A CATEGORICAL QUANTUM TOROIDAL ACTION ON THE HILBERT SCHEMES

Abstract We categorify the commutation of Nakajima’s Heisenberg operators 
$P_{\pm 1}$
 and their infinitely many counterparts in the quantum toroidal algebra 
$U_{q_1,q_2}(\ddot {gl_1})$
 acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical 
$U_{q_1,q_2}(\ddot {gl_1})$
 action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

Given a smooth quasi-projective surface S over k = C, let S [n]   be the Hilbert schemes of points on S. Schiffmann-Vasserot [34], Feigin-Tsymbaliuk [11] and Negut ¸ [29] constructed the U q1,q2 ( gl 1 ) action on the Grothendieck group of M. It generalizes the action of • the Heisenberg algebra (Nakajima [25] and Grojnowski [12]) • the W algebra (Li-Qin-Wang [22]) on the cohomology of Hilbert schemes.There are already some applications of this action in algebraic geometry, like the Beauville-Voison conjecture for the Hilbert schemes of points on K3 surfaces [23].The purpose of this paper is to categorify the above quantum toroidal algebra action.We prove that Date: November 6, 2021.
where S [n,n+1] is the nested Hilbert scheme S [n,n+1] := {(I n , I n+1 , x) ∈ S [n] × S [n+1] × S|I n+1 ⊂ I n , I n /I n+1 = k x } and the line bundle L on S [n,n+1] has fibers equal to I n /I n+1 .Then (1) For every two integers m and r, there exists natural transformations if m < 0 f r e −r = e r f −r ⊕ O ∆ [1] where ∆ is the diagonal of M × M × S × S.
(2) When m = 0, the cone of the natural transformations in (1.1) has a filtration with associated graded object where h + m,k , h − m,k ∈ D b (M × S) are defined in Definition 3.1 and are complexes of universal sheaves on M × S.
(3) (Theorem 3.2) At the level of Grothendieck groups, we have the formula: where ω S is the canonical line bundle of S and h ± m is defined in [26] (also see Definition 2.8) .
Note that the non-triviality of the extension is a feature of the derived category statement, which is not visible at the level of Grothendieck groups.Proposition 7.1 provides a precise extension formula.
The positive part of the quantum toroidal algebra action was already categorified in [29].Theorem 1.1 categorifies the commutation of the positive part and the negative part, and thus accounts for the action of the whole quantum toroidal algebra.
Remark 1.2.We have not categorified all the relations defining the elliptic Hall algebra of [2] (see [29] for more discussions).
Remark 1.3.Our categorification is a geometric categorification in the sense of [3].The 2-categorification of the quantum toroidal algebra is still unclear to us.1.2.Outline of The Proof.Let us first review the categorification of the commutation of e k and e l for different k and l in [29].Consider the moduli space Z 2 , Z ′ 2 which parameterize diagrams (1.2) x y (1.3) respectively, of ideal sheaves where each successive inclusion is colength 1 and supported at the point indicated on the diagrams.Then e k e l and e l e k are the derived pushfoward of line bundles on Z 2 and Z ′ 2 to S [n−1] × S [n+1] × S × S respectively.In order to compare e k e l and e l e k , [29] introduced the quadruple moduli space Y which parameterize diagrams (1.4) of ideal sheaves where each successive inclusion is colength 1 and supported at the point indicated on the diagrams.Z + is smooth and induces resolutions of Z 2 and Z ′ 2 .Proposition 2.30 of [29] proved that Z 2 and Z ′ 2 are rational singularities, based on the fact that any fiber of the resolution has dimension ≤ 1.Thus e k e l and e l e k could be compared through line bundles on Y. Now in order to compare f r e m−r and e m−r f r , we introduce the triple moduli spaces Z + , Z − which parameterize diagrams (1.5) Then f r e m−r and e m−r f r are the derived push forward of line bundles on Z + , Z − respectively.The quadruple moduli space Y still induces resolutions of Z − , but Z + have two irreducible components.One irreducible component is S [n,n+1] , denoted by W 1 and the other irreducible component is denoted by W 0 in Section 7. Y induces a resolution of W 0 .
In order to compare f r e m−r and e m−r f r through line bundles on Y, one must prove that Z − and W 0 are rational singularities.The approach in [29] did not work here, as the fiber could have pretty large dimensions.Instead, we study the singularity structure of Z + and Z − through the viewpoint of the minimal model program (MMP) [19,18].We prove that Proposition 1.4 (Proposition 5.6 and Proposition 5.13).The pair (Z + , 0) is semidlt.Z − and W 0 are canonical singularities.
We prove Proposition 1.4 by explicitly computing the discrepancy (see Section 7 for the definitions of semi-dlt, canonical singularities and the discrepancy).Canonical singularities are always rational singularities by Theorem 4.7.
1.3.Higher Rank Stable Sheaves and the Deformed W-algebra.The quantum toroidal algebra also acts on the Grothendieck group of higher rank stable sheaves [29].The action factors through the deformed W -algebra [28], which leads to the AGT correspondences for algebraic surfaces.
We will pursue the categorification of the above algebra action in the future.Different to the case of Hilbert schemes, Z + is no longer equi-dimensional and the obstruction bundle has to be accounted for.1.4.Categorical Heisenberg Actions.Khovanov [17] defined the Heisenberg category through graphical calculus.Cautis-Licata [5] constructed a categorical Heisenberg action on the derived category of Hilbert schemes of points of the minimal resolution of the type ADE singularities.Krug [21] constructed the weak categorical Heisenberg action on the derived category of Hilbert schemes of points on smooth surfaces.Our categorification is different from those above, as it is given in terms of explicit correspondences and independent of the derived McKay correspondence.
Although the higher Nakajima operators were categorified by the objects e (0,...,0) of [29], the relations between them (as well as the morphisms between them in Khovanov's Heisenberg category) are still unclear to us.1.5.Double Categorified Hall Algebra.The study of Cohomological Hall algebra was initiated by Kontsevich-Soibelman [20] and Schiffmann-Vasserot [32].and the author [39] constructed the K-theoretic Hall algebra on surfaces, which was categorified by Porta-Sala [30].It also categorified the positive half of U q ( gl 1 ) when S = A 2 .The relation between the categorified Hall algebra of minimal resolution of type A singularities and quivers was studied by Diaconescu-Porta-Sala [6].On the other hand, the Drinfeld double of the categorified Hall algebra is still mysterious.As an attempt to understand the action of the "double Categorified Hall algebra", it is natural to expect that our approach could be generalized to categorifications in other settings, like those of Toda [38] and Rapcak-Soibelman-Yang-Zhao [31].
1.6.Other Related Work.Recently, Addington-Takahashi [37] studied certain sequences of moduli spaces of sheaves on K3 surfaces and show that these sequences can be given the structure of a geometric categorical sl 2 action in the sense of [3].It would be interesting to explore the interactions between their action and ours.
Another related work is Jiang-Leung's projectivization formula [15].Through this formula, they obtained a semi-orthogonal decomposition of the derived category of the nested Hilbert schemes.1.7.The Organization of The Paper.The proof of the main theorem is in Section 6 and the extension formula is in Section 7. The other sections are organized as follows: Section 2: we review the action of U q1,q2 ( gl 1 ) on the Grothendieck group of Hilbert schemes [11,34,26,29]; Section 3: we define h + m,k ∈ D b (M × S) and prove the third part of Theorem 1.1; Section 4: we make a review of the singularity of the minimal model program.Section 5: we study the singularity structures of Z − and Z + through the singularity theory of the minimal model program.2. The Quantum Toroidal Algebra U q1,q2 ( gl 1 ) and the K-theory of Hilbert scheme of points on surfaces In this section, we will review the action of U q1,q2 ( gl 1 ) on the K-theory on Hilbert scheme of points on surfaces from [11,34,26,29].It will be formulated in Theorem 2.10.
In this paper, we will denote K(X) the Grothendieck group of coherent sheaves on X for a scheme X.
2.1.Hilbert Schemes and Nested Hilbert Schemes.Given an integer n > 0 and a smooth quasi-projective surface S over k = C, let S [n] := {I n ⊂ O|O/I n is dimension 0 and length n} be the Hilbert scheme of n points on S.There is a universal ideal sheaf on S [n] × S, still denoted by I n , and a universal closed subscheme Z n ⊂ S [n] × S. ' Proposition 2.1 (Proposition 2.11 of [29]).There exists a resolution of I n by where W n and V n are locally free coherent sheaves with the same determinant.Let w n and v n be the rank of W n and V n , respectively.Then v n − w n = 1.
Definition 2.2.The nested Hilbert scheme S [n,n+1] is defined to be with natural projection maps (2.2) and let (2.4) which consists of There are two tautological line bundles L 1 , L 2 on S [n−1,n,n+1] whose fibers are I n /I n+1 , I n−1 /I n respectively.We denote the projection morphism.
We define to be the Hilbert schemes of points on S.
Example 2.4.Let ∆ S : S → S × S be the diagonal embedding and I ∆S be the ideal sheaf of the diagonal.Then S [1,2] = Bl ∆S (S × S) = P S×S (I ∆S ) S [2] = Bl ∆S (S × S)/Z 2 where the Z 2 action on Bl S×S (I ∆S ) is induced by the Z 2 action i : S × S → S × S i(x, y) = (y, x).
By [35], the projection morphism The Quantum Toroidal algebra U q1,q2 ( gl 1 ).We follow the notation of [29] for the definition of the quantum toroidal algebra U q1,q2 ( gl 1 ).Given two formal parameters q 1 and q 2 , let q = q 1 q 2 .Let where Sym refers to polynomials which are symmetric in q 1 and q 2 .
Definition 2.5.The quantum toroidal algebra U q1,q2 ( gl 1 ) is the K-algebra with generators: together with the opposite relations for F (z) instead of E(z), as well as: where (2.9) We will set H + 0 = q and H − 0 = 1.2.3.The Quantum Toroidal Algebra Action on the K-theory of Hilbert Schemes.We will write S 1 , S 2 for two copies of S, in order to emphasize the factors of S × S. Definition 2.6 (Definition 4.10 and Definition 4.11 of [29]).For any group homomorphisms x, y : K(M) → K(M × S), we define: where ∆ S : S → S × S is the diagonal embedding.Also define: The definition is unambiguous, since ∆ S * : K(S) → K(S × S) is injective, and so z is unique.
Definition 2.7.For a two term complex of locally free sheaves as elements in K(X) and define det(U ) := det(V ) det(W ) .We define U ∨ to be the two term complex as elements in K(S [n] × S).Here we abuse the notation to denote in the short exact sequence (2.1).
Remark 2.9.Definition 2.8 is equivalent to the definition of h ± m in [26].We will prove it in Appendix A.
Theorem 2.10 (Theorem 1.2 of [26]).Let T * S be the cotangent bundle of S and ω S be the canonical bundle of S. The morphism: Regarding through the K-theoretic correspondences, where we regard S [n] × S as a closed subscheme of S [n] × S [n] × S through the diagonal embedding, then there exists a unique K-homomorphism For all x, y ∈ U q1,q2 ( gl 1 ), we have ).

Nested Hilbert Schemes and h
Given any scheme X, let D b (X) be the bounded derived category of coherent sheaves on X.In this section, we will define objects Theorem 3.2.At the level of Grothendieck groups, and (3.1) holds.
Theorem 3.2 will be proved later in this section.

3.1.
Projectivization and A Categorical Projection Lemma.In this paper, for any complex 0} we will assume that C 0 has cohomological degree 0 unless explicitly pointing out the cohomological degree.Definition 3.3.Given a two term complex of locally free sheaves U := {W s − → V }, we define the symmetric product and the wedge product complexes: − → V } be a two term complex of locally free sheaves over a scheme X such that W has rank w and V has rank v. Let Z ⊂ P X (V ) be the closed subscheme such that O Z is the cokernel of the composition of morphisms where ρ : P X (V ) → X is the projection morphism.We define Z to be the projectivization of U over X, denoted by When Z is a projectivization of U over X, we have a categorical projection lemma for Rρ * (O Z (k)): Lemma 3.5 (Categorical Projection Lemma).If Z is the projectivization of U over X in Definition 3.4, then the tensor contraction induces a morphism of complexes and Consider the following two complexes Then the morphism By Exercise III.8.4 of Hartshorne [13],

and thus
3.2.Nested Hilbert Schemes as Projectivization.Recall the short exact sequence (2.1): 0 → W n sn −→ V n → I n → 0. Nested Hilbert schemes can be realized as projectivizations, as in the following Propositions: Proposition 3.6 (Proposition 2.2 of [8]).The nested Hilbert scheme S [n,n+1] is the blow up of Z n in S [n] × S: and is smooth of dimension 2n + 2.Moreover, S [n,n+1] is the projectivization of Corollary 3.7.L is the exceptional divisor of S [n,n+1] as the blow up of Z n , i.e. we have the short exact sequence: Proof.It is obvious from the Proposition 7.13 of [13].
Let V n be the kernel of the surjective morphism [28]).The scheme S [n−1,n,n+1] is smooth of dimension 2n + 1.Moreover, it is the projectivization of for the total complex of the double complex C • .In this subsection, we still abuse the notation to denote We have the following formula for the derived push-forward Rp n * L j : Proof.By Proposition 3.6, S [n,n+1] is the projectivization of Thus (3.8) follows from Lemma 3.5.
Lemma 3.10.We have the following formula for Rq n * L k 2 : (3.9) ) is the derived pull back morphism.Proof.By Proposition 3.8, S [n−1,n,n+1] is the projectivization of [n,n+1] .We notice that p * n (W ∨ n ) ⊗ ω S and V n ∨ ⊗ ω S have the same rank and • When k > 0, we have we have By Lemma 3.9 and Lemma 3.10, we have Corollary 3.11.We have the following formula for R( Remark 3.12.When m = k < 0, same as the computation in the Grothendieck group, the complex } is quasi-isomorphic to 0 and thus we have It follows from Definition 2.8, Lemma 3.9 and Corollary 3.11.

Singularity of the Minimal Model Program
In this section, we will review the singularities in the minimal model program from [19,18].We use the notation D to replace ∆ in [19,18], as ∆ is already used to denote the diagonal embedding in our paper.For a normal variety, we will denote K X the canonical Weil divisor.We will denote by ω X the dualizing sheaf when X is Cohen-Macaulay.They coincide when X is Gorenstein.[19] or Definition 2.4 of [18], Discrepancy).Let (X, D) be a pair where X is a normal variety and D = a i D i , a i ∈ Q is a sum of distinct prime divisors.Assume that m(K X + D) is Cartier for some m > 0. Suppose f : Y → X is a birational morphism from a normal variety Y .Let E ⊂ Y denote the exceptional locus of f and E i ⊂ E the irreducible exceptional divisors.The two line bundles are naturally isomorphic.Thus there are rational numbers a(E i , X, D) such that ma(E i , X, D) are integers, and a(E, X, D) is called the discrepancy of E with respect to (X, D).We define the center of E in X by center X (E) := f (E).
When D = 0, then a(E i , X, D) depends only on E i but not on f .Lemma 4.2 (Lemma 2.29 of [19]).Let X be a smooth variety and D = a i D i a sum of distinct prime divisors.Let Z ⊂ X be a closed subvariety of codimension k.Let p : Bl Z X → X be the blow up of Z and E ⊂ Bl Z X the irreducible component of the exceptional divisor which dominates Z, (if Z is smooth, then this is the only component).Then where mult Z D i is the multiplicity of D i in Z. Definition 4.3 (Definition 2.34 and 2.37 of [19], or Definition 2.8 of [18]).Let (X, D) be a pair where X is a normal variety and D = a i D i is a sum of distinct prime divisors where a i ∈ Q and a i ≤ 1. Assume that m(K X + D) is Cartier for some m > 0. We say that (X, D) is for every E Here klt is short for "Kawamata log terminal", plt for "pure log terminal" and lc for "log canonical".Above, non-snc(X, D) denotes the set of points where (X, D) is not simple normal crossing(snc for short).
We say that X is terminal (canonical,etc) if and only if (X, 0) is terminal (canonical,etc).
Each class contains the previous one, except canonical does not imply klt if D contains a divisor with coefficient 1. Theorem 4.4 (Theorem 5.50 of [19], or Theorem 4.9 of [18], Inversion of adjunction).Let X be normal and S ⊂ X a normal Weil divisor which is Cartier in codimension 2. Let B be an effective Q-divisor and assume that K X + S + B is Q-Cartier.Then (X, S + B) is plt near S iff (S, B| S ) is klt.4.2.Rational Singularities.Definition 4.5 (Definition 5.8 of [19]).Let X be a variety of a field of characteristic 0. We say that X is a rational singularity if there exists one resolution of singularities f : Y → X, Remark 4.6.By Theorem 5.10 of [19], if X is a rational singularity, then for all resolution of singularities f : Y → X, Theorem 4.7 (Theorem 5.22 of [19]).Let X be a normal variety over a field of characteristic 0. If X is a canonical singularity, then X is a rational singularity.If X is Gorenstein, then X is a canonical singularity if X is a rational singularity.
Here are also some examples of rational singularities: Example 4.8.By [35], the universal closed subscheme Z n is a rational singularity.
Example 4.9.The morphism q n : S [n−1,n,n+1] → S [n,n+1]  (1) Definition 4.12 (Definition 5.1 of [18], Demi-normal schemes).A scheme X is called demi-normal if it is S 2 and codimension 1 points are either regular points or nodes.Here we say a scheme X has a node at a point x ∈ X if its local ring O x,X can be written as R/(f ) where (R, m) is a regular local ring of dimension 2, f ∈ m 2 and f is not a square in m 2 /m 3 .Definition 4.13 (Section 5.2 of [18], Conductor).Let X be a reduced scheme and π : X → X its normalization.The conductor ideal is the largest ideal sheaf on X that is also an ideal sheaf on X.We write it as cond X when we view the conductor as an ideal sheaf on X.The conductor subschemes are defined as Definition 4.14 (Definition-Lemma 5.10 of [18]).Let X be a demi-normal scheme with normalization π : X → X and conductors T ⊂ X and T ⊂ X.Let D be an effective Q-divisor whose support does not contain any irreducible components of T and D the divisorial part of π −1 (D).The pair (X, D) is called semi log canonical or slc if (1) ( X, D + T ) is lc.
Definition 4.15 (Definition 5.19 of [18]).An slc pair (X, D) is semi-divisorial log terminal or semi-dlt if a(E, X, D) > −1 for every exceptional divisor E over X such that (X, D) is not snc at the generic point of center X E.
Proposition 4.17 (Proposition 5.20 of [18]).Let (X, D) be a demi-normal pair over a field of characteristic 0. Assume that the normalization ( X, T + D) is dlt and there is a codimension

The Quadrulple/Triple Moduli Spaces and the Minimal Model Program
In this section we introduce the triple moduli spaces Z + , Z − and the quadruple moduli space Y which parameterize diagrams: (5.1) (5.2) respectively, of ideal sheaves where each successive inclusion is colength 1 and supported at the point indicated on the diagrams.We consider line bundles We consider the Cartesian diagram: (5.4) (2.4) is the restriction of (5.4) to the diagonal ∆ : Z + has two irreducible components and each one is isomorphic to Bl ∆ (S × S).
The main purpose of this section is to compute Rα + * O Y and Rα − * O Y explicitly.
Proposition 5.2.We have the formula where W 0 will be defined in Section 5.3.
Proposition 5.2 follows from Corollary 5.7 and Corollary 5.14, which will be proved later in this section.

The Geometry of Y.
Theorem 5.3 (Proposition 2.28 and Proposition 5.28 of [29]).The scheme Y is smooth of dimension 2n + 2. The closed embedding: is a regular closed subscheme of codimension 1.If we abuse the notation to denote S [n−1,n,n+1] by ∆ Y , then the morphism of coherent sheaves over Y × S: Remark 5.4.From now on we will abuse the notation to denote S [n−1,n,n+1] by ∆ Y .

5.2.
The Geometry of Z − .
Proposition 5.6.The scheme Z − is an irreducible 2n + 2 dimensional locally complete intersection scheme and is a canonical singularity.
Proof.By Lemma 5.5, smooth open subscheme and the complement is the 2n dimensional closed subscheme S [n−1,n] by (5.4).Hence Z − is a 2n + 2 dimensional R 1 scheme in the sense of Serre's condition.
On the other hand, has the expected dimension also equal to 2n + 2. Thus Z − is a locally complete intersection scheme and thus is Cohen-Macaulay and normal.Now we prove that Z − is a canonical singularity.The complement of ,n,n+1] and thus there exists a ∈ Q such that We define and the morphism where all horizontal morphisms are natural projections.By Example 5.1, Y 2 = Bl ∆S (S × S).Thus by Lemma 4.2, a = 1 and Z − is a canonical singularity.
Corollary 5.7.We have the formula

5.3.
The Geometry of Y + .The geometry of Y + is more complicated, as it is no longer irreducible.We define W 0 as a closed subscheme of Z + by ] .Proposition 5.8.The scheme Z + is a locally complete intersection scheme (and hence Cohen-Macaulay) of dimension 2n + 2, with two irreducible components W 0 and W 1 .
Theorem 1.1 follows from Theorem 6.2 and Theorem 3.2.We will prove Theorem 6.2 later in this section.[1].(6.1) 6.2. a k m,r and A k m,r .We recall the short exact sequence (6.
Definition 6.3.For any integer r, we define We define the exact traingles Recalling the short exact sequence in Corollary 3.7: We have the following commutative diagram: where all rows and columns are short exact sequences.Thus we have short exact sequences:  ) [1] to (6.4) and (6.5) respectively, we get the exact triangles When m = 0, We have the short exact sequence: and ×S } by Lemma 3.9 = 0 By (6.4) and (6.12), we have isomorphisms When m > 0, the composition of B k m,r and B k+1 m,r in Theorem 6.2 induces an extension of R∆ * h + m,k and R∆ * h + m,k+1 , i.e. an extension class of Hom(R∆ * h + m,k+1 , R∆ * h + m,k [1]).By the Hochschild-Kostant-Rosenberg theorem ( [36], also Remark 7.4),

Hom(R∆
When m < 0, there is also be an extension class of

Hom(R∆
In this section, we will explicitly computing the above extension classes and prove that Proposition 7.1.The extension class in Hom k and H − m,k will be defined in Definition 7.6.Proposition 7.1 will be proved later in this section.

7.1.
A Splitting Formula and HKR isomorphism.Let j : Y ⊂ X be a regular embedding of smooth schemes and N Y /X be the normal bundle.In a first step, we will suppose that Y is given as the zero locus of a regular section s ∈ H 0 (X, E) of a locally free sheaf E of rank r.In this case, the structure sheaf O Y can be resolved by the Koszul complex: with isomorphism given by contraction with s.The normal bundle N Y /X is E| Y .
Proposition 7.2 (Proposition 11.1 of [14]).There exists a canonical isomorphism: A stronger version of Proposition 7.2 was studied in [1]: Theorem 7.3 (Theorem 1.8, Lemma 3.1 of [1]).Let j : Y → X be a closed embedding of smooth varieties.A choice of splitting of determines an isomorphism: Remark 7.4.Let X be a smooth variety and ∆ X : X → X × X the diagonal embedding.Then T X×X | X = T X ⊕ T X has a canonical splitting and thus which is the HKR isomorphism in [36].

A morphism s
In this subsection, we first construct two canonical morphisms: ,n,n+1] be the projection morphism and Γ be the graph of the projection map to S. Then there exists short exact sequences: which induces two global sections: n] be the projection map and Γ n be the graph of the projection map from S ) by the flat base change theorem.With the composition of the following two natural homomorphisms: T S, we get the two canonical morphisms in (7.3)  which induces r 1 ∈ Hom(I n , k x ) and r 2 ∈ Ext 1 (k x , I n ).Let V be the vector space of pairs {(w 0 , w 1 ) ∈ Ext 1 (I n−1 , I n−1 ) ⊕ Ext 1 (I n , I n )} such that w 0 , w 1 map to the same element of Ext 1 (I n , I n−1 ).By the proof of Proposition 5.28 of [29], elements in V are in 1-1 correspondence with the commutative diagrams of short exact sequences: For any element u ∈ Hom(I n , k x ), (ŝ 1 (u ⊗ r 2 ), 0) ∈ V .Moreover, by diagram chasing (we left it to interested readers) (7.6) dp 1 S (ŝ 1 (u ⊗ r 2 ), 0) = ŝ2 (u ⊗ r 2 ).Simlilarly, for any element v ∈ Ext 1 (k x , I n ), (0, ŝ1 (r 1 ⊗ v)) ∈ V ′ and (7.7) dp 2 S (0, ŝ1 (r 1 ⊗ v)) = ŝ2 (r 1 ⊗ v).By Proposition 5.28 of [29], the tangent space T t Y is the space of (w 0 , w 1 , w ′ 1 , w 2 , u, v) in Ext 1 (I n−1 , I n−1 )⊕Ext 1 (I n , I n )⊕Ext 1 (I n , I n )⊕Ext 1 (I n+1 , I n+1 )⊕Ext 1 (k x , k x )⊕Ext 1 (k x , k x ) such that

Another way of inducing the morphism is restricting
(1) (w 0 , w 1 ) and (w 0 , w ′ 1 ) are in A.

Definition 2 . 3 .
The tautological line bundle L is the line bundle whose fibers are I n /I n+1 .Define the nested Hilbert scheme S [n−1,n,n+1] by the Cartesian diagram:

4. 1 .
Discrepancy and Classifacation of Singularities.Definition 4.1 (Definition 2.25 of factors through p −1 n Z n and induces a resolution of p −1 n Z n .By Lemma 3.10 and Corollary 3.7, p −1 n Z n is a Gorenstein rational singularity and thus is a canonical singularity by Theorem 4.7.4.3.Semi-dlt Pairs.Definition 4.10 (Definition 1.10 of [18], Semi-snc pairs).Let W be a regular subscheme and i∈I E i a snc divisor on W . Write I = I Y ∪ I D as a disjoint union.Set Y := i∈IV E i as a subscheme and D Y := i∈ID a i E i | Y as a divisor on Y for some a i ∈ Q.We call (Y, D Y ) an embedded semi-snc pair.A pair (X, D) is called semi-snc if it is Zariski locally isomorphic to an embedded semi-snc pair.Example 4.11.We have the following three examples of semi-snc pairs (X, D):

Example 5 . 1 .
When n = 1, then Y = Bl ∆S (S × S), where ∆ S : S → S × S is the diagonal embedding.Z − = S × S and Z + is induced by the Cartesian diagram (5.5)

6. 1 .
e m−r f r and f r e m−r Revisited.As Z − and Z + are both Cohen-Macaulay of expected dimension, we have the following formula:

1 ≤m + 1
r → a 0 m,r → R∆ * (Rp n * L m )[2].(6.7) 6.4.B k m,r and C k m,r .Definition 6.4.When m > 0, we define b k m.r ∈ D b (S [n] × S [n] × S × S) and natural transforms B k m,r : b k m,r → b k+1 m,r by (6.8) b k m,r := f r e m−r k = 0 a kk ≤ m − 1.We have b 0 m,r = f r e m−r and b m m,r = e m−r f r .When m < 0, we define c k m,r ∈ D b (S [n] × S [n] × S × S) and C k m,r : c k−1 m,r → c k ≤ k ≤ −1 We have f r e m−r = c 0 m,r and e m−r f r = c m m,r .Proof of Theorem 6.2.When m > 0, we only need to prove that the cone of B k m,r is R∆ * (h + m,k ) and the cone of C k m,r is R∆ * (h − m,k ).It follows from (6.3), (6.6) and (6.7).