Moduli of Framed Sheaves on Projective Surfaces

Ugo Bruzzo and Dimitri Markushevish^{1}

Received: October 19, 2009 Revised: November 11, 2010

Communicated by Thomas Peternell

Abstract. We show that there exists a fine moduli space for torsionfree sheaves on a projective surface which have a “good framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space and discuss some examples on rational surfaces.

2010 Mathematics Subject Classification: 14D20; 14D21;14J60 Keywords and Phrases: Framed sheaves, moduli spaces, stable pairs, instantons

1. Introduction

There has been recently some interest in the moduli spaces of framed sheaves. One reason is that they are often smooth and provide desingularizations of the moduli spaces of ideal instantons, which in turn are singular [17, 19, 18]. For this reason, their equivariant cohomology under suitable toric actions is relevant to the computation of partition functions, and more generally expectation values of quantum observables in topological quantum field theory [20, 2, 19, 6, 3]. On the other hand, these moduli spaces can be regarded as higher-rank generalizations of Hilbert schemes of points, and as such they have interesting connections with integrable systems [12, 1], representation theory [26], etc. While it is widely assumed that such moduli spaces exist and are well behaved, an explicit analysis, showing that they are quasi-projective schemes and are fine moduli spaces, is missing in the literature. In the present paper we provide such a construction for the case of framed sheaves on smooth projective surfaces under some mild conditions. We show that if D is a big and nef curve in a smooth projective surface X, there is a fine quasi-projective moduli space for sheaves that have a “good framing” on D (Theorem 3.1). The point here is that the sheaves under consideration are not assumed a priori to be semistable, and the basic idea is to show that there exists a stability condition making all of them stable, so that our moduli space is an open subscheme of the moduli space of stable pairs in the sense of Huybrechts and Lehn [8, 9]. In the papers [21, 22] T. Nevins constructed a scheme structure for these moduli spaces, however we obtain a stronger result, showing that these schemes are quasi-projective, and in particular are separated and of finite type. Moreover we compute the obstruction to the smoothness of these moduli spaces (Theorem 4.3). In fact, the tangent space is well known, but we provide a more precise description of the obstruction space than the one given by Lehn [14]. We show that it lies in the kernel of the trace map, thus extending a previous result of Lübke [15] to the non-locally free case. In some cases there is another way to give the moduli spaces 𝔐(r, c, n) a structure of algebraic variety, namely, by using ADHM data. This was done for vector bundles on ℙ^{2} by Donaldson [5], while (always in the locally free case) the case of the blowup of ℙ^{2} at a point is studied in A. King’s thesis [13], and ℙ^{2} blown-up at an arbitrary number of points was analyzed by Buchdahl [4]. The general case (i.e., including torsion-free sheaves) is studied by C. Rava for Hirzebruch surfaces [24] and A.A. Henni for multiple blowups of ℙ^{2} at distinct points [7]. The equivalence between the two approaches follows from the fact that in both cases one has fine moduli spaces. On the ADHM side, this is shown by constructing a universal monad on the moduli space [23, 7, 25]. In the final section we discuss some examples, i.e. framed bundles on Hirzebruch surfaces with “minimal invariants", and rank 2 framed bundles on the blowup of ℙ^{2} at one point. In the present article, all the schemes we consider are separated and are of finite type over ℂ, and “a variety” is a reduced irreducible scheme of finite type over ℂ. A “sheaf” is always coherent, the term “(semi)stable” always means “μ-(semi)stable”, and the prefix μ- will be omitted. Framed sheaves are always assumed to be torsion-free.

1This research was partly supported by prin “Geometria delle varietà algebriche” and by the European Science Foundation Programme Misgam.

2. Framed sheaves

Let us characterize the objects that we shall study.

Definition 2.1. Let X be a scheme over ℂ, D ⊂ X an effective Weil divisor, and E_{D} a sheaf on D. We say that a sheaf E on X is (D, E_{D})-framable if E is torsion-free and there is an epimorphism E → E_{D} of O_{X}-modules inducing an isomorphism E_{|D} →^{∼} E_{D}. An isomorphism ϕ: E_{|D} →^{∼} E_{D} will be called a (D, E_{D})-framing of E. A framed sheaf is a pair (E, ϕ) consisting of a (D, E_{D})- framable sheaf E and a framing ϕ. Two framed sheaves (E, ϕ) and (E′, ϕ′) are isomorphic if there is an isomorphism f: E → E′ and a nonzero constant λ ∈ ℂ such that ϕ′ ∘ f_{|D} = λϕ.

Let us remark that our notion of framing is the same as the one used in [14, 22, 21], but is more restrictive than that of [8], where a framing is any homomorphism α : E → E_{D} of O_{X}-modules, not necessarily factoring through an isomorphism E_{|D} →^{∼} E_{D}. To distinguish between the two definitions, we will call such a pair (E, α) a framed pair, whilst the term framed sheaf will refer to the notion introduced in Definition 2.1 Our strategy to show that framed sheaves on a projective variety make up “good” moduli spaces will consist in proving that, under some conditions, the framed sheaves (E, ϕ) are stable according to a notion of stability introduced by Huybrechts and Lehn [8, 9]. The definition of stability for framed pairs depends on the choice of a polarization H on X and a positive real number δ (in our notation, δ is the leading coefficient of the polynomial δ in the definition of (semi)stability in [8]).

Definition 2.2 ([8, 9]). A framed pair (E, α) on an n-dimensional projective variety X, consisting of a torsion-free sheaf E and its framing α : E → E_{D}, is said to be (H, δ)-stable, if for any subsheaf G ⊂ E with 0 < rk G ≤ rk E, the following inequalities hold:

(1) \frac{c_{1}(G) · H^{n−1}}{rk(G)} < \frac{c_{1}(E) · H^{n−1} − δ}{rk(E)} when G is contained in ker α;

(2) \frac{c_{1}(G) · H^{n−1} − δ}{rk(G)} < \frac{c_{1}(E) · H^{n−1} − δ}{rk(E)} otherwise.

Remark, that according to this definition, any rank-1 framed sheaf is (H, δ)- stable for any ample H and any 0 ≤ δ < deg D. For any sheaf F on X, P_{F}^{H} denotes the Hilbert polynomial P_{F}^{H}(k) = χ(F ⊗ O_{X}(kH)). For a non-torsion sheaf F on X, μ^{H} denotes the slope of F: μ^{H}(F) = \frac{c_{1}(F)·H^{n−1}}{rk F}.

Theorem 2.3 ([8, 9]). Let X be a smooth projective variety, H an ample divisor on X and δ a positive real number. Let D ⊂ X be an effective divisor, and E_{D} a sheaf on D. Then there exists a fine moduli space 𝔐 = 𝔐_{X}^{H}(P) of (H, δ)-stable (D, E_{D})-framed sheaves (E, ϕ) with fixed Hilbert polynomial P = P_{E}^{H}, and this moduli space is a quasi-projective scheme.

Since we are using slope stability, and a more restrictive definition of framing with respect to that of [8, 9], our moduli space 𝔐_{X}^{H}(P) is actually an open subscheme of the moduli space constructed by Huybrechts and Lehn. Another general result on framed sheaves we shall need is a boundedness theorem due to M. Lehn. Given X, D, E_{D} as above, a set M of (D, E_{D})-framed pairs (E, ϕ) is bounded if there exists a scheme of finite type S over ℂ together with a family (G, ϕ) of (D, E_{D})-framed pairs over S such that for any (E, ϕ) ∈ M, there exist s ∈ S and an isomorphism (G_{s}, ϕ_{|D×s}) ≃ (E, ϕ).

Definition 2.4. Let X be a smooth projective variety. An effective divisor D on X is called a good framing divisor if we can write D = ∑ n_{i}D_{i}, where D_{i} are prime divisors and n_{i} > 0, and there exists a nef and big divisor of the form ∑ a_{i}D_{i} with a_{i} ≥ 0. For a sheaf E_{D} on D, we shall say that E_{D} is a good framing sheaf, if it is locally free and there exists a real number A_{0}, 0 ≤ A_{0} < \frac{1}{r}D^{2} · H^{n−2}, such that for any locally free subsheaf F ⊂ E_{D} of constant positive rank, \frac{1}{rk F} deg c_{1}(F) ≤ \frac{1}{rk E_{D}} deg c_{1}(E_{D}) + A_{0}.

Theorem 2.5. Let X be a smooth projective variety of dimension n ≥ 2, H an ample divisor on X, D ⊂ X an effective divisor, and E_{D} a vector bundle on D. Assume that D is a good framing divisor. Then for every polynomial P with coefficients in ℚ, the set of torsion-free sheaves E on X that satisfy the conditions P_{E}^{H} = P and E_{|D} ≃ E_{D} is bounded.

This is proved in [14], Theorem 3.2.4, for locally free sheaves, but the proof goes through also in the torsion-free case, provided that E_{D} is locally free, as we are assuming.

3. Quasi-projective moduli spaces

Using the notions introduced in the previous section, we now can state the main existence result for quasi-projective moduli spaces:

Theorem 3.1. Let X be a smooth projective surface, D ⊂ X a big and nef curve, and E_{D} a good framing sheaf on D. Then for any c ∈ H^{∗}(X, ℚ), there exists an ample divisor H on X and a real number δ > 0 such that all the (D, E_{D})-framed sheaves E on X with Chern character ch(E) = c are (H, δ)- stable, so that there exists a quasi-projective scheme 𝔐_{X}(c) which is a fine moduli space for these framed sheaves.

Proof. Let us fix an ample divisor C on X. Set O_{X}(k) = O_{X}(kC) and E(k) = E ⊗O_{X}(k) for any sheaf E on X and for any k ∈ ℤ. Recall that the Castelnuovo-Mumford regularity ρ(E) of a sheaf E on X is the minimal integer m such that h^{i}(X, E(m − i)) = 0 for all i > 0. According to Lehn’s Theorem (Theorem 2.5), the family M of all the sheaves E on X with ch(E) = c and E_{|D} ≃ E_{D} is bounded. Hence ρ(E) is uniformly bounded over all E ∈ M. By Grothendieck’s Lemma (Lemma 1.7.9 in [10]), there exists A_{1} ≥ 0, depending only on E_{D}, c and C, such that μ^{C}(F) ≤ μ^{C}(E) + A_{1} for all E ∈ M and for all nonzero subsheaves F ⊂ E. For n > 0, denote by H_{n} the ample divisor C + nD. We shall verify that there exists a positive integer n such that the range of positive real numbers δ, for which all the framed sheaves E from M are (H_{n}, δ)-stable, is nonempty. Let F ⊂ E, 0 < r′ = rk F ≤ r = rk E. Assume first that F ⊄ ker (E → E_{|D}). Then we may only consider the case r′ < r, and the (H_{n}, δ)-stability condition for E reads:

(1)  μ^{H_{n}}(F) < μ^{H_{n}}(E) + (\frac{1}{r′} − \frac{1}{r}) δ.

Saturating F, we make μ^{H_{n}}(F) bigger, so we may assume that F is a saturated subsheaf of E, and hence that it is locally free. Then F_{|D} ⊂ E_{|D} and we have:

(2)  μ^{H_{n}}(F) = \frac{n}{r′} deg c_{1}(F_{|D}) + μ^{C}(F) ≤ μ^{H_{n}}(E) + nA_{0} + A_{1}.

Thus we see that (2) implies (1) whenever

(3)  \frac{rr′}{r − r′} (nA_{0} + A_{1}) < δ.

Assume now that F is a saturated, and hence a locally free subsheaf of ker (E → E_{|D}) ≃ E(−D). Then the (H_{n}, δ)-stability condition for E is

(4)  μ^{H_{n}}(F) < μ^{H_{n}}(E) − \frac{1}{r}δ,

and the inclusion F(D) ⊂ E yields:

(5) μ^{H_{n}}(F) < μ^{H_{n}}(E)−H_{n}D +nA_{0} +A_{1} = μ^{H_{n}}(E)−(D^{2} −A_{0})n+A_{1} −DC.

We see that (5) implies (4) whenever

(6)  δ < r[(D^{2} − A_{0})n − A_{1} + DC].

The inequalities (3), (6) for all r′ = 1, …, r − 1 have a nonempty interval of common solutions δ if

n > max \{\frac{rA_{1} − CD}{D^{2} − rA_{0}} , 0\} .

□

Remark 3.2. Grothedieck’s Lemma is stated in [10] in terms of the so called μ̂ slope. However, for torsion-free sheaves, the μ̂ slope and the usual slope differ by constants depending only on (X, O_{X}(1)), see Definition 1.6.8 in [10] and the following remark. △

Note that up to isomorphism, the quasi-projective structure making 𝔐_{X}(c) a fine moduli space is unique, which follows from the existence of a universal family of framed sheaves over it. If D is a smooth and irreducible curve and D^{2} > 0, then our definition of a good framing sheaf with A_{0} = 0 is just the definition of semistability. The following is thus an immediate consequence of the theorem:

Corollary 3.3. Let X be a smooth projective surface, D ⊂ X a smooth, irreducible, big and nef curve, and E_{D} a semistable vector bundle on D. Then for any c ∈ H^{∗}(X, ℚ), there exists a quasi-projective scheme 𝔐_{X}(c) which is a fine moduli space of (D, E_{D})-framed sheaves on X with Chern character c.

4. Infinitesimal study

Let X be a smooth projective variety, D an effective divisor on X, E_{D} a vector bundle on D. We shall consider sheaves E on X framed to E_{D} on D. We recall the notion of a simplifying framing bundle introduced by Lehn.

Definition 4.1. E_{D} is simplifying if for any two vector bundles E, E′ on X such that E_{|D} ≃ E′_{|D} ≃ E_{D}, the group H^{0}(X, Hom(E, E′)(−D)) vanishes.

An easy sufficient condition for E_{D} to be simplifying is H^{0}(D, End(E_{D}) ⊗ O_{X}(−kD)_{|D}) = 0 for all k > 0. Lehn [14] proved that if D is good and E_{D} is simplifying, there exists a fine moduli space 𝔐 of (D, E_{D})-framed vector bundles on X in the category of separated algebraic spaces. Lübke [15] proved a similar result: if X is a compact complex manifold, D a smooth hypersurface (not necessarily “good”) and if E_{D} is simplifying, then the moduli space 𝔐 of (D, E_{D})-framed vector bundles exists as a Hausdorff complex space. In both cases the tangent space T_{[E]}𝔐 at a point representing the isomorphism class of a framed bundle E is naturally identified with H^{1}(X, End(E)(−D)), and the moduli space is smooth at [E] if H^{2}(X, End(E)(−D)) = 0. Lübke gives a more precise statement about smoothness: [E] is a smooth point of 𝔐 if H^{2}(X, End_{0}(E)(−D)) = 0, where End_{0} denotes the traceless endomorphisms. Huybrechts and Lehn in [9] define the tangent space and give a smoothness criterion for the moduli space of stable pairs that are more general objects than our framed sheaves. In this section, we adapt Lübke’s criterion to our moduli space 𝔐_{X}(c), parametrizing not only vector bundles, but also some non-locally-free sheaves. When we work with stable framed sheaves, we do not need the assumption that E_{D} is simplifying. We shall use the notions of the trace map and traceless exts, see Definition 10.1.4 from [10]. Assuming X is a smooth algebraic variety, F any (coherent) sheaf on it, and N a locally free sheaf (of finite rank), the trace map is defined

(7)  tr : Ext^{i}(F, F ⊗ N) → H^{i}(X, N) , i ∈ ℤ,

and the traceless part of the ext-group, denoted by Ext^{i}(F, F ⊗ N)_{0}, is the kernel of this map. We shall need the following property of the trace:

Lemma 4.2. Let 0 ⟶ F ⟶^{α} G ⟶^{β} E ⟶ 0 be an exact triple of sheaves and N a locally free sheaf. Then there are two long exact sequences of ext-functors giving rise to the natural maps

μ_{i} : Ext^{i}(F, E ⊗ N) → Ext^{i+1}(E, E ⊗ N) ,
τ_{i} : Ext^{i}(F, E ⊗ N) → Ext^{i+1}(F, F ⊗ N) ,

and we have tr ∘ μ_{i} = (−1)^{i}tr ∘ τ_{i} as maps Ext^{i}(F, E ⊗ N) → H^{i+1}(X, N).

Proof. This is a particular case of the graded commutativity of the trace with respect to cup-products on Homs in the the derived category (see Section V.3.8 in [11]): if ξ ∈ Hom(F, E ⊗ N [i]), η ∈ Hom(E, F[j]), then tr (ξ ∘ η) = (−1)^{ij}tr ((η ⊗ id_{N}) ∘ ξ). This should be applied to ξ ∈ Hom(F, E ⊗ N [i]) and η = ∂ ∈ Hom(E, F[1]), where ∂ is the connecting homomorphism in the distinguished triangle associated to the given exact triple:

E[−1] ⟶^{−∂} F ⟶^{α} G ⟶^{β} E ⟶^{∂} F[1].

□

Theorem 4.3. Let X be a smooth projective surface, D ⊂ X an effective divisor, E_{D} a locally free sheaf on D, and c ∈ H^{∗}(X, ℚ) the Chern character of a (D, E_{D})-framed sheaf E on X. Assume that there exists an ample divisor H on X and a positive real number δ such that E is (H, δ)-stable, and denote by 𝔐_{X}(c) the moduli space of (D, E_{D})-framed sheaves on X with Chern character c which are (H, δ)-stable. Then the tangent space to 𝔐_{X}(c) is given by

T_{[E]}𝔐_{X}(c) = Ext^{1}(E, E ⊗ O_{X}(−D)),

and 𝔐_{X}(c) is smooth at [E] if the traceless ext-group

Ext^{2}(E, E ⊗ O_{X}(−D))_{0} = ker [ tr : Ext^{2}(E, E ⊗ O_{X}(−D)) → H^{2}(X, O(−D)) ]

vanishes.

Proof. We prove this result by a combination of arguments of Huybrechts-Lehn and Mukai, so we just give a sketch, referring to [9, 16] for details. As in Section 4.iv) of [9], the smoothness of 𝔐 = 𝔐_{X}(c) follows from the T^{1}-lifting property for the complex E → E_{D}. Let A_{n} = k[t]/(t^{n+1}), X_{n} = X × SpecA_{n}, D_{n} = D × SpecA_{n}, E_{D_{n}} = E_{D} ⊠ A_{n}, and let E_{n} ⟶^{α_{n}} E_{D_{n}} be an A_{n}-flat lifting of E → E_{D} to X_{n}. Then the infinitesimal deformations of α_{n} over k[ϵ]/(ϵ^{2}) are classified by the hyper-ext 𝔼xt^{1}(E_{n}, E_{n} ⟶^{α_{n}} E_{D_{n}}), and one says that the T^{1}-lifting property is verified for E → E_{D} if all the natural maps

T_{n}^{1} : 𝔼xt^{1}(E_{n}, E_{n} ⟶^{α_{n}} E_{D_{n}}) → 𝔼xt^{1}(E_{n−1}, E_{n−1} ⟶^{α_{n−1}} E_{D_{n−1}})

are surjective whenever (E_{n}, α_{n}) ≡ (E_{n−1}, α_{n−1}) mod (t^{n}). In loc. cit., the authors remark that there is an obstruction map ob on the target of T_{n}^{1} which embeds the cokernel of T_{n}^{1} into 𝔼xt^{2}(E, E → E_{D}), so that if the latter vanishes, the T^{1}-lifting property holds. In our case, E is locally free along D, so the complex E → E_{D} is quasi-isomorphic to E(−D) and 𝔼xt^{i}(E, E → E_{D}) = Ext^{i}(E, E(−D)). It remains to prove that the image of ob is contained in the traceless part of Ext^{2}(E, E(−D)). This is done by a modification of Mukai’s proof in the non-framed case. First we assume that E is locally free. Then the elements of Ext^{1}(E_{n−1}, E_{n−1}(−D_{n−1})) can be given by Čech 1-cocycles with values in End(E_{n−1})(−D_{n−1}) for some open covering of X, and the image of such a 1-cocycle (a_{ij}) under the obstruction map Ext^{1}(E_{n−1}, E_{n−1}(−D_{n−1})) → Ext^{2}(E, E(−D)) is a Čech 2-cocycle (c_{ijk}) with values in End(E)(−D). A direct calculation shows that (tr c_{ijk}) is a Čech 2-cocycle with values in O_{X}(−D) which is the obstruction to the lifting of the infinitesimal deformation of the framed line bundle det E_{n−1} from A_{n−1} to A_{n}. As we know that the moduli space of line bundles, whether framed or not, is smooth, this obstruction vanishes, so the cocycle (tr c_{ijk}) is cohomologous to 0. Now consider the case when E is not locally free. Replacing E, E_{D} by their twists E(n), E_{D}(n) for some n > 0, we may assume that H^{i}(X, E) = H^{i}(X, E(−D)) = 0 for i = 1, 2 and that E is generated by global sections. Then we get the exact triple of framed sheaves

0 → (G, γ) → (H^{0}(X, E) ⊗ O_{X}, β) → (E, α) → 0,

where G is locally free (at this point it is essential that dim X = 2 and X is smooth). Then we verify the T^{1}-lifting property for the exact triples

0 → (G_{n}, γ_{n}) → (O_{X_{n}}^{N}, β_{n}) → (E_{n}, α_{n}) → 0.

The infinitesimal deformations of such exact triples are classified by Hom(G_{n}, E_{n}(−D_{n})), and the obstructions lie in Ext^{1}(G, E(−D)). We have two connecting homomorphisms μ_{1} : Ext^{1}(G, E(−D)) → Ext^{2}(E, E(−D)) and τ_{1} : Ext^{1}(G, E(−D)) → Ext^{2}(G, G(−D)). Our hypotheses on E imply that: 1) every infinitesimal deformation of (E_{n}, α_{n}) lifts to that of the triple, and 2) μ_{1} is an isomorphism, that is, the infinitesimal deformation of E_{n} is unobstructed if and only if that of the triple is. By Lemma 4.2, tr (μ_{1}(ξ)) = −tr (τ_{1}(ξ)) in H^{2}(X, O_{X}(−D)). As in 1.10 of [16], τ_{1}(ξ) is the obstruction ob(G_{n−1}, γ_{n−1}) to lifting (G_{n−1}, γ_{n−1}) from A_{n−1} to A_{n}. As G_{n−1} is locally free, we can use the Čech cocycles as above and see that tr (τ_{1}(ξ)) ∈ H^{2}(X, O_{X}(−D)) is the obstruction to lifting (det G_{n−1}, det γ_{n−1}), hence it is zero and we are done. □

The following Corollary describes a situation where the moduli space 𝔐_{X}(c) is smooth (hence, every connected component is a smooth quasi-projective variety).

Corollary 4.4. In addition to the hypothesis of Theorem 4.3, let us assume that D is irreducible, that (K_{X} + D) · D < 0, and choose the framing bundle to be trivial. Then the moduli space 𝔐_{X}(c) is smooth.

This happens for instance when X is a Hirzebruch surface, or the blowup of ℙ^{2} at a number of distinct points, taking for D the inverse image of a generic line in ℙ^{2} via the birational morphism X → ℙ^{2}. In this case one can also compute the dimension of the moduli space, obtaining dim 𝔐_{X}(c) = 2rn, with r = rk(E) and

c_{2}(E) − \frac{r − 1}{2r}c_{1}(E)^{2} = n ϖ,

where ϖ is the fundamental class of X. When X is the p-th Hirzebruch surface 𝔽_{p} we shall denote this moduli space by 𝔐^{p}(r, k, n) if c_{1}(E) = kC, where C is the unique curve in 𝔽_{p} having negative self-intersection.

The next example shows that the moduli space may be nonsingular even if the group Ext^{2}(E, E ⊗ O_{X}(−D)) does not vanish.

Example 4.5. For r = 1 the moduli space 𝔐(1, 0, n) is isomorphic to the Hilbert scheme X_{0}^{[n]} parametrizing length n 0-cycles in X_{0} = X\D. Of course this space is a smooth quasi-projective variety of dimension 2n. Indeed in this case the trace morphism Ext^{2}(E, E ⊗ O_{X}(−D)) → H^{2}(X, O(−D)) is an isomorphism.

5. Examples

5.1. Bundles with small invariants on Hirzebruch surfaces. Let X be the p-th Hirzebruch surface 𝔽_{p}, and normalize the Chern character by twisting by powers of the line bundle O_{𝔽_{p}}(C) so that 0 ≤ k ≤ r − 1. It has been shown in [3] that the moduli space 𝔐^{p}(r, k, n) is nonempty if and only if the bound

n ≥ N = \frac{pk}{2r} (r − k)

is satisfied. The moduli spaces 𝔐^{p}(r, k, N) can be explicitly characterized: 𝔐^{p}(r, k, N) is a rank k(r−k)(p−1) vector bundle on the Grassmannian G(k, r) of k-planes in ℂ^{r} [25]; in particular, 𝔐^{1}(r, k, N) ≃ G(k, r), and 𝔐^{2}(r, k, N) is isomorphic to the tangent bundle of G(k, r). This is consistent with instanton counting, which shows that the spaces 𝔐^{p}(r, k, N) have the same Betti numbers as G(k, r) [3].

5.2. Rank 2 vector bundles on 𝔽_{1}. We study in some detail the moduli spaces 𝔐^{1}(2, k, n). As [27] and [28] show, the non-locally free case turns out to be very complicated as soon as the value of n exceeds the rank. So we consider only locally free sheaves. To simplify notation we call this moduli space M̂(k, n), where n denotes now the second Chern class. We normalize k so that it will assume only the values 0 and −1. Moreover we shall denote by M(n) the moduli space of rank 2 bundles on ℙ^{2}, with second Chern class n, that are framed on the “line at infinity” ℓ_{∞} ⊂ ℙ^{2} (which we identify with the image of D via the blow-down morphism π: 𝔽_{1} → ℙ^{2}).

Let us start with the case k = −1. We introduce a stratification on M̂(−1, n) according to the splitting type of the bundles it parametrizes on the exceptional line E ⊂ 𝔽_{1}

M̂(−1, n) = Z_{0}(−1, n) ⊃ Z_{1}(−1, n) ⊃ Z_{2}(−1, n) ⊃ …

defined as follows: if Z_{k}^{0}(−1, n) = Z_{k}(−1, n) \ Z_{k+1}(−1, n) then

Z_{k}^{0}(−1, n) = {E ∈ M̂(−1, n) | E_{|E} ≃ O_{E}(−k) ⊕ O_{E}(k + 1)} .

Proposition 5.1. There is a map

F_{1} : M̂(−1, n) → ∐_{k=0}^{n} M(n − k)

which restricted to the subset Z_{k}^{0}(−1, n) yields a morphism

Z_{k}^{0}(−1, n) → M(n − k)

whose fibre is an open set in Hom(σ^{∗}E_{|E}, O_{E}(k))/ℂ^{∗} ≃ ℙ^{2k+1}, made by k-linear forms that have no common zeroes on the exceptional line.

Proof. We start by considering Z_{0}^{0}(−1, n). The morphism Z_{0}^{0}(−1, n) → M(n) is given by E_{1} ↦ E = (π_{∗}E)^{∗∗}. The fibre of this morphism includes a ℙ^{1}. To show that this is indeed a ℙ^{1}-fibration we need to check that E_{1} has no other deformations than those coming from the choice of a point in M(n) and a point in this ℙ^{1}. This follows from the equalities

dim Ext^{1}(E_{1}, E_{1}(−E)) = dim Ext^{1}(E, E(−ℓ_{∞}) + 1
Ext^{2}(E_{1}, E_{1}(−E)) = 0

Note that this result is compatible with the isomorphism 𝔐^{1}(r, k, N) ≃ G(k, r) mentioned in Section 5.1. In general, if E_{1} ∈ Z_{k}^{0}(−1, n) with k ≥ 1, so that E_{1}_{|E} ≃ O_{E}(k+1)⊕O_{E}(−k), the direct image π_{∗}(E_{1}(kE)) is locally free. This defines the morphism Z_{k}^{0}(−1, n) → M(n − k). □

We consider now the case k = 0. One has Z_{0}^{0}(0, n) ≃ M(n). We study the other strata by reducing to the odd case. If E_{1} ∈ Z_{k}^{0}(0, n), there is a unique surjection α: E_{1} → O_{E}(−k); let F be the kernel. Restricting 0 → F → E_{1} → O_{E}(−k) → 0 we get an exact sequence

0 → O_{E}(1 − k) → F_{|E} → O_{E}(k) → 0

so that

F_{|E} ≃ O_{E}(a + 1) ⊕ O_{E}(−a) with − k ≤ a ≤ k − 1.

A detailed analysis shows that a = k − 1. As a result we have:

Proposition 5.2. For all k ≥ 1 there is a morphism

Z_{k}^{0}(0, n) → M(n − 2k + 1)

whose fibres have dimension 2k − 1.

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Ugo Bruzzo Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265 34136 Trieste and Istituto Nazionale di Fisica Nucleare Sezione di Trieste Italia bruzzo@sissa.it Dimitri Markushevich Mathématiques — Bât. M2 Université Lille 1 F-59655 Villeneuve d’Ascq Cedex France markushe@math.univ.lille1.fr