is a tensor A in the vector space ℜ(α) = (A_{1}^{∗} ⊗ A_{2}) ⊕ (W^{∗} ⊗ A_{2}^{∗} ⊗ A_{3}).

A homomorphism f from representation A to representation A′ is a triple (f_{1}, f_{2}, f_{3}) of linear maps f_{i} : A_{i} → A′_{i} such that for any z ∈ ℂ and w ∈ W the following diagram commutes:

A_{1} ⟶^{a_{12}(z)} A_{2} ⟶^{a_{23}(w)} A_{3}
f_{1}↓  ^{f_{2}}↓  ^{f_{3}}↓
A′_{1} ⟶^{a′_{12}(z)} A′_{2} ⟶^{a′_{23}(w)} A′_{3}

A subrepresentation in A is a triple of subspaces A′_{i} ⊂ A_{i} which is closed under the maps a_{12}(z) and a_{23}(w) for all z ∈ ℂ, w ∈ W.

2.2. Fix some α. Let GL(α) = GL(α_{1}, α_{2}, α_{3}) = GL(α_{1}) × GL(α_{2}) × GL(α_{3}) and ℂ^{∗} = Δ ⊂ GL(α) is the diagonal subgroup. The group GL(α) naturally acts on the space ℜ(α) and Δ is the kernel of the action. The classes of isomorphism of α-dimensional representations of Q is nothing but the orbits of GL(α) in ℜ(α).

Let us denote by χ = (c_{1}, c_{2}, c_{3}) a character of GL(α) defined as follows

χ(g_{1}, g_{2}, g_{3}) = (det g_{1})^{c_{1}} · (det g_{2})^{c_{2}} · (det g_{3})^{c_{3}}.

Note that for λ ∈ Δ we have χ(λ) = λ^{⟨χ,α⟩} = λ^{c_{1}α_{1}+c_{2}α_{2}+c_{3}α_{3}}.

The number μ_{χ}(A) = ⟨χ, α⟩ is called the slope of A with respect to χ. If χ_{|Δ} = id then the slope of A is zero. The representation A is called semistable (stable) with respect to χ if its slope is zero and for all proper subrepresentations A′ ⊂ A we have μ_{χ}(A′) ≥ 0 (> 0 respectively).

The χ-semistable representations A and A′ are called S-equivalent if they have the same composition factors in the category of χ-semistable representations of Q (note that simple objects in this category are but χ-stable representations).

Theorem 2.3 (A. King, [3]). There exists a coarse moduli space M_{Q}(α, χ) of χ-semistable α-dimensio-nal representations of Q. The space M_{Q}(α, χ) is a projective variety, its points are in one-one correspondence with S-equivalence classes of χ-semistable representations. The space M_{Q}(α, χ) contains an open smooth subspace M_{Q}^{s}(α, χ) whose points correspond to isomorphism classes of χ-stable representations. If α is indivisible vector then M_{Q}^{s}(α, χ) is a fine moduli space for families of χ-stable representations of Q.

Remark 1. A universal family A of representations of Q on the fine moduli space M^{s}(α, χ) is not unique (it can be twisted by any invertible sheaf). Hence it is not generally true that for any variety X equipped with a family of α-dimensional χ-stable representations G there exists a map f : X → M^{s}(α, χ) such that G = f^{∗}A. The correct statement is that under the same conditions there exists a map f : X → M^{s}(α, χ) such that G = f^{∗}(A ⊗ L) for some invertible sheaf L over M^{s}(α, χ). In other words, for given G we can chose universal family so that the desired map f will exist.

However, in the case when α is indivisible vector there is a possibility to avoid indeterminacy in choice of the universal family. Namely, fix a triple of integers (m_{1}, m_{2}, m_{3}), such that m_{1}α_{1}+m_{2}α_{2}+m_{3}α_{3} = ±1. Then there exists a unique universal family A over M^{s}(α, χ) such that

m_{1} · c_{1}(A_{1}) + m_{2} · c_{1}(A_{2}) + m_{3} · c_{1}(A_{3}) = 0  (1)

(c_{1}(A_{i}) is the first Chern class of A_{i}). If both the family G over X and the universal family A over M^{s}(α, χ) satisfy the same condition (1) then there exists a map f : X → M^{s}(α, χ) such that G = f^{∗}A.

In the cases when it is not important we do not specify which universal family we use. In other cases, if it is possible (i.e. when α_{2} − α_{3} = ±1), we will normalize universal families by condition

c_{1}(A_{2}) = c_{1}(A_{3}).  (2)

2.4. Consider also the quiver Q_{small}

• \underrightarrow{W} •

All definitions can be repeated for Q_{small} (as well as for any other quiver) hence we can speak about moduli spaces of representations of Q_{small}.

Note also, that there is two embeddings of Q_{small} into Q, namely k_{13}, which sends first and second vertices of Q_{small} to the first and third vertices of Q, and k_{23}, which sends first and second vertices of Q_{small} to the second and third vertices of Q. The corresponding restriction functors F_{13} and F_{23} from the category of representations of Q to the category of representations of Q_{small} induce (generally rational) maps of the corresponding moduli spaces.

3. The moduli spaces of (1, 2, 3)-dimensional representations of Q.

From now on we will consider the case α = (1, 2, 3). The group GL(α)/Δ will be denoted by G.

Let χ_{0} = (−6, 0, 2) and χ_{1} = (0, −3, 2) be the characters of the group GL(α). For both these characters we have ⟨χ, α⟩ = 0, hence we can speak about χ_{0} and χ_{1}-(semi)stability of α-dimensional representations of Q.

Proposition 3.1. M(α, χ_{0}) ≅ ℙ(W).

Proof. Every (1, 2, 3)-dimensional representation A = (A_{1}, A_{2}, A_{3}) of the quiver Q has a canonical (1, 1, 3)-dimensional subrepresentation

A^{0} = (A_{1}, a_{12}(A_{1}), A_{3}).

Its slope μ_{χ_{0}}(A^{0}) is zero. It is clear, that A^{0} is semistable with respect to the character χ_{0} considered as character of the group GL(1, 1, 3). Moreover, every semistable (1, 1, 3)-dimensional representation A of Q is stable: if A′ were its proper α′-dimensional subrepresentation with zero slope, then we should have

3α′_{1} = α′_{3},  0 ≤ α′_{1} ≤ 1,  0 ≤ α′_{3} ≤ 3

and that is possible only if

α′_{1} = α′_{3} = 0 or α′_{1} = 1, α′_{3} = 3.

In the first case we should also have α′_{2} = 0 (otherwise A should have the subrepresentation (A_{1}, A_{2}, 0) with negative slope), i.e. A′ = 0.

In the second case we should have α′_{2} = 1 (otherwise A should have the subrepresentation (A_{1}, 0, 0) with negative slope), i.e. A′ = A.

This implies that 0 ⊂ A^{0} ⊂ A is the Jordan-Hölder composition series of A. The composition factors are A^{0} and the trivial (0, 1, 0)-dimensional representation. Therefore α-dimensional representations A and A′ are S-equivalent iff A^{0} ≅ A′^{0}.

This means that there is a canonical map

ψ_{0} : M(α, χ_{0}) → M((1, 1, 3), χ_{0}) = M^{s}((1, 1, 3), χ_{0}).

and it is an isomorphism.

Consider the restriction functor F_{13} from the category of (1, 1, 3)-dimensional representations of Q to the category of (1, 3)-dimensional representations of Q_{small}, induced by the embedding k_{13}:

F_{13}(A_{1}, A_{2}, A_{3}) = (A_{1}, A_{3}).

This functor certainly maps the subcategory of χ_{0}-stable representations of Q to the subcategory of χ_{0}- stable representations of Q_{small} (now we consider χ_{0} as the character of the group GL(1) × GL(3)) and is an equivalence of the categories. Hence the induced map

ψ̂_{0} : M((1, 1, 3), χ_{0}) → M_{Q_{small}}((1, 3), χ_{0})

is an isomorphism.

Now it is remained to note that the latter space is nothing else then ℙ(W) equipped with the universal family of representations

(O_{ℙ(W)}(1), T_{ℙ(W)}).

and the structure of representation is given by the canonical map W ⊗ O_{ℙ(W)}(1) → T_{ℙ(W)}.

Remark 2. Since (O_{ℙ(W)}(1), T_{ℙ(W)}) is universal family over ℙ(W) we have

ψ̂_{0}^{∗}O_{ℙ(W)}(1) = A_{1}^{0},  ψ̂_{0}^{∗}T_{ℙ(W)} = A_{3}^{0},  (3)

where A^{0} = (A_{1}^{0}, A_{2}^{0}, A_{3}^{0}) is a certain universal family over M((1, 1, 3), χ_{0}).

Let 𝔚^{∘} be the space of smooth rational twisted cubic curves in ℙ(W) and 𝔚 be its Ellingsrud-Piene-Strømme compactification (see [2]). Recall that 𝔚 comes equipped with a pair of vector bundles E and K such that

rank E = 2, rank K = 3, det E^{∗} = det K^{∗} is the ample generator of Pic 𝔚 = ℤ

and a canonical map W ⊗ E → K.

Proposition 3.2. M(α, χ_{1}) ≅ 𝔚.

The proof goes parallel with the proof of the Proposition 3.1. Proof. Every (1, 2, 3)-dimensional representation A = (A_{1}, A_{2}, A_{3}) of the quiver Q has a canonical (0, 2, 3)-dimensional subrepresentation A^{1} = (0, A_{2}, A_{3}). Its slope μ_{χ_{1}}(A^{1}) is zero. It is clear that A^{1} is semistable with respect to the character χ_{1} considered as character of the group GL(0, 2, 3) = GL(2) × GL(3). Moreover, every χ_{1}-semistable (0, 2, 3)-dimensional representation A of Q is stable: if A′ were its α′-dimensional subrepresentation with zero slope, then we should have

3α′_{2} = 2α′_{3},  0 ≤ α′_{2} ≤ 2,  0 ≤ α′_{3} ≤ 3.

but that is possible only if

α′_{2} = α′_{3} = 0 or α′_{2} = 2, α′_{3} = 3

i.e. if A′ = 0 or A′ = A.

This means that 0 ⊂ A^{1} ⊂ A is the Jordan-Hölder composition series of A. The composition factors are A^{1} and the trivial (1, 0, 0)-dimensional representation. Therefore α-dimensional representations A and A′ are S-equivalent iff A^{1} ≅ A′^{1}.

This means that there is a canonical map

ψ_{1} : M(α, χ_{1}) → M((0, 2, 3), χ_{1}) = M^{s}((0, 2, 3), χ_{1}).

and it is an isomorphism.

The same arguments as in the proof of the Proposition 3.1 show that the map

ψ̂_{1} : M((0, 2, 3), χ_{1}) → M_{Q_{small}}((2, 3), χ_{1}),

induced by the restriction functor F_{23} : (0, A_{2}, A_{3}) ↦ (A_{2}, A_{3}) is an isomorphism.

But the latter moduli space is nothing but 𝔚 (see [2]) and the pair (E, K) of universal bundles over 𝔚 form universal family of representations over 𝔚.

Remark 3. Since (E, K) is universal family over 𝔚 we have

ψ̂_{1}^{∗}E = A_{2}^{1},  ψ̂_{1}^{∗}K = A_{3}^{1},  (4)

where A^{1} = (A_{1}^{1}, A_{2}^{1}, A_{3}^{1}) is a certain universal family over M((0, 2, 3), χ_{1}). Note that the family A^{1} satisfies the norming condition (2).

Consider the following linear family of fractional characters of the group GL(α)

χ_{t} = (1 − t)χ_{0} + tχ_{1} = (6t − 6, −3t, 2), t ∈ ℚ.  (5)

Note that for all t we have ⟨χ_{t}, α⟩ = 0, hence we can speak about χ_{t}-(semi)stability of α-dimensional representations of the quiver Q.

Let ℜ_{t}^{ss} ⊂ ℜ(α) denote the set of all points which correspond to α-dimensional χ_{t}-semistable representations of Q and let M_{t} = ℜ_{t}^{ss}//GL(α) = M(α, χ_{t}) be the corresponding moduli space.

Theorem 3.3 (Thaddeus, [7]). There exists a finite number of rationals t_{0} < t_{1} < ⋯ < t_{N} such that 1) For t < t_{0} or t > t_{N} we have ℜ_{t}^{ss} = M_{t} = ∅ and for t_{0} ≤ t ≤ t_{N} we have ℜ_{t}^{ss} ≠ ∅.

2) For all t_{k} < t, t′ < t_{k+1} we have ℜ_{t}^{ss} = ℜ_{t′}^{ss} and M_{t} ≅ M_{t′}, (k = 0, …, N − 1).

3) Let us denote by t_{k} + 0 (t_{k} − 0) any t such that t_{k} < t < t_{k+1} (resp. t_{k−1} < t < t_{k}). Then we have an embeddings ℜ_{t_{k}±0}^{ss} ⊂ ℜ_{t_{k}}^{ss} inducing proper maps p_{t_{k}}^{±} : M_{t_{k}±0} → M_{t_{k}} (k = 0, …, N).

4) If k = 1, …, N − 1 then maps p_{t_{k}}^{±} are birational.

Lemma 3.4. In our case N = 2, t_{0} = 0, t_{1} = 2/3, t_{2} = 1.

Proof. Let A be an α-dimensional representation of Q. As was pointed in the proofs of the Propositions 3.1 and 3.2 it contains canonical subrepresentations A^{0} ((1, 1, 3)-dimensional) and A^{1} ((0, 2, 3)- dimensional). Since

μ_{χ_{t}}(A^{0}) = ⟨χ_{t}, (1, 1, 3)⟩ = 3t, μ_{χ_{t}}(A^{1}) = ⟨χ_{t}, (0, 2, 3)⟩ = 6 − 6t

it follows that the representation A can be χ_{t}-semistable only if 0 ≤ t ≤ 1. Since M_{0} and M_{1} are nonempty (see Propositions 3.1 and 3.2) we have t_{0} = 0 and t_{N} = 1.

Now we need to determine all values 0 < t < 1 such that ℜ_{t}^{ss} ≠ ℜ_{t+0}^{ss} or ℜ_{t}^{ss} ≠ ℜ_{t−0}^{ss}. If one of these holds then there exist a representation A and its proper subrepresentation A′, such that A is χ_{t}-semistable, but μ_{χ_{t±0}}(A′) < 0. Since μ_{χ_{t}}(A′) is continuous function of t this means that

μ_{χ_{t}}(A′) = 0.  (6)

Let dim A′ = α′ = (α′_{1}, α′_{2}, α′_{3}). Then (6) is equivalent to

2α′_{3} − 6α′_{1} = 3t(α′_{2} − 2α′_{1}).

This equation under constraints

0 ≤ α′_{1} ≤ 1, 0 ≤ α′_{2} ≤ 2, 0 ≤ α′_{3} ≤ 3, α′ ≠ (0, 0, 0), α′ ≠ (1, 2, 3), 0 < t < 1

has only the following solutions

t = 1/3 α′ = (1, 0, 2)
α′ = (0, 2, 1)
t = 2/3 α′ = (1, 0, 1)
α′ = (0, 2, 2)
α′ = (1, 1, 2)
α′ = (0, 1, 1)

The solutions from first to forth must be put aside for the following reason. If A has (1, 0, 1)-dimensional or (1, 0, 2)-dimensional subrepresentation then it also has (1, 0, 0)-dimensional subrepresentation, but ⟨χ_{t}, (1, 0, 0)⟩ = 6t − 6 < 0 for all 0 < t < 1, i.e. A is unstable. Similarly, if A has (0, 2, 1)-dimensional or (0, 2, 2)-dimensional subrepresentation then it also has (1, 2, 2)-dimensional subrepresentation, but ⟨χ_{t}, (1, 2, 2)⟩ = −2, i.e. A is unstable.

This means that in the interval (0, 1) there is unique critical value t_{1} = 2/3 and the corresponding character is χ_{2/3} = (−2, −2, 2).

Corollary 3.5. The set ℜ_{2/3}^{ss} \ ℜ_{2/3+0}^{ss} consists of all representations A containing (0, 1, 1)-dimensional subrepresentation A′. The set ℜ_{2/3}^{ss} \ ℜ_{2/3−0}^{ss} consists of all representations A containing (1, 1, 2)-dimen-sional subrepresentation A′′. The set ℜ_{2/3}^{ss} \ (ℜ_{2/3−0}^{ss} ∪ ℜ_{2/3+0}^{ss}) consists of all representations A which are the direct sum

A = A′ ⊕ A′′  (7)

of (0, 1, 1)-dimensional and (1, 1, 2)-dimensional subrepresentations. The decomposition (7) is unique.

Proof. The first and second parts follow immediately from the calculations of Lemma 3.4. In the third part we need to prove that if χ_{2/3}-semistable representation A contains (0, 1, 1) and (1, 1, 2)-dimensional subrepresentations A′ and A′′ then A = A′ ⊕ A′′. To this end consider exact sequence

0 → A′ → A → A/A′ → 0  (8)

induced by the embedding A′ → A. Since both A′ and A′′ are χ_{2/3}-stable and non-isomorphic we have Hom(A′′, A′) = 0. Hence the composition of the embedding A′′ → A and of the projection A → A/A′ can not be zero and therefore it is an isomorphism (because A′′ is χ_{2/3}-stable and dim A/A′ = dim A′′). This means that exact sequence (8) splits and A = A′ ⊕ A′′.

It is remained to prove the uniqueness of the decomposition (7). Assume that A = A′⊕A′′ = A′′′⊕A′′′′. Since the composition of the embedding A′ → A and of the projection A → A′′′′ must be zero it follows that A′ = A′′′. By the same reasons A′′ = A′′′′.

Remark 4. The proof of the Lemma 3.4 implies also that for any 0 < t < 2/3 and 2/3 < t < 1 all χ_{t}-semistable representations are χ_{t}-stable and therefore all varieties M_{t} are fine moduli spaces. We will denote by A^{±} = (A_{1}^{±}, A_{2}^{±}, A_{3}^{±}) the universal families of representations normalized by the condition (2) over M_{t}, (t ∈ (2/3, 1)) and M_{t}, (t ∈ (0, 2/3)) respectively, by

a_{12} : A_{1}^{±} → A_{2}^{±}, a_{13} : W ⊗ A_{1}^{±} → A_{3}^{±} and a_{23} : W ⊗ A_{2}^{±} → A_{3}^{±}

the structural maps and by

a_{13}^{T} : A_{1}^{±} → W^{∗} ⊗ A_{3}^{±}, a_{23}^{T} : A_{2}^{±} → W^{∗} ⊗ A_{3}^{±}

the transposed maps.

3.6. By the Theorem 3.3 we have the following picture:

[figure: M_{1/3}; M_{5/6}; ℙ(W) =M_{0}; M_{2/3}; M_{1}= 𝔚; p_{0}; p^{−}; p^{+}; p_{1}]

Here the values t = 1/3 and t = 5/6 are used as typical representatives in the intervals (0, 2/3) and (2/3, 1).

Maps p^{±}, p_{0}, p_{1} are proper, and maps p^{±} are birational. Below we will describe these maps more precisely.

4. The transformations of M_{t} and the space M̃

Consider the natural morphism O_{ℙ(W)}(1) → W^{∗} ⊗ T_{ℙ(W)} of vector bundles over ℙ(W). This is an embedding of vector bundles, therefore its cokernel is a vector bundle over ℙ(W). Let us denote it by R:

0 ⟶ O_{ℙ(W)}(1) ⟶ W^{∗} ⊗ T_{ℙ(W)} ⟶ R ⟶ 0.  (9)

Remark 5. The above exact sequence is nothing but the right mutation of the exceptional vector bundle O_{ℙ(W)}(1) through the exceptional vector bundle T_{ℙ(W)}. Therefore the bundle R is also exceptional.

Proposition 4.1. Let ℙ(R) be the projectivization of R and π_{0} : ℙ(R) → ℙ(W) be the natural projection. There exists an isomorphism f : M_{1/3} → ℙ(R) making the below diagram commutative

M_{1/3} ⟶_{f}^{∼} ℙ(R)
p_{0}↓  ^{π_{0}}↓
M_{0} ⟶_{ψ̂_{0}·ψ_{0}}^{∼} ℙ(W)

Proof. Both M_{1/3} and M((1, 1, 3), χ_{0}) are fine moduli spaces. Let A^{−} = (A_{1}^{−}, A_{2}^{−}, A_{3}^{−}) and A^{0} = (A_{1}^{0}, A_{2}^{0}, A_{3}^{0}) be the corresponding universal families. By the arguments of the proof of Proposition 3.1 the family A^{−} contains subfamily (A_{1}^{−}, a_{12}(A_{1}^{−}), A_{3}^{−}), which is a family of χ_{0}-stable (1, 1, 3)-dimensional representations. Hence there exists a map ψ_{1/3} : M_{1/3} → M((1, 1, 3), χ_{0}) such that ψ_{1/3}^{∗}A_{1}^{0} = A_{1}^{−}, ψ_{1/3}^{∗}A_{3}^{0} = A_{3}^{−}. Combining this with (3) we get

ψ_{1/3}^{∗}ψ̂_{0}^{∗}O_{ℙ(W)}(1) = A_{1}^{−}, ψ_{1/3}^{∗}ψ̂_{0}^{∗}T_{ℙ(W)} = A_{3}^{−}.

On the other hand, it is evident that ψ_{0} · p_{0} = ψ_{1/3}, therefore ψ̂_{0} · ψ_{0} · p_{0} = ψ̂_{0} · ψ_{1/3}. Let us denote the latter map by p (this is the composition of the left and the bottom arrows of the above diagram). We have

p^{∗}O_{ℙ(W)}(1) = A_{1}^{−}, p^{∗}T_{ℙ(W)} = A_{3}^{−}.

Applying p^{∗} to (9) we get

0 ⟶ A_{1}^{−} ⟶^{a_{13}^{T}} W^{∗} ⊗ A_{3}^{−} ⟶ p^{∗}R ⟶ 0.

The map a_{13}^{T} factors as A_{1}^{−} ⟶^{a_{12}} A_{2}^{−} ⟶^{a_{23}^{T}} W^{∗} ⊗ A_{3}^{−} and a_{12} is an embedding of vector bundles, therefore p^{∗}R contains A_{2}^{−}/A_{1}^{−} as a linear subbundle. This means that there exists a map f : M_{1/3} → ℙ(R) such that π_{0} · f = p and A_{2}^{−}/A_{1}^{−} = f^{∗}L_{R} is the pull-back of the tautological linear subbundle L_{R} ⊂ π_{0}^{∗}R over ℙ(R).

On the other hand the embedding L_{R} ⊂ π_{0}^{∗}R and the pull-back by π_{0} of (9) give rise to the commutative diagram:

0 ⟶ π_{0}^{∗}O_{ℙ(W)}(1) ⟶  E^{−}  ⟶ L_{R} ⟶ 0
∥  ↓  ↓
0 ⟶ π_{0}^{∗}O_{ℙ(W)}(1) ⟶ W^{∗} ⊗ π_{0}^{∗}T_{ℙ(W)} ⟶ π_{0}^{∗}R ⟶ 0

where E^{−} is a 2-dimensional vector bundle over ℙ(R). This diagram allows to define a structure of (1, 2, 3)-dimensional representation of Q in the family (π_{0}^{∗}O_{ℙ(W)}(1), E^{−}, π_{0}^{∗}T_{ℙ(W)}). It is easy to check that all representations in this family are χ_{1/3}-stable. Therefore there is unique map g : ℙ(R) → M_{1/3} such that this family is the pull-back of the universal family over M_{1/3}.

The map g is certainly inverse to f, hence both of them are isomorphisms.

Remark 6. Since for the universal families we have chosen the normalization (2) (see Remark 1) we should twist the previous family by L_{R} ⊗ O_{ℙ(W)}(−3). Below by the universal family on M_{1/3} = ℙ(R) we will always mean the family

A^{−} = (L_{R} ⊗ π_{0}^{∗}O_{ℙ(W)}(−2), E^{−} ⊗ L_{R} ⊗ π_{0}^{∗}O_{ℙ(W)}(−3), L_{R} ⊗ π_{0}^{∗}T_{ℙ(W)}(−3)).  (10)

Remark 7. The extension

0 ⟶ π_{0}^{∗}O_{ℙ(W)}(1) ⟶ E^{−} ⟶ L_{R} ⟶ 0

defining E^{−} can be interpreted as left mutation of the exceptional bundle π_{0}^{∗}O_{ℙ(W)}(1) through the exceptional vector bundle L_{R}. Hence E^{−} is a 2-dimensional exceptional vector bundle over ℙ(R).

Proposition 4.2. Let ℙ(E) be the projectivization of the universal 2-dimensional vector bundle E on 𝔚 and π_{1} : ℙ(E) → 𝔚 be the natural projection. There exists an isomorphism f : M_{5/6} → ℙ(E) making the below diagram commutative

M_{5/6} ⟶_{f}^{∼} ℙ(E)
p_{1}↓  ^{π_{1}}↓
M_{1} ⟶_{ψ̂_{1}·ψ_{1}}^{∼} 𝔚

Proof. The proof goes parallel with the proof of the Proposition 4.1 . Both M_{5/6} and M((0, 2, 3), χ_{1}) are fine moduli spaces. Let A^{+} = (A_{1}^{+}, A_{2}^{+}, A_{3}^{+}) and A^{1} = (0, A_{2}^{1}, A_{3}^{1}) be the corresponding universal families. By the arguments of the proof of Proposition 3.2 the family A^{+} contains the subfamily (0, A_{2}^{+}, A_{3}^{+}), which is a family of χ_{1}-stable (0, 2, 3)-dimensional representations. Hence there exists a map ψ_{5/6} : M_{5/6} → M((0, 2, 3), χ_{1}) such that ψ_{5/6}^{∗}A_{2}^{1} = A_{2}^{+}, ψ_{5/6}^{∗}A_{3}^{1} = A_{3}^{+}. Combining this with (4) we get

ψ_{5/6}^{∗}ψ̂_{1}^{∗}E = A_{2}^{+}, ψ_{5/6}^{∗}ψ̂_{1}^{∗}K = A_{3}^{+}.

On the other hand, it is evident that ψ_{1} · p_{1} = ψ_{5/6}, therefore ψ̂_{1} · ψ_{1} · p_{1} = ψ̂_{1} · ψ_{5/6}. Let us denote the latter map by p (this is the composition of the left and bottom arrows of the above diagram). We have

p^{∗}E = A_{2}^{+}.

Therefore the bundle p^{∗}E contains A_{1}^{+} as a linear subbundle. This means that there exists a map f : M_{5/6} → ℙ(E) such that π_{1} · f = p and A_{1}^{+} = f^{∗}L_{E} is the pull-back of the tautological linear subbundle L_{E} ⊂ π_{1}^{∗}E over ℙ(E).

On the other hand, consider the following family of (1, 2, 3)-dimensional representations of Q over ℙ(E):

A^{+} = (L_{E}, π_{1}^{∗}E, π_{1}^{∗}K).  (11)

It is easy to check that all representations in this family are χ_{5/6}-stable. Therefore there is unique map g : ℙ(E) → M_{5/6} such that this family is the pull-back of universal family over M_{5/6}.

The map g is certainly inverse to f, hence both of them are isomorphisms.

Remark 8. The family (11) satisfies the norming condition (2).

It is left to describe the birational transform M_{1/3} −−→ M_{5/6}.

Accordingly to Theorem 3.3 we have embeddings ℜ_{1/3}^{ss} ⊂ ℜ_{2/3}^{ss} ⊃ ℜ_{5/6}^{ss}. Let

ℜ^{+} = ℜ_{2/3}^{ss} \ ℜ_{1/3}^{ss},  ℜ^{−} = ℜ_{2/3}^{ss} \ ℜ_{5/6}^{ss},  ℜ^{0} = ℜ^{−} ∩ ℜ^{+}.

Remark 9. By the Corollary 3.5 the set ℜ^{+} consists of all representations containing (1, 1, 2)-dimensional subrepresentation, ℜ^{−} consists of all representations containing (0, 1, 1)-dimensional subrepresentation and ℜ^{0} consists of all representations which are the direct sum of (0, 1, 1)-dimensional and (1, 1, 2)- dimensional representations.

Lemma 4.3. The stabilizer G_{A} of A ∈ ℜ^{0} in the group G = GL(α)/Δ is isomorphic to ℂ^{∗}. In an appropriate bases (e_{i}^{j}) of the spaces A_{i} (i = 1, 2, 3) we have

G_{A} = Ĝ_{A}/±1,  Ĝ_{A} = {(λ, diag(λ^{−1}, λ), diag(λ^{−1}, λ, λ))}, λ ∈ ℂ^{∗}.

Proof. First of all we will describe the stabilizer G̃_{A} of A in GL(α). Let A = A′ ⊕ A′′ be the decomposition into the sum of (0, 1, 1) and (1, 1, 2)-dimensional representations. Then

A_{1} = A′′_{1},  A_{2} = A′_{2} ⊕ A′′_{2},  A_{3} = A′_{3} ⊕ A′′_{3}.

Let (0, (e_{2}^{1}), (e_{3}^{1})) be a basis in A′ and ((e_{1}^{1}), (e_{2}^{2}), (e_{3}^{2}, e_{3}^{3})) be a basis in A′′. Let g be an element in G_{A}. Then A = gA = gA′ ⊕ gA′′ is a decomposition of A. By the Corollary 3.5 we have gA′ = A′, gA′′ = A′′, i.e. g = g′ ⊕g′′, where g′ ∈ GL(0, 1, 1) and g′′ ∈ GL(1, 1, 2). Since all automorphisms of the simple object over the algebraically closed field are dilations we have g′ = (0, λ′, λ′) and g′′ = (λ′′, λ′′, diag(λ′′, λ′′)). This means that

G̃_{A} = {(λ′′, diag(λ′, λ′′), diag(λ′, λ′′, λ′′))}, λ′, λ′′ ∈ ℂ^{∗}.

Now the Lemma follows from G_{A} = G̃_{A}/Δ.

Remark 10. The group Hom(G_{A}, ℂ^{∗}) of characters of G_{A} is a free cyclic group generated by

v((λ, diag(λ^{−1}, λ), diag(λ^{−1}, λ, λ)) = λ^{2}.

Remark 11. It is easy to see that χ_{1}_{|G_{A}} = v. This implies that the restriction map Hom(G, ℂ^{∗}) → Hom(G_{A}, ℂ^{∗}) is surjective.

Let M^{+} be the image of ℜ^{+} \ ℜ^{−} in M_{5/6}, M^{−} be the image of ℜ^{−} \ ℜ^{+} in M_{1/3} and M^{0} be the image of ℜ^{0} in M_{2/3}.

Remark 12. The Remark 9 implies that every element of ℜ^{+} is S-equivalent (with respect to χ_{2/3}- stability) to some element of ℜ^{0}. This means that p^{+}(M^{+}) = M^{0}. Similarly p^{−}(M^{−}) = M^{0}. Moreover, sets ℜ_{1/3}^{ss} \ ℜ^{−} and ℜ_{5/6}^{ss} \ ℜ^{+} contain no strictly semistable points, therefore the maps p^{+} : ℙ(R) = M_{1/3} → M_{2/3} and p^{−} : ℙ(E) = M_{5/6} → M_{2/3} are isomorphisms on the complements of p^{±} : M^{±} → M^{0} (compare [7, 1.4]).

The Lemma 4.3 and Remark 11 allow to apply the theory of Thaddeus (see [7], Theorem 4.7). This implies the following.

Corollary 4.4. The restriction p_{|M^{±}}^{±} : M^{±} → M^{0} is a locally trivial fibration with fiber at the point A ∈ M^{0} the weighted projective space ℙ(|w_{i}^{±}|) in the sense of Bialynicki-Birula [1], where w_{i}^{±} are positive (resp. negative) weights of the action G_{A} in the fiber of the normal bundle N_{G·A/ℜ}.

The following Proposition describes the weight decomposition of the normal bundle.

Proposition 4.5. Let A ∈ ℜ^{0}. The weight decomposition of the fiber of the normal bundle N_{G·A/ℜ} with respect to the action of Ĝ_{A} is

N_{G·A/ℜ} = N^{−2} ⊕ N^{0} ⊕ N^{+2}, dim N^{−2} = 2, dim N^{0} = 7, dim N^{+2} = 5.

Proof. Let 𝔤𝔩(α) and 𝔤̃_{A} be the tangent Lie algebras of the groups GL(α) and G̃_{A} respectively. We have natural exact sequence of Ĝ_{A}-modules

0 ⟶ 𝔤̃_{A} ⟶ 𝔤𝔩(α) ⟶ ℜ ⟶ (N_{G·A/ℜ})_{A} ⟶ 0.

Let A = A′ ⊕ A′′ be the decomposition of A into the sum of (1, 1, 2) and (0, 1, 1)-dimensional representations and recall that ℜ = (A_{1}^{∗} ⊗ A_{2}) ⊕ (A_{2}^{∗} ⊗ A_{3} ⊗ W^{∗}). The action of Ĝ_{A} splits the above exact sequence into the direct sum of exact sequences of weight spaces:

0 ⟶ 𝔤̃_{A}^{w} ⟶ 𝔤𝔩(α)^{w} ⟶ ℜ^{w} ⟶ N^{w} ⟶ 0.  (12)

The Lemma 4.3 implies that the weight spaces are following:

w 𝔤̃_{A}^{w}  𝔤𝔩(α)^{w}  ℜ^{w}
−2  0 (A′_{2}^{∗} ⊗ A′′_{2}) ⊕ (A′_{3}^{∗} ⊗ A′′_{3}) (A′_{1}^{∗} ⊗ A′′_{2}) ⊕ (A′_{2}^{∗} ⊗ A′′_{3} ⊗ W^{∗})
ℂ  ⊕_{i=1}^{3}(A′_{i}^{∗} ⊗ A′_{i})  (A′_{1}^{∗} ⊗ A′_{2}) ⊕ (A′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗})
0  ⊕  ⊕  ⊕
ℂ  ⊕_{i=2}^{3}(A′′_{i}^{∗} ⊗ A′′_{i})  (A′′_{2}^{∗} ⊗ A′′_{3} ⊗ W^{∗})
2  0 (A′′_{2}^{∗} ⊗ A′_{2}) ⊕ (A′′_{3}^{∗} ⊗ A′_{3})  A′′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗}
others 0  0  0

and Proposition follows.

Remark 13. The weight 2 of the group Ĝ_{A} corresponds to the weight 1 of the group G_{A} (see Remark 10), hence the group G_{A} acts on N^{±2} with weights equal to ±1.

Consider the linear bundle L = A′_{2}^{∗} ⊗ A′′_{2} over ℜ^{0}. It is easy to see that the group G_{A} acts on L with weight −1, hence the bundles N^{±2} ⊗ L^{±1} are acted upon trivially by G_{A}. Therefore they descent to some vector bundles S^{±} over M^{0}. Another Theorem of Thaddeus (loc. cit., Theorem 4.8) implies the following.

Corollary 4.6 (And the definition of M̃ and M̃). a) The spaces M^{±} are naturally isomorphic to the projective bundles ℙ_{M^{0}}(S^{±}) and the restrictions of the maps p^{±} (see 3.6) on M^{±} are but the natural projections p^{±} : ℙ_{M^{0}}(S^{±}) → M^{0}.

b) The normal bundles of M^{±} in M_{5/6} and M_{1/3} respectively are naturally isomorphic to p^{±}^{∗}S^{∓} ⊗ L^{±}, where L^{±} ⊂ p^{±}^{∗}S^{±} are the tautological linear subbundles on M^{±}.

c) The blow-ups of M_{1/3} at M^{−}, of M_{5/6} at M^{+} and of M_{2/3} at M^{0} are naturally isomorphic to the fiber product M̃ = M_{1/3} ×_{M_{2/3}} M_{5/6}. The fiber product M̃ = M^{−} ×_{M^{0}} M^{+} ⊂ M_{1/3} ×_{M_{2/3}} M_{5/6} is the common exceptional divisor of all these blow-ups. Its normal bundle is naturally isomorphic to L^{−} ⊠ L^{+}.

The Corollary 4.6 together with the Propositions 4.1 and 4.2 shows that we have the following Theorem.

Theorem 4.7. There exists a smooth algebraic variety M̃ equipped with a pair of proper birational maps π̂_{0} : M̃ → ℙ(R) and π̂_{1} : M̃ → ℙ(E) such that

1) the map π̂_{0} is the blow-up of ℙ(R) along the smooth subvariety M^{−} ⊂ ℙ(R);

2) the map π̂_{1} is the blow-up of ℙ(E) along the smooth subvariety M^{+} ⊂ ℙ(E);

3) M^{±} ≅ ℙ_{M^{0}}(S^{±}), where M^{0} is smooth and S^{±} are vector bundles over M^{0};

4) the blow-ups π̂_{0} and π̂_{1} have common exceptional divisor M̃ ↪^{i} M̃;

5) M̃ ≅ M^{−} ×_{M^{0}} M^{+} and the restrictions of π̂_{0} (resp. π̂_{1}) on M̃ are nothing but projections on the first (resp. second) factors.

The following picture illustrates the Theorem.

[figure: M̃; ℙ(R); ℙ(E); ℙ(W); 𝔚; π̂_{0}; π̂_{1}; π_{0}; π_{1}; ↩^{i}; M̃ = M^{−}×_{M^{0}}M^{+}; M^{−} = ℙ_{M^{0}}(S^{−}); ℙ_{M^{0}}(S^{+}) = M^{+}; M^{0}; π̂_{0}; π̂_{1}; p^{−}; p^{+}]

The complete description of the variety M^{0} and of the vector bundles S^{±} will be given in Propositions 5.1 and 5.2.

Let π̃_{0} = π_{0} · π̂_{0} and π̃_{1} = π_{1} · π̂_{1} be the projections from M̃ to ℙ(W) and 𝔚 respectively. Let π̃ =

π̃_{0} ×π̃_{1} : M̃ → ℙ(W)×𝔚. Let 𝔚^{∘} ⊂ 𝔚 be the locus of smooth curves, I^{∘} = {(x, C) | x ∈ C} ⊂ ℙ(W)×𝔚^{∘}

be the point-curve incidence subspace, π′_{0}, π′_{1} projections of ℙ(W) × 𝔚^{∘} on ℙ(W) and 𝔚^{∘} respectively and

let I be the closure of I^{∘} in ℙ(W) × 𝔚.

Proposition 4.8. The image π̃(M̃) lies in I. The map π̃ : M̃ → I is birational.

Proof. Over ℙ(W) × 𝔚^{∘} we have the following resolvent of I^{∘}:

0 → π′_{1}^{∗}E ⊗ π′_{0}^{∗}O_{ℙ(W)}(−3) → π′_{1}^{∗}K ⊗ π′_{0}^{∗}O_{ℙ(W)}(−2) → O_{ℙ(W)×𝔚^{∘}} → OI^{∘} → 0.

This is nothing but the universal extension of the usual resolvent of the structure sheaf of the twisted cubic curve C ⊂ ℙ(W):

0 → O_{ℙ(W)}(−3)^{⊕2} → O_{ℙ(W)}(−2)^{⊕3} → O_{ℙ(W)} → O_{C} → 0.

Multiplying the first resolvent on OI^{∘} and twisting by π′_{0}^{∗}O_{ℙ(W)}(3) we see that the bundle π′_{1}^{∗}E has a

subsheaf Tor^{2}(OI^{∘}, OI^{∘}) ⊗ π′_{0}^{∗}O_{ℙ(W)}(3). Since I^{∘} is a smooth subvariety in ℙ(W) × 𝔚^{∘} of codimension 2

this subsheaf is locally free of rank 1 and gives rise to a map f : I^{∘} → ℙ𝔚^{∘}(E) = π_{1}^{−1}(𝔚^{∘}) ⊂ ℙ_{𝔚}(E). We claim that this map is an isomorphism. In order to prove this it suffices to check that the degree of the restriction of our linear bundle on the fibers of π′_{1} : I^{∘} → 𝔚^{∘} is equal to −1 for all points c ∈ 𝔚^{∘}.

Let C ⊂ ℙ(W) be a smooth rational twisted cubic curve and c ∈ 𝔚^{∘} the corresponding point. Then (π′_{1})^{−1}(c) = C and

Tor^{2}(O_{C}, O_{C}) ⊗ O_{ℙ(W)}(3) = Λ^{2}N_{C/ℙ(W)}^{∗} ⊗ O_{C}(9) = O_{C}(−10) ⊗ O_{C}(9) = O_{C}(−1).

But (Tor^{2}(OI^{∘}, OI^{∘}) ⊗ O_{ℙ(W)}(3))_{|(π′_{1})^{−1}(c)} = Tor^{2}(O_{C}, O_{C}) ⊗ O_{ℙ(W)}(3), hence the degree of the restric-

tion is equal to −1 and the claim follows.

Thus we see, that the open subset π̃_{1}^{−1}(𝔚^{∘}) ⊂ M̃ maps isomorphically on I^{∘}. Hence M̃ maps to its closure I birationally.

The Theorem 4.7 and the Proposition 4.8 show that M̃ is a desingularization of I. This desingularization is evidently GL(W)-equivariant where the group GL(W) acts upon moduli spaces via automorphisms of the quiver Q.

5. The exceptional divisor and universal families

We will give here an explicit description of the exceptional divisor M̃ ⊂ M̃ and of the connection between the pull-backs of universal families on M̃. Let A^{−} be the universal family (10) over M_{1/3} = ℙ(R) and A^{+} be the universal family (11) over M_{5/6} = ℙ(E). Both of them satisfy the norming condition (2).

Proposition 5.1. The space M^{0} is naturally isomorphic to

M^{0} ≅ M((1, 1, 2), χ_{2/3}) × M((0, 1, 1), χ_{2/3}) ≅ Gr(2, W) × ℙ(W^{∗}).

Proof. The first isomorphism follows immediately from Remark 9. It remains to prove the isomorphisms M((1, 1, 2), χ_{2/3}) ≅ Gr(2, W) and M((0, 1, 1), χ_{2/3}) ≅ ℙ(W^{∗}).

It is easy to see that for (1, 1, 2)-dimensional and (0, 1, 1)-dimensional representations of Q the stability and semistability with respect to χ_{2/3} coincide, therefore both M((1, 1, 2), χ_{2/3}) and M((0, 1, 1), χ_{2/3}) are fine moduli spaces. Let A′ and A′′ be the universal families over them.

Since family A′ consists of stable representations the map a_{23} : W ⊗ A′_{2} → A′_{3} is surjective, hence there exists a map f : M((1, 1, 2), χ_{2/3}) → Gr(2, W) such that

f^{∗}(W/U) = A′_{2}^{∗} ⊗ A′_{3},  f^{∗}U = A′_{2}^{∗} ⊗ Ker a_{23},  (13)

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

Vice versa, the family

(O_{Gr(2,W)}, O_{Gr(2,W)}, W/U)  (14)

over Gr(2, W) is naturally the family of χ_{2/3}-stable representations of Q, hence there exists a map g : Gr(2, W) → M((1, 1, 2), χ_{2/3}) such that the family (14) is the pull-back of the universal family over M((1, 1, 2), χ_{2/3}). Certainly, the maps f and g are mutually inverse.

Similarly, the map a_{23} : W ⊗ A′′_{2} → A′′_{3} is surjective, hence there exists a map f : M((0, 1, 1), χ_{2/3}) → ℙ(W^{∗}) such that

f^{∗}η^{∗} = A′′_{2}^{∗} ⊗ A′′_{3},  f^{∗}(W^{∗}/η)^{∗} = A′′_{2}^{∗} ⊗ Ker a_{23},  (15)

where η stands for the tautological linear subbundle in the trivial bundle W^{∗} ⊗ O_{ℙ(W^{∗})} over ℙ(W^{∗}). Vice versa, the family

(0, η, O_{ℙ(W^{∗})})  (16)

over ℙ(W^{∗}) gives rise to a map g : ℙ(W^{∗}) → M((0, 1, 1), χ_{2/3}) such that the family (16) is the pull-back of the universal family over M((0, 1, 1), χ_{2/3}). Certainly, the maps f and g are mutually inverse.

Below we will identify the space M^{0} with Gr(2, W) × ℙ(W^{∗}) and consider bundles U and η as bundles over M^{0}.

Proposition 5.2. There are natural isomorphisms of vector bundles over M^{0}:

S^{−} ≅ η^{∗} ⊗ U^{∗}, S^{+} ≅ \frac{W/U ⊗ W^{∗}/η}{O_{M^{0}}} .

Proof. The exact sequence (12) corresponding to w = −2 can be included into the commutative diagram

A′_{3}^{∗} ⊗ A′′_{3} ⟶^{a_{23}^{T}⊗id} A′_{2}^{∗} ⊗ A′′_{3} ⊗ W^{∗}
↓  ↓
0 ⟶ 𝔤𝔩(α)^{−2} ⟶  ℜ^{−2}  ⟶ N^{−2} ⟶ 0
↓  ↓
A′_{2}^{∗} ⊗ A′′_{2} ⟶^{a_{12}^{T}⊗id} A′_{1}^{∗} ⊗ A′′_{2}

with exact columns. Since a_{12} is an isomorphism and Ker a_{23} = A′_{2} ⊗ U (see (13)) we have

N^{−2} = A′_{2}^{∗} ⊗ A′′_{3} ⊗ U^{∗}.

Twisting on L^{−1} = A′_{2} ⊗ A′′_{2}^{∗} we get

S^{−} = A′′_{2}^{∗} ⊗ A′′_{3} ⊗ U^{∗}.

The first isomorphism now follows from (15).

The cokernel of the morphism

(A′′_{2}^{∗} ⊗ A′_{2}) ⊕ (A′′_{3}^{∗} ⊗ A′_{3}) ⟶^{(id ⊗a_{23}^{T})⊕(a_{23}^{∗}⊗id)} A′′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗}

from the sequence (12) corresponding to w = 2 is isomorphic to the cokernel of the induced morphism from A′′_{2}^{∗} ⊗ A′_{2} to the cokernel of the morphism

A′′_{3}^{∗} ⊗ A′_{3} ⟶^{a_{23}^{∗}⊗id} A′′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗}

which by (15) is isomorphic to A′′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗}/η. This means that we have an exact sequence

0 ⟶ A′′_{2}^{∗} ⊗ A′_{2} ⟶ A′′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗}/η ⟶ N^{+2} ⟶ 0.

Twisting on L we get

0 ⟶ O_{M^{0}} ⟶ A′_{2}^{∗} ⊗ A′_{3} ⊗ W^{∗}/η ⟶ S^{+} ⟶ 0.

The second isomorphism now follows from (13).

Now we will describe the restriction of the universal family A^{+} on the subvariety M^{+}. Consider the following commutative diagram over M^{0}:

0 ⟶ W ⊗ η ⟶ W ⊗ η ⊗ (W/U ⊗ W^{∗}/η) ⟶ W ⊗ η ⊗ S^{+} ⟶ 0
↓  ↓  ↓
0 ⟶ W/U ⊗ η ⟶  \frac{W/U ⊗ W^{∗}}{O}  ⟶  S^{+}  ⟶ 0

Here the top row is the definition of S^{+} multiplied on W ⊗ η, the bottom row is evident, the vertical arrows are following: the left arrow is induced by projection W → W/U; the middle arrow is induced by the embedding η ⊗ W^{∗}/η ↪ Λ^{2}W^{∗} ⊗ O_{ℙ(W^{∗})} and by pairing W ⊗ Λ^{2}W^{∗} → W^{∗}; the right arrow is induced by the pairing W ⊗ η → O_{ℙ(W^{∗})}. One can easily check that everything here commutes and the diagram is correct.

The pull-back via p^{+} and the embedding L^{+} ⊂ p^{+}^{∗}S^{+} transform this diagram as follows:

\begin{aligned}0 & ⟶ & W ⊗ p^{+}^{∗}η & ⟶ & W ⊗ p^{+}^{∗}η ⊗ E^{+} & ⟶ & W ⊗ p^{+}^{∗}η ⊗ L^{+} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & p^{+}^{∗}(W/U ⊗ η) & ⟶ & K^{+} & ⟶ & L^{+} & ⟶ & 0 \end{aligned}  (17)

Here E^{+} and K^{+} are certain 2-dimensional and 3-dimensional vector bundles (defined by these sequences) on ℙ_{M^{0}}(S^{+}).

The maps p^{+}^{∗}η → p^{+}^{∗}η ⊗ E^{+} and W ⊗ p^{+}^{∗}η ⊗ E^{+} → K^{+} define the structure of (1, 2, 3)-dimensional representation of Q in the family

(p^{+}^{∗}(Λ^{2}U ⊗ η), p^{+}^{∗}(Λ^{2}U ⊗ η) ⊗ E^{+}, p^{+}^{∗}Λ^{2}U ⊗ K^{+}).  (18)

Remark 14. The additional twisting on p^{+}^{∗}Λ^{2}U is needed to satisfy the norming condition (2).

All representations in the family (18) are χ_{5/6}-stable, hence there exists a map f : ℙ_{M^{0}}(S^{+}) → M_{5/6} such that this family is the pull-back of the universal family over M_{5/6}. Since the family (18) has (1, 1, 2)-dimensional subfamily

(p^{+}^{∗}(Λ^{2}U ⊗ η), p^{+}^{∗}(Λ^{2}U ⊗ η), p^{+}^{∗}(Λ^{2}U ⊗ η ⊗ W/U))

(see (17)) the image of the map f is contained in M^{+}. Since the above subfamily as well as the corresponding quotient family

(0, p^{+}^{∗}(Λ^{2}U ⊗ η) ⊗ L^{+}, p^{+}^{∗}Λ^{2}U ⊗ L^{+})

are nothing but the universal families (14) and (16) over the moduli spaces M((1, 1, 2), χ_{2/3}) and M((0, 1, 1), χ_{2/3}) twisted by p^{+}^{∗}(Λ^{2}U ⊗ η) and p^{+}^{∗}Λ^{2}U ⊗ L^{+} respectively, we have

p^{+} · f = id_{M^{0}} .

This means that f : ℙ(S^{+}) → M^{+} is an isomorphism. We have proved the following.

Proposition 5.3. The restriction of the universal family A^{+} on M^{+} is the family (18). There exists an exact sequence of families over M^{+}:

0 → p^{+}^{∗}A′ → A_{|M^{+}}^{+} → p^{+}^{∗}A′′ ⊗ L^{+} → 0,

where

A′ = (Λ^{2}U ⊗ η, Λ^{2}U ⊗ η, Λ^{2}U ⊗ η ⊗ W/U), A′′ = (0, Λ^{2}U ⊗ η, Λ^{2}U).

are universal families over M((1, 1, 2), χ_{2/3}) and M((0, 1, 1), χ_{2/3}) respectively.

Now we will describe the restriction of the universal family A^{−} on the subvariety M^{−}. For convenience we will consider M^{−} not as ℙ(S^{−}) but as ℙ(Λ^{2}U ⊗ η ⊗ S^{−}) = ℙ(U). Let L′ = p^{−}^{∗}(Λ^{2}U ⊗ η) ⊗ L^{−} be the tautological linear subbundle in p^{−}^{∗}U. Consider the commutative diagram of vector bundles over ℙ(U):

0 ⟶ W ⊗ (p^{−}^{∗}η ⊗ \frac{p^{−}^{∗}U}{L′} ) ⟶ W ⊗ (p^{−}^{∗}η ⊗ \frac{p^{−}^{∗}U}{L′} ) ⊕ W ⟶ W  ⟶ 0
↓  ↓  ↓
0 ⟶  \frac{p^{−}^{∗}U}{L′}  ⟶  \frac{W}{L′}  ⟶ p^{−}^{∗}W/U ⟶ 0

and note that p^{−}^{∗}η ⊗ \frac{p^{−}^{∗}U}{L′} ≅ L^{−}^{∗}. The isomorphism on the second summand O ⟶ L^{−}^{∗} ⊕ O and

the map W ⊗ (L^{−}^{∗} ⊕ O) ⟶ \frac{W}{L′} from the above diagram define the structure of (1, 2, 3)-dimensional representation in the family

(p^{−}^{∗}(Λ^{2}U ⊗ η), p^{−}^{∗}(Λ^{2}U ⊗ η) ⊗ (L^{−}^{∗} ⊕ O) , p^{−}^{∗}(Λ^{2}U ⊗ η) ⊗ \frac{W}{L′} )  (19)

Remark 15. The additional twisting on p^{−}^{∗}(Λ^{2}U ⊗ η) is needed to satisfy the norming condition (2).

All representations in the family (19) are χ_{1/3}-stable, hence there exists a map f : ℙ_{M^{0}}(S^{−}) → M_{1/3} such that this family is the pull-back of the universal family A^{−} over M_{1/3}. Since family (19) has (0, 1, 1)-dimensional subfamily

(0, p^{−}^{∗}(Λ^{2}U ⊗ η) ⊗ L^{−}^{∗}, p^{−}^{∗}Λ^{2}U ⊗ L^{−}^{∗})

the image of the map f is contained in M^{−}. Since the above subfamily as well as the corresponding quotient family

(p^{−}^{∗}(Λ^{2}U ⊗ η), p^{−}^{∗}(Λ^{2}U ⊗ η), p^{−}^{∗}(Λ^{2}U ⊗ η) ⊗ p^{−}^{∗}(W/U))

are nothing but the universal families over the moduli spaces M((1, 1, 2), χ_{2/3}) and M((0, 1, 1), χ_{2/3}) twisted by p^{−}^{∗}Λ^{2}U ⊗ L^{−}^{∗} and p^{−}^{∗}(Λ^{2}U ⊗ η) respectively, we have

p^{−} · f = id_{M^{0}} .

This means that f : ℙ(S^{−}) → M^{−} is an isomorphism. We have proved the following.

Proposition 5.4. The restriction of the universal family A^{−} on M^{−} is the family (19). There is an exact sequence of families over M^{−}:

0 → p^{−}^{∗}A′′ ⊗ L^{−}^{∗} → A_{|M^{−}}^{−} → p^{−}^{∗}A′ → 0,

where A′ and A′′ are the same as in Proposition 5.3.

Let p := p^{−} · π̂_{0} = p^{+} · π̂_{1} : M̃ → M^{0} be the natural projection.

Proposition 5.5. On the space M̃ there is an exact sequence of coherent sheaves

0 ⟶ π̂_{0}^{∗}A^{−} ⟶ π̂_{1}^{∗}A^{+} ⟶ i_{∗}(p^{∗}A′′ ⊗ π̂_{1}^{∗}L^{+}) ⟶ 0.  (20)

The restriction of this sequence on M̃ factors as

p^{∗}A′′ ⊗ π̂_{0}^{∗}L^{−}^{∗} ↪ (π̂_{0}^{∗}A^{−})_{|M̃} ↠ p^{∗}A′ ↪ (π̂_{1}^{∗}A^{+})_{|M̃} ↠ p^{∗}A′′ ⊗ π̂_{1}^{∗}L^{+}.

Proof. Morphism A_{|M^{+}}^{+} → p^{+}^{∗}A′′ ⊗L^{+} from Proposition 5.3 induces the surjective morphism π̂_{1}^{∗}A^{+} ↠ i_{∗}(p^{∗}A′′ ⊗ π̂_{1}^{∗}L^{+}). Denote its kernel by G. By reasons of homological dimension G is locally free. We claim that actually G is a pull-back from M_{1/3}. In order to prove this we should check that the restriction of G on all fibers of π̂_{0} is trivial. Since π̂_{0} is an isomorphism on the complement of M̃ we may restrict ourselves to the latter divisor. The definition of G and Proposition 5.3 implies that the restriction of G on M̃ admits the following exact sequence:

0 → p^{∗}A′′ ⊗ π̂_{1}^{∗}L^{+} ⊗ N_{M̃/M̃}^{∗} → G_{|M̃} → p^{∗}A′ → 0.

Since by Corollary 4.6 we have N_{M̃/M̃} = π̂_{0}^{∗}L^{−} ⊗ π̂_{0}^{∗}L^{+} we get

0 → π̂_{0}^{∗}(p^{−}^{∗}A′′ ⊗ L^{−}^{∗}) → G_{|M̃} → π̂_{0}^{∗}p^{−}^{∗}A′ → 0

and the claim follows, i.e. G = π̂_{0}^{∗}G′ with G′ locally free. Identifying the complements of M^{0}, M^{−}, M^{+} and M̃ in M_{2/3}, M_{1/3}, M_{5/6} and M̃ respectively via the maps π̂_{0}, π̂_{1} and p^{±} we get the following isomorphisms:

G′_{|M_{1/3}−M^{−}} ≅ G_{|M̃−M̃} ≅ π̂_{1}^{∗}A_{|M̃−M̃}^{−} ≅ A_{|M_{5/6}−M^{+}}^{−} ≅ A^{0} ≅ A_{|M_{1/3}−M^{−}}^{+},

where by A^{0} we denote the universal family over M_{2/3}^{s} = M_{2/3} − M^{0} satisfying the norming condition (2). The first and the last terms in the above sequence of isomorphisms are locally free sheaves on M_{1/3} and isomorphism is defined on the complement of the subvariety M^{−} of big codimension (namely codim_{M_{1/3}} M^{−} = 5) hence it can be continued to all M_{1/3} (Harthog’s Theorem). This means that G = π̂_{0}^{∗}A^{−} and the Proposition follows.

Exact sequence (20) is an exact sequence of representations of Q. It consists of three components corresponding to three vertices of Q. They relate the pull-backs of universal bundles E and K with pull-backs of bundles over ℙ(W). Now we will write them down:

Corollary 5.6. Over the space M̃ there exist the following exact sequences of coherent sheaves:

0 → π̂_{0}^{∗}L_{R} ⊗ π̃_{0}^{∗}O_{ℙ(W)}(−2)  → π̂_{1}^{∗}L_{E} → 0,
0 → π̂_{0}^{∗}(E^{−} ⊗ L_{R}) ⊗ π̃_{0}^{∗}O_{ℙ(W)}(−3) → π̃_{1}^{∗}E → i_{∗}(p^{∗}(Λ^{2}U ⊗ η) ⊗ π̂_{1}^{∗}L^{+}) → 0,
0 → π̂_{0}^{∗}L_{R} ⊗ π̃_{0}^{∗}T_{ℙ(W)}(−3)  → π̃_{1}^{∗}K → i_{∗}(p^{∗}Λ^{2}U ⊗ π̂_{1}^{∗}L^{+})  → 0.

Now we will write down the sequences describing restrictions of pull-backs of L_{E}, E and K on M̃ (that is just the pull-backs of the components of the exact sequence of Proposition 5.3).

Corollary 5.7. Over the divisor M̃ there exist the following exact sequences of coherent sheaves:

0 → p^{∗}(Λ^{2}U ⊗ η)  → (π̂_{1}^{∗}L_{E})_{|M̃} → 0,
0 → p^{∗}(Λ^{2}U ⊗ η)  → (π̃_{1}^{∗}E)_{|M̃} → p^{∗}(Λ^{2}U ⊗ η) ⊗ π̂_{1}^{∗}L^{+} → 0,
0 → p^{∗}(Λ^{2}U ⊗ η ⊗ W/U) → (π̃_{1}^{∗}K)_{|M̃} → p^{∗}Λ^{2}U ⊗ π̂_{1}^{∗}L^{+}  → 0.

Here we gather some information about the Picard group of M̃. Let h = c_{1}(O_{ℙ(W)}(1)) ∈ Pic ℙ(W) be the class of hyperplane on ℙ(W), H = c_{1}(E^{∗}) = c_{1}(K^{∗}) ∈ Pic 𝔚, r = c_{1}(L_{R}^{∗}) ∈ Pic ℙ_{ℙ(W)}(R), e = c_{1}(L_{E}^{∗}) ∈ Pic ℙ_{𝔚}(E) and d = [M̃] ∈ Pic M̃. The natural embeddings of these Picard groups into Pic M̃ allows to consider all above classes as elements of Pic M̃.

Corollary 5.8. We have the following relations in Pic M̃

e = r + 2h, H = 3r + 5h − d,  r = 2H − 5e + 2d, h = 3e − H − d.  (21)

Proof. The first relation follows from the first sequence of Corollary 5.6. The second relation follows from the third sequence of Corollary 5.6. The third and the fourth relations follows from the first and the second.

Corollary 5.9. The following formulas express canonical classes in terms of the above generators of the Picard groups.

K_{ℙ(W)} = −4h,  K_{𝔚} = −4H
K_{ℙ_{ℙ(W)}(R)} = −19h − 11r,  K_{ℙ_{𝔚}(E)} = −3H − 2e,
KM̃ = −19h − 11r + 4d = −3H − 2e + d.

Proof. Evident.

Let s^{−} = c_{1}(L^{−}^{∗}) ∈ Pic M^{−} and s^{+} = c_{1}(L^{+}^{∗}) ∈ Pic M^{+}.

Corollary 5.10. We have the following equalities in Pic M̃

i^{∗}h = s^{−} + …, i^{∗}r = −2s^{−} + …,  i^{∗}H = s^{+} + …,  i^{∗}e = ….

where … stands for the pull-back of some class from M^{0}.

Proof. Follows from the Propositions 5.3 and 5.4.

The Corollary 5.6 and Corollary 5.7 give a way to compute the cohomology of any vector bundle over 𝔚 which can be produced from the universal bundles E and K by tensor operations and of its restriction on M̃. In particular, if someone could invent, how from these bundles can be constructed an exceptional collection on 𝔚, then its exceptionality could be verified by such computations. Also the fullness of this collection could be deduced. One only need to produce from this collection an exceptional collection on M̃ via Orlov’s procedure (see [6]) and compare it with another collection, produced from the standard exceptional collection on ℙ(W).

Unfortunately, we have no idea of how to construct an exceptional collection on 𝔚. But the method works for Fano threefolds V_{22}. Such threefold X can be embedded in the space 𝔚 (see [5]) as zero locus of a certain section of the vector bundle K^{∗} ⊕ K^{∗} ⊕ K^{∗}. But the most amazing, that taking zero locus of the pull-back of this bundle on the space M̃ we will get for the threefold X almost the same picture as we have for 𝔚, namely the blow-up of the projectivization of 2-dimensional vector bundle on X is isomorphic to the blow-up of the projectivization of 2-dimensional vector bundle over ℙ(W). The main difference is that in the case of threefold X the exceptional collection can be constructed and we can get the description of the derived category of coherent sheaves on X. This will be done in the forthcoming paper.

References

[1] A. Bialynicki-Birula, Some theorems on action of algebraic groups, Ann. Math. 98 (1973), pp. 480–497.

[2] G. Ellingsrud, R. Piene, S. A. Strømme. On the variety of nets of quadrics defining twisted cubics, in Space curves, Lect. Notes Math. 1266 (1987), pp. 84–96.

[3] A. King. Moduli of representations of finite dimensional algebras, Q. J. Math., Oxf. II. Ser. 45, No. 180 (1994), pp. 515– 530.

[4] A. Kuznetsov. An exceptional collection of vector bundles on Fano threefolds V_{22}, Vestnik MGU, Ser. 1, Mat. Mekh., 1996, No. 3, pp. 41–44.

[5] S. Mukai. Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 (1989), No. 9, pp. 3000–3002.

[6] D. Orlov. Projective bundles, monoidal transformations and derived categories of coherent sheaves, Izv. Akad. Nauk SSSR, Ser. Mat., 56 (1992), pp. 852–862; Math USSR Izv., 38 (1993), pp. 133-141.

[7] M. Thaddeus. Geometric invariant theory and flips, Journal of the American Mathematical Society, Vol. 9 (1996), No. 3, pp. 691–724.

Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow 121002 Russia

   E-mail address: sasha@ium.ips.ras.ru