Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima-Yoshioka's diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.

In these examples, the moduli spaces appearing in wall crossing diagrams are smooth and birational to each other. In fact, in the case studied in [10,11], the moduli spaces are connected by standard flips. In the case of [21,22,23], their geometry is more complicated. Indeed, Nakajima-Yoshioka proved that the contracted loci of the morphisms ξ ± m have stratifications (called Brill-Noether stratifications) such that each stratum has the structure of a Grassmannian bundle.
The aim of this paper is the further study of birational geometric properties of the diagram (1.1). In particular, we show that it is an instance of the minimal model program.
Theorem 1.1 (Theorem 4.7). The diagram (1.1) realizes a minimal model program for the moduli space of framed torsion free sheaves on the blow-up P 2 . The program ends with the minimal model, the moduli space of framed torsion free sheaves on P 2 , which is a hyper-Kähler manifold.
We will also verify Bondal-Orlov [5], Kawamata's [15] D/K equivalence conjecture for these moduli spaces: Theorem 1.2 (Theorem 4.9). For each integer m ∈ Z ≥0 , we have a fully faithful embedding ) between the derived categories. In particular, we have an embedding where M P 2 , MP 2 denote the moduli spaces of torsion free framed sheaves on P 2 ,P 2 , respectively.
So we get an interesting relationship among wall-crossing formulas for Donaldson type invariants, birational geometry, and derived categories.
We can also consider the moduli space of Gieseker stable sheaves on a smooth projective surface and that on its blow-up (see Theorem 4.10 for the precise statement): Theorem 1.3. Let S be a smooth projective surface,Ŝ the blow-up of S at a point. Under certain numerical conditions, the MMP for the moduli space MŜ of Gieseker stable sheaves onŜ is reduced to MMP for the moduli M S on S. Furthermore, there exists a fully faithful embedding between their derived categories.
For instance, we can apply the above theorem when S is a del Pezzo surface.
1.2. Strategy of the proof. To prove our main result Theorem 1.1, we will compute the normal bundles of the fibers explicitly, following the idea from [10,11]. Although the geometry of the diagram (1.1) is more complicated compared to the one considered in [10,11], it turns out their method still works in our setting. Actually, we are able to describe the normal bundle of each Brill-Noether stratum explicitly. Then we will see that the diagram realizes the MMP when we decrease the stability parameter m ∈ Z ≥0 . Furthermore, the normal bundle computation enables us to reduce the construction of the fully faithful embedding between derived categories to the formal local case; the latter is already handled in the paper [8] and hence we can prove Theorem 4.9.
1.3. Relation to existing works. In [17], the author studied birational geometry of the Hilbert scheme of two points on blow-ups. The main result of the present paper is an extension to the completely general setting.
There are several works investigating birational geometry and derived categories of moduli spaces. For a standard flip between moduli spaces obtained in [10,11], Ballard [4] constructed a semi-orthogonal decomposition (SOD) of their derived categories.
Recently, Toda [24,25] introduced the notion of d-critical birational geometry, which is a certain virtual analogue of usual birational geometry. It is shown that if two smooth varieties are connected by a simple d-critical flip, then we have an SOD of their derived categories. See [18,25] for interesting examples of d-critical flips.
The SODs obtained in the papers [4], [25] can be considered as categorifications of wall crossing formulas for Donaldson type invariants, Donaldson-Thomas type invariants, respectively. It would be interesting to describe the semi-orthogonal complements of the embedding in our Theorem 4.9, which would give a categorification of Nakajima-Yoshioka's wall crossing formula.
1.4. Organization of the paper. The paper is organized as follows. In Section 2, we collect some terminology and useful lemmas from birational geometry and derived categories. In Section 3, we recall the result of Nakajima-Yoshioka. In Section 4, we prove our main results. In Section 5, we give some explicit examples.
Acknowledgement. I would like to thank Professors Jim Bryan and Yukinobu Toda for fruitful discussions. This work was supported by Grant-in-Aid for JSPS Research Fellow 17J00664.
Finally, I would like to thank the referee for various suggestions and comments to the previous version of this paper.
Notation and Convention. In this paper, we always work over the complex number field C.
• For a variety X, we denote by D b (X) := D b (Coh(X)) the bounded derived category of coherent sheaves on X. • For a proper morphism f : M → N between varieties and objects E, F ∈ D b (M ), we denote by Ext q f (E, F ) the q-th derived functor of Hom f (E, F ) := f * Hom(E, F ).
• For coherent sheaves E, F on a variety, we define hom(E, F ) := dim Hom(E, F ) and ext i (E, F ) := dim Ext i (E, F ). • For a vector bundle V on a variety and an integer i > 0, we denote by Gr(i, V) the Grassmann bundle of i-dimensional subbundles of V.

Preliminaries
2.1. Terminologies from birational geometry. In this subsection, we recall some notions from birational geometry. The standard reference for this subsection is [16].
Definition 2.1. Let φ : X → Z be a projective morphism between normal quasi-projective varieties. We say that φ is a K-positive (resp. K-negative) contraction if the following conditions hold: the canonical divisor K X (resp. the anti-canonical divisor −K X ) is φ-ample.
(1) φ is called a divisorial (resp. flipping) contraction if it is birational and the φ-exceptional locus has codimension one (resp. at least two).
(3) Assume that φ is a flipping contraction. Then a flip of φ is a Kpositive birational contraction φ + : X + → Z. We also call the birational map X X + a flip.
Definition 2.3. Let X be a quasi-projective variety with at worst terminal singularities. A minimal model program of X is a sequence of birational maps (1) each birational map X i X i+1 is either a divisorial contraction or a flip, (2) the variety X N is either a minimal model (i.e. K X N is nef) or has a structure of a Mori fiber space.
We do not give the definition of a terminal singularity, as we only consider smooth varieties in this paper. For the precise definition, see [16,Definition 2.34].
The following lemma is useful for our purpose: . Let X be a smooth variety, φ : X → Z a K-negative contraction, and F ⊂ X a smooth φ-fiber. Assume that the following conditions hold: Then the formal neighborhood of F in X is isomorphic to that of F in the total space of N , embedded as the zero section.

2.2.
Fourier-Mukai functors. In this subsection, we recall the definition and basic properties of Fourier-Mukai functors. The standard reference is [13].
Definition 2.5. Let X, Y be smooth quasi-projective varieties, and P ∈ D b (X × Y ) be an object whose support is proper over Y . The Fourier-Mukai (FM) functor with kernel P is the functor Φ P : Note that, in the above definition, we assume the object P has proper support over Y to ensure the associated FM functor Φ P preserves bounded complexes.
Let us give a trivial example: Then the associated FM functor Φ O ∆ X coincides with the identity functor id D b (X) .
be a FM functor with kernel P. By [13, Proposition 5.9], the right adjoint functor Φ R : D b (Y ) → D b (X) is given by the FM functor with kernel Similarly, the composition of two FM functors is again a FM functor: [13,Proposition 5.10]. Now let us consider the adjoint map The following result tells us that it lifts to the morphism between FM kernels: be a FM functor. Then there exists a morphism , which induces the adjoint map (2.1).
We will use the following lemma in the proof of the main result: 2). The following statements hold: (1) The FM functor Φ = Φ P is fully faithful if and only if Q = 0.
(2) For a point p ∈ U , denote byÛ p the completion of U at p. Denote byX p the base change of X → U alongÛ p → U . Then Q = 0 if and only if Q ⊗ O X×X OX p ×Xp = 0 for every p ∈ U . In particular, the FM functor Φ P is fully faithful if and only if the FM functors Φ P ∧ p are fully faithful for all p ∈ U , where we put P ∧ p := P ⊗ O X×Y OX p×Ŷp . Proof.
(1) First observe that the functor Φ is fully faithful if and only if the adjoint map (2.1) is an isomorphism. Let us take an object E ∈ D b (X). By applying Φ (−) (E) to the exact triangle , we obtain an exact triangle For the converse, assume that Q = 0. Then there exists a point x ∈ X such that Q x := Q| {x}×X = 0. In particular, 3), i.e., the functor Φ is not fully faithful.
(2) The second assertion follows from the fact that the completionÛ p → U is faithfully flat.

Grassmannian flip.
Here we recall about geometry and the derived categories of Grassmannian flips, which play important roles in this paper. We refer [3, Chapter II] for the details (see also [8,Section 1]). Let W ± be vector spaces. Take a positive integer i ≤ min{dim W ± }.
Then the determinantal variety is defined to be and its Springer resolution is defined as We have the following projections: The fiber of h + : Y + → Gr(i, W + ) over a point V ∈ Gr(i, W + ) is

Hence we have an isomorphism
where S + denotes the universal subbundle on Gr(i, W + ). This shows that the variety Y + is smooth of dimension i(dim W + + dim W − − i). Moreover, the other projection φ + : Y + → Z is isomorphism over the open locus On the other hand, we have canonical isomorphisms hence by replacing the roles of W ± in the above construction, we get a second resolution of singularities of the variety Z: where S − denotes the universal subbundle of Gr(i, W − ).
Hence we obtain the diagram which we call the Grassmannian flip. The following lemma justifies this notion: Then the following statements hold: (1) The canonical bundles of Y ± are given as In particular, the morphism φ + (resp. φ − ) is a K-negative (resp. K-positive) contraction. (2) When i = dim W − , the morphism φ + is a divisorial contraction and the morphism φ − is an isomorphism. Proof.
(1) We have the following exact sequence and the tangent bundle of the Grassmannian variety is given by T Gr(i,W ± ) ∼ = S ± * ⊗ Q ± , where Q ± denotes the universal quotient bundle of Gr(i, W ± ). Hence we obtain (2.4) Moreover, the tautological sequence

Now the equation (2.4) becomes
as required.
To prove (2) and (3), we need to determine the dimensions of the φ ±exceptional loci. First note that we have a sequence of closed immersions and the assertions (2) and (3) hold.
For the derived categories of Y ± , we have the following result: Remark 2.11. When dim W − = dim W + , the canonical bundles of Y ± are trivial, and we call the birational map Y + Y − the Grassmannian flop. In this case, the FM functor Φ O W is an equivalence, which is treated also in [9].
2.4. Bott type vanishing. For later use, we prove some vanishing results on cohomology groups of certain vector bundles on the Grassmannian varieties. Let W be a vector space of dimension n, and i ∈ Z >0 be a positive integer with i < n. We consider the Grassmannian variety G = Gr(i, W ). Denote by S, Q the universal sub and quotient bundles, respectively.
3. Framed sheaves on the blow-up 3.1. ADHM description and wall crossing. Let us consider the projective plane P 2 = C 2 ∪ l ∞ and its blow-up f :P 2 → P 2 at the origin 0 ∈ C 2 . Denote by C ⊂ P 2 the f -exceptional curve. In this subsection, we recall the notion of m-stable sheaves onP 2 with framing at l ∞ studied by Nakajima-Yoshioka [21].
Definition 3.1. Fix a non-negative integer m ∈ Z ≥0 . Let (E, Φ) be a framed sheaf onP 2 , i.e., let E be a coherent sheaf onP 2 and Φ : E| l∞ a framing at l ∞ . We say that (E, Φ) is m-stable if the following conditions hold: Remark 3.2. Let E be an m-stable framed sheaf. By [22, Proposition 1.9 (1)], the condition (1) in the above definition implies that the sheaf E(−mC) is an object of the category Per(P 2 /P 2 ) of perverse coherent sheaves introduced by Bridgeland [6]. Hence we may also think of m-stable sheaves as stable objects in the category Per(P 2 /P 2 ) ⊗ O(mC).
We denote by M m (v) the fine moduli space of m-stable framed sheaves on P 2 with Chern character v. Let us recall the ADHM description of framed sheaves onP 2 ; Consider the following quiver Q For a given Chern character we associate the dimension vector d = (d 0 , d 1 , d ∞ ) by the following formula:
(3) There exists an integer m 0 ∈ Z ≥0 such that for every integer m ≥ m 0 , we have an isomorphism where MP 2 (v) denotes the moduli space of torsion free framed sheaves onP 2 . (4) For each integer m ∈ Z ≥0 and a stability condition ζ ∈ W m , there exists a set-theoretic bijection where we put c m := ch(O C (−m − 1)).
Proof. The classification of walls is explained in [21,Section 4.3]; The statements (1), (2), (3) are proved in Theorems 1.5 and 2.5, Proposition 7.4, and Proposition 7.1 in [21], respectively. Let us now consider the statement (4). By [21, Section 4.3, Proposition 5.3], the wall W m corresponds to the destabilizing object O C (−m − 1) in the following sense: for any stability condition ζ ∈ W m on the wall and any ζsemistable object E, its S-equivalence class is of the form O C (−m−1) ⊕i ⊕E ′ for some non-negative integer i ≥ 0 and m-stable and (m + 1)-stable sheaf E ′ with ch(E ′ ) = v − ic m . Hence as a closed point in the moduli space, we have i.e., the closed point [E] ∈ M ζ (Q, I; d) is uniquely determined by the sheaf Hence we have the bijection as stated.
By the above theorem, we have the diagram as in (1.1) connecting the moduli spaces M P 2 (r, 0, ch 2 ) and MP 2 (r, −kC, ch 2 ).

3.2.
Brill-Noether loci. Next we recall the Brill-Noether stratifications on the moduli spaces and the determination of the fibers over ξ ± m . For each integer i ∈ Z ≥0 , let us consider the following locally closed subschemes: We call them as the Brill-Noether strata. We also denote as Let us take an object E ∈ M m (v). By [22,Proposition 3.15], we have the exact sequence and it is an open immersion.
Let us denote by the universal family, and let p : be the projections. The following theorem shows the structure of the morphism ξ ± m in terms of the Brill-Noether strata: In particular, every fiber of the morphisms ξ ± m is the Grassmannian variety.

Birational geometry of moduli spaces
In this section, we will prove that the diagram (1.1) realizes the MMP.
The key ingredient is to compute the normal bundles of the fibers of ξ ± m , following the arguments of Ellingsrud-Göttsche [10] and Friedman-Qin [11]. We keep the notations as in the previous section. Fix integers m, i ∈ Z ≥0 . Let be the universal families, and let which are isomorphic to the Brill-Noether loci M m (v) i , M m+1 (v) i , respectively, by Theorem 3.4. On G ± i , we have the following tautological sequences: We start with the following lemma.

Now, applying the functor Hom
) to the exact sequence (4.5), we get the morphism On the other hand, applying the functor ) to the exact sequence (4.5), we get the morphism

By the above arguments, we have a morphism
In the following lemma, we show that our morphism δ is surjective, by checking it on the fibers of the morphism π − : G − i → M m,m+1 (v) i in (4.3). By abuse of notation, we also denote by δ 1 , δ 2 their restrictions to the π −fibers.
Proof. By replacing the object E ′ ∈ M m,m+1 (v) i with E ′ ⊗ OP 2 (−mC) ∈ M m,m+1 (v.e −mC ) i , we may assume m = 0. Let us take an object E ′ ∈ M 0,1 (v) i and an i-dimensional subspace V ⊂ Ext 1 (E ′ , O C (−1)). Let be the associated universal extension. By applying the functor Hom(E − , (−)⊗ O(−l ∞ )), we have the exact sequence Note that we have used the fact that O C (−l ∞ ) = O C . By Serre duality and the 0-stability of E − , we have the vanishing and hence δ 1 is surjective. Furthermore, by applying the functor Hom(−, O C (−1)) to the exact sequence (4.7) we have the exact sequence Hence we have Similarly, applying the functor Hom(−, E ′ (−l ∞ )) to the exact sequence (4.7), we obtain By Lemma 4.5 below, we have the vanishing Ext 2 (E ′ , E ′ (−l ∞ )) = 0 and hence δ 2 is surjective. Proof. By Serre duality, we have
By applying the functor Hom(G, −) to the exact sequence On the other hand, as G is a framed sheaf on P 2 , we have G| l∞ ∼ = O ⊕ ch 0 (G) P 2 . Hence we have Combining with the exact sequence (4.9), we conclude that Hom(G, G(−2)) ∼ = Hom(G, G(−3)).
By tensoring the exact sequence (4.8) with O P 2 (−i) and repeating the above argument, we can inductively prove the isomorphism for all i ≥ 0. By Serre duality, we obtain and the right hand side vanishes for sufficiently large i > 0, since O P 2 (1) is ample. We conclude that Hom(G, G(−2)) = 0 as required.
We also need the following: We only prove the first equality. By the dimension formula for the framed moduli space in Theorem 3.3 (1), we have where the vector (d 0 , d 1 , d ∞ ) is defined as in (3.1). On the other hand, for any object E ′ ∈ M m,m+1 (v) i , we have where the first equality follows from the fact that π − : G − i → M m,m+1 (v) i is a Grassmannian bundle defined as in (4.3). For the second equality, first observe that we have is an open immersion and hence the second equality in (4.11) holds. Similarly, by using the m-stability and (m + 1)-stability of E ′ , we have (4.12) rk (see (4.2) and (4.4) for the definitions of W + i and S − i ). Combining the equalities (4.10), (4.11) and (4.12), we obtain where the second equality holds since we have and for the third equality, we use the Serre duality

Now we begin the proof of Theorem 4.2.
Proof of Theorem 4.2. We only prove the first assertion. By Lemma 4.3 and Lemma 4.4, we have a surjective morphism Let the notations be as in Lemma 4.4. By Lemma 4.6, the vector spaces T E − G − i and ker(δ E − ) are of the same dimension. Hence it is enough to show that the composition is zero. Indeed, if this is the case, then we have T E − G − i = ker(δ E − ), which induces a surjection T G − i ։ ker(δ) between torsion free sheaves of the same rank: it should be an isomorphism.
We have the exact sequence We can see that the composition coincides with the morphism α 1 in the exact sequence (4.6). In particular, it becomes zero after composing with δ 1 . Hence the morphism which is zero by the second exact sequence in (4.6). We conclude that T E − G − i = ker(δ E − ) as required.
Now we have the following theorem: Theorem 4.7. Fix a Chern character of the form v = (r, 0, ch 2 ) ∈ H 2 * (P 2 , Q). Then the diagram (1.1) is a minimal model program for the moduli space MP 2 (v) of framed torsion free sheaves on the blow-upP 2 . The program ends with the minimal model, the moduli space M P 2 (r, 0, ch 2 ) of framed torsion free sheaves on P 2 , which is a hyper-Kähler manifold.
Proof. We claim that for each m ∈ Z ≥0 , the morphism ξ + m (resp. ξ − m ) is a K-negative (resp. K-positive) contraction (cf. Definition 2.1). By Lemma 2.9 and Theorem 4.2, it is enough to show the inequality Now the assertion directly follows from the Riemann-Roch theorem. Explicitly, we have (4.13) To see that the moduli space M P 2 (r, 0, ch 2 ) is hyper-Kähler, recall that the space M P 2 (r, 0, ch 2 ) is isomorphic to Nakajima's quiver variety associated with the quiver with one vertex and one loop [20,Chapter 2]. By the general fact that Nakajima's quiver varieties are hyper-Kähler [19,12], so is the variety M P 2 (r, 0, ch 2 ).
From the arguments above, we can also deduce the following result: Proposition 4.8. Let us take a Chern character v = (r, 0, ch 2 ) ∈ H 2 * (P 2 , Q). The following statements hold.
Proof. By Lemma 4.6 and the equation (4.13), we have the inequality dim Hence it is enough to estimate the dimension of G + i . By Lemma 4.6, we have It follows that for each m ≥ 1, the morphism ξ ± m is a small contraction, while ξ + 0 is a divisorial contraction.  Then for each integer m ∈ Z ≥0 , we have the fully faithful functor whose Fourier-Mukai kernel is O W m . In particular, we have a fully faithful embedding D b (M P 2 (r, 0, ch 2 )) ֒→ D b (MP 2 (r, 0, ch 2 )).
Proof. By Lemma 2.8, we can reduce the statement to the formal completion at a point [ We claim that the diagram (1.1) is formally locally isomorphic to the Grassmannian flip appered in Section 2.3, by using Lemma 2.8. Let us take an object [E ′ ] ∈ M m,m+1 (v) i and put U : Recall that the fibers of ξ ± m are the Grassmannian varieties By Theorem 4.2, their normal bundles are given as Here, S ± denotes the tautological subbundles on G ± (E ′ ). First note that the conormal bundle N ∨ G + (E ′ )/M m+1 (v) is nef as it is globally generated. Combining with Lemma 2.14, we can apply Lemma 2.8 to G + (E) ⊂ M m+1 (v). Moreover, its flip is unique by [16,Corollary 6.4]. We conclude that the formal completion of the diagram (1.1) is isomorphic to the formal completion of the diagram (4.14) (see Section 2.3). By Theorem 2.10, we have the fully faithful functor whose Fourier-Mukai kernel is the structure sheaf of the fiber product Y − × Z Y + . Hence globally, the functor Φ : Projective case. Let S be a smooth projective surface, f :Ŝ → S be the blow-up at a point. Let H be an ample divisor on S. In this setting, we can consider the m-stability for coherent sheaves E onŜ (cf. [22]), by replacing the condition (3) in Definition 3.1 with Let us fix a cohomology class w = (w 0 , w 1 , w 2 ) ∈ H 2 * (S, Q) which is in the image of the Chern character map, and v := f * w ∈ H 2 * (Ŝ, Q). Assume the following conditions hold: In the projective setting, we can also prove the results in the previous subsections by using the properties listed above. We omit the proof, since the arguments are very similar.
between their derived categories.

Examples
In this section, we give some explicit examples. 5.1. Simplest example. As the first example, we consider the case when the Chern character is (r, 0, −1) with r ≥ 1. In this case, the corresponding quiver representation is We begin with the following easy observation:  Furthermore, we can describe these moduli spaces explicitly: We have the diagram M 0 (r, 0, −1) P P P P P P P P P P P P P P P P P P P P P P P P where P (r−1) is embedded into M 0 (r, 0, −1) as the zero section.
Before starting the proof, let us recall the ADHM description of the framed sheaves on P 2 (see [21,Section 1] and [20,Capter 2] for the details). Let V, W be vector spaces. An ADHM data is the data X := (B 1 , B 2 , i, j), where B α ∈ End(V ), i ∈ Hom(W, V ), and j ∈ Hom(V, W ) satisfying the relation An ADHM data X = (B 1 , B 2 , i, j) is called stable if there is no proper subspace T ⊂ V such that B α (T ) ⊂ T for α = 1, 2, and Image(i) ⊂ T . We have the moduli space of stable ADHM data as the quotient of the stable locus inside the affine space modulo the natural GL(V )-action. Then the moduli space M P 2 (r, 0, ch 2 ) of torsion free framed sheaves on P 2 is isomorphic to the moduli space of stable ADHM data with dim V = − ch 2 , dim W = r (cf. [20, Chapter 2]) Proof of Lemma 5.2. Let us put dim V = 1, dim W = r, and let X = (B 1 , B 2 , i, j) be a stable ADHM data. Since dim V = 1, the relation (5.3) becomes ij = 0, and the stability condition becomes i = 0. Hence the stable locus is given as is the moment map. As GL(V ) = C * acts trivially on End(V ), and on W by weight −1, we have Next we determine the variety M 1 (r, 0, −1). Recall that we have an isomorphism (cf. [21,Proposition 7.4 The locus blown-up by ξ + 0 is given by under the isomorphism (5.4), which is nothing but the zero section.

5.2.
Hilbert scheme of points. In this subsection, we consider the moduli spaces with Chern character (1, 0, −n). In this case, the diagram (1.1) connects the Hilbert schemes of points Hilb n (C 2 ) and Hilb n (Ĉ 2 ). We first analyze the stability of ideal sheaves I Z ∈ Hilb n (Ĉ 2 ) (it is an ideal sheaf of a length n subscheme Z ⊂P 2 with Z ∩ l ∞ = ∅).
Lemma 5.3. Let us take a point I Z ∈ Hilb n (Ĉ 2 ) and let k be a length of Z ∩ C. Then I Z is k-stable but not (k − 1)-stable. Furthermore, its destabilizing sequence for (k − 1)-stability is given as for some length (n−k) zero dimensional subscheme W ⊂P 2 with W ∩C = ∅.
Proof. The first statement follows from the proof of [ Hence the second assertion follows.

5.2.2.
Next we consider the case when n = 3. First let us analyze geometry of (5. Next we analyze the geometry of the morphisms ξ ± 1 . We have just seen that (1) an ideal sheaf I Z , where Z ⊂Ĉ 2 is a length 3 zero dimensional subscheme with Z ∩ C = {pt}. (2) an object E 1 ∈ P(Ext 1 (I p (−C), O C (−2))), where p ∈Ĉ 2 .
Let us consider a sheaf E 1 of type (2). By a computation similar as above, we can see that hom(E 1 , O C (−1)) ≤ 2. Again, by a simple diagram chasing, we have the following possibilities: • When hom(E 1 , O C (−1)) = 1, E 1 fits into a exact sequence for some q ∈ C with q = p.