Framed sheaves on projective space and Quot schemes

We prove that, given integers m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document}, r≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\ge 1$$\end{document} and n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document}, the moduli space of torsion free sheaves on Pm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}^m$$\end{document} with Chern character (r,0,…,0,-n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r,0,\ldots ,0,-n)$$\end{document} that are trivial along a hyperplane D⊂Pm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D \subset {\mathbb {P}}^m$$\end{document} is isomorphic to the Quot scheme QuotAm(O⊕r,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Quot}_{{\mathbb {A}}^m}({\mathscr {O}}^{\oplus r},n)$$\end{document} of 0-dimensional length n quotients of the free sheaf O⊕r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {O}}^{\oplus r}$$\end{document} on Am\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {A}}^m$$\end{document}. The proof goes by comparing the two tangent-obstruction theories on these moduli spaces.


Introduction
This paper builds an identification between two classical moduli spaces in algebraic geometry: the moduli space of framed sheaves on projective space P m and Grothendieck's Quot scheme. Unless stated otherwise, we work over an algebraically closed field k of characteristic 0. If D ⊂ Y is a divisor on a projective variety Y , a D-framed sheaf on Y is a pair (E, φ) where E is a torsion free sheaf on Y and φ is an isomorphism E| D → O ⊕r D , where r = rkE. Such pairs (E, φ) are a special case of the more general notion of framed modules introduced by Huybrechts-Lehn [16].
For a fixed coherent sheaf V on Y , the Quot scheme Quot Y (V , P) parametrises quotients V Q such that Q has Hilbert polynomial P. If P is a constant polynomial, the Quot scheme also exists (as a quasiprojective scheme) for quasiprojective varieties. For instance, if P = n ∈ Z ≥0 , we have a natural open immersion Quot A m (O ⊕r , n) → Quot P m (O ⊕r , n).
The following is the main result of this paper, proved in Theorem 2.7 in the main body of the text. The map η, constructed in Proposition 2.3, is defined on closed points by where Q is a 0-dimensional coherent sheaf on P m supported away from D. The fact that η is not an isomorphism for m = 2 (unless r = 1) ultimately depends on the fact that on P 2 there are nontrivial vector bundles that are trivial on a line: this says that given a framed sheaf (E, φ) of rank r > 1 on P 2 , one may not be able to reconstruct an embedding i : E → O ⊕r P 2 , and this prevents η from being surjective. In fact, the moduli space Fr r ,n (P 2 ) is a smooth variety of dimension 2nr containing Quot A 2 (O ⊕r , n) as an irreducible subvariety of dimension (r + 1)n, which is singular as soon as r , n > 1 (Example 3.3).
Donaldson [10] constructed a canonical identification between the moduli space of instantons on S 4 = R 4 ∪{∞} with SU (r )-framing at ∞ and the moduli space of rank r holomorphic vector bundles on P 2 trivial on a line ∞ . He defined a partial compactification of the moduli space on the 4-manifold side of the correspondence by allowing connections acquiring singularities. This in turn corresponds to considering torsion free sheaves on the algebro-geometric side, leading to the study of Fr r ,n (P 2 ).
The 3-dimensional analogue of Donaldson's construction has attracted lots of attention in string theory and hence, after translating in the language of algebraic geometry, in Donaldson-Thomas theory. For instance, in the work of Cirafici-Sinkovics-Szabo [8,Sec. 4.1], the authors construct a correspondence between non-commutative U (r )-instantons on A 3 and the 3-dimensional analogue of Donaldson's construction, namely the moduli space Fr r ,n (P 3 ). They relate the construction to the quiver gauge theory of the 'r -framed 3-loop quiver' (Fig. 1), which corresponds to Quot A 3 (O ⊕r , n) in a precise sense [2]. We briefly review this story in Sect. 3. Moreover, the very same quiver gauge theory can be derived from the rank r Donaldson-Thomas theory of A 4 , as shown by Nekrasov and Piazzalunga in [19]. Theorem A formalises this correspondence from an algebraic perspective in the 3-dimensional case, and extends it to higher dimensions.
Framed sheaves and framed modules were mostly studied on surfaces. We do not aim at giving an exhaustive list of references, but we refer the reader to [4,26] for a more complete bibliography. Framed sheaves were also studied on 3-folds by Oprea [20], where a symmetric obstruction theory on their moduli space is constructed-we end Sect. 2 with a conjecture suggesting that Oprea's obstruction theory might take a very explicit form (Conjecture 2.12). Quot schemes also received a lot of attention lately in enumerative geometry [13,21,23,25], and in the context of motivic invariants [9,17,24].

Framed modules and framed sheaves
In this section we briefly review the notion of stability on framed modules introduced by Huybrechts-Lehn [16], and we show that D-framed sheaves on P m (Definition 1.4) are stable with respect to a suitable choice of stability parameters (Lemma 1.7). This implies the representability of their moduli functor.

Framed modules after Huybrechts-Lehn
Let Y be a smooth projective variety over an algebraically closed field k of characteristic 0, and let H be an ample divisor on The map α is called the framing, whereas ker α (resp. rkE) is called the kernel (resp. the rank) of the framed module. Set (α) = 1 if α = 0 and (α) = 0 otherwise.
The Hilbert polynomial of a coherent sheaf E, with respect to H , is defined as with positive leading coefficient. The framed Hilbert polynomial of a framed module (E, α), depending on the pair (H , δ), is defined as rank r , with induced framing α , one has r P (E ,α ) ≤ r P (E,α) . We say that (E, α) is δ-stable if the same holds with '<' replacing '≤'.

Huybrechts and Lehn defined moduli functors
parametrising isomorphism classes of flat families of δ-(semi)stable framed modules with framing datum G and framed Hilbert polynomial P ∈ Q[k].
As proved in [16,Lemma 1.7], if deg δ ≥ m = dim Y then in every semistable framed module (E, α) the framing α either vanishes or is injective, thus the study of δ-semistable framed modules reduces to Grothendieck's theory of the Quot scheme. Thus one focuses on the case deg δ = m − 1, writing Huybrechts and Lehn defined the (H , δ)-slope of a framed module (E, α) with positive rank as the ratio For framed modules of positive rank, such as those studied in this paper, one has that μstability with respect to δ 1 implies δ-stability. Also note that a rank 1 framed module (E, α) with E torsion free is μ-stable for any choice of (H , δ).
The notion which behaves best in the sense of moduli is δ-stability. We now recall the part of the main theorem of [16] which is relevant for our paper.

Framed sheaves on projective spaces
Fix a hyperplane ι : D → P m , with m ≥ 2, and the polarisation H = O P m (1). Of course D is linearly equivalent to H , so in particular we have D · H m−1 = 1, but we distinguish them as they play different roles.

Definition 1.4
Let m ≥ 2 be an integer. A D-framed sheaf of rank r on P m is a framed module (E, α) on P m with framing datum G = ι * O ⊕r D , such that E is torsion free with Chern character ch(E) = v r ,n for some n ≥ 0, and the morphism φ α : E| D → O ⊕r D induced by the framing α is an isomorphism.
Note that, for a D-framed sheaf (E, α), the torsion free sheaf E is locally free in a neighborhood of D, and the canonical map E → E ∨∨ is an isomorphism in a neighborhood of D.
We will make crucial use of the following result due to Abe and Yoshinaga.
where Q has finite support contained in A m = P m \D.
Proof Since E is torsion free, the natural map E → E ∨∨ to its double dual is injective.
Moreover, E ∨∨ is reflexive and α induces a canonical isomorphism E ∨∨ | D ∼ = O ⊕r D . By Theorem 1.5 we have that E ∨∨ splits as a direct sum of line bundles, and it is immediate to see that these line bundles are necessarily trivial. This yields an isomorphism E ∨∨ ∼ = O ⊕r P m , and since E| D ∼ = O ⊕r D it follows that the quotient Q = O ⊕r P m /E is supported on finitely many points lying in P m \D.
In the case of projective surfaces it has been proved by Bruzzo and Markushevich that μ (H ,δ) -stability is automatically implied when considering a "good framing" [4, Thm. 3.1]. The strategy of the proof does not extend in full generality to higher dimensional varieties, as observed by Oprea [20]. We shall now provide a new argument for the particular case at hand, but it is still an open question whether it is possible to extend the result to more general settings.

Lemma 1.7 Fix integers m ≥ 3, and r ≥ 1. Let (E, α) be a D-framed sheaf of rank r on P m , and consider a polynomial δ as in
Clearly ker α → E is torsion free because E is torsion free by definition. Moreover, by means of the diagram We now have to distinguish two cases: We have the sought after inequality . This means (α ) = 0. We compute the ordinary H -slope by our assumption δ 1 < r and Eq. (1.5).
The proof is complete.

The moduli functor of framed sheaves
Fix integers m ≥ 2, r ≥ 1, and n ≥ 0. Also fix a hyperplane ι : D → P m . Consider the moduli functor of D-framed sheaves of rank r on P m with Chern character v r ,n = (r , 0, . . . , 0, −n), i.e. the functor Fr r ,n (P m ) : Sch We have defined the functor using the map where P r ,n (k) = χ(E(k)) is the Hilbert polynomial of a coherent sheaf E with Chern character v r ,n .
Proof The case of P 2 is well known [4,18]. Hence, we can restrict to the case m ≥ 3. The locus of framed modules (E, α) ∈ M st δ (P m ; G, P) such that E is torsion free, and the map φ α : E| D → O ⊕r D induced by the framing α is an isomorphism, is open. But by Lemma 1.7, all D-framed sheaves are δ-stable.

Moduli of framed sheaves and Quot schemes
In this section we review the notion of tangent-obstruction theory on a deformation functor [12], and we compare the tangent-obstruction theory on the local Quot functor with that on the D-framed sheaves local moduli functor. This leads to the proof of Theorem A.

Comparing tangent-obstruction theories
We refer the reader to [12,Ch. 6] for a thorough exposition on tangent-obstruction theories on deformation functors.
Let Art k be the category of local artinian k-algebras with residue field k. 1 A deformation functor is a covariant functor D : Art k → Sets such that D(k) is a singleton. A tangentobstruction theory on a deformation functor D is defined to be a pair (T 1 , T 2 ) of finite dimensional k-vector spaces such that for any small extension I → B A in Art k there is an 'exact sequence of sets' is the projection and Q is a coherent sheaf on Y × k B, flat over B, whose fibres Q b = Q| Y × k {b} have Hilbert polynomial P. Two surjections are 'isomorphic' if they have the same kernel. The Quot functor is represented by a projective k-scheme Q = Quot Y (V , P). We refer the reader to [12,Ch. 5] for a complete, modern discussion on Quot schemes. Fix a point x 0 ∈ Q(k) corresponding to a quotient V Q with kernel E. One can consider the local Quot functor at x 0 , namely the subfunctor Q x 0 ⊂ Q| Art k : Art k → Sets sending a local artinian k-algebra A to the set of families x ∈ Q(Spec A) such that x| m = x 0 , where m is the closed point of Spec A. By representability of Q, the functor Q x 0 is prorepresentable, isomorphic to Hom k-alg (O Q,x 0 , −). By [12,Thm. 6.4.9], the pair of k-vector spaces form a tangent-obstruction theory on the deformation functor Q x 0 .
The proof of the following result is included for the sake of completeness (and for lack of a suitable reference). (T 1 , T 2 ) and (T 1 , T 2 ), respectively. Let η : D → D be a natural transformation inducing a k-linear isomorphism d : T 1 → T 1 and a k-linear embedding T 2 → T 2 . Then η is a natural equivalence.

Proposition 2.2 Let D and D be two pro-representable deformation functors carrying tangent-obstruction theories
Proof We already know that η B : D(B) → D (B) is bijective when B = k and when B = k[t]/t 2 , by assumption. We then proceed by induction on the length of the artinian rings A ∈ Art k . Fix a small extension I → B A in Art k and form the commutative diagram where the leftmost vertical map is d ⊗ k id I and the isomorphism D(A) → D (A) is the induction hypothesis. We have to show that η B is bijective. The statement is reminiscent of the Five Lemma, but since we are dealing with the (non-standard) concept of short exact sequence of sets, we include full details.
To prove injectivity, pick two elements β 1 = β 2 ∈ D(B). We may assume their images in D(A) agree, for otherwise there is nothing to prove. Then, by pro-representability of D, we have β 2 = v · β 1 for a unique nonzero v ∈ T 1 ⊗ k I . Then, after setting To prove surjectivity, pick β ∈ D (B). It maps to 0 ∈ T 2 ⊗ k I , and its image α in D (A) lifts uniquely to an element α ∈ D(A) such that ob(α) goes to 0 ∈ T 2 ⊗ k I . But by the injectivity assumption, we have ob(α) = 0, i.e. α lifts to some β ∈ D(B). But η B (β) is a lift of α ∈ D(A), so β = v · η B (β) for a unique v , as above. Then, if v ∈ T 1 ⊗ k I is the preimage of v , we conclude that v · β ∈ D(B) is a preimage of β under η B .

Relating Quot scheme and framed sheaves
Let k be an algebraically closed field of characteristic 0. Let M = M st δ (Y ; G, P) be a fine moduli space of δ-stable framed modules (with framing datum G and framed Hilbert polynomial P) on a smooth projective k-variety Y , as in Theorem 1.3. Fix a closed point y 0 ∈ M(k) corresponding to a framed module (E, α). Consider the deformation functor Proof Fix a k-scheme B. Consider a short exact sequence . This means that the image of Such a map is easily seen to be injective on geometric points, by definition of the Quot functor. If m ≥ 3, we can construct the inverse of η k as follows. Given a D-framed sheaf (E, φ), with trivialisation φ : E| D → O ⊕r D , we know by the proof of Corollary 1.6 how to construct a canonical isomorphism E ∨∨ → O ⊕r P m . Thus the inverse of η k will send (E, φ) to the isomorphism class of the surjection The same argument works in the isolated case (m, r ) = (2, 1). Indeed, in that case E = I Z is an ideal sheaf of a 0-dimensional subscheme Z ⊂ A 2 = P 2 \D of length n, and again we have I ∨∨ Z → O P 2 , canonically. The proof is complete.
We will use an infinitesimal method based on Proposition 2.2 to prove that the map η of Proposition 2.3 is an isomorphism as long as m ≥ 3.

Infinitesimal method
Let y 0 = η(x 0 ) ∈ Fr r ,n (P m ) be the image of a point x 0 ∈ Quot A m (O ⊕r , n) under the morphism η. We obtain an induced natural transformation η 0 : Q x 0 → Fr y 0 between the local moduli functors-Q x 0 was defined in Example 2.1. Both functors are prorepresentable and carry a tangent-obstruction theory, cf. (2.2) for the case of the Quot scheme. Our next goal is to show that η 0 is an equivalence when m ≥ 3, using Proposition 2.2. This will be achieved by means of the following two lemmas.

Lemma 2.4 Fix m ≥ 3 and a hyperplane D ⊂ P m . Let E be a torsion free sheaf on P m such that E| D ∼ = O ⊕r P m . Then
If the strict inequality m > 3 holds, then Proof Consider the short exact sequence of sheaves obtained from the ideal sheaf short exact sequence of the hyperplane D ⊂ P m . The map is injective because it is locally given as multiplication by the defining equation of D, and the sheaf E is torsion-free. Notice first that The first vanishing follows by our assumption m ≥ 3. For any k > −m we then deduce the following isomorphisms from the long exact sequence in cohomology associated to (2.3): Since both cohomology groups on the right hand side of the isomorphisms vanish and a k-linear inclusion E(−D)).

If the strict inequality m > 3 holds, the k-linear inclusion is in fact an isomorphism.
Proof Twisting the exact sequence (1.4) by O(−D) and applying the Hom(E, −) functor we obtain a long cohomology sequence and by Serre duality we have for i = 1, 2, so that the result follows from the vanishings of Lemma 2.4.
We have thus essentially obtained the proof of the following result.

Proposition 2.6
If m ≥ 3, the natural transformation η 0 : Q x 0 → Fr y 0 of local moduli functors induces an isomorphism on tangent spaces and an injection on obstruction spaces. Hence, η 0 is a natural equivalence.
Proof The first statement follows from Lemma 2.5. The conclusion follows from Proposition 2.2.
We can now finish the proof of our main result. Assume m ≥ 3 for the rest of the proof. The morphism η is locally of finite type, since the Quot scheme is of finite type. Next, we check that η is formally étale, using the infinitesimal criterion. Consider a square zero extension S → S of fat points (i.e. spectra of objects A, B of Art k ), denote by m the closed point of S and form a commutative diagram where the dotted arrow u is the unique extension of h we have to find in order to establish formal étaleness of η at x 0 = h(m) → y 0 = h(m). We shall use the notation Hom p (T , Y ), for T a fat point and p a point on a scheme Y , to indicate the set of morphisms T → Y sending the closed point to p ∈ Y . Using pro-representability of Q x 0 and Fr y 0 , the condition that η 0 is a natural equivalence (proved in Proposition 2.6) translates into a commutative diagram Hom y 0 (S, Fr r ,n (P m )) Hom y 0 (S, Fr r ,n (P m )) where the vertical maps are the isomorphisms η 0,S and η 0,S respectively. Since h ∈ Hom y 0 (S, Fr r ,n (P m )) lifts to a map u ∈ Hom x 0 (S, Quot A m (O ⊕r , n)) and both u • i and h map to η • h ∈ Hom y 0 (S, Fr r ,n (P m )), they must be equal, since the vertical map on the right is also a bijection. Thus u is the unique lift we wanted to find. We conclude that η is étale. Since it is bijective by Proposition 2.3, it is an isomorphism.

Remark 2.9
We thank A. Henni for suggesting that it might also be possible to give a proof of Theorem A combining the formalism of perfect extended monads [14,15] with the result of Abe-Yoshinaga (Theorem 1.5). The 3-dimensional case is also studied along these lines in [ , which works over an arbitrary algebraically closed field of characteristic 0. The pair (U r ,n,3 , f ) will be given in Remark 3.1.

Remark 2.11
Another Quot scheme on A 3 that has been recently proven to be a global critical locus is Quot A 3 (I L , n), where I L ⊂ C[x, y, z] is the ideal sheaf of a line L ⊂ A 3 [9]. This was the starting point for the motivic refinement of the local DT/PT (or, ideal sheaves/stable pairs) correspondence around a smooth curve in a 3-fold [22,23].
Set k = C. By Oprea's construction [20,Thm. 1 and Sec. 4.4], there exists a symmetric perfect obstruction theory on Fr r ,n (P 3 ), where π : P 3 × C Fr r ,n (P 3 ) → Fr r ,n (P 3 ) is the projection, (E , ) is the universal framed sheaf, and L denotes the truncated cotangent complex. On the other hand, the critical locus structure on the Quot scheme [2, Thm. 2.6] induces a canonical 'critical' symmetric perfect obstruction theory See [3] for background on symmetric obstruction theories. See also [21,25] for the construction of virtual fundamental classes on several Quot schemes for varieties of dimension at most 3.
We propose the following conjecture, essentially a higher rank version of [13,Conj. 9.9].

Relation to quiver gauge theories
In this section we set k = C, essentially to be coherent with the literature on the subject. We start by recalling the explicit description of the Quot scheme as a closed subscheme of a nonsingular variety, the so-called non-commutative Quot scheme, which can be seen as the moduli space of stable r -framed representations on a quiver (Fig. 1); the relations cutting out Quot A m (O ⊕r , n) are precisely given by annihilating the commutators between all the matrices arising from the m loops in the quiver. This story is particularly rich in the case m = 3, where such relations agree with a single vanishing relation 'd f = 0' (Remark 3.1). We emphasise this since it is the starting point of higher rank Donaldson-Thomas theory of points in all its flavours: enumerative [2,25], motivic [6,24], K-theoretic [13]. We conclude this final section by stressing the dichotomy between the case m = 2 and the case m ≥ 3. More precisely, in Sect. 3.2 we exhibit the equations cutting out Quot A 2 (O ⊕r , n) inside the moduli space Fr r ,n (P 2 ) of framed sheaves on P 2 . In the case of higher rank r > 1, the describes Quot A 2 (O ⊕r , n) as a closed singular subvariety of Fr r ,n (P 2 ) of codimension n(r − 1).

Embedding in the non-commutative Quot scheme
The Quot scheme can be embedded in a smooth quasiprojective variety U r ,n,m , called the non-commutative Quot scheme in [2,13], as follows. Consider the m-loop quiver, i.e. the quiver L m with one vertex '0' and m loops. Now consider the quiver L m obtained by adding one additional vertex '∞' along with r edges ∞ → 0 (see Fig. 1). This construction is called r -framingfor m = 3 it has some relevance in motivic Donaldson-Thomas theory [6,7] and K-theoretic Donaldson-Thomas theory [13]. It is also performed with care in [15] in the r = 1 case and in [14] for arbitrary r .
The space of representations of L m of dimension vector (n, 1) is the affine space Of course, W r ,n,m could be defined without reference to quivers, but it is interesting to notice that there exists a quiver stability condition θ on L m such that the open subscheme of Rep (n,1) ( L m ) consisting of θ -stable representations is precisely W r ,n,m . The gauge group GL n acts freely on the smooth quasi-affine scheme W r ,n,m , by conjugation on the matrices and via the natural action on the vectors. Therefore the quotient U r ,n,m = W r ,n,m /GL n is a smooth quasiprojective variety, of dimension (m −1)n 2 +rn. The Quot scheme is realised as the closed subscheme cut out as the locus where the m matrices commute, i.e. by the vanishing relations

The 2-dimensional case
The following example shows that the Quot scheme of a surface, such as A 2 , is often singular. Then the tangent space to Quot S (O ⊕r S , r ) at ξ is given by Hom(I ⊕r p , O ⊕r p ) = Hom(I p , O p ) ⊕r 2 ∼ = C 2r 2 , using that Hom(I p , O p ) is 2-dimensional, being the tangent space to the smooth scheme Hilb 1 S = S at p. On the other hand, the Quot scheme Quot A 2 (O ⊕r , n) is irreducible of dimension (r + 1)n, as was proven by Ellingsrud and Lehn [11]. Since 2r 2 > (r + 1)r , the point ξ is a singular point.
In the case of P 2 , we already mentioned that Theorem A does not hold (unless r = 1). In this case, we do have a closed immersion Quot A 2 (O ⊕r , n) → Fr r ,n (P 2 ) (3.2) of codimension n(r − 1), which is an isomorphism if and only if r = 1. The moduli space of framed sheaves is smooth and irreducible of dimension 2nr, and can be realised as where B i ∈ End(C n ), i ∈ Hom(C r , C n ) and j ∈ Hom(C n , C r ). See [18, Thm. 2.1] and the references therein. The inclusion (3.2) is obtained as the locus j = 0. In particular, Quot A 2 (O ⊕r , n) is a (singular) scheme, cut out as the zero locus of a section of a tautological bundle of rank nr on the smooth quiver variety Fr r ,n (P 2 ). topics throughout the years. A.C. thanks AREA Science Park and CNR-IOM for support and the excellent working conditions. A.R. thanks Dipartimenti di Eccellenza for support and SISSA for the excellent working conditions.
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