Hecke correspondences for smooth moduli spaces of sheaves

We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain commutation relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K$\end{document}-theory of the moduli space ℳ, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.


Introduction
1.1. Let S be a smooth projective surface over an algebraically closed field of characteristic 0 (henceforth denoted by C). An important family of moduli spaces associated to S are the Hilbert schemes of points on S, which we will denote by Hilb n (S): these are smooth 2n-dimensional algebraic varieties, which parameterize colength n ideal sheaves I ⊂ O S . The Betti numbers of Hilbert schemes were computed in [10,16], and this lead to a very interesting observation: the generating series of the Betti numbers of Hilb n (S), as n goes from 0 to ∞, matches the Poincaré polynomial of (an appropriately graded version of) the Fock space of the infinite-dimensional Heisenberg algebra. Coupled with expectations from mathematical physics, the way to understand this interesting connection is to consider all Hilbert schemes together: Hilb n (S) and construct an action This was achieved by Grojnowski in [17] and Nakajima in [23]. We will mostly use the formulation of loc. cit., in which the generators {a ±n } n∈N of the Heisenberg algebra act on H * (Hilb(S)) by the so-called Hecke correspondences where I ⊂ nx I means that the ideal sheaves I, I ⊂ O S are contained inside each other, and their quotient is a length n sheaf supported at a single (but arbitrary) closed point x ∈ S. Nakajima considered the projection maps The main result of [23] is that the operators above satisfy the defining relations of the Heisenberg algebra, with H * (S) as a "parameter space". Under this action, H * (Hilb(S)) is isomorphic to the Fock space of the Heisenberg algebra, which explains in a representation theoretic way the classical formulas for the Betti numbers.

1.2.
After constructing the action (1.1), one might wonder how to generalize it. One direction would be to replace singular cohomology by other homology theories. While the construction extends almost verbatim to Chow groups, in algebraic K-theory things are not so simple. The reason for this is that the correspondences C • n are very badly behaved for n > 1 (one does not even know if they are equidimensional in general) so the initial question is to find suitable replacements.
Problem: define analogues of the operators (1.3) in K-theory, which satisfy the defining relations of the deformed Heisenberg algebra One approach toward this problem was undertaken in [11,31] for S = A 2 , where the authors used equivariant localization computations to construct an action (1.4) A K Hilb(S) = ∞ n=0 K Hilb n (S) (see Section 3.3 for our conventions on K-theory). In the elliptic Hall algebra formulation of [31], the algebra A is generated by elements {e ±n,k } (n,k)∈N×Z , with the subalgebra generated by the e ±n,0 's isomorphic to the deformed Heisenberg algebra, thus explaining our interest in constructing an action (1.4). In [25], we realized the operators e ±n,k using certain K-theory classes on the correspondence which can thus be thought of as a resolution of C • n of (1.2). While loc. cit. only proved results for S = A 2 , the correspondence (1.5) is well-defined for any smooth surface S, and we are thus poised to solve the Problem stated above. Even more so, the operators e ±n,k naturally lift to the derived category, and are thus related to various categorifications of Hecke algebras (see [15] for an overview).

1.3.
Another direction for generalizing (1.1) is to replace Hilbert schemes by other moduli spaces of coherent sheaves on S. An important example is given by moduli spaces of stable sheaves (see Section 2.8 for more details), in which the analogue of the action (1.1) was worked out in [2]. The main purpose of the present paper is to construct an action (1.4) in the aforementioned setup of stable sheaves. Thus, let S be a smooth projective surface with an ample divisor H, and also fix (r, c 1 ) ∈ N × H 2 (S, Z). Consider the moduli space M of H-stable sheaves on the surface S with the numerical invariants r, c 1 and any c 2 ∈ Z. We make the following two assumptions throughout the present paper: In the following formulas, we will denote by F the coherent sheaves on S that are parameterized by the moduli space M. With this in mind, let us recall the following moduli spaces of flags of sheaves that we studied in [27] and [28], respectively: where F ⊂ x F means that F ⊂ F and F/F is the length 1 skyscraper sheaf at the closed point x ∈ S. We recall the scheme structures of Z 1 and Z • 2 in Section 2, and in particular, the fact that they are both smooth. This implies that the maps induce direct and inverse image functors between the derived categories D M , D Z 1 , D Z • 2 . We may combine these spaces into more complicated diagrams of the form where the number of copies of Z 1 contained in the middle row is denoted by n. Then for an arbitrary sequence d 1 , . . . , d n ∈ Z, we consider the functor given by the composition where L is the line bundle on Z 1 with fibers (S, F 1 /F 0 ). 1 Note that e d := e (d) are the derived category versions of the operators e 1,d of Section 1.2, but it is crucial to us that e (d 1 ,...,d n ) is different from the composition e d 1 • · · · • e d n (see Propositions 2.27 and 3.4). With this in mind, our main result is the following.  Theorem 1.4 proves the m = 1 case of Conjecture 1.8, for any n. It would be interesting to compare Conjecture 1.8 with other categorifications of Heisenberg algebras in the literature. For example, such a categorification was proposed in [6] when the surface is of ADE type (generalized to all surfaces in [19]), but using the language of derived categories of symmetric powers of the surface S. It is thus not easy, but would be very interesting, to compare our results with those of loc. cit.
One can generalize Conjecture 1.8 by recalling that the generators e −n,k ∈ A satisfy where the coefficients of the linear combination match the analogous structure constants in the algebra A (see (4.6) and (4.8)). We expect such formulas to categorify. . Indeed, following the principles laid out in [15], the natural transformations that appear in Theorem 1.4 can be interpreted as yielding skein exact triangles between certain objects in the category C of affine Soergel bimodules (in [14], we make this precise by realizing Proposition 2.39 in C ).
The construction of the functors (1.8) was given in terms of derived schemes in [28], and Corollary 1.6 was conjectured therein. In the present paper, we show that Assumption S implies that many of the spaces featured in loc. cit. are local complete intersections. Therefore, one can work in classical algebraic geometry rather than derived algebraic geometry. By combining our Corollary 1.6 with the results of [29], we obtain the identification of the Carlsson-Okounkov operator K M → K M with a W-algebra intertwiner (see loc. cit. for definitions). When the surface is S = A 2 (and K-theory is understood equivariantly), this is enough to prove a mathematical incarnation of the deformed AGT correspondence for U(r) gauge theory with matter, see [26] (inspired by [21,32] and the physics literature).
We observe that Theorem 1.4 also holds when the moduli space of stable sheaves is replaced by the Hilbert scheme of points on an arbitrary smooth quasiprojective surface (we do not need the surface to be projective, as all the push-forward maps in Section 1.3 are proper). The modifications required to make the argument work are minimal, and we leave the details to the interested reader. When the surface S is K3, we revisit some of the methods in the present paper in [20] in order to prove that the cycle class map (1.14) A * taut → H * (Hilb (K3)) is injective, where A * taut denotes the subring of the Chow ring of the Hilbert scheme of arbitrarily many points on S that is generated by the tautological classes. This is a partial version of the Beauville-Voisin conjecture [34] for hyperkähler manifolds.
1.11. The structure of the present paper is the following. In Section 2 we recall the moduli space of stable sheaves M, introduce the moduli spaces of flags of sheaves Z λ and several versions of the moduli space of quadruples Y, and state a number of geometric properties of Z λ and Y (such as dimension, irreducibility, smoothness, l.c.i.ness, Cohen-Macaulay-ness and normality). In Section 3, we use these properties to set up and prove Theorem 1.4. In Section 4, we go from derived categories to K-theory and prove Corollary 1.6. Finally, in Sections 5 and 6 we prove all the aforementioned geometric properties: dimension estimates and irreducibility in Section 5 and all other properties in Section 6.
2. The moduli space of sheaves 2.1. All our schemes will be projective over an algebraically closed field of characteristic 0, henceforth denoted by C. A scheme will be called smooth if all of its local rings are regular. A closed embedding Z → X is called regular if the ideal of Z in X is locally generated by a regular sequence. In the present paper, we will often encounter closed embeddings which arise as the zero loci of sections where W is a locally free sheaf on the scheme X. We will often abuse terminology and refer to either σ or σ ∨ : W ∨ → O X as "the section", with the latter having the advantage that the zero locus is defined as the scheme If X is a Cohen-Macaulay scheme, we have the well-known inequality Since all schemes in our paper will be Cohen-Macaulay, we give the following. Indeed, the usual definition of regularity (that the coordinates of σ in any local trivialization of W form a regular sequence) is equivalent to that of Definition 2.2 over Cohen-Macaulay schemes. In this case, we have a quasi-isomorphism (the maps in the complex above are given by contraction with σ ∨ ) and we call Z(σ ) → X a complete intersection.

2.3.
Given a map of schemes η : X → X, we may form the fiber square If σ is a regular section and X, X are Cohen-Macaulay schemes, then (2.2) implies that η * (σ ) is regular if and only if Definition 2.5. -With the notation as above, assume that there exists a subsheaf such that both L and η * (W ∨ ) L are locally free. In this case, if the induced section 3) a derived fiber square with excess.
In the setting of Definition 2.5, the locally free sheaf L is called the excess bundle, and its rank is precisely the difference between the two sides of (2.5).

2.6.
Given a locally free sheaf V on a scheme X, consider the projective bundle P X (V) = Proj symmetric algebra of V Projective bundles are among the easiest examples of moduli spaces in algebraic geometry, in the sense that they represent the functor of line bundle quotients of the vector bundle V. More specifically, this means that there is a natural identification Assume that we have a map W → V of locally free sheaves on X. In the present paper, we will encounter local complete intersection morphisms of the form where the map ι is cut out by the section (1) and the map denoted by taut is the tautological morphism on P X (V).
Remark 2.7. -We note a convenient abuse of terminology in calling (2.9) a "section", as it will appear again throughout the present paper. Given a line bundle L and a vector bundle E on a scheme X, whenever we refer to a map L σ − → E as a section, we are implicitly referring to the induced section O → E ⊗ L −1 . The same terminology will apply to the dual map E ∨ → L −1 in relation to E ∨ ⊗ L → O.
Since the map ρ in (2.8) is smooth, the considerations of the preceding subsection apply to the composed map Z(σ ) → X. More specifically, given any map η : X → X, the fiber square analogous to (2.3) is derived iff σ and η * (σ ) are regular.

2.8.
Consider a smooth projective surface S and an ample divisor H ⊂ S. The Hilbert polynomial of a coherent sheaf F on S is given by where a, b, c are rational numbers that one can compute from the Hirzebruch-Riemann-Roch theorem. One can find formulas for these numbers in the Appendix to [27], but the only thing we will need in the present paper is that they can be expressed in terms of S, H and the rank r and Chern classes c 1 , c 2 of F . If F is torsion free, then a > 0, and one defines the reduced Hilbert polynomial as A torsion-free coherent sheaf F on S is called (Gieseker H-) stable if for all proper subsheaves G ⊂ F , we have the inequality whenever n 0. Since the reduced Hilbert polynomials are monic and quadratic, stability is determined by checking certain inequalities for the linear term and constant term coefficients. Note that stability depends on the ample divisor H, but we will fix a choice throughout this paper.
Definition 2.9 (See [18]). -Let M (r,c 1 ,c 2 ) denote the quasiprojective variety which corepresents the moduli functor of stable sheaves on S with the invariants r, c 1 , c 2 .
It is straightforward to compute the tangent spaces to the variety M (r,c 1 ,c 2 ) , and the interested reader can refer to Section 6.10 in the Appendix. A simple consequence of this computation is the following well-known fact. where const only depends on S, H, r, c 1 (the specific formula can be found in [27]).

2.11.
Recall Assumption A of (1.6), which states that gcd(r, c 1 · H) = 1. An important consequence of this assumption is the existence of a universal sheaf which is flat over M (r,c 1 ,c 2 ) , and has the universal property (2.13). In what follows, given a coherent sheaf F on a scheme T × S, we will denote by F t its fiber over {t} × S, for any closed point t ∈ T. Then we have the following fact (see [18]).
is a projective variety, which represents the moduli functor of stable sheaves on S with the invariants r, c 1 , c 2 .
The functorial description of M (r,c 1 ,c 2 ) entails the existence of natural bijections Maps(T, M (r,c 1 ,c 2 ) ) 1-to-1 ← − → coherent sheaves F on T × S, flat over T, (2.12) such that F t is stable with invariants (r, c 1 , c 2 ), ∀t ∈ T ∼ for all schemes T. Above, we write F ∼ F if F = F ⊗ π * 1 (L) for some line bundle L on T, and π 1 : T × S → T denotes the standard projection. The word "represents" in Proposition 2.12 means that the 1-to-1 correspondence (2.12) is given by Because of the equivalence relation ∼ in (2.12), the universal sheaf U (r,c 1 ,c 2 ) is not unique, but may be tensored with any line bundle pulled back from M (r,c 1 ,c 2 ) .

2.13.
From now on, we will fix (r, c 1 ) ∈ N × H 2 (S, Z) and set Note that the Bogomolov inequality implies that c 2 is bounded below by r−1 2r c 2 1 .
We will write U for the universal sheaf on M × S obtained as the disjoint union of the universal sheaves (2.11) over all c 2 ; we assume that the latter are compatible with each other as c 2 varies, as explained in [27,Section 5.9]. This implies that the moduli spaces of flags of sheaves that we will introduce in the next subsection also have universal sheaves which are contained inside each other in the obvious way.
Because the universal sheaf U is flat over M, it inherits certain properties from the stable sheaves it parameterizes, such as having homological dimension 1. Indeed, any stable sheaf of rank r > 0 is torsion free, and any torsion free sheaf on a smooth projective surface has homological dimension 1, see [18,Example 1.1.16]. The fact that this also holds in families is made explicit by the following result.
Proposition 2.14 ( [27]). -There exists a short exact sequence with W and V locally free sheaves on M × S. In fact, we can take for n 0, where π 1 : M × S → M and π 2 : M × S → S are the standard projections.

2.15.
A set partition is an equivalence relation on a finite ordered set. We will represent set partitions symbolically, for example (x, y, z) will refer to the partition of a 3-element set into distinct 1-element subsets, while (x, y, x) (respectively (x, x, x)) refers to the equivalence relation which sets the first and the last element (respectively all elements) equivalent to each other. The size of a set partition λ, which is denoted by |λ|, is the number of elements of the underlying set.
Definition 2.16. -For a set partition λ of size n, we will consider the scheme We will often write and Z • n = Z (x,...,x) Definition 2.16 should be read as "the scheme Z λ represents the functor" on T × S, flat over T, such that F 0,t , . . . , F n,t are stable with invariants (r, c 1 ), ∀t ∈ T, together with the data in (a) and (b) below ∼ where ∼ is the equivalence relation induced by tensoring F 0 , . . . , F n by one and the same line bundle pulled back from T. In the formula above, (a) and (b) refers to (a) maps x 1 , . . . , x n : T → S such that where i : T → T × S denotes the graph of the map x i : T → S from (a) Proposition 2.19 will establish (by induction on n) that the functor (2.17) is indeed representable, which implies the existence of the scheme stipulated in Definition 2.16. However, to keep our notation simple, we will denote points of this scheme as (2.16) instead of the more complicated notation (2.17). In other words, all the constructions from now on will be done at the level of closed points, and the family versions are left as exercises to the interested reader.

2.17.
If λ is a partition on an n element ordered set, then we will use the notation |λ and λ| for the partition on the n − 1 element ordered set obtained by dropping the first (respectively last) element of λ. Then we may consider the maps given by forgetting the first (respectively last) sheaf in the flag (2.16). The number # is 0 or 1 depending on whether the first (respectively last) element of λ is or is not equivalent to some other element of λ. Therefore, the map π − (resp. π + ) remembers the point x 1 ∈ S (resp. x n ∈ S) if and only if # = 1. For example: As is clear from the example above, in the case of the arrow on the left, there is no reason to remember the point x ∈ S "forgotten" from the set partition λ = (x, y, x, z), since it can be recovered from |λ = (y, x, z). In other words, our convention when # = 0 is that π − (resp. π + ) composed with the graph of the map that remembers x 1 (resp. x n ) yields the analogous map Z λ → Z |λ × S (resp. Z λ → Z λ| × S) to the one we would have defined in the # = 1 case.

2.18.
There exists a natural map p i : Z λ → M which only remembers the sheaf F i in the flag (2.16). We will write U i = (p i × Id S ) * (U ), and note that Proposition 2.14 implies that there exists an exact sequence In the following Proposition, we consider any set partition λ of size n, and let Z λ denote the moduli space (2.16), whose points will be denoted by F 0 ⊂ · · · ⊂ F n .
Proposition 2.19. -The map π − of (2.18) can be realized as the diagonal arrow in (in the latter case, x : Z |λ → Z |λ × S is the graph of the function Z |λ → S that remembers the support point we forget when going from λ to |λ) where → is the closed embedding cut out by the following composition of maps of locally free sheaves on the projectivization: The line bundle L 1 on Z λ is the restriction of O(1) from the projectivization.
Similarly, the map π + of (2.18) can be realized as the diagonal arrow in (we write ω S for the canonical line bundle on S, and its various pullbacks) where → is the closed embedding cut out by the following composition of maps of locally free sheaves on the projectivization: The line bundle L n on Z λ is the restriction of O(−1) from the projectivization.
Proof. -When λ has size 1, the Proposition was proved in [27, Definition 2.5, Proposition 2.8, Proposition 2.10], and the general case will follow the same logic. We will prove the statements pertaining to π − , and leave those pertaining to π + as exercises to the interested reader. Moreover, we will only prove the case # = 1, as # = 0 is analogous. We interpret the Proposition as follows: having constructed Z |λ which represents the functor (2.17) for the set partition |λ, we must show that the scheme Z λ defined as the closed embedding (2.20) represents the functor (2.17) for the set partition λ.
Recall that a map φ : T → Z |λ × S consists of a flag (2.28) F 1 ⊂ F 2 ⊂ · · · ⊂ F n of coherent sheaves on T × S which are flat over T, together with maps consists of the data ofφ and x 1 as above, together with a line bundle L 1 on T and a surjective homomorphism Finally, a map from T to the subscheme Z λ of (2.20) consists ofφ, x 1 , L 1 and τ as above, such that the following composition vanishes: Since pull-back is right-exact, this is equivalent to a surjective map By adjunction, this datum is equivalent to a surjection F 1 1 * (L 1 ) on T × S. Letting F 0 be the kernel of this surjection, this precisely completes the flag (2.28) to the flag (2.17). By [27,Proposition 5.5], F 0 is stable if and only if F 1 is stable.

2.20.
Consider an arbitrary set partition λ of size n and the scheme Z λ of Definition 2.16. Let π 1 : Z λ × S → Z λ and π 2 : Z λ × S → S denote the standard projections. For each i ∈ {1, . . . , n}, we have the following short exact sequence on Z λ × S: , and i ⊂ Z λ × S denotes the graph of the map that remembers the i-th support point x i . If i and j are equivalent elements of the set partition λ, then i = j . We may upgrade (2.29) to the following commutative diagram of sheaves on Z λ × S with all rows and columns exact: where I i denotes the ideal of the graph i inside Z λ × S, and the line bundle In the Proposition below, we will restrict the diagram (2.30) to 1 and n . (−) If p 1 S is flat, then there exists an injective map of sheaves L 1 ⊗ p 1 * S (ω S ) → W 0 | 1 with locally free quotient, such that the following composition vanishes: If p n S is flat, then there exists an injective map of sheaves L −1 n → (V n | n ) ∨ with locally free quotient, such that the following composition vanishes: In both formulas above, the second arrows are induced by (2.19).
Remark 2.22. -The flatness of p 1 S , . . . , p n S is proved in Proposition 6.2 for any λ of size n ≤ 3, but we expect it to hold for any set partition λ.
Proof. -Let us first prove (−). Upon restricting (2.30) (for i = 1) to 1 : Z λ → Z λ × S, we obtain the following diagram of Tor sheaves with exact rows and columns: The injectivity of the maps denoted by → in the diagram above is due to the fact that W 1 , V 0 , V 1 , B 1 are locally free, except for the injectivity of the horizontal map in the bottom left corner, which follows from the fact that U 1 has homological dimension 1 (see (2.19)). We have the following fiber square: The assumption that p 1 S is a flat morphism implies that T or 2 The latter equality in the equation above is a standard and straightforward exercise, which follows from the fact that S → S × S is a smooth embedding of codimension 2. If we recall the fact that B 1 | 1 ∼ = L 1 , then we conclude that as can be seen from diagram (2.34)), this establishes the vanishing of the composition (2.32). The fact that the sheaf is locally free follows from the facts that W 1 is locally free, and I 1 | 1 has homological dimension 1 (since it is the pull-back of the defining ideal of the codimension 2 regular embedding : S → S × S under the flat morphism p 1 * S ).
To deal with the (+) case, let us restrict (2.30) (for i = n) to n and then dualize: The top-most row is exact, due to the fact that restriction and dualization preserve short exact sequences of locally free sheaves. Because p n S : Z λ → S is flat, we have and the right-most vertical map in (2.36) is zero. Therefore, the rank 1 sub-bundle (B n | n ) ∨ ∼ = L −1 n embeds in the kernel of (V n | n ) ∨ → (W n | n ) ∨ , thus implying the vanishing of the composition (2.33). Moreover, the quotient is locally free by construction.

2.24.
We will now state certain geometric properties of several schemes Z λ with n = |λ| ≤ 4, to be proved in Section 6. Let k be the number of equivalence classes of λ.
Moreover, consider the map π : In what follows, whenever we say that Z λ is irreducible, we actually mean that π −1 (C) is irreducible for every connected component C of M × S n .
The excess bundle is Proposition 2.28. -Z (x,y) is l.c.i. and irreducible of expected dimension. Moreover, the following fiber square is derived: i. and irreducible of expected dimension. Moreover, the following fiber square is derived: x,y) and Z (y,x,x) are l.c.i. and irreducible of expected dimension. Moreover, the following fiber squares are derived: Finally, we will need the normality of some of the varieties Z λ .
Proposition 2.34. -The scheme Z λ is normal for any and where the notation is applicable, line bundles Note that obtained by remembering only the middle and top part of (2.40)-(2.43), and obtained by remembering only the middle and bottom part of (2.40)-(2.43). In Proposition 2.37 we will show that the maps (2.46) and (2.47) are representable, which together with Proposition 2.19 shows that the functors Y, Y − , Y + , Y −+ are themselves representable. Let us first introduce some notation pertaining to Y. The short exact sequence consists of coherent sheaves on Z 2 × S which are flat over Z 2 = Z (x,y) . Here and in what follows, x and y denote the graphs of the maps which record the points x and y, respectively. Let us write for the standard projection. Then the short exact sequence consists of locally free sheaves of ranks 1, 2, 1, respectively, on Z 2 . The composition vanishes (on account of I y being the ideal sheaf of functions which vanish on any coherent sheaf of the form y * (. . . )), and therefore induces a map By adjunction, this gives rise to a map which can be rewritten as Proposition 6.2 asserts that the vertical maps are flat, hence I y = (p y S × Id) * (I ). The preceding discussion applies equally well to the spaces Proposition 2.37. -With the notation above, we have where ι ↑ and ι ↓ are cut out by the following section: Proof. -We will only prove the required statement for Y, as the cases of Y − , Y + , Y −+ are analogous (equivalently, they follow from base change in the derived fiber squares (6.40)-(6.41)). Also, we will only prove the case of the map π ↑ , as the situation is symmetric by replacing (↑, x, y, L 1 , L 2 ) with (↓, y, x, L 1 , L 2 ). If we define the scheme Y by (2.51), then maps T → Y are in one-to-one correspondence with (1) a map ρ : T → Z 2 , i.e. a flat family of stable coherent sheaves on T × S, together with line bundles L 1 , L 2 on T and points x, y :

) a line bundle O(1) on T and a surjective homomorphism
where E = pr * (F 2 /F 0 ) with pr : T × S → T the standard projection. If we let H denote the kernel of (2.52), then one requires that the composition We must show that any datum as in items (1)-(2) above gives rise to a square (2.40). Item (1) yields the commutative diagram with exact rows and column on T × S. The kernel of (2.52) gives rise to a sub line bundle H ⊂ E, and the question is when y * (H) is a subsheaf of F 2 /F 0 (this subsheaf would be F 1 /F 0 , and this would complete the square (2.40) by constructing the bottom sheaf). It is easy to see that this happens precisely when the following composition vanishes: , so the question is when the map vanishes. By adjunction, this happens when (as a consequence of the sentence preceding diagram (2.50)), the vanishing of (2.54) is equivalent to the vanishing of (2.53).

2.38
. Now that we have defined the schemes Y, Y − , Y + , Y −+ , let us consider their basic properties. An important fact is the following geometric observation.
induces the following map of line bundles on Y: If we interpret this map as a section of L 2 ⊗ L −1 1 , then its zero locus consists of and is isomorphic to Proof. -By the functor-of-points description, a map from a scheme T into the zero locus of (2.55) consists of a square (2.40) (of coherent sheaves on T × S, flat over T) such that the corresponding induced map vanishes, where pr : T × S → T denotes the projection (here we have used the facts that and pr • x = Id). Because the sheaves F 1 /F 0 , and F 2 /F 1 are flat of length 1 over T, the map pr * is an isomorphism on local sections, hence the induced map F 1 /F 0 → F 2 /F 1 vanishes. This implies that F 1 ⊂ F 1 as subsheaves of F 2 , hence we obtain short exact sequences , which implies that x = y as maps T → S (the statement is local on both S and T). Since x is a closed embedding, x * is an exact functor and so the homomorphisms f and g induce line bundle homomorphisms is only possible if the aforementioned line bundle homomorphisms are isomorphisms (in other words, if ab is a unit in a certain ring, then both a and b are units). Therefore, we must have L 1 = L 1 as subsheaves of E, hence F 1 = F 1 as subsheaves of F 2 , which precisely yields a point of Z (x,x) . The analogous results for Y − , Y + , Y −+ instead of Y are proved by the same argument.

2.40.
The following results will be proved in Section 6. The expected dimension of the varieties Y, Y − , Y + , Y −+ is defined to be the dimension of the respective spaces on the bottom of (2.46) or (2.47) (which is equal to the actual dimension of these spaces by Propositions 2.28, 2.30, 2.31, 2.33). Using the results above, we will prove the following fact in Section 6.
Our main interest in the above geometric properties is the following result.
The analogous properties hold for π ↓ . Moreover, the analogous properties hold with the scheme Y replaced by the schemes Y − , Y + , Y −+ of (2.46) and (2.47).
Proof. -Recall from Proposition 2.37 that π ↑ = ρ • ι ↑ , where ι ↑ is a closed embedding and ρ : P Z 2 (E) Z 2 is a P 1 -bundle. Therefore, we have a short exact sequence (the latter equality would actually hold for any coherent sheaf on P Z 2 (E)). Therefore, in the long exact sequence in cohomology associated to (2.59) the spaces on the left and on the right are 0 for any i ≥ 1. This implies (2.58) for i ≥ 1. As for the case i = 0, let us recall the following well-known fact.

proper morphism with all fibers connected and non-empty. Then f
Proof. -Stein factorization implies that we can express f as and h a finite morphism. However, because the fibers of f = h • g are connected, so are the fibers of h. Since h is finite, this implies that all the fibers of h consist of a single point, and the normality of Y implies that h is an isomorphism. We conclude To obtain formula (2.58) for i = 0, we apply Claim 2.46 for X = Y (which is reduced by Proposition 2.44), Y = Z 2 (which is normal by Proposition 2.34) and f = π ↑ (which is proper by Proposition 2.37). Moreover, the proof of Proposition 2.37 shows that the fiber of π ↑ above a closed point x , and is a point in all other cases.
Remark 2.47. -All the results in Section 2 hold with the moduli space M replaced by the Hilbert schemes of points Hilb(S): all one needs to do is to replace stable sheaves F by ideal sheaves I everywhere. In particular, Proposition 2.45 shows that the nested Hilbert scheme Z 2 = {(I 0 ⊂ I 1 ⊂ I 2 )} has rational singularities, and Y is the blow-up of its singular locus (see the proof of Proposition 6.9).

3.1.
Recall that all our schemes are projective over C. Consider the following: D b (Coh(X)) = the bounded derived category of coherent sheaves on X Perf(X) = the derived category of perfect complexes on X We recall that a complex is perfect if it is quasi-isomorphic to a finite complex of locally free sheaves of finite rank on X (this notion is usually called "strictly perfect", although it is equivalent to the more general notion of "perfect" on projective varieties). The natural inclusion functor is fully faithful, and is an equivalence if the scheme X is smooth. We will write The functors between derived categories associated to a morphism f : The existence of the push-forward on the second line is non-trivial, and we refer the reader to [33] for an overview. As for the pull-back on the third line, we note that we always have a left derived functor on the derived categories of bounded above complexes of coherent sheaves over projective schemes. However, in order to ensure that the functor f * takes bounded complexes to bounded complexes, one needs a strong assumption, such as f having finite Tor di- where η is proper. Suppose that Z → X and Z → X are closed embeddings cut out by regular sections σ and η * (σ ) of locally free sheaves W and η * (W), respectively. Then we have equivalences The same equivalences hold if the regular embeddings ι and ι are replaced by smooth morphisms such as the projectivization of a locally free sheaf coming from X.
Proof. -The equivalences (3.3) and (3.4) are well-known to hold in the derived category of unbounded complexes, as long as the maps η and ι are Tor independent: for all points z ∈ Z and x ∈ X which map to the same point x ∈ X. Since ι is a regular embedding, we may replace O Z,z by the Koszul complex of the section σ ∨ : being exact everywhere except at the right-most place. However, the above is none other than the Koszul complex of the section η * (σ ) of the locally free sheaf η * (W), and its exactness follows from our assumption that the section η * (σ ) is regular.
Having established (3.3) and (3.4) at the level of unbounded complexes, the desired conclusion follows from the fact that Perf and D b (Coh) are full subcategories of the derived category of unbounded complexes, together with the fact that the pull-back and push-forward functors associated to ι, ι , η, η are well-defined on these subcategories due to items (1)-(4) that precede the statement of Proposition 3.2.

3.3.
Associated to a scheme X, we have the K-theory groups If X is smooth, these groups are isomorphic and we will write Associated to a morphism f : X → Y, we have group homomorphisms f * , f * between the various K-theory groups, as in the items (1) for the excess bundle L.
Equality (3.6) is called the "excess intersection formula", and it does not generally lift to the derived category. One reason for this is that formula (2.6) does not naturally give rise to a map s : L → O whose Koszul complex categorifies ∧ • (L). The case when (3.6) lifts to the derived category, with the section being s = 0, is treated in [1] and linked with the relation between Z and the derived fiber product Z × L X X .

3.5.
Let us recall the schemes Z 1 and Z • 2 of Section 2.15. It was shown in [27,28] that these schemes are smooth (see also Propositions 2.26 and 2.27). Because of Proposition 2.19, the maps p ± and π ± of Section 1.3 are all proper and l.c.i., which allow us to define functors for all d 1 , . . . , d n ∈ Z, according to the composition immediately after (1.8).
Remark 3.6. -Analogously, one can define the transposed functor by the composition going the other way (left-to-right): and shifting complexes by rn.
In Section 4, we will show that the functors e (d 1 ,...,d n ) categorify the elliptic Hall algebra introduced in [4]. Our main tool is the commutation relation between the functors (3.7) that we stipulated in Theorem 1.4. To set up the relation, let us consider the compositions (we will often write e k = e (k) ) and denote them by e (d 1 ,...,d n ) • e k and e k • e (d 1 ,...,d n ) , respectively. In formulas (3.9) and (3.10), we set S 1 = S 2 = S, but we use different notations for the two copies of S to emphasize the fact that the functor e (d 1 ,...,d n ) takes values in the first factor of S × S, while the functor e k takes values in the second factor.
Proof of Theorem 1.4. -We will begin with the case n = 1, then n = 2 and finally general n. The fact that the fiber square (2.38) is derived, combined with Proposition 3.2, implies that the compositions e d • e k and e k • e d are given by the following correspondences, respectively: Above, we write p x S , p y S for the maps Z 2 → S that remember the points x, y, respectively. The way to interpret the diagrams above as functors is the following: start from D M , pull-back to Z 2 , tensor by the line bundle on top, and then push-forward to D M×S×S . For example, in the case of the diagram on the left, e d • e k equals D M×S×S Note that the middle arrow consists of tensoring with a line bundle, followed by the fully faithful functor (3.1). This aspect is a necessary technicality, because the scheme Z 2 is not smooth (in the case at hand, we could have gotten away with only using the category D b (Coh) since the map p − is an l.c.i. morphism, but we will apply the notation above in situations where the morphism which plays the role of p − will not be l.c.i.). Consider the scheme Y, together with the maps (2.46) and (2.47): in the derived category. Therefore, the compositions e d • e k and e k • e d can be alternatively given by the following correspondences: (the map that points left, respectively right, remembers the sheaf F 0 , respectively F 2 in the notation of (2.40)). Recall from (2.44) the identity L 1 L 2 = L 1 L 2 on Y. The map of line bundles L 1 → L 2 of (2.55) gives rise to maps which induces a natural transformation between the functors e d • e k and e k • e d , as required in item (2) of Theorem 1.4. In general, the cone of a composition of maps in any triangulated category has a filtration with associated graded object given by the direct sum of the cones of the individual maps. 2 Thus, the cone of the maps denoted by → and ← in (3.11) has a filtration with associated graded object given by the sum of the cones By Proposition 2.39, the cone of the maps denoted by → and ← above is precisely the where a goes over the indexing sets featured in either of the two situations above. The correspondence Let us now deal with the case n = 2. The fact that the fiber squares in Proposition 2.30 are derived, combined with Proposition 3.2, implies that the compositions g 0 = e (d 1 ,d 2 ) • e k and g 2 = e k • e (d 1 ,d 2 ) are given by the following correspondences, respectively: The maps denoted by p + and p − only remember the sheaves F 0 and F 3 , respectively, from points (F 0 ⊂ · · · ⊂ F 3 ) of either Z (x,x,y) or Z (y,x,x) . We will also consider the functor g 1 : D M → D M×S×S given by the following correspondence Because of Proposition 2.45, the functors g 0 and g 1 can be given by the following correspondences, respectively: while the functors g 1 and g 2 can be given by the following correspondences, respectively: Depending on whether d 2 > k or d 2 < k or d 2 = k, we obtain natural transformations g 0 → g 1 or g 0 ← g 1 or g 0 ∼ = g 1 by using the map (2.55) on the space Y + (cf. (3.11)). Similarly, depending on whether d 1 > k or d 1 < k or d 1 = k, we obtain natural transformations g 1 → g 2 or g 1 ← g 2 or g 1 ∼ = g 2 by using the map (2.55) on the space Y − . This establishes item (2) of Theorem 1.4. To prove item (3), one needs to proceed as in the n = 1 case, and we will present the detailed argument for the natural transformation g 0 → g 1 in the case d 2 > k. The cone of this natural transformation arises from the cone of the map of line bundles on Y + , which in turn has a filtration whose associated graded object is the direct sum of the cones The analogue of Proposition 2.39 for Y + instead of Y implies that the cone of the map (3.12) is the structure sheaf of the subscheme Z (x,x,x) → Y + given by the condition x = y and F 2 = F 2 . Therefore, the cone in (3.12) is the correspondence where we consider the maps ρ ± and the line bundle L mid as in the following diagram: The required statements about the natural transformations g 0 ↔ g 1 and g n−1 ↔ g n are proved just like in the case n = 2 in the previous paragraph, so we will not review them. Similarly, the required statements about the natural transformations g i−1 ↔ g i are all proved just like in the case n = 3 and i = 2, which we will now explain. According to the diagram above, g 1 is given by the correspondence One should read the diagram above as inducing a functor D M → D M×S×S by composing the pull-back and push-forward functors corresponding to the solid arrows, read rightto-left, all the while at every step tensoring by the line bundle which is displayed with a dotted line above every space. Similarly, g 2 is given by the correspondence Because the fiber squares in Proposition 2.33 are derived, we may invoke Proposition 3.2 in order to conclude that g 1 and g 2 are given by the following correspondences: respectively. However, Proposition 2.45 applied to Y −+ implies that we may replace the line bundles above on Z (x,x,y,x) and Z (x,y,x,x) by their pullbacks to Y −+ without changing the correspondences. Therefore, the required natural transformation between g 1 and g 2 is induced by a map of line bundles on Y −+ . The construction of this map, as well as the proof of the fact that its cone has the properties stipulated in item (3)  We also consider the power series of maps defined as the composition Above, recall that U is the universal sheaf on M × S, and q = [ω S ] ∈ K S (we abuse notation and use the symbol q also for the pull-back of the canonical line bundle to K M×S ). The meaning of the wedge power is the following. Proposition 2.14 allows one to write [U ] = [V] − [W] ∈ K M×S , where V and W are locally free sheaves. Since the total exterior power is supposed to be multiplicative, we define We showed in [27] that the operators (3.13), (3.14), (3.15) satisfy the relations where O is the K-theory class of the diagonal inside S × S. We will now work out some relations among the operators (3.13) and (3.14). Proof. -We will prove (3.19) for n = n = 1, and then the general case will be clear, in light of the fact that the operators (3.13) arise as successive push-forwards and pull-backs between Z 1 and Z • 2 (cf. the diagram after (1.8)). Consider the diagram with the notation as in Section 1.3. By a simple diagram chase, we have By Propositions 2.27 and 3.4 (the latter Proposition applies because all the maps in (3.20) are closed embeddings followed by smooth maps), we have and it is easy to see that the right-hand side is e (d,d ) − q · e (d−1,d +1) .
As an immediate consequence of Theorem 1.4, we conclude that the operators (1.11) satisfy the identities (1.12) of operators K M → K M×S×S , as well as the opposite Lie bracket relations for the commutator of f (d 1 ,...,d n ) with f k .
together with the opposite relations for F(z) instead of E(z), as well as We will set H + 0 = 1 and H − 0 = c −1 , and note that c is a central element of A.

4.3.
The algebra A is known by many names, including the Ding-Iohara-Miki algebra ( [8,22]), the double shuffle algebra ( [12]), the stable limit of trigonometric DAHA ( [7]), and many other incarnations in representation theory and mathematical physics. However, the main description we will be concerned with is that of the elliptic Hall algebra of [4]. Specifically, the following is the main theorem of [30].  ([4], see [28,Theorem 3.5] for our conventions) to be generated by elements {E n,k } (n,k)∈Z 2 \(0,0) , modulo the relations for any (n , k ) (n, k) (the notation means that (n, k) can be reached clockwise from (n , k ) without crossing the negative y axis) and certain scalars coeff ∈ K.
Then upon localization with respect to the element (1 − q 1 )(1 − q 2 ) ∈ K, the algebra A becomes isomorphic to the elliptic Hall algebra via the assignment We will henceforth abuse notation and refer to E n,k as elements of A. If the triangle with vertices (0, 0), (n, k), (n + n , k + k ) has no lattice points inside and on any of the edges, then (4.6) takes the following simple form: Formula (4.8) shows how, after inverting (1 − q 1 )(1 − q 2 ), any E n,k for gcd(n, k) = 1 can be obtained from successive commutators of E ±1,k (there is a similar formula in [4] which deals with the non coprime case, see [28] for our conventions).
-Another important case of (4.6) is when (n, k) and (n , k ) are proportional. In order to state the relation, let us set for any coprime integers n, k Then the elements P n,k ∈ A defined by the formula above are multiples of the generators considered in [4] (see [26,Section 2.6] for a formula of the precise multiple). In terms of the P n,k 's, formula (4.6) when (n, k) and (n , k ) are proportional reads where s = gcd(n, k). In all representations of A considered in the present paper, we will set c = q r for a natural number r (called the "level" of the representation) and therefore the fraction in (4.9) is an element of K. As a consequence of this fact, we conclude that the algebra A contains a deformed Heisenberg algebra for each slope k n .  E (d 1 ,...,d n ) by replacing E −n,k with E n,k in (4.12).

In
The elements (4.10) were constructed using the isomorphism between A and a double shuffle algebra (see [24]), and this allows us to prove the following.
Proposition 4.8. -For any d 1 , . . . , d n There is no summand corresponding to any i for which d i = k.
Proposition 4.8 may be proved by embedding the negative half of A (i.e. the subalgebra generated by {E k } k∈Z ) in the algebra A q,t of [5] via the last formula of loc. cit. Alternatively, we recall that the elliptic Hall algebra A acts on the K-theory group of the moduli space of rank r framed sheaves on P 2 ( [11,31], see [25] for our conventions). This statement may be interpreted as an equivariant version of the results in the present paper, and our proof carries through in the equivariant case. Therefore, (1.12) implies that relation (4.14) holds in the level r representation of A. Since any non-zero element of A acts non-trivially in some level r representation for r large enough, this implies that relations (4.14) hold in A. Proof. -Taking the coefficient of z −m w −n in relation (4.2) shows it is equivalent to for all m, n ∈ Z. We may rewrite the relation above as In the right-hand side, we may convert every E m E n into E (m,n) − qE (m−1,n+1) using relation (4.11), and so the formula above becomes equivalent to It is easy to check that the formula above follows by applying (4.14) to the four Lie brackets in the left-hand side. As for (4.4), it follows from the fact that both of which are special cases of (4.14).

4.10.
We would like to compare the abstract algebra elements E (d 1 ,...,d n ) ∈ A of the previous Subsections with the explicit maps e (d 1 ,...,d n ) : K M → K M×S from Section 3.7. The goal would be to say that we have an action of the algebra A on K M , but this statement requires care: taken literally, having an action means that any element of A gives rise to a homomorphism K M → K M . In formula (3.13), we see that this is not quite the case, so we must adapt the notion of "action".

but we use different labels in order to emphasize the fact that x takes values in the first factor of S and y takes values in the second factor.
The notions in Definition 4.11 satisfy associativity, in the sense that (4.15) (xy| )z| = x(yz| )| as homomorphisms K M → K M×S . Therefore, we will unambiguously use the notation xyz| . We also have the following version of the Jacobi identity

4.13.
For the remainder of the paper, the parameters q 1 , q 2 over which the algebra A is defined will be reinterpreted such that While q 1 , q 2 are not themselves well-defined in K S , any symmetric Laurent polynomial in q 1 , q 2 is indeed well-defined. In other words, we obtain a ring homomorphism where K is the ring (4.1). In order for the ring homomorphism above to be well-defined, we need the quantity 1 + q −1 + · · · + q −n+1 to be invertible in K S , and the solution to achieving this is to work with K-theory with Q coefficients (indeed, q is unipotent in K S , so 1 + q −1 + · · · + q −n+1 is equal to n times a unit). Define for all X, Y ∈ A, and note that the right-hand side is well-defined due to the fact that all commutators in the algebra A are multiples of (1 − q 1 )(1 − q 2 ), see (4.6).  such that for all X, Y ∈ A, we have The compatibility between relations (4.23) and (4.24) is visible when restricting the latter formula to the diagonal : M × S → M × S × S, as follows: Because the normal bundle N :S →S×S is isomorphic to the tangent bundle to S, hence the exterior algebra of its dual equals Moreover, the series H ± (z) ∈ A of (4.7) act by (3.15).
Proof. -Recall from [4] (see [28] for our notations) that a K-basis of A is given by k 1 ), . . . , (n t , k t )} goes over all convex paths of lattice points, i.e. such that the slopes k i /n i are ordered clockwise starting from the negative y axis (formula (4.9) implies that E n,k and E n ,k commute if (n, k) ∈ R >0 (n , k ), so we may order them arbitrarily).
Then requirement (4.23) forces us to set Together with K-linearity, this completely determines . [ where the coefficients in the right-hand side match the analogous ones in the right-hand side of (4.6), but with q 1 and q 2 interpreted as the Chern roots of 1 S .
To conclude the proof of Theorem 4.15, we must show that relations (4.29) imply that satisfies properties (4.23) and (4.24); we will only take care of the former of these, as the latter is analogous and so left to the interested reader. We will use the "straightening" argument of [4]: one may define the elements (4.26) of A for an arbitrary path v of lattice points. The convexification of v is the path consisting of the same collection of lattice points, but ordered in clockwise order of slope, starting from the negative y axis. The area of a path, denoted by a(v), is defined as the area between the path v and its convexification. Therefore, we will show that holds for an arbitrary path v, by induction on a(v) (this claim will establish (4.23)). The base case is when a(v) = 0, i.e. v is already convex, in which case (4.30) is simply the definition of (E v ). A non-convex path v may be written as v = . . . , (n i , k i ), (n i+1 , k i+1 ), . . .
for some i such that k i /n i < k i+1 /n i+1 . Let v denote the path obtained from v by switching the lattice points (n i , k i ) and (n i+1 , k i+1 ). It is elementary to observe that a(v ) < a(v), and moreover the straightening argument of [4] shows that where the sum goes over paths with a(v ) < a(v), and the coefficients are induced from (4.6). Because of (4.29) and the induction hypothesis, we have with the coefficients in the RHS being the same ones as in (4.31) (indeed, the two computations (4.31) and (4.32) take as inputs (4.6) and (4.29), respectively, so it should be no surprise that they produce the same coefficients). Formula (4.32) yields the induction step of (4.30), which as we have seen establishes (4.23).
Proof of Claim 4.16. -Let us first give a slightly imprecise argument. By Theorem 4.4, the commutation relations (4.6) can be deduced from relations (4.2)-(4.5). As the latter hold with E k , F k , H ± k replaced by e k = (E k ), f k = (F k ), h ± k = (H ± k ) of Section 3.7 (this fact was proved in [27]), one might conclude the claim.
The imprecision in the previous paragraph is that, strictly speaking, Theorem 4.4 requires one to invert the scalar (1 − q 1 )(1 − q 2 ). When q 1 and q 2 are set equal to the Chern roots of 1 S , this scalar is manifestly a zero divisor, so inverting it is off the table. However, the ace up our sleeve is the operation (x, y) → [x, y] red for homomorphisms x, y : K M → K M×S , and its counterpart (4.21) for elements of A. We thus revisit the argument of [30], which consisted of the following main steps: (1) Define E n,k ∈ A as iterated reduced commutators (4.21) of E ±1,l ∈ A via (4.8) (and its generalization to the case when gcd(n, k) > 1, see [28]). In the geometric case, we emulate this procedure and define (E n,k ) : K M → K M×S as iterated reduced commutators (4.18) of (E ±1,l ) : K M → K M×S . (2) Show that relations (4.6) among the elements E n,k ∈ A thus constructed can be obtained from relations (4.2)-(4.5). While combinatorially quite involved, this part of the argument of [30] only uses the Lebniz rule and Jacobi identities. By (4.19) and (4.20), the same argument therefore applies equally well to show that relations (4.29) among the operators (E n,k ) : K M → K M×S can be obtained from the geometric counterparts of relations (4.2)-(4.5), which we know to hold.
The only thing that remains to be proved is that the operators (E n,k ) constructed as in item (1) above match the operators in the right-hand sides of (4.27) and (4.28). This happens because relations (4.11) and (4.14) in A are matched by relations (3.19) and (1.12) between operators K M → K M×S .

5.1.
The main purpose of the present Section is to obtain certain estimates on the dimensions of various strata of the schemes Z λ of Definition 2.16, when n = |λ| ≤ 4. In what follows, the dimension of a scheme will refer to the maximum of the dimensions of its irreducible components. Consider the map Z λ π − → M × S n which takes a flag (2.16) to (F n , x 1 , . . . , x n ). The target of this map is smooth but disconnected, with the dimension of its connected components depending linearly on the second Chern class of F n . Thus, the phrase "dimension (resp. number of irreducible components) of Z λ " will refer to the dimension (resp. number of irreducible components) of π −1 (C) for any fixed connected component C of M × S n . The analogous terminology will apply to locally closed subsets of Z λ .

5.2.
Any stable sheaf F of rank r > 0 on a smooth surface is torsion-free, hence it injects into its double dual F → F ∨∨ . The double dual is a reflexive sheaf, hence locally free (over a smooth surface). Therefore, we have an injection Proposition 5.5]), and since stable sheaves over an algebraically closed field are simple, this implies that the inclusion (5.1) is unique up to constant multiple. The quotient V/F is supported at finitely many distinct closed points x 1 , . . . , x k ∈ S, hence where Q i is a finite length sheaf supported at the closed point x i ∈ S. The length d i of the sheaf Q i will be called the defect of F at the point x i , since it measures how far F is from being locally free in the vicinity of x i . As the notation suggests, the variety Quot d does not depend (up to isomorphism) on the choices of V, x, S, as long as the latter is a smooth surface. In order to produce a stable sheaf F with prescribed defects at distinct points x 1 , . . . , x k ∈ S, one must start from a stable locally free sheaf V with invariants (r, c 1 ), and modify it according to a point in the irreducible algebraic variety Quot d 1 × · · · × Quot d k . This means that the locally closed subscheme which parameterizes sheaves with defects d 1 , . . . , d k at given distinct points x 1 , . . . , x k (and no defect anywhere else) has codimension r(d 1 + · · · + d k ) + k in M × S k .

5.5.
Recall the schemes introduced in Definition 2.16, namely where λ is a set partition of size n. Assume that the set partition λ consists of k distinct parts with multiplicities n 1 , . . . , n k , and we will use the notation x 1 , . . . , x k for the points of S corresponding to the distinct parts of λ. All the sheaves in a flag (5.3) have isomorphic double duals, hence they are all contained in one and the same locally free sheaf V. Therefore, we let We will give an explicit construction of the schemes Quot d,...,d+n in the next subsection, as well as some dimension estimates when n ≤ 4 (as opposed from the scheme in Definition 5.3, their dimensions grow wildly for large n, and they are far from being irreducible). The construction will not depend on V, x, S, and it will be clear that the same principle as in the previous subsection applies. Namely, in order to construct a flag More rigorously, the discussion above implies the following.
As a consequence of Lemma 5.7, we obtain the following result (recall the conventions of Section 5.1 when discussing dimension and the number of irreducible components of locally closed subsets of Z λ , such as Z def d 1 ,...,d k λ ).
Corollary 5.8. -For any sequence of defects d 1 , . . . , d k ≥ 0 and any set partition λ whose parts have multiplicities n 1 , . . . , n k ∈ N, we have where c first 2 , c last 2 are the second Chern classes of the sheaves labeled F 0 and F n in (5.3), and const is the same constant that appears in (2.10).
Proof. -Because the second Chern class of the stable sheaf V (defined in item (2) before the statement of Lemma 5.7) is The choice of the support points x 1 , . . . , x k adds 2k dimensions, and the choice of flag F 0 ⊂ · · · ⊂ F n ⊂ V with the given defects adds k i=1 dim Quot d i ,...,d i +n i dimensions to our estimate (according to item (3) above Lemma 5.7).

5.9.
Let us now describe the schemes Quot d and Quot d,...,d+n explicitly. Since these do not depend on the choice of the locally free sheaf V, or of a closed point x on a smooth surface S, we may as well consider V = O ⊕r , S = A 2 and x = (0, 0). Therefore, we have As vector spaces, we may consider isomorphisms Q i ∼ = C i , and the O A 2 = C[x, y] module structure on Q i can be packaged by providing two nilpotent commuting endomorphisms X, Y ∈ End(C i ). A surjective homomorphism O ⊕r A 2 Q i is the same datum as r vectors v 1 , . . . , v r ∈ C i which are cyclic. Recall that cyclicity means that the vector space C i is generated by polynomials in X and Y acting on linear combinations of v 1 , . . . , v r . In other words, let us construct the i × r matrix v = (v 1 , . . . , v r ) ∈ Hom(C r , C i ) and let End 0 (C i ) denote the variety of nilpotent endomorphisms of C i . Then where we take the quotient by the G = GL d action in order to factor out the ambiguity of the chosen isomorphism Q ∼ = C d . Explicitly, g ∈ G acts by (X, Y, v) (gXg −1 , gYg −1 , gv) and the cyclicity of v implies that the G action is free. The quotient by G is geometric, because the fact that we restrict to the open subscheme of cyclic triples (X, Y, v) is precisely equivalent to restricting to the locus of stable points (as in geometric invariant theory) with respect to the inverse of the determinant character. Similarly, where End 0 (C d+n · · · C d ) denotes the variety of nilpotent endomorphisms of C d+n which preserve a fixed flag of quotients of the specified dimensions (and P ⊂ GL d+n denotes the subgroup of automorphisms with the same property). If one is apprehensive about taking the quotient of a quasiprojective algebraic variety by the non-reductive group P, then one must employ the following trick. Instead of fixing a flag of quotients, let it vary and replace the datum (X, Y, v) by where dim V i = i and the maps X, Y ∈ End 0 (C d+n ) are required to preserve the flag {V i } d≤i<d+n . Then the P quotient should be replaced by a GL d+n quotient. We will refrain from making this modification, so as to keep the presentation clear.

5.10.
We fix a basis of C i so that the endomorphism X takes the form where the block in the top left is a d × d matrix, and the lower triangular block in the bottom right is n × n. We will use the notations X 0 , x, X for the three blocks, as pictured above, and employ the analogous notations for Y. By analogy with the constructions in Section 5.9, we define the affine algebraic variety Comm n = (X, Y) ∈ End 0 (C n · · · C 1 )× × End 0 (C n · · · C 1 ) s.t. [X, Y] = 0 and the stack where B ⊂ GL n denotes the Borel subgroup of lower triangular matrices. The variety Comm n has been studied in the literature, and we refer the reader to [3] for a survey, where the author proves that the dimension and number of irreducible components of Comm n grow wildly as n increases. However, we are interested only in the particular cases n ∈ {1, 2, 3, 4}, when this variety is behaved rather mildly.
Proposition 5.11. -When n ≤ 3, the variety Comm n is irreducible of dimension When n = 4, it has the dimension above, but consists of two irreducible components.
For example, we have (2) the subvariety Z 2 ⊂ Comm 4 defined as the closure of the locus (x 32 , y 32 ) = (0, 0), for which it is not hard to observe that dim Z 2 = 12 − 3 = 9 It is straightforward to prove that Comm 4 is generically reduced. Its singular locus is precisely the intersection Z 1 ∩ Z 2 , which has codimension 1 in Comm 4 .

5.12.
Consider the natural map (notation as in (5.7)) where v 0 ∈ Hom(C r , C d ) is the projection of the cyclic vector v ∈ Hom(C r , C d+n ) onto the quotient C d+n C d (it is easy to see that projection preserves cyclicity).  To see this, recall that Quot 0,...,n is an open subset of a rank rn affine bundle over Stack n . As shown in Proposition 5.11, the latter stack is irreducible if n ≤ 3, while if n = 4 its two irreducible components were described in items (1) and (2) of the previous subsection. The open condition that determines Quot 0,...,n in the said affine bundle is given by requiring that v of (5.6) be cyclic, and it is easy to see that this open condition does not "miss" any of the two components.
Proof of Proposition 5.13. -Consider the locally closed subset To prove the proposition, it suffices to show that for all n, d, r, i. The fiber of ζ over a point (5.13) consists of a choice of (1) n × d blocks x, y as in the bottom left corner of (5.7), which satisfy The data in (1)-(2) above provide 2dn + rn affine coordinates, and the quotient by the unipotent part of the group P (i.e. the block strictly lower triangular matrices, in the notation of (5.7)) subtracts dn dimensions, on account of the P-action being free. After accounting for all the generators and relations above, we see that (5.15) dimζ −1 (point (5.13)) = n(r + d) − # of independent equations in (5.14) Therefore, condition (5.12) is trivial unless Given that we fixed X 0 , Y 0 , X , Y , one may think of (5.14) as a linear map The number of independent equations in (5.14) is equal to dim Im μ.
Claim 5.15. -With the notation as above, we have Proof. -Let us consider the flag of quotients C n π n C n−1 π n−1 · · · π 1 C 1 which is preserved by the nilpotent maps X and Y , and consider the commutative diagram Above, the horizontal arrows are defined by the same formula as (5.17), and the vertical arrows are the operators of composition with π i . Let K = Ker π i ∼ = C. Since all the maps in the commutative diagram are linear, it suffices to show that Because X | C i and Y | C i annihilate K, then for any we have μ i (x, y) = xY 0 − yX 0 . Then we fix a basis of Im X 0 + Im Y 0 consisting of linearly independent vectors in Im X 0 ∪ Im Y 0 . We may define x, y ∈ Hom(C d , K) ∼ = (C d ) ∨ to take any values on these basis vectors, which precisely implies (5.19).
The essential feature which allowed the proof of Claim 5.15 to work was that X and Y are strictly lower triangular. Since the commuting nilpotent matrices X 0 and Y 0 also preserve a full flag of subspaces of C d , a completely analogous argument yields so we obtain the estimate # of independent equations in (5.14) ≥ ≥ max(n · dim(Im X 0 + Im Y 0 ), d · dim(Im X + Im Y )) Claim 5.16. -For any 4 ≥ n > r, the locus of points (5.13) such that max(n · dim(Im X 0 + Im Y 0 ), d · dim(Im X + Im Y )) = j ∈ N has codimension ≥ d(n − r) − j in Quot d × Stack n , with the following exception. There is an irreducible component G of the locus above for which has codimension = 2(4 − 1) − 5 instead of ≥ 2(4 − 1) − 4. However, outside of codimension 1, points of G have the property that dim Im μ ≥ 5 instead of ≥ 4.
It is straightforward to see that formula (5.15) and Claim 5.16 imply the required inequality (5.12) (even the presence of G does not violate this fact, because even though G has dimension one larger than expected, outside of codimension 1 points of G have the property that dim Im μ is one larger than expected).
Proof of Claim 5.16. -We may assume d > 0, since the d = 0 case is trivial. Because the matrices X 0 and Y 0 have a cyclic vector v 0 ∈ Hom(C r , C d ), then dim(Im X 0 + Im Y 0 ) ≥ d − r Indeed, the existence of a proper subspace of C d that contains Im X 0 , Im Y 0 and the r columns of v 0 contradicts the cyclicity of v 0 . The fact that n(d − r) ≥ d(n − r) establishes Claim 5.16 when n ≤ d. Therefore, we are left to deal with the case Taken together with the inequality (5.16), as well as the fact that we only consider n ∈ {1, 2, 3, 4}, this means that we are left with finitely many cases.
Claim 5.17. -The locally closed substack L n,λ ⊂ Stack n consisting of (X , Y ) with the property that dim(Im X + Im Y ) = λ has codimension The proof of Claims 5.17 and 5.18 are straightforward exercises, which we leave to the interested reader. For example, the μ = 0 case of Claim 5.18 holds because the subscheme {X 0 = Y 0 = 0} ⊂ Quot d has codimension d 2 − 1. Indeed, this subscheme is nothing but the Grassmannian of d dimensional quotients of r dimensional space, so it has dimension d(r − d), whereas Quot d has dimension rd − 1. Note that Claim 5.16 reduces to the inequality for all 0 ≤ λ < n, max(0, d − r) ≤ μ < d and 1 ≤ d, r < n ≤ 4. We leave it to the interested reader to show that inequality (5.22) follows from Claims 5.17 and 5.18, with one exception: the inequality fails for n = 4, d = 2, r = 1, λ = 2, μ = 1, and in fact, in this case the difference between the LHS and RHS of (5.22) is equal to −1. The failure of the inequality is due to the locally closed subset G ⊂ Quot 2 × Stack 4 consisting of points Y ) such that X 0 , Y 0 are generic, but Im X + Im Y has dimension 2. After a change of basis, points of G take the form with (s, t) = (0, 0) and y 1 x 2 = x 1 y 2 , y 1 x 4 + y 2 x 5 = x 1 y 4 + x 2 y 5 . It is easy to show that G is irreducible (compare it with Z 1 of Section 5.10). Moreover, G contains the point x 1 = · · · = x 5 = y 1 = · · · = y 5 = 1, for which it is elementary to show that dim Im μ ≥ 5. The fact that the latter inequality holds on G outside of codimension 1 is a consequence of lower semicontinuity of rank. This establishes the final sentence of Claim 5.16.

5.19.
Since the locally closed subsets Z def d 1 ,...,d k λ stratify the scheme Z λ , Corollary 5.14 implies that Let us now consider Z (x,y) , which parameterizes flags (F 0 ⊂ x F 1 ⊂ y F 2 ). It was shown in [28,Claim 4.22] that the singular locus of Z (x,y) is given by where split means that the length 2 sheaf F 2 /F 0 supported at x is a direct sum of length 1 skyscraper sheaves C x . We will now estimate the dimension of (5.24). Proof. -Consider the stratification into locally closed subsets where the d-th stratum parameterizes those flags as in (5.24) where F 2 has defect d at x. It suffices to show that To prove the inequality above, one follows the proof of Proposition 5.13 closely, but with X = Y = 0 throughout. In particular, (5.16) and (5.21) reduce the problem to the case r = d = 1, in which case Claim 5.16 is trivial. As for the statement of a single irreducible component of top dimension, this is because one only has equality in (5.25) for d = 0, and one argues as in the proof of Corollary 5.14.
6. Geometric properties 6.1. We will now use the dimension estimates from the previous Section to obtain various geometric properties of the varieties Z λ , which we have invoked in Section 2. Proof. -Since S is smooth and Z λ is Cohen-Macaulay, the Miracle Flatness Theorem asserts that all we need to show is that the fibers of the morphism p i S : Z λ → S all have dimension equal to dim Z λ − 2. By upper semicontinuity, it suffices to show that all the fibers have dimension ≤ dim Z λ − 2. This is proved in the same way as Corollary 5.8, since fixing one of the support points of the flag merely replaces the number k in the right-hand side of (5.5) by k − 2.
where : Z (x) → Z (x) × S is the graph of the map p S : Z (x) → S. The rightmost fiber square in (6.1) is already derived, because the maps ρ and ρ are smooth. However, the leftmost fiber square in (6.1) is not derived, because (this follows from the dimension statements of Propositions 2.26 and 2.27, which have already been proved). Recall from Proposition 2.19 that ι is cut out by the regular section hence is a complete intersection of codimension equal to the rank of W 1 . On the other hand, ι is cut out by the section In Proposition 2.21, we showed that the section above factors through where the left-hand side is a locally free sheaf of rank 1 less than the rank of W 1 . By (6.2), the section (6.3) is therefore regular, and this establishes the claim that the square (2.37) is derived with excess bundle Proof of Propositions 2.28, 2.29, 2.30. -See Corollary 5.20 for the dimension and irreducibility statements. The fact that the schemes in question are l.c.i. is proved by showing that the fiber squares in the statements of the Propositions are derived. The latter statement is an immediate consequence of Proposition 2.19 together with the fact that the spaces in the four corners of each square satisfy the analogue of (2.5) (itself an immediate consequence of the already proved fact that the northeast, southeast and southwest corners of these squares are l.c.i. of expected dimension, while the northwest corner has expected dimension, see Corollary 5.20).

6.3.
We will now prove that the variety Z (x,y,x) is Cohen-Macaulay.
Proof of Proposition 2.31. -See Corollary 5.20 for the dimension and irreducibility statements. As a warm-up to proving the fact that Z (x,y,x) is Cohen-Macaulay, let us recall the explicit realization of Z (x,y) as a local complete intersection. We do so not only because the computation will serve as valuable illustration, but we will need a specific realization of Z (x,y) as the zero locus of a regular section. Therefore, consider the smooth scheme Z (x) × S and take the map A particular case of Proposition 2.19 says that this map can be realized as where the closed embedding ι is cut out by the section and O(1) denotes the tautological line bundle on the projective bundle in (6.4). Above and hereafter, we abuse notation by writing W 1 for the vector bundle on Z (x) × S, as well as for its pull-back to P. The dimension of Z (x) × S is equal to and therefore the dimension of the projective bundle (6.4) is equal to To obtain the closed embedding ι in (6.4), we impose as many equations as the number of coordinates of the section (6.5), so the expected dimension of Z (x,y) is Since this is equal to the actual dimension of Z (x,y) by (5.23), we conclude that the coordinates of the section σ form a regular sequence (6.6) = regular sequence formed by the coordinates of σ In order to study in detail, recall that the vector bundle V 1 is given by (2.15): from now on we will set n = 0 in (2.15), in order to keep our formulas legible (otherwise, one would have to often tensor our formulas by line bundles coming from S, but this has no substantial effect on our argument). We have a map of locally free sheaves φ : V 1 → L 1 on Z (x) that is a particular case of the middle row of diagram (2.30) when i = 1. Putting all of these constructions together, we may consider the map of line bundles on P. Since Ker φ = V 0 is locally free, the map of line bundles φ • σ may be thought of as one of the coordinates of the section σ in any local trivialization. Therefore, it is one of the entries of the regular sequence , so let us compute it explicitly on a closed point of the projective bundle (6.4). We may work locally, so we assume that x and y are points of A 2 = Spec C[s 1 , s 2 ] (alternatively, we need to take generators of the maximal ideals at the closed points x and y, but the explanation is analogous) given by coordinates (x 1 , x 2 ) and (y 1 , y 2 ), respectively. As we have seen in the proof of Proposition 2.21, the map taut in (6.7) is identified with where y : P → P × S is the graph of the map that remembers the point y. The Tor groups above may be computed using the following length 2 resolution of O y : Therefore, the map denoted taut in (6.7) takes a generator v of the one-dimensional vector space (S, of v ∈ (S, F 2 /F 1 ) (global sections exist due to our assumption that n = 0 in (2.15), otherwise we would need to tensor F 1 and F 2 by a very ample line bundle). Therefore, the entire composition (6.7) takes the generator v to the image of ( In coordinates, if we identify (S, F 2 /F 0 ) with C 2 , then the O S = C[s 1 , s 2 ]-module structure on this 2-dimensional vector space is given by matrices (6.8) and so the assignment (6.7) takes Therefore, we conclude that is one of the elements of the regular sequence (6.6). Since over local Noetherian rings, one may permute the order of elements of a regular sequence, we will assume (6.9) to be the last element of the regular sequence.
Claim 6.4. -Before imposing Equation (6.9), the other elements of the regular sequence cut out a regular local ring. In more mathematical terms, the local rings of Z (x,y) are quotients of regular local rings by the single equation (6.9).
Indeed, the claim follows from the fact (proved in [28]) that the tangent spaces to Z (x,y) have expected dimension, except at a closed point such that x = y and F 2 /F 0 is split, where the dimension of the tangent space jumps by 1. In the local coordinates (6.9), this corresponds to a 1 = a 2 = x 1 − y 1 = x 2 − y 2 = 0: therefore, Equation (6.9) fails to cut down the dimension of the tangent spaces on the split locus. Since Z (x) is smooth, this means that all other elements in the regular sequence do cut out regular subschemes, and regularity is broken precisely by (6.9).
Armed with the discussion above, we are ready to analyze the scheme Z (x,y,x) : A particular case of Proposition 2.19 says that this map can be realized as (6.10) Z (x,y,x) where x : Z (x,y) → Z (x,y) × S is the graph of the map that remembers the point x, and the closed embedding ι is cut out by the section S ) (we abuse notation by identifying the vector bundles on Z (x,y) above with their pull-backs to P ) where O(1) denotes the tautological line bundle on P . By (5.23), the dimension of Z (x,y) is equal to const + r(c 2 (F 0 ) + c 2 (F 2 )) + 2 and therefore the dimension of the projective bundle (6.10) is equal to const + r(c 2 (F 0 ) + c 2 (F 2 )) + 1 + rank W 2 To obtain the closed embedding ι in (6.10), we impose as many equations as the number of coordinates of (6.11), so the expected dimension of Z (x,y,x) is However, according to (5.23), the actual dimension of Z (x,y,x) is 1 bigger than the above expected dimension. Therefore, Z (x,y,x) is an almost complete intersection, i.e. cut out by one more equation than its dimension. We have the following criterion for when such schemes are Cohen-Macaulay. The claim is proved by induction and the well-known fact that if an element f in a Cohen-Macaulay local ring R has the property that dim R/(f ) = dim R − 1, then f is a non-zero divisor and R/(f ) is Cohen-Macaulay. We will apply Claim 6.5 to our situation by constructing a map of locally free sheaves on P : (6.12) φ : x * (V 2 ⊗ ω −1 S ) N such that N has rank 2 and the closed subscheme is Cohen-Macaulay of codimension 1. Once we do so, the fact that Z (x,y,x) is Cohen-Macaulay of the expected dimension follows from Claim 6.5 with i = 1.
To construct the map (6.12), we go back to diagram (2.30) for i = 2. Its middle row consists of locally free sheaves, so we may restrict it to x : (the right-most term is L 2 because we assumed n = 0 in (2.15), otherwise we could have had to twist by a line bundle). The push-out of the short exact sequence above with respect to the tautological map x * (V 1 ) L 1 that we have on Z (x,y) yields where N is defined by the short exact sequence on the bottom row. This defines the map (6.12). We will now write out the entries of the map φ explicitly in coordinates, just as we did in the discussion immediately preceding Claim 6.4. We still work locally, so assume that x and y are points of A 2 = Spec C[s 1 , s 2 ] given by coordinates (x 1 , x 2 ) and (y 1 , y 2 ), respectively. Then we identify (S, F 3 /F 0 ) with C 3 , and then the O S = C[s 1 , s 2 ]-module structure on this 3-dimensional vector space is given by matrices (6.14) The matrices (6.8) are precisely the bottom right 2 × 2 blocks of (6.14). Therefore, the composition φ • σ : O(−1) → N is given explicitly in coordinates by ⎛ Therefore, we conclude that the subscheme (6.13) is obtained by imposing the equations b 1 (x 2 − y 2 ) = b 2 (x 1 − y 1 ) and b 1 a 2 = a 1 b 2 on the local rings of a projective bundle over the scheme Z (x,y) . As we have seen in Claim 6.4, the local rings of the scheme Z (x,y) were obtained by imposing the equation a 1 (x 2 − y 2 ) = a 2 (x 1 − y 1 ) of (6.9) in a regular local ring. Therefore, we are in a particular case of the following general situation.
Claim 6.6. -Suppose R is a regular ring with given elements a 1 , b 1 , a 2 , b 2 , d 1 , d 2 , and let Using Claim 6.4, let us show that the hypotheses of Claim 6.6 hold with a 1 , b 1 , a 2 , b 2 as in (6.14) and d 1 = x 1 − y 1 , d 2 = x 2 − y 2 . For item (1), we observe that imposing d 1 = d 2 = 0 has the effect of setting the support points x, y equal to each other, which cuts out the smooth subscheme Z (x,x) → Z (x,y) . Since P | Z (x,x) is smooth, it does not have any non-trivial zero-divisors, so must show that in the local rings of P | Z (x,x) . If (6.15) failed to hold, then # equations cutting out ι| Z (x,x) ≤ # equations cutting out ι − 1 and so which would contradict (5.23). As for item (2), we must show that b 1 d 2 − b 2 d 1 is not a zero-divisor in the local rings of P . This is the case because b 1 , b 2 are linear coordinates in a projective bundle over Z (x,y) , so the only way b 1 d 2 − b 2 d 1 could be a zero divisor is if d 1 and d 2 were zero-divisors. If this were the case, then Z (x,x) would have the same dimension as Z (x,y) , which would contradict (5.23).
Proof of Claim 6.6. -By the Auslander-Buchsbaum formula, it is enough to show that the ring R/I has projective dimension 2 as an R-module. In fact, we claim that a projective resolution is given by Exactness at R and R/I is obvious. As for exactness at R 2 , if the first map failed to be injective then a 1 b 2 − a 2 b 1 , d 1 b 2 − b 1 d 2 and d 1 a 2 − a 1 d 2 would all be zero-divisors (hence zero in the regular ring R) which is not allowed by property (1). It remains to prove exactness at R 3 . Assume that we have (x, y, z) ∈ R 3 such that , so property (1) implies z = d 1 m + d 2 n. Therefore, we may rewrite the relation above as n and y = y + a 1 m + a 2 n. The relation above reduces to and then item (2) implies that we have which shows that the triple (x, y, z) came from the image of the 3 × 2 matrix.

6.7.
We will now prove the normality of some of the schemes Z λ for |λ| ≤ 4.
Proof of Proposition 2.34. -Since all the schemes in question are Cohen-Macaulay, it suffices to show that they are singular in codimension ≥ 2. Let us note that the required statement for λ = (x, y) follows from Proposition 5.21, which shows that the singular locus Z split (x,x) ⊂ Z (x,y) has codimension 3. As for the other λ's in (2.39), we will only prove the case λ = (x, x, y, x), as the analysis in the other cases is analogous and no more difficult. Below, we will list certain locally closed subsets of (6.16) which form a stratification of the variety Z (x,x,y,x) . To show normality of this variety, one may ignore all strata of codimension 2 and higher.
(1) when x = y and F 4 has non-zero defect > 0 at x, Corollary 5.14 shows that the corresponding locally closed subset of (6.16) has codimension ≥ 2, hence can be ignored; (2) when x = y and F 4 is locally free near x, the scheme Z (x,x,y,x) is locally isomor- Compare with the scheme Quot 0,1,2,3,4 of Definition 5.6. We must prove that is normal. Since we may work locally, we assume that the base surface is S = A 2 , and we will normalize x = (0, 0) and y = (a, b). By analogy with Section 5.9, the scheme parameterizes triples (X, Y, v) where where (∂ X , ∂ Y ) runs over the 14-dimensional affine space of pairs of matrices with the same pattern of zeroes as (X, Y). Thus, is smooth at a point (X, Y) iff μ has 10-dimensional kernel, so let us see when this happens.
Assume first that (a, b) = (0, 0), and by taking appropriate linear combinations of X, Y we may assume (a, b) = (1, 0). Then one can successively solve the equation Ker μ = 0 for the entries of the matrices (∂ X , ∂ Y ) one by one, concluding that the kernel of μ fails to be 10-dimensional when x 43 = y 21 = y 31 = y 32 = y 42 = y 42 = x 31 − x 32 x 21 = 0. The dimension of this locus is 5, and after we add 2 dimensions to reverse the choice (a, b) = (1, 0), we conclude that is smooth in codimension 3 on the locus (a, b) = (0, 0). Now assume (a, b) = (0, 0). The corresponding subvariety is precisely Comm 4 ⊂ , and we have seen at the end of Section 5.10 that it is 9dimensional and has two irreducible components Z 1 and Z 2 . Thus, it suffices to take a generic point (X, Y) in each of these components, and show that the map μ has 10-dimensional kernel at the chosen point. It is straightforward to show that the following choices will do: (3) when x = y and F 4 has non-zero defect at both x and y, Corollary 5.14 shows that the corresponding locally closed subset has codimension ≥ 2, hence can be ignored; (4) when x = y and F 4 is locally free near x, the scheme Z λ is locally isomorphic to Z 1 × Quot 0,1,2,3 , and thus normal (indeed, Quot 0,1,2,3 is open in an affine bundle over Stack 3 , and the latter is easily seen to be normal); (5) when x = y and F 4 has non-zero defect at x but is locally free near y, the scheme Z λ is locally isomorphic to Z (x,x,x) × Quot 1 , so it suffices to show that Z (x,x,x) is normal near any point with defect ≥ 1. To this end, consider the stratification in terms of the defect of F 4 at x. As shown in Corollary 5.14, the open subset Z def 0 (x,x,x) is the only top-dimensional stratum; as it is locally isomorphic to M × Quot 0,1,2,3 , it is normal. Tracing through the proof of Proposition 5.13 shows that other strata can have codimension 1 only if r = 1 and d = 1, in which case the moduli space of stable sheaves may be replaced with the Hilbert scheme of points on S. Therefore, it suffices to show that the scheme parameterizing flags of ideals (I 0 ⊂ x I 1 ⊂ x I 2 ⊂ x I 3 ) is normal near any ideal I 3 of defect precisely 1 at x. Since the problem is local, we may assume S = A 2 and x = (0, 0), in which case may be described by analogy with Section 5.9 as the space of triples (X, Y, v) such that and v ∈ C 4 is cyclic for X, Y. As before, one may explicitly write down the quadratic equations among the {x ij , y ij , a, b} 4≥i>j≥1 and conclude that is normal. The method of proof is analogous to that in item (2) above, so we leave it as an analogous exercise to the interested reader.

6.8.
We will now study the schemes Y, Y − , Y + , Y −+ of Section 2.35. Recall that when we say that Y (or any of the other 3 schemes) has # irreducible components of expected dimension, what we actually mean is that it has # such irreducible components over each connected component of the moduli space M. ,y) , the fiber of this map is either a single point, or a copy of P 1 . The latter happens if and only if x = y and F 2 /F 0 is split, so we conclude that the only points where the fibers jump are those of Z split (x,x) . Since the locus Z split (x,x) has codimension 3 in Z (x,y) (see Proposition 5.21), and the dimensions of the fibers above such points are all 1, this implies that Y will have the same dimension as Z 2 (in Section 6.11, we will show that Y is actually smooth, so the map π ↑ can be thought of as the blow-up of the singular locus of Z (x,y) ). The fact that Y has a single irreducible component of top dimension is clear, since Z (x,y) is irreducible.
The proof for the schemes Y − , Y + , Y −+ is analogous. For example, the fibers of consist of a single point or a copy of P 1 , with the latter situation only happening over closed points Since the locus of such points is contained in Z (x,x,x,x) , which has dimension 1 less than Z (x,x,y,x) by Propositions 2.32 and 2.33, the dimension of Y −+ is the same as that of Z (x,x,y,x) . As for the irreducible components of top dimension in Y −+ , we note that one of them is the closure of the locus x = y. But recall that where V 1 and V 2 lie above the irreducible components Z 1 and Z 2 of Stack 4 (see items (1)-(2) in Section 5.10). The map π ↑ has inverse image a single point over the generic point of V 2 . However, over any point of V 1 we have F 3 /F 1 split, and so there exists a whole P 1 in Y −+ above points of V 1 . As dim V 1 = dim Z (x,x,x,x) = dim Y −+ − 1, this contributes an irreducible component of top dimension to Y −+ .

6.10.
In the next subsection, we will prove that the scheme Y is smooth. To do so, we will explicitly describe the tangent space to a closed point (2.40) of Y and compute its dimension. Let us recall that the tangent space to the moduli space M at a point F ∈ Coh(S) is given by (6.17) T a n F M = Ext 1 (F , F) Indeed, the functor-of-points description (2.12) implies that a tangent vector at F ∈ M is a coherent sheaf on S × Spec C[ν]/(ν 2 ) which is flat over the second factor and restricts to F when one sets ν = 0. In other words, a tangent vector is a coherent sheaf G on S with an morphism ν : G → G that squares to 0, such that is a short exact sequence, which precisely gives rise to an element of Ext 1 (F , F). The moduli space M is smooth precisely when the dimensions of the tangent spaces (6.17) are locally constant in F (for a more rigorous presentation of the smoothness of the moduli space via obstruction theory, we refer the reader to [18]). We have Since stable sheaves are simple (see [18]), we have dim Hom(F , F) = 1 because Hom(F , F) ∼ = C (6.20) where the latter isomorphism is Serre duality, and the number ε is 1 or 0 depending on which situation of Assumption S we are in (ε = 1 for ω S ∼ = O S and ε = 0 for c 1 (ω S ) · H < 0). Therefore, we conclude that where χ(F , F) = −γ − 2rc 2 can be computed using the Hirzebruch-Riemann-Roch theorem, and the constant γ only depends on S, H, r, c 1 .
6.11. Following [27, relation (2.23)], the tangent space to Z 1 at a closed point is the vector space of pairs of the form: where the arrow is the difference of the two natural maps induced by the inclusion F 0 ⊂ F 1 . These maps fit into the diagram below with exact rows and columns: where we write F 1 /F 0 = C x for the skyscraper sheaf at the closed point x ∈ S. The dimensions of the Ext spaces in the diagram above may be computed as in the previous subsection: where c first 2 and c last 2 are the second Chern classes of the sheaves denoted by F 0 and F 1 , respectively. The fact that the kernel of ψ in (6.23) has the expected dimension 1 + ε + γ + r(c first 2 + c last 2 ) + 1 then follows from the elementary facts below: (1) the image of ψ coincides with the kernel of the dotted arrow (2) the target Ext 2 (C x , C x ) of the dotted arrow is 1 dimensional (3) the dotted arrow is non-zero if and only if ε = 0 These facts were proved in [27]. As shown in loc. cit., a pair as in (6.23) contains precisely the same information as a commutative diagram with exact rows: In this language, the differential of the map p S : Z 1 → S is given by where we use the fact that F 1 /F 0 ∼ = C x and the fact that there exists a canonical isomorphism Tan x S = Ext 1 (C x , C x ). A diagram in the kernel of dp S is one in which the extension G 1 /G 0 splits, which precisely means that the diagram (6.25) allows one to insert an extra row, as follows: The ability to insert the middle row into the diagram above is equivalent to saying that the pair (6.23) comes from one and the same element in the vector space Ext 1 (F 1 , F 0 ) situated in the top left corner of diagram (6.24). We conclude that (6.27) E x t 1 (F 1 , F 0 ) Ker dp S is a surjective map. Since the dimension of Ext 1 (F 1 , F 0 ) is 2 less than that of the tangent space to Z 1 , this implies that the map dp S is surjective.
Proof of Proposition 2.41. -Since we already proved the dimension and irreducibility statements in Proposition 6.9, it remains to prove that Y is smooth. By analogy with the discussion above, we showed in [28, relation (4.37)] that the tangent space to Z 2 at a closed point (F 0 ⊂ x F 1 ⊂ y F 2 ) consists of triples of the form where the arrow is the alternating sum of the four natural maps induced by the inclusions In loc. cit., we also showed that dim space of triples (6.28 and c last 2 are the second Chern classes of the sheaves denoted by F 0 and F 2 , respectively (meanwhile, the Kronecker δ symbol is 1 if x = y and F 2 /F 0 is split, i.e. ∼ = C ⊕2 x , and 0 otherwise). We conclude that the dimensions of the tangent spaces to Z 2 jump by 1 precisely on the split locus. The differential of the map admits a presentation analogous to (6.26). It was shown in [28] that the differential dp 1 S × dp 2 S is surjective if and only if either x = y or x = y and F 2 /F 0 is split. By combining the discussion above with the moduli functor presentation of the scheme Y, we see that Tan (F 0 ⊂F 1 ,F 1 ⊂F 2 ) Y is the space of quadruples (w 0 , w 1 , w 1 , w 2 ) ∈ Ext 1 (F 0 , F 0 ) ⊕ Ext 1 (F 1 , F 1 ) ⊕ Ext 1 (F 1 , F 1 )⊕ ⊕ Ext 1 (F 2 , F 2 ) which satisfy the four properties below: (1) w 0 and w 1 (or w 1 ) map to the same element of Ext 1 (F 0 , F 1 ) (or Ext 1 (F 0 , F 1 )) (2) w 1 (or w 1 ) and w 2 map to the same element of Ext 1 (F 1 , F 2 ) (or Ext 1 (F 1 , F 2 )) (3) dp S (w 0 , w 1 ) = dp S (w 1 , w 2 ) ∈ Ext 1 (C x , C x ) (4) dp S (w 0 , w 1 ) = dp S (w 1 , w 2 ) ∈ Ext 1 (C y , C y ) By analogy with (6.23) and (6.24), consider the vector space (6.30) A= Ker Ext 1 (F 0 , F 0 ) ⊕ Ext 1 (F 2 , where ψ is the difference of the two natural maps in the diagram below with exact rows and columns: (here F 2 /F 0 = Q is a length 2 sheaf which is filtered by C x and C y ). By analogy with our analysis of (6.24), it is easy to show that the image of the map ψ in (6.30) coincides with the kernel of the dotted arrow, and so (6.32) d i mA= 1 + ε + γ + r(c first 2 + c last 2 ) + dim Ext 2 (Q, Q) The dimension of Ext 2 (Q, Q) is 4 if x = y and Q is split, and 2 otherwise. Items (1)-(4) above imply that we have a Cartesian diagram of vector spaces: (b,dp 2 S ,dp 1 S ) (a,dp 1 S ,dp 2 S ) A ⊕ Tan x S ⊕ Tan y S where the maps a, a forget w 1 and the maps b, b forget w 1 .
-The map a is injective, unless x = y in which case Ker a is one-dimensional and spanned by (0, w 1 , 0), where w 1 represents the following extension: (the middle space requires fixing isomorphisms F 2 /F 1 ∼ = F 1 /F 0 ∼ = C x ). The image of the extension (6.34) under dp 1 S × dp 2 S is equal to (v, v) ∈ Tan x S ⊕ Tan x S, where v ∈ Ext 1 (C x , C x ) is the class of the extension 0 → F 1 /F 0 → F 2 /F 0 → F 2 /F 1 → 0.
We will first show how Claim 6.12 allows us to prove that all tangent spaces to Y have dimension ≤ than = 1 + ε + γ + r(c first 2 + c last 2 ) + 2 (which would conclude the proof of Proposition 2.41) and then prove the claim. Case 1: when x = y or x = y and F 2 /F 0 is not split (i.e. we are at a smooth point of Z 2 ), Claim 6.12 implies that (a, dp 1 S , dp 2 S ) is injective. Then the fact that diagram (6.33) is Cartesian implies that the map a is also injective, which implies Because of Proposition 6.9 and the fact that we are on the smooth locus of Z 2 , the dimensions of the two tangent spaces must be equal.
Case 2: when x = y and F 2 /F 0 is split, the dimensions of the vector spaces in (6.33) are ? a b C d+1 (b,dp 2 S ,dp 1 S ) C d+1 (a,dp 1 S ,dp 2 S ) where d is the number in the right-hand side of (6.35). The goal is to show that the Cartesian product of the diagram, namely the vector space ?, has dimension d. If F 1 = F 1 , then the northeast and southwest corners are naturally identified, as are the maps a and b. By Claim 6.12, we can decompose C d+1 = C ⊕ C d , where a(C) = 0 and a| C d is injective. Moreover, dp 1 S (C) = dp 2 S (C) = 0, which implies that dp 1 S × dp 2 S | C d is surjective (this follows from the surjectivity of dp 1 S × dp 2 S , proved in [28, relation (4.47)]). We conclude that a point of ? is of the form (l, l , v) ∈ C ⊕ C ⊕ C d such that dp 1 S (v) = dp 2 S (v). The latter equality imposes two non-trivial linear conditions on v, so we conclude that the dimension of ? is 1 + 1 + d − 2 = d.
If F 1 = F 1 , then it suffices to prove that Im a and Im b are transversal ddimensional subspaces of A. Consider a point of A given by a pair of extensions The diagram above induces an extension at the level of quotients (6.37) 0→ Q → H → Q → 0 where Q = F 2 /F 0 and H = G 2 /G 0 . The pair of extensions (6.36) lies in Im a iff (6.38) H has a length 2 subscheme compatible with F 1 /F 0 ⊂ Q (and similarly for Im b, if we replace F 1 by F 1 ). Fix a vector space isomorphism F 2 /F 0 ∼ = C 2 with respect to which F 1 /F 0 is the first standard coordinate line and F 1 /F 0 is the second coordinate line. The rank 4 coherent sheaf H is determined on the local neighborhood of x ∈ S by two commuting 4 × 4 matrices X and Y, whose only non-zero entries are allowed to be in the bottom left 2 × 2 block, as below: is surjective. To this end, note that the dimension of A is given by (6.32), dim Ext 2 (Q, Q) = 4, dim Ext 1 (Q, Q) = 8. Therefore, it suffices to show that dim Ker α ≤ 1 + ε + γ + r(c first 2 + c last 2 ) − 4 By analogy with (6.27), we have a surjective map

Ker α
A simple application of Hirzebruch-Riemann-Roch shows that dim Ext 1 (F 2 , F 0 ) = ε + γ + r(c first 2 + c last 2 ). Therefore, it suffices to show that the kernel of the map β has dimension ≥ 3. To this end, consider the following piece of the Ext long exact sequence corresponding to the short exact sequence 0 → F 0 → F 2 → Q → 0: Consider the 4-dimensional subspace V ⊂ Hom(F 2 , Q) which consists of a fixed homomorphism with kernel F 0 , composed with an arbitrary endomorphism of Q ∼ = C 2 x . Any element of V maps to 0 in both Ext 1 (F 0 , F 0 ) and Ext 1 (F 2 , F 2 ), so ρ(V) lies inside the kernel of β. However, the map ρ has a 1-dimensional kernel since Hom(F 2 , F 2 ) = C, so we conclude that Ker β has dimension at least 3.
Proof of Claim 6.12. -It is enough to show that any triple (0, w 1 , 0) satisfying 0 and w 1 map to the same element in Ext 1 (F 0 , F 1 ) w 1 and 0 map to the same element in Ext 1 (F 1 , F 2 ) must have w 1 equal to a multiple of the extension (6.34). Indeed, the natural long exact sequences imply that it suffices to show that any w 1 ∈ Ext 1 (F 1 , F 1 ) which lies in the intersection of the images of Ext 1 (C x , F 1 ) and Hom(F 1 , C y ) is a multiple of (6.34), where C x = F 1 /F 0 and C y = F 2 /F 1 . In other words, if we have a diagram (the maps denoted "taut" are the projection maps F 1 C x and F 2 C y that give rise to the flag F 0 ⊂ F 1 ⊂ F 2 ) where the middle short exact sequence is the pull-back of both the top and the bottom short exact sequences, we must show that the middle short exact sequence is a multiple of (6.34). As the sheaves F 0 , F 1 , F 2 , H all have the same reflexive hull, and since the reflexive hull is stable, we may regard F 0 , F 1 , F 2 , H as subsheaves of the same stable vector bundle V.
(1) If H = F 2 (as subsheaves of V), then the two sides of the inclusion have the same colength as subsheaves of V, so the inclusion above is an equality. Similarly, the two sides of the inclusion have the same colength as subsheaves of V ⊕ V, so the inclusion above is an equality. But then the short exact sequence 0 → F 1 → G → F 1 → 0 is split, so w 1 = 0. (2) if H = F 2 (as subsheaves of V), then x = y. Since Hom(F 1 , F 2 ) is one dimensional, any two injections F 1 → F 2 are scalar multiples of each other, which implies that β = λ · taut for some λ ∈ C × (notations as in (6.39)). Then the extension G is equal to λ times the extension (6.34), as we needed to show.

6.13.
Let us now consider the schemes Y − , Y + , Y −+ in relation to Y.
Proof of Propositions 2.42 and 2.43. -Since we already proved the dimension and irreducibility statements in Proposition 6.9, it remains to prove that Y − , Y + and Y −+ are l.c.i. This is a consequence of the claim that the following squares are all derived, which we will now prove: In all cases above, the arrow on the left is the only map one can write which forgets a single sheaf (in the notation of (2.40)-(2.43)), while the arrow on the right is the unique map which forgets the same sheaf as the arrow on the left. We will only prove the fact that the first square in (6.40) is derived, since all other cases are analogous. Consider the map on the left of the square (6.42) Y + −→ Y With the notation in (2.40) and (2.42), we note that the fibers of this map consist of all ways to append a sheaf F −1 ⊂ x F 0 to diagram (2.40). Just like in Proposition 2.19, one sees that the map (6.42) factors as Y where x : Y → Y × S is the graph of the map p x S that records the support point x ∈ S. The closed embedding ι is cut out by the composition (1) and just like in Proposition 2.21, one may show that the section σ factors through a locally free sheaf of rank 1 less, as follows: (the argument requires the fact that p x S is flat, which is proved by estimating the dimensions of its fibers, akin to the proof of Proposition 6.2). Because of (2.2), we obtain: However, Proposition 2.41 implies that Y is smooth, while Proposition 6.9 implies that dim Y + = dim Y + r. Therefore, we actually have equality in the inequality above. This is a particular case of Definition 2.2, hence the section σ is regular. However, this is precisely the same section that describes the map Z (x,x) → Z (x) . By Definition 2.4, this precisely says that the first fiber square in (6.40) is derived.
Proof of Proposition 2.44. -The scheme Y is reduced because it is smooth. As for the other schemes, they are local complete intersections, so it suffices to prove that their generic points are reduced. In the case of Y − and Y + , they are irreducible, and the generic point corresponds to a diagram (2.41)-(2.42) with x = y. Near such a point, Y − and Y + are isomorphic to Z (y,x,x) and Z (x,x,y) , respectively. Since the latter schemes are normal (due to Proposition 2.34), reducedness follows.
The same argument applies to the irreducible component of Y −+ which is the closure of the locus of diagrams (2.43) with x = y. As for the other irreducible component, we recall that it corresponds to diagrams (2.43) with x = y and F 3 /F 1 a split length 2 sheaf. Therefore, the second component is locally isomorphic to V 1 × P 1 where V 1 ⊂ Z (x,x,x,x) is the irreducible component consisting of such that F 3 /F 1 is a split length 2 sheaf. It suffices to show that V 1 is generically reduced. As a consequence of Corollary 5.14, the generic point of V 1 corresponds to F 4 locally free. Lemma 5.7 implies that near such a point, V 1 is locally isomorphic to the smooth moduli space M 4 = {F 4 } times the component Z 1 /B ⊂ Comm 4 /B that we studied in Section 5.10. As we noted therein, Z 1 is generically reduced, so we are done.