Exceptional collections on certain Hassett spaces

We construct an $S_2\times S_n$ invariant full exceptional collection on Hassett spaces of weighted stable rational curves with $n+2$ markings and weights $(\frac{1}{2}+\eta, \frac{1}{2}+\eta,\epsilon,\ldots,\epsilon)$, for $0<\epsilon, \eta\ll1$ and can be identified with symmetric GIT quotients of $(\mathbb{P}^1)^n$ by the diagonal action of $\mathbb{G}_m$ when $n$ is odd, and their Kirwan desingularization when $n$ is even. The existence of such an exceptional collection is one of the needed ingredients in order to prove the existence of a full $S_n$-invariant exceptional collection on $\overline{\mathcal{M}}_{0,n}$. To prove exceptionality we use the method of windows in derived categories. To prove fullness we use previous work on the existence of invariant full exceptional collections on Losev-Manin spaces.


Introduction
A conjecture of Manin and Orlov states that Grothendieck-Knudsen moduli space M 0,n of stable, rational curves with n markings admits a full, exceptional collection which is invariant (as a set) under the action of the symmetric group S n permuting the markings. The conjecture has been proved by the authors in [CT20b] by reducing it to the similar statement for several Hassett spaces, one of which is the space under consideration in this paper. While the proof presented in [CT20b] for other needed Hassett spaces is valid in this particular case as well, it was not discussed in [CT20b] and we prefer to give a different and much simpler proof here.
For a vector of rational weights a = (a 1 , . . . , a n ) with 0 < a i ≤ 1 and a i > 2, the Hassett space M a is the moduli space of weighted pointed stable rational curves, i.e., pairs (C, a i p i ) with slc singularities, such that C is a genus 0, at worst nodal, curve and the Q-line bundle ω C ( a i p i ) is ample. For example, M 0,n = M (1,...,1) . There exist birational reduction morphisms M a → M a every time the weight vectors are such that a i ≥ a i for every i.
Understanding the derived categories of the Hassett spaces M a was considered in the work of Ballard, Favero and Katzarkov [BFK19], and earlier, for M 0,n in the work of Manin and Smirnov [MS13] (see also [Smi13,MS14]). However, here we consider a modified question. If Γ a ⊆ S n denotes the stabilizer of the set of weights a, we ask whether there exists a full, Γ a -invariant exceptional collection on M a . Theorem [CT20b, Theorem 1.5] reduces the existence of such collections on M 0,n , as well as many other Hassett spaces M a , to the following cases: (II) The Hassett spaces M p,q , for p + q = n (q ≥ 0, p ≥ 2) having p heavy weights and q light weights with the following properties: (1.1) a 1 = . . . = a p = a + η, a p+1 = . . . = a n = , pa + q = 2, where 0 < η, 1.
To reduce to the above cases, the authors were inspired by results of Bergstrom and Minabe [BM13,BM14] that used reduction maps between Hassett spaces. The existence of a full, invariant, exceptional collection in case (I) was proved in [CT20a]. The work in [CT20b] proves the statement for the spaces M p,q in (II) with p ≥ 3 and is the most difficult part of the argument. The current paper treats the spaces M p,q in (II) with p = 2. We emphasize that this case is not explicitly proved in [CT20b]. However, the proof for p > 2 is valid even when p = 2. The proof for p > 2 requires a lot of different comparisons between different Hassett spaces. Here we prove that this can be avoided when p = 2. More precisely, the main space under consideration when p = 2 is the following: Notation 1.1. Let Z N denote the Hassett moduli space of rational curves with markings N ∪ {0, ∞} with weights of markings 0 and ∞ equal to 1 2 + η and the markings from N equal to 1 n , where 0 < η 1. We also write Z n := Z N for n = |N | when there is no ambiguity.
The condition on the weights is equivalent to the condition (1.1) for p = 2 (in which case, a = 1 − (n−2) 2 ). Explicitly, all light points may coincide with one another and one heavy point may coincide with at most n−1 2 heavy points. We have the following description: Theorem 1.2. When n is odd, the space Z n is isomorphic to the symmetric GIT quotient Z n = (P 1 ) n / / O(1,...,1) G m , with respect to the diagonal action of G m on (P 1 ) n , coming from G m acting on P 1 by z · [x, y] = [zx, z −1 y]. When n is even, Z n is isomorphic to the Kirwan desingularization of the same GIT quotient.
Theorem 1.2 for n odd is stated in [Has03] within a more general set-up. Theorem 1.2 for n even is a direct consequence of [Has03]. For the reader's convenience, we give the proofs in Lemma 3.4 and Lemma 4.3.
The group S 2 × S n acts on Z n by permuting 0, ∞, and the markings from N respectively. In a similar fashion, the Losev-Manin space LM N (or LM n , for n = |N |) of dimension (n − 1) is the Hassett space with weights (1, 1, , . . . , ), with markings from N ∪ {0, ∞} with the weights of 0, ∞ equal to 1, while markings from N are equal to , with 0 < 1. The space LM N is isomorphic to an iterated blow-up of P n−1 along points q 1 , . . . , q n in linearly general position, and all linear subspaces spanned by {q i }. In particular, LM n is a toric variety. The action of S n permuting the markings from N corresponds to a relabeling of the points {q i }, while the action of S 2 , permuting 0, ∞, corresponds, at the level of P n−1 , to a Cremona transformation with center at the points {q i }. There is a birational S 2 × S N -equivariant morphism, reducing the weights of 0 and ∞: p : LM N → Z N . In particular, Z N is also a toric variety. Our main theorem is the following: Theorem 1.3. The Hassett space Z n has a full exceptional collection which is invariant under the action of (S 2 × S n ). In particular, the K-group K 0 (Z n ) is a permutation (S 2 × S n )-module. Theorem 1.3 is the immediate consequence of Theorem 1.6 (case of n odd) and Theorem 1.8 (case of n even). We now describe the collections. Definition 1.4. If (π : U → M, σ 1 , . . . , σ n ) is the universal family over the Hassett space M, one defines tautological classes ψ i := σ * i ω π , δ ij = σ * i σ j . Note that when n is odd, we have ψ 0 + ψ ∞ = 0 on Z n . For other relations, including the case when n is even, see Section 2. Definition 1.5. Assume n is odd. Let E ⊆ N and p ∈ Z, such that if e = |E| we have that p + e is even. We define line bundles on Z n as follows: As sums of Q-line bundles, L E,p = p 2 ψ ∞ + 1 2 j∈E ψ j = − p 2 ψ 0 + 1 2 j∈E ψ j . In particular, the action of S 2 exchanges L E,p with L E,−p . The line bundles L E,p are natural from the GIT point of view, see (3.1).
The line bundles are ordered by decreasing e, and for a fixed e, arbitrarily.
The collection in Theorem 1.6 is the dual of the collection in [CT20b, Theorem 1.10] for p = 2, with some of the constraints on the order removed. See also Remark 3.7 for a more precise statement.
Consider now the case when n = 2s + 2 ≥ 2 is even. In this case the universal family over Z n has reducible fibers. For each partition N = T T c , |T | = |T c | = s + 1, we denote δ T ∪{∞} ⊆ Z n the boundary component parametrizing nodal rational curve with two components, with markings from T ∪ {∞} on one component and T c ∪ {0} on the other. Moreover, δ T ∪{∞} = P s × P s and we have that Z n → (P 1 ) n / / O(1,...,1) PGL 2 is a Kirwan resolution of singularities with exceptional divisors δ T ∪{∞} . Definition 1.7. Assume n is even. Let E ⊆ N and p ∈ Z, such that if e = |E| we have that p + e is even. We define line bundles on Z n as follows: The line bundles L E,p are natural from the GIT point of view, see Definition 4.6 and the discussion thereafter. From this point of view, it is also clear that the action of S 2 exchanges L E,p with L E,−p .
Theorem 1.8. Assume n = 2s + 2 is even, s ≥ 0. The following form a full, (S 2 × S n ) invariant exceptional collection in D b (Z n ): • The torsion sheaves O(−a, −b) supported on δ T ∪{∞} = P s × P s , for all T ⊆ N , |T | = |T c | = s + 1, such that one of the following holds: 2 . • The line bundles {L E,p } (Definition 1.7) under the following condition: The order is as follows: all torsion sheaves precede the line bundles, the torsion sheaves are arranged in order of decreasing (a + b), while the line bundles are arranged in order of decreasing e, and for a fixed e, arbitrarily.
The torsion part of the collection in Theorem 1.8 is the same as the torsion part of the collection in [CT20b, Theorem 1.15] for p = 2. However, the remaining parts are not the same, nor are they dual to each other, as in the case of Theorem 1.6. There is a relationship between the dual collection {L ∨ E,p } and the torsion free part of the collection in [CT20b, Theorem 1.15] for p = 2, but this is more complicated -see Remark 4.23 for a precise statement.
To prove that our collections are exceptional, we use the method of windows [HL15,BFK19]. We then use some of the main results of [CT20a, Proposition 1.8, Theorem 1.10] to prove fullness, by using the reduction map p : LM n → Z n in order to compare our collections on Z n with with the push forward of the full exceptional collection on the Losev-Manin space. We emphasize that while in [CT20b] we prove exceptionality and fullness on spaces like Z N indirectly, by working on their contractions (small resolutions of the singular GIT quotient when n is even), in this paper we prove both exceptionality and fullness directly, by using the method of windows (for n even on the Kirwan resolution, the blow-up of the strictly semistable locus).
As remarked in [CT20a], we do not know any smooth projective toric varieties X with an action of a finite group Γ normalizing the torus action which do not have a Γ -equivariant exceptional collection {E i } of maximal possible length (equal to the topological Euler characteristic of X). From this point of view, the Losev-Manin spaces LM N and their birational contractions Z N provide evidence that this may be true in general. The existence of such a collection implies that the K-group K 0 (X) is a permutation Γ -module. In the Galois setting (when X is defined over a field which is not algebraically closed and Γ is the absolute Galois group), an analogous statement was conjectured by Merkurjev and Panin [MP97]. Of course one may further wonder if {E i } is in fact full, which is related to (non)-existence of phantom categories on X, another difficult open question.
We refer to [CT15,CT13,CT12] for background information on the birational geometry of M 0,n , the Losev-Manin space and other related spaces.
Organization of paper. In Section 2 we discuss preliminaries on Hassett spaces and prove some general results on how tautological classes pull back under reduction morphisms. These results are of independent interest and have been already used in a crucial way in [CT20b]. In Section 3, we discuss the GIT interpretation of the Hassett spaces Z n in the n odd case and prove Theorem 1.6. In Section 4, we do the same for the n even case and prove Theorem 1.8. Section 5 serves as an appendix, recalling results on Losev-Manin spaces from [CT20a] and calculating the push forward to Z n of the full exceptional collection on the Losev-Manin space LM n . These results are used in Sections 3 and 4 to prove fullness in Theorems 1.6 and 1.8. Throughout the paper, we do not distinguish between line bundles and the corresponding divisor classes.

Acknowledgements.
We are grateful to Alexander Kuznetsov for suggesting the problem about the derived categories of moduli spaces of pointed curves in the equivariant setting. We thank Daniel Halpern-Leistner for his help with windows in derived categories. We thank Valery Alexeev and the anonymous referee for useful comments.

Preliminaries on Hassett spaces
We refer to [Has03] for background on the Hassett moduli spaces. Recall that for a choice of weights a = (a 1 , . . . , a n ), a i ∈ Q, 0 < a i ≤ 1, we denote by M a the fine moduli space of weighted rational curves with n markings which are stable with respect to the set of weights a. Moreover, M a is a smooth projective variety of dimension (n − 3). Note that the polytope of weights has a chamber structure with walls i∈I a i = 1 for every subset I ⊆ {1, . . . , n}. One obtains the Losev-Manin space LM N by considering weights on the set of markings {0, ∞} ∪ N : 1, 1, 1 n , . . . , 1 n , n = |N |.
Replacing the weights equal to 1 n with some ∈ Q, for some 0 < 1, defines the same moduli problem, hence, gives isomorphic moduli spaces.
If a = (a 1 , . . . , a n ) and a = (b 1 , . . . , b n ) are such that a i ≥ b i , for all i, there is a reduction morphism ρ : M a → M a . This is a birational morphism whose exceptional locus consists of boundary divisors δ I (parametrizing reducible curves with a node that disconnects the markings from I and I c ) for every subset I ⊆ N such that i∈I a i > 1, but i∈I b i ≤ 1. For us a special role will be played by the reduction map p : LM N → Z N which reduces the weights of {0, ∞} from 1 to the minimum possible.
For a Hassett space M = M a , with universal family (π : U → M, {σ i }), recall that we define ψ i := σ * i ω π , δ ij = σ * i σ j . Since the sections σ i lie in the locus where the map π is smooth, the identity σ i · ω π = −σ 2 i holds on U . Therefore, Lemma 2.1. Assume M is a Hassett space whose universal family π : U → M is a P 1 -bundle. Then the identity −ω π = 2σ i + π * (ψ i ) holds on U , and therefore, on M we have for all i j: Proof. Indeed, −ω π − 2σ i restricts to the fibers of the P 1 -bundle trivially, and therefore is of the form π * (L) for some line bundle on M. Pulling back by σ i shows that L = ψ i .
When n is odd, the universal family U → Z N is a P 1 -bundle and the sections σ 0 and σ ∞ are distinct. Lemma 2.1 has the following: Corollary 2.2. The following identities hold on Z N when n is odd: Furthermore, identifying U with a Hassett space Mã, whereã = (a 1 , . . . , a n , 0) (with an additional marking x with weight 0) [Has03, 2.1.1], we have: Proof. The spaces U and U are smooth [Has03, Propositions 5.3 and 5.4]. The existence of the commutative diagram follows from semi-stable reduction [Has03, Proof of Theorem 4.1]. The map v is obtained by applying the relative MMP for the line bundle ω π ( b i σ i ). Concretely, the relative MMP results in a sequences of blow-downs, followed by a small crepant map: (all over M ). The resulting map v : U → V is a birational map which contracts divisors in U to codimension 2 loci in V (as the relative dimension drops from 1 to 0). Note that V is generically smooth along these loci.
The v-exceptional divisors can be identified via U Mã with boundary divisors δ I∪{x} (I ⊆ N ), with the property that i∈I a i > 1, i∈I b i ≤ 1.
For a flat family of nodal curves u : C → B with Gorenstein base B (in our case smooth) the relative dualizing sheaf ω u is a line bundle on C with first Chern class K C − u * K B , where K C and K B denote the corresponding canonical divisors. In particular: Since the map v on an open set is the blow-up of codimension 2 loci in V , it follows that K U = v * K V + E, by the blow-up formula. Hence, v * ω ρ = ω π − E, where the sum runs over all prime divisors E which are v-exceptional. This proves the first identity. For the second, we identify the sections σ i (resp., σ i ) with the boundary divisors δ ix in U (resp., in U ). Note that the proper transform of the section s i is σ i and s i contains v(δ I∪{x} ) (|I| ≥ 2), for δ I∪{x} v-exceptional if and only if i ∈ I. Moreover, in this case, v(δ I∪{x} ) is contained in s i (with codimension 1) and s i is smooth (since M is). The second identity follows. By Definition 1.4 and the diagram, The last two formulas now follow using the first two and the fact that σ * i δ I∪{x} = δ I if i ∈ I and is 0 otherwise.   When n = |N | is even, the Hassett space Z N = M ( 1 2 +η, 1 2 +η, 1 n ,..., 1 n ) of Notation 1.1 is closely related to the following Hassett spaces: , with weights assigned to (∞, 0, p 1 , . . . , p n ). There exist p : Z N → Z N , p : Z N → Z N , reduction maps that contract the boundary divisors using the two different projections. The universal families over Z N and Z N are P 1 -bundles. Lemma 2.3 applied to the reduction maps p , p leads to: Lemma 2.6. Assume n = |N | is even. The following relations hold between the tautological classes on the Hassett space Z N : Proof. The second relation follows from the first using the S 2 symmetry, while the third follows by adding the first two. To prove the first relation, consider the reduction map p : Z N → Z N . To avoid confusion, we denote by ψ i , δ ij (resp., ψ i , δ ij ) the tautological classes on Z N (resp., on Z N ). The universal family C → Z N is a P 1 -bundle. By Lemma 2.1, we have ψ ∞ = δ i0 − δ i∞ (since δ 0∞ = 0). The relation follows, as by Lemma 2.3, we have

Proof of Theorem 1.6
We start with a few generalities on GIT quotients (P 1 ) n ss / / G m . For n odd, we first show that the Hassett space Z N introduced in (1.1) can be identified with symmetric GIT quotients (P 1 ) n ss / / G m . We use the method of windows from [HL15] to prove exceptionality of the collections in Theorem 1.6. We then prove that the collection is full, by using the full exceptional collection on the Losev-Manin spaces LM N (see Section 5).
We use concepts of "linearized vector bundles" and "equivariant vector bundles" interchangeably. For (complexes of) coherent sheaves, we prefer "equivariant". We endow the line bundle O P 1 (−1) with a G m -linearization induced by the above action of G m on its total space V O P 1 (−1) ⊂ P 1 × A 2 .
Consider the diagonal action of G m on (P 1 ) n . Forj = (j 1 , . . . , j n ) in Z n , we denote O(j) the line bundle O(j 1 , . . . , j n ) on (P 1 ) n with G m -linearization given by the tensor product of linearizations above. We denote O ⊗ z k the trivial line bundle with G m -linearization given by the character G m → G m , z → z k . For every equivariant coherent sheaf F (resp., a complex of sheaves F • ), we denote by F ⊗ z k (resp., F • ⊗ z k ) the tensor product with O ⊗ z k . Note that O(j) ⊗ z k is P G m -linearized iff j 1 + . . . + j n + k is even.
There is an action of S 2 × S n on (P 1 ) n which normalizes the G m action. Namely, S n permutes the factors of (P 1 ) n and S 2 acts on P 1 by z → z −1 . This action permutes linearized line bundles O(j) ⊗ z k as follows: S n permutes components ofj and S 2 flips k → −k.
Notation 3.1. Consider the GIT quotient Σ n := (P 1 ) n ss / / L G m , L = O(1, . . . , 1), with respect to the ample line bundle L (with its canonical G m -linearization described above). Here (P 1 ) n ss denotes the semi-stable locus with respect to this linearization. Let φ : (P 1 ) n ss → Σ n denote the canonical morphism.
As GIT quotients X / / L G are by definition Proj R(X, L) G , where R(X, L) G is the invariant part of the section ring R(X, L), we may replace L with any positive multiple. As the action of P G m on (P 1 ) n is induced from the action of G m , Σ n is isomorphic to the GIT quotient (P 1 ) n ss / / P G m (with respect to any even multiple of L). The action of S 2 × S n on (P 1 ) n descends to Σ n .
By the Hilbert-Mumford criterion, a point (z i ) in (P 1 ) n is semi-stable (resp., stable) if ≤ n 2 (resp., < n 2 ) of the z i equal 0 or equal ∞.

The space Z N as a GIT quotient when n is odd
When n is odd, there are no strictly semistable points and the action of P G m on (P 1 ) n ss is free. In particular, Σ n is smooth and by Kempf's descent lemma, any P G m -linearized line bundle on (P 1 ) n ss descends to a line bundle on Σ n . Furthermore, Σ n can be identified with the quotient stack [(P 1 ) n ss /P G m ] and its derived category D b (Σ n ) with the equivariant derived category D b P G m ((P 1 ) n ss ). Consider the trivial P 1 -bundle on (P 1 ) n with the following sections: where pr i : (P 1 ) n → P 1 is the i-th projection. The sections s 0 , resp., s ∞ are induced by the projection is the zero locus of the section x i , or the locus in (P 1 ) n where s i = s 0 . Similarly, let ∆ i∞ the zero locus of the section y i .
Denote δ i0 the locus in Σ n where σ i = σ 0 . This is the zero locus of the section giving the map L ∞ → L i on Σ n , i.e., the section whose pull-back to (P 1 ) n is the section x i . Similarly, we let δ i∞ the locus in Σ n where σ i = σ ∞ . Hence, the sections x i , y i of pr * i O(1) ⊗ 1 defining ∆ i0 , ∆ i∞ descend to global sections of the corresponding line bundle on Σ n and define δ i0 , δ i∞ . Lemma 3.3. Assume n is odd. We have the following dictionary between line bundles on the GIT quotient Σ n and P G m -linearized line bundles on (P 1 ) n : Proof. The first two formulas follows from the previous discussion: The remaining formulas follow from Lemma 3.4 and the identities (2.1).
Proof. The trivial P 1 -bundle ρ : (P 1 ) n ss × P 1 → (P 1 ) n ss with sections s 0 , s ∞ , s i is the pull-back of the P 1bundle π : P(E) → Σ n and sections σ 0 , σ ∞ , σ i . Since the former is a family of A-stable rational curves, where A = ( 1 2 + η, 1 2 + η, 1 n , . . . , 1 n ), we have an induced morphism f : Σ n → Z N . Clearly, every A-stable pointed rational curve is represented in the family over (P 1 ) n ss (hence, Σ n ). Furthermore, two elements of this family are isomorphic if and only if they belong to the same orbit under the action of G m . It follows that f is one-to-one on closed points. As both Z N and Σ n are smooth, f must be an isomorphism. Alternatively, there is an induced morphism F : (P 1 ) n ss → Z N which is G m -equivariant (with G m acting trivially on Z N ). As Σ n is a categorical quotient, it follows that F factors through Σ n and as before, the resulting map f : Σ n → Z N must be an isomorphism.

Exceptionality
When n is odd, Σ n is a smooth polarized projective toric variety for the torus G n−1 m and its polytope is a cross-section of the n-dimensional cube (the polytope of (P 1 ) n with respect to L) by the hyperplane normal to and bisecting the big diagonal. In particular, the topological Euler characteristic e(Σ n ) is equal to the number of edges of the hypercube intersecting that hyperplane: By Lemma 3.3, the line bundles {L E,p } in Theorem 1.6 correspond to restrictions to (P 1 ) n ss of P G m linearized line bundles on (P 1 ) n Proof of Theorem 1.6 -exceptionality. Let G := P G m . We use the method of windows [HL15]. We describe the Kempf-Ness stratification [HL15, Section 2.1] of the unstable locus (P 1 ) n us with respect to L. The G-fixed points are for every subset I ⊆ {1, . . . , n}. Let σ I : Z I → (P 1 ) n be the inclusion map. The stratification comes from an ordering of the pairs (λ, Z), where λ : G m → G is a 1-PS and Z is a connected component of the λ-fixed locus (the points Z I in our case). The ordering is such that the function is decreasing. Here |λ| is an Euclidean norm on Hom(G m , G) ⊗ Z R. We refer to [HL15, Section 2.1] for the details. As µ(λ, Z) = µ(λ k , Z) for any integer k > 0, it follows that, in our situation, one only has to consider pairs (λ, Z I ) and (λ , Z I ), for the two 1-PS λ(z) = z and λ (z) = z −1 . Recall that The unstable locus is the union of the following Kempf-Ness strata: The destabilizing 1-PS for S I (resp. for S I ) is λ (resp. λ ). The 1-PS λ (resp., λ ) acts on the conormal bundle N ∨ S I |(P 1 ) n (resp., N ∨ S I |(P 1 ) n ) restricted to Z I with positive weights and their sum η I (resp., η I ) can be computed as To see this, note that the sum of λ-weights of N ∨ Note that S I can be identified with A |I c | and the point Z I ∈ S I with the point 0 ∈ A |I c | . The action of G on For the Kempf-Ness strata S I and S I we make a choice of "weights" w I = w I = −2s, where n = 2s + 1.
By the main result of [HL15, Theorem 2.10], D b G ((P 1 ) n ss ) is equivalent to the window G w in the equivariant derived category D b G ((P 1 ) n ), namely the full subcategory of all complexes of equivariant sheaves F • such that all weights (with respect to corresponding destabilizing 1-PS) of the cohomology sheaves of the complex We prove that the window G w contains all linearized line bundles L E,p = O(−E) ⊗ z p from Theorem 1.6. Recall that n = 2s + 1. Since the collection is S 2 invariant and S 2 flips the strata S I and S I , it suffices to check the window conditions for It is straightforward to check that the maximum of this quantity over all E is equal to 2s + 2|I| − n + 1 when s is odd, or 2s + 2|I| − n − 1 when s is even, and the minimum to −2s, hence the claim. Since our collection of linearized line bundles is clearly an exceptional collection on D b G ((P 1 ) n ), it follows it is an exceptional collection in D b G (Z n ).

Fullness
We will prove the following general statement.
Theorem 3.5. The collection in Theorem 1.6 generates all line bundles Proof of Theorem 1.6 -fullness. By Theorem 3.5, the collection in Theorem 1.6 generates all the objects Rp * (π * IĜ ) from Corollary 5.11. Fullness then follows by Corollary 5.5. Alternatively, it is easy to see that line bundles L E,p generate the derived category of the stack [(P 1 ) n /P G m ] and we can finish as in [CT20b,Proposition 4.1].
Proof Theorem 3.5. For simplicity, denote by C the collection in the theorem. We introduce the score of a pair (E, p), with e = |E| as s(E, p) := |p| + min{e, n − e}.
The collection C consists of L E,p with s(E, p) ≤ s. We prove the statement by induction on the score s(E, p), and for equal score, by induction on |p|.
Let (E, p) be any pair as in Theorem 3.5. If s(E, p) ≤ s, there is nothing to prove. Assume s(E, p) > s. Using S 2 -symmetry, we may assume w.l.o.g. that p ≥ 0. We will use the two types of P G m -equivariant Koszul resolutions from Lemma 3.6 to successively generate all objects.
Case e ≤ s. The sequence (1) in Lemma 3.6 for a set I with |I| = s + 1 followed by tensoring with We prove that each term L E∪J,p−j is generated by C for all j > 0. Note that s(E, p) = |p| but as p − j < p, we are done by induction on |p|. If p − j < 0 then since we assume e + p > s. In particular, L E∪J,p−j is in C.
Case e ≥ s + 1. Let I ⊆ E, with |I| = s + 1. The sequence (2) in Lemma 3.6 for the set I, followed by As p + j − s − 1 ≤ p with equality if and only if J = I, we are done by induction on |p|. (1) The restriction to (P 1 ) n ss of the Koszul complex of the intersection of the divisors ∆ i0 (Notation 3.2) for i ∈ I, which takes the form (2) The restriction to (P 1 ) n ss of the Koszul complex of the intersection of the divisors ∆ i∞ (Notation 3.2) for i ∈ I, which takes the form Each of these maps in the sequence is a direct sum of maps of the form obtained by multiplication with a canonical section corresponding to the effective divisor D j t . This can be made into a G-equivariant map: .3 and the discussion preceding it). The Lemma follows by restriction to (P 1 ) n ss . Note that Y I ∩ (P 1 ) n ss = ∅. The proof of (2) is similar, with the only difference that multiplication with y i , the canonical section of ∆ i∞ corresponds to a G- Remark 3.7. We explain the connection with case p = 2, q = n = 2s + 1 of [CT20b, Theorem 1.10]. The collection there is the following: (i) The line bundles F 0,E := − 1 2 j∈E ψ j (e = |E| is even) in the so-called group 1 (group 1A and group 1B of the theorem coincide in this case).
(ii) The line bundles in the so-called group 2: This collection is the dual of the one in Theorem 1.6. The elements in group 2 with l = p − 1, u = ∞ recover the dual of the collection in Theorem 1.6 when p > 0. Similarly, elements in group 2 with l = −p − 1, u = 0 recover the dual of the collection in Theorem 1.6 when p < 0. The elements of group 1 recover the dual of the collection in Theorem 1.6 when p = 0.

Proof of Theorem 1.8
We employ a similar strategy as in Section 3. We identify the Hassett space Z N (see (1.1)) when n = |N | is even with the Kirwan resolution of the symmetric GIT quotient Σ n . We use the method of windows [HL15] to prove the exceptionality part of Theorem 1.8. We prove fullness using previous results on Losev-Manin spaces LM N (see Section 5).

The space Z N as a GIT quotient, n even
Assume n = 2s + 2. There are n s+1 strictly semistable points {p T } ∈ (P 1 ) n ss one for each subset T ⊆ N , |T | = s + 1. More precisely, the point p T is obtained by taking ∞ for spots in T and 0 for spots in T c . Instead of the GIT quotient Σ n , which is singular at the images of these points, we consider its Kirwan resolutioñ Σ n constructed as follows.
Let W = W n be the blow-up of (P 1 ) n at the points {p T } and let {E T } be the corresponding exceptional divisors. The action of G m lifts to W . To describe this action locally around a point p T , assume for simplicity T = {s + 2, . . . , n} around the point p T . Consider the affine chart In the new coordinates, we have p T = 0 = (0, . . . , 0). We let ((x i ), (y i )), resp., ((t i ), (u i )), for i = 1, . . . , s, be coordinates on A n , resp., P n−1 . Then W is locally the blow-up Bl 0 A n , with equations The action of G m on W is given by The fixed locus of the action of G m on E T consists of the subspaces  . . . , j n )( α T E T ) on W n (where j i , α T integers and π : W n → (P 1 ) n is the blow-up map), with the G m -linearization given by the tensor product of the canonical linearizations above. As before, for every equivariant coherent sheaf F , we denote by F ⊗ z k the tensor product with O ⊗ z k . For a subset E ⊆ N , we denote (Lemma 4.4).
Consider the GIT quotient with respect to a (fractional) polarization where 0 < 1, ∈ Q, and the sum is over all exceptional divisors (with the canonical polarization described above):Σ n = (W n ) ss / / L G m .
Proof. By Kempf's descent lemma, a G-linearized line bundle L descends to the GIT quotient if and only if the stabilizer of any point in the semistable locus acts trivially on the fiber of L at that point, or equivalently, weight λ L |q = 0, for any semistable point q and any 1-PS λ : G m → G. By definition, weight λ L |q = weight λ L |p 0 , where p 0 is the fixed point lim t→0 λ(t) · q.
For any point q in (P 1 ) n \ {p T } such that q = (z i ) has z i = ∞ for i ∈ I and z i ∞ for i ∈ I c , we have for λ(z) = z that lim t→0 λ(t) · q is the point with coordinates z i = ∞ for i ∈ I and z i = 0 for i ∈ I c , and hence: Note that such a point q is semistable if and only if |I| < s + 1. Similarly, if q has z i = 0 for i ∈ I c and z i 0  Proof. The trivial P 1 -bundle (P 1 ) n × P 1 → (P 1 ) n has sections s 0 , s ∞ , s i . We still denote by s 0 , s ∞ , s i the induced sections of the pull back W ss × P 1 → W ss . The family is not A-stable at the points p T , where s i = s ∞ for all i ∈ T and s i = s 0 for all i ∈ T c (markings in T are identified with ∞, and markings in T c with 0). Here A = ( 1 2 + η, 1 2 + η, 1 n , . . . , 1 n ). Let C be the blow-up of W × P 1 along the codimension 2 loci Denote byẼ 0 T andẼ ∞ T the corresponding exceptional divisors in C . The resulting family π : C → W has fibers above points p ∈ E T a chain of P 1 's of the form C 0 ∪F ∪ C ∞ , whereF is the proper transform of the fiber of W × P 1 → W andF meets each of C 0 (the fiber ofẼ 0 T → E T at p) and C ∞ (the fiber ofẼ ∞ T → E T at p). The proper transforms of s i for i ∈ T (resp., i ∈ T c ) intersect C ∞ (resp., C 0 ) at distinct points. The dualizing sheaf ω π is relatively nef, with degree 0 onF. It follows that ω π induces a morphism C → C over W ss which contracts the componentF in each of the above fibers, resulting in an A-stable family. Therefore, we have an induced morphism F : W ss → Z N . Clearly, the map F is G m -equivariant (where G m acts trivially on Z N ). As the GIT quotientΣ n is a categorical quotient, there is an induced morphism f :Σ n → Z N . Two elements of the family C → W ss are isomorphic if and only if they belong to the same orbit under the action of G m . Hence, the map f is one-to-one on closed points (as there are no strictly semistable points in W ss , Σ n is a good categorical quotient [Dol03,p. 94]). It follows that f is an isomorphism.
Lemma 4.4. Assume n = 2s + 2 is even. We have the following dictionary between tautological line bundles on the Hassett space Z N (idenitified with the GIT quotientΣ n ) and G m -linearized line bundles on W n : around the point p T (markings in T = {s + 2, . . . , n} are identified with ∞, and markings in T c with 0). We have coordinates x 1 , . . . , x s+1 , y 1 , . . . , y s+1 . The GIT quotient map (P 1 ) n ss → Σ is locally at p T given by The morphism F : W ss →Σ n =Σ induced by the universal family over W ss (proof of Lemma 4.3) is locally the restriction to the semistable locus of the rational map (which we still call F) Consider coordinates ((x i , y i ), [t i , u i ]) (with x i t j = x j t i , x i u j = y j t i , x i t j = x j t i ) on Bl 0 A n ⊆ A n × P n−1 and coordinates (z ij , [w ij ]) on Bl 0 A (s+1) 2 (with z ij w kl = z kl w ij ). Consider the affine charts U 1 = {t 1 0} ⊆ Bl 0 A n and V 1j = {w 1j 0} ⊆ Bl 0 A r 2 . The map F |U 1 is the rational map (Note that all other relations will then follow by S 2 -symmetry and Lemma 2.6.) Clearly, for some integer k. The pull-back of the canonical section of the effective divisor δ i0 (which is x i ) must be an invariant section. The section x i of O P 1 (1) becomes the constant section 1 in the open chart U : x i 0.
Considering a point q = (q 1 , . . . , q n ) in U , with q i = ∞ and q j ∈ P 1 general for j i, it follows that for the 1-PS λ(z) = z we have weight λ pr * i O(1) |q = −1, weight λ O ⊗ z k |q = k, hence, the constant section 1 becomes z −1+k under the action of λ and we must have k = 1 for the section to be invariant.
We identify δ T = M × M = P s × P s , where M , resp., M are Hassett spaces with weights ( 1 2 + η, 1 n , . . . , 1 n , 1), with the attaching point x having weight 1. We identify M = P s via the isomorphism |ψ x | : M → P s . We have δ i∞|δ T The identity δ T |δ T = O(−1, −1) follows now from the previous ones by restricting to δ T any of the identities in Lemma 2.6.

Exceptionality
Note that W n is a polarized toric variety with the polytope ∆ obtained by truncating the n-dimensional cube at vertices lying on the hyperplane H normal to and bisecting the big diagonal. ThenΣ n is a smooth polarized projective toric variety for the torus G n−1 m and its polytope is ∆ ∩ H. In particular, the topological Euler characteristic e(Σ n ) is equal to the number of edges ∆ intersecting H: e(Z n ) = (s + 1) 2 n s + 1 = s 2 n s + 1 + (n − 1) n s + 1 (n = 2s + 2).
Note that (s + 1) n s+1 = n n 0 + (n − 2) n 1 + (n − 4) n 2 + . . . + 2 n s . Definition 4.6. For E ⊆ N , e = |E|, p ∈ Z such that p + e is even, let α T ,E,p := −|x T ,E,p |, x T ,E,p := |E ∩ T | − e − p 2 i.e., the descent toΣ n of the restriction to (W n ) ss of the above G m -linearized line bundle on W n . By Lemma 4.4 we recover Definition 1.7: We write x T if there is no ambiguity. Note that x T ,E,p = −x T c ,E,−p .
We describe the Kempf-Ness stratification of the unstable locus in W n . Let G = G m . As before, we consider (λ, Z), with a 1-PS λ : G m → G and Z a connected component of the λ-fixed locus. It suffices to consider λ(z) = z and λ (z) = z −1 . The G-fixed locus in W = W n consists of the points while for all subsets T ⊆ N with |T | = s + 1 we have: . As in the n odd case, we define for any subset I ⊆ N affine subsets: The unstable locus arises from the pairs with negative weight: (λ, Z I ) (for |I| > s + 1), (λ , Z I ) (for |I| < s + 1), (λ, Z + T ), (λ , Z − T ) (for |T | = s + 1) : S I A |I c | (for |I| > r), S I A |I| (for |I| < s + 1), . The destabilizing 1-PS for S I (resp., for S I ) is λ (resp. λ ). The 1-PS λ (resp., λ ) acts on the restriction to Z I of the conormal bundle N ∨ S I |(P 1 ) n (resp., N ∨ S I |(P 1 ) n ) with positive weights. Their sum η I (resp., η I ) is: η I = 2|I|, resp., η I = 2|I c |.
Letting m := n 4 = s+1 2 , we make a choice of windows G w : which by (4.5) lies in [w + T , w + T + η + T ). Hence, all {L E,p } in Theorem 1.8 are contained in the window G w . We now check exceptionality. Consider two line bundles as in Theorem 1.8: where α T := α T ,E,p , α T := α T ,E ,p . Assume that e = |E| ≥ e = |E |. Hence, E E unless E = E . By the main result of [HL15, Theorem 2.10], we have that RHom(L E ,p , L E,p ) equals the weight (p − p) part (with respect to the canonical action of G) of Hence, letting we need to understand the weight (p − p) part of Note that M 0 is a pull-back from (P 1 ) n ; hence, by the projection formula, RΓ Consider a simplified situation. For a line bundle M on W , G := E T , β := β T > 0 consider the exact sequences: 0, 1, . . . , β − 1), have no weight (p − p) part. Put an arbitrary order on the subsets T with β T > 0 (T 1 , T 2 , . . .). Applying the above observation successively, first for M 0 , E T 1 , then inductively for M 0 (−β 1 T 1 − . . . − β i T i ), E T i+1 , it suffices to prove that for all T , the following spaces We We now continue with RΓ (M 0 (−iE T ) |E T ). By the projection formula, where M 0|p T is the fiber of M 0 at p T (we denote M 0 both the line bundle on (P 1 ) n and its pull back to W ). By (4.2), the action of G on M 0|p T has weight Consider coordinates t i , u i on E = P n−1 , such that t i (resp., u i ) have weight 2 (resp., weight −2). There is a canonical identification with the weight of t a k k u b k k equal to 2 a k − 2 b k . As 2 a k − 2 b k ranges through all even numbers between −2i and 2i, it follows that the possible weights of elements in RΓ (M 0 (−iE T ) |E T ) are for all the values of j between −i and i.
Assume now that for some Using the definition of α T , α T , it follows that ±2α T ± 2α T = −2j.
Proof. By symmetry, it is enough to prove that none of ±α T ± α T lies in the interval [0, (α T − α T − 1)]. As α T , α T ≤ 0 and α T > α T . Hence, it remains to prove that This finishes the proof that the collection in Theorem 1.8 is exceptional. • a = 0, a = s, b > b, Proof. As any line bundle on δ is the restriction of a line bundle on M, we have that O δ (a − a, b − b)).
Applying RHom (−, O δ (a − a, b − b)) to the canonical sequence it follows that there is a long exact sequence on M

It is clear now that if any of the conditions in the Lemma hold, then
Assume now that none of the conditions holds. Then either a < a or where m := n 4 = s + 1 2 .
In particular, when (E, p) are as in Theorem 1.8, the coefficients α T ,E,p in (4.3) satisfy The proof is straightforward and we omit it.

Fullness
Let C be the collection in Theorem 1.8. We denote by A ⊂ C the collection of torsion sheaves in Theorem 1.8. We prove more generally: Proof of Theorem 1.8 -fullness. By Theorem 4.11, the collection C generates all the objects Rp * (π * IĜ ) from Corollary 5.11. Fullness then follows by Corollary 5.5.
Remark 4.12. As S(E, p) is even, the range of (E, p) in Theorem 1.8 is precisely: • If s is even: S(E, p) ≤ s, • If s is odd: S(E, p) ≤ s + 1 if e ≤ s + 1 and S(E, p) ≤ s − 1 if e ≥ s + 2.
Using notation (4.7), (E, p) is not in the range of Theorem 1.8 if q ≥ 1 when s is even or s is odd and e ≥ s + 2, and if q ≥ 2 when s is odd and e ≤ s + 1.
To prove Theorem 4.11 we introduce three other types of line bundles.
where the last equality follows from (4.9) and (4.10).
We recall for the reader's convenience that using Notation 4.3 we have x T ,E,p δ T ∪{∞} . Therefore, |x T ,E,p |δ T ∪{∞} and by using Lemma 2.6, we have also (4.10) We remark that using Lemma 4.4, we have: Remark 4.14. It is clear by the definition that by the S 2 symmetry (i.e., exchanging 0 with ∞) the line bundle R E,p is exchanged with Q E,−p . The line bundles R E,p , Q E,p will be crucial for the proof of Theorem 4.11. We note that the line bundles V E,p are used only in the proof of Corollary 4.18.
For every divisor δ T := δ T ∪{∞} , we have by Lemma 4.5 that (ii) L E,p and Q E,p are related by quotients which are direct sums of type (iv) Q E,p and V E,p are related by quotients which are direct sums of type where we denote for simplicity δ T := δ T ∪{∞} and x T := x T ,E,p . In particular, all pairs are related by quotients of type Proof. This follows immediately from (4.11), (4.9), (4.10) and (4.8). The last statement follows by Lemma 4.17. The proof is straightforward and we omit it. Note, (4.5) is a particular case. with p = 2q − 1, q ≥ 1 if s is even, and p = 2q − 2, q ≥ 2 if s is odd. Assume the following objects are generated by C: (i) All torsion sheaves O δ T (−a, 0) for all 0 < a < s 2 + q and all T , (ii) The line bundles R E,p , Q E,p .
As C is invariant under the action of S 2 , it follows from Corollary 4.18 that a similar statement holds when replacing O δ T (−a, 0) with O δ T (0, −a).
Proof. We claim that V E,p is generated by C. Since R E,p is generated by C by assumption, using Lemma 4.16(iii), it suffices to prove that when x T < 0, O(−x T − i, −i) is generated by A, for all 0 < i ≤ |x T |, i.e., |x T | < s+1 2 . Since the assumptions on q imply that p > 0, we have that and the claim follows. By Lemma 4.16(iv), the quotients relating Q E,p and V E, 2 ) are already by assumption generated by C. Note that this quotient appears exactly once. Since Q E,p , V E,p are generated by C, it follows that this quotient is also.
Corollary 4.19. Let q ∈ Z, q > 0. Assume that R E,p , Q E,p are generated by C whenever S(E, p) = 2 s 2 + 2q , with 0 < q ≤ q, and e = s + 1. Then for all T , δ T := δ T ∪{∞} , the following torsion sheaves are generated by C: Proof. By the S 2 symmetry, it suffices to prove the statement for O δ T (−a, 0). For any q > 0, taking E ⊆ N with e = s+1 and p = 2q−1 when s is even, or p = 2q−2 when s is odd, gives a pair (E, p) with S(E, p) = 2 s 2 +2q. If s is even, or if s is odd and q ≥ 2, the assumptions of Corollary 4.18 are satisfied. By induction on q > 0, , 0) is generated by C when T = E, a = s 2 + q. The only case left is when s is odd and q = 1 (p = 0). By assumption R E,0 , Q E,0 are generated by C if e = s + 1 (S(E, 0) = s + 1). We have to prove that O δ T (− s+1 2 , 0) is generated by C. Taking Proof. We have i∈I δ i∞ = ∅ and the boundary divisors {δ i∞ } i∈I intersect transversely (the divisors intersect properly and the intersection is smooth, being a Hassett space). It follows that there is a long exact sequence Tensoring this long exact sequence by − i∈E\I δ i∞ − e−p 2 ψ ∞ , gives the second long exact sequence in the lemma. The first long exact sequence is obtained in a similar way by considering the Koszul resolution of the intersection of the boundary divisors {δ i0 } i∈I . Proof. We prove (1). We have S(E, p) = p + e. If p − j ≥ 0, then and clearly |p − j| = p − j < p if j 0. If p − j < 0, we prove that the inequality on slopes is strict. We have since S(E, p) = e + p = 2 s 2 + 2q > s + 1. We prove (2). We have S(E, p) = p + (n − e). If p − j ≥ 0, then and clearly |p − j| = p − j < p if j 0. If p − j < 0, we prove that the inequality on slopes is strict. We have since e − p < s + 1, as S(E, p) = p + n − e = 2 s 2 + 2q > s + 1.
Proof of Theorem 4.11. Case s even. For any (E, p) write the score S(E, p) as ≤ s 2 < s+1 2 , such quotients are in A. We prove by induction on q ≥ 0, and for equal q, by induction on |p|, that R E,p , Q E,p with S(E, p) = s + 2q are generated by C. By Corollary 4.19, it follows that all O δ T (−a, −b) are generated by C. Then Lemma 4.16 implies then that all line bundles L E,p are generated by C.
We now prove the inductive statement. For q ≤ 0, we already proved that R E,p , Q E,p are generated by C. Assume q ≥ 1. Take a pair (E, p) with score S(E, p) = s + 2q. Using the S 2 symmetry, we may assume p ≥ 0. For any (E , p ) with strictly smaller score than s + 2q, or equal score and strictly smaller |p|, we have by induction that Q E ,p , R E ,p are generated by C.
If e ≤ s + 1, we apply Lemma 4.20 and get a resolution for Q E,p . Using Lemma 4.21(i), all terms in the resolution are generated by C by induction. Hence, Q E,p is generated by C if e ≤ s + 1. Similarly, using Lemma 4.20, Lemma 4.21(ii) and induction, R E,p is generated by C if e ≥ s + 1.
We have that both Q E,p , R E,p are generated by C if e = s + 1. By Corollary 4.19 and the induction assumption, O δ T (−a, 0), O δ T (0, −a) if 0 < a ≤ s 2 + q are generated by C. By Lemma 4.16 we have that L E,p is related to each of Q E,p , R E,p by quotients which are direct sums of O δ T (−a, * ), O δ T ( * , −a) with 0 < a ≤ S(E,p) 2 = s 2 + q. Since for any e s + 1, one of Q E,p , R E,p is generated by C, it follows that L E,p is generated by C.
We prove by induction on q ≥ 0, and for equal q, by induction on |p|, that the line bundles R E,p and Q E,p with S(E, p) = (s − 1) + 2q are generated by C. This proves the theorem, as Corollary 4.19 gives that all torsion sheaves supported on boundary are generated by C. The inductive argument we did for s even goes through verbatim if q ≥ 2 (the assumption is used in Lemma 4.21). Hence, we only need to prove that R E,p , Q E,p are generated by C for q = 0 and q = 1. We may assume w.l.o.g. that p ≥ 0.
Assume q = 0. Fix a pair (E, p) with S(E, p) = s − 1. Then (E, p) is in the range of Theorem 1.8 and L E,p is in C. As in the previous case, by Lemma 4.16, the line bundles R E,p , Q E,p are related to L E,p by quotients generated by A. Hence, R E,p , Q E,p are generated by C.
Assume now q = 1 and fix a pair (E, p) with S(E, p) = s + 1.
are generated by C.
Proof. By Corollary 4.19, it suffices to prove that R E,0 , Q E,0 are generated by C for some E with e = |E| = s +1. Take such an E. By Remark 4.14, R E,0 and Q E,0 are exchanged by the action of S 2 . Hence, by symmetry, it suffices to prove that R E,0 is generated by C.
The claim follows. It follows that for j > 0, R E\J,−j is generated by C. Using the resolution, it follows that R E,0 is generated by C.
Assume that e ≤ s + 1. Assume now that e > s + 1. Then (E, p) is not in the range of Theorem 1.8. Note that it suffices to prove that R E,p is generated by C, since by Lemma 4.16 R E,p , L E,p are related by quotients which are direct sums of O δ T (−a, * ) with 0 < a ≤ S(E,p) 2 = s+1 2 (generated by C by Claim 4.22). To prove R E,p is generated by C, we do an induction on e ≥ s + 1 (for (E, p) of fixed score s + 1) by using a resolution as in Lemma 4.20 for R E,p .
The line bundles F 0,E are defined in [CT20b] as Rπ * (N 0,E ), for certain line bundles N 0,E on the universal family over Z n . One checks directly (or see the proof of [CT20b, Lemma 5.8]) that N 0,E restrict trivially to every component of any fiber of the universal family π : U → Z n . Hence, for any marking u. In particular, for u ∈ {0, ∞}, we obtain formula (4.15).
The formula generalizing both expressions in (ii) and (iii) is

Pushforward of the exceptional collection on the Losev-Manin space LM N to Z N
We refer to [CT20a] for background on Losev-Manin spaces. Recall that the Losev-Manin moduli space LM N is the Hassett space with markings N ∪ {0, ∞} and weights (1, 1, 1 n , . . . , 1 n ), where n = |N |. The space LM N parametrizes nodal linear chains of P 1 's marked by N ∪ {0, ∞} with 0 is on the left tail and ∞ is on the right tail of the chain. Both ψ 0 and ψ ∞ induce birational morphisms LM N → P n−1 (Kapranov models) which realize LM N as an iterated blow-up of P n−1 in n points (standard basis vectors) followed by blowing up n 2 proper transforms of lines connecting points, etc. In particular, LM N is a toric variety of dimension n − 1. Its toric orbits (or their closures, the boundary strata as a moduli space) are given by partitions N = N 1 . . . N k , |N i | > 0 for all i, which correspond to boundary strata which parametrizes (degenerations of) linear chains of P 1 's with points marked by, respectively, N 1 ∪ {0}, N 2 ,. . . , N k−1 , N k ∪ {∞}. We can identify where the left node of every P 1 is marked by 0 and the right node by ∞.
There are forgetful maps π K : LM N → LM N \K , for all K ⊆ N , 1 ≤ |K| ≤ n − 1, given by forgetting points marked by K and stabilizing. Consider now the case when I ∅. To compute Rp * (π * I T ), consider the boundary divisors D 1 , D 2 , . . . , D t−1 whose intersection is Z, denoting For the remaining part of the proof, we denote for simplicity K = K 1 , K = K 2 . . . K t and consider the canonical inclusions We resolve i 1 * O Z using the Koszul complex for the corresponding boundary divisor on LM K : By Lemma 5.10, we may choose a line bundle M on LM K such that the restriction of M to the massive We now: (1) Tensor the Koszul sequence with L, (2) Apply Ri D 1 * (−), and (3) Apply Lπ * I (−). Since π I is flat, we obtain a resolution for π * I T with sheaves whose support is contained in π −1 I (D 1 ). To prove (4) and the remaining part of (2), it suffices to show that for all 2 ≤ i 1 < . . . < i k ≤ t Rp * π * I Ri D 1 * L ⊗ O(−D i 1 − . . . − D i k ) |D 1 is 0 when n is odd, or generated by the sheaves O(−a) O(−b) (a, b > 0) supported on the divisors P n 2 −1 × P n 2 −1 as in (4), when n is even. Here we need the same statement also for Rp * π * I Ri D 1 * L (i.e., k = 0). Note that There is a commutative diagram where i π −1 I (D 1 ) is the canonical inclusion map and ρ I is the restriction of π I to π −1 I (D 1 ). Let q = p • i π −1 (D 1 ) . As π I is flat, we have The preimage π −1 I (D 1 ) has several components B I 1 ,I 2 : B I 1 ,I 2 = LM K ∪I 1 × LM K ∪I 2 for every partition I = I 1 I 2 We order the set {B I 1 ,I 2 } as follows: B I 1 ,I 2 must come before B J 1 ,J 2 if |I 1 | > |J 1 | and in a random order if |I 1 | = |J 1 |. Hence, if B I 1 ,I 2 comes before B J 1 ,J 2 , then B I 1 ,I 2 ∩ B J 1 ,J 2 ∅ if and only if J 1 I 1 , in which case, the intersection takes the form For simplicity, we rename the resulting ordered sequence as B 1 , B 2 , . . . , B r . A consequence of the ordering is that B r is the component B I 1 ,I 2 with I 1 = ∅, I 2 = I, and if 1 ≤ i ≤ r − 1 and B i is B I 1 , where the first sum runs over all J I 1 (including J = ∅), while for the second sum we use the identification As the restriction of the map ρ I to a component B i of the form B I 1 ,I 2 for some partition I = I 1 I 2 , is the product of forgetful maps π I 1 × π I 2 , it follows that, if i r, then We claim that both components of all the above sheaves are acyclic. To prove the claim, recall that M⊗O(−D i 1 −. . .−D i k ) is acyclic by the choice of M. We are left to prove that π * I 1 (G ∨ a 1 )⊗O(− ∅ S⊆I 1 δ S∪{x} ) is acyclic when I 1 ∅. Since we may rewrite the line bundle G ∨ a 1 using the x marking, we are done by the following: Claim 5.7. Consider the forgetful map π I : LM N ∪I → LM N for some subset I ∅. For all 1 ≤ b ≤ |N | − 1, the line bundle π * I (G ∨ b ) ⊗ O(− ∅ S⊆I δ S∪{0} ) is acyclic. Proof. Using the Kapranov model with respect to the 0 marking, we have As b − |J ∩ N | − 1 ≥ 0, the result follows by Lemma 5.8.
Recall that the map p either contracts B I 1 ,I 2 = LM K ∪I 1 × LM K ∪I 2 by mapping LM K ∪I 1 to a point if |I 1 + K | < n 2 , or by mapping LM K ∪I 2 to a point if |I 2 + K | < n 2 ), or, we have |I 1 + K | = |I 2 + K | = n 2 and p(B I 1 ,I 2 ) is a divisor in Z N which is isomorphic to P as above (in particular, n is even). In the latter case, writing n = 2s + 2, as both components of the above sheaves are acyclic, such objects are generated by O(−a) O(−b) for 0 < a, b ≤ s. We use here that if A is an object in D b (LM s+1 ) with RΓ (A) = 0 and f : LM s+1 → P s is a Kapranov map, then Rf * A has the same property, therefore it is generated by O(−a), for 0 < a ≤ s. Using the above exact sequences, Rq * ρ * I L ⊗ O(−D i 1 − . . . − D i k ) |D 1 is either 0 or, when n is even, generated by O(−a) O(−b) (0 < a, b ≤ s) on P s × P s . Proposition 5.6 now follows. Assume now t ≥ 2 and at least two of the T i 's are non-trivial. Consider π N 1 : LM N → LM N \N 1 and let Z = π N 1 (Z). Then Z can be identified with LM N 2 × . . . × LM N t and the map π N 1 : Z → Z is the second projection. Let T = T 2 . . . T t . By induction, there is an acyclic line bundle L on LM N \N 1 such that L |Z = T and whose restriction to every stratum containing Z is also acyclic. If T 1 = O, we let L = π * N 1 L and clearly all of the properties are satisfied. If T 1 = G ∨ a , we define L = G ∨ a ⊗ π * N 1 L . Clearly, L |Z = T . By the projection formula, Rπ N 1 * (L) = L ⊗ Rπ N 1 * (G ∨ a ). As Rπ i * (G ∨ a ) = 0 for all i, it follows that Rπ N 1 * (L) = 0,i.e., L is acyclic.
The same argument applies to show that the restriction of L to a stratum W containing Z is acyclic. Consider such a stratum: and let W = LM M 2 × . . . × LM M s , considered as a stratum in in LM N \M 1 . If M 1 = N 1 , the restriction L |W equals G ∨ a (L |W ) and is clearly acyclic. If M 1 N 1 , then M 1 = N 1 + . . . + N i , with i ≥ 2, and π N 1 (W ) is the stratum LM M 1 \N 1 × W in LM N \N 1 . The restriction of L to this stratum has the form L 1 L 2 . Then L |W = (G ∨ a ⊗ π * N 1 L 1 ) L 2 , where π N 1 : LM M 1 → LM N 1 is the forgetful map. Again, by the projection formula, L |W is acyclic.
We now prove the last assertion in the lemma. As L, L D i are both acyclic, L(−D i ) is acyclic (case k = 1). The statement follows by induction on k using the Koszul resolution for the intersection ∩ j∈I D j , I = {i 1 , . . . , i k }: .
where |a − |E ∩ T c || denotes the absolute value of (a − |E ∩ T c |). Moreover, , All other pushforwards are either 0 or are generated by the above torsion sheaves.
When n = 4, the map p : LM N → Z N is an isomorphism. In particular, the objects in Rp * π * IĜ form a full exceptional collection. However, it is straightforward to see that this is different than the collection in Theorem 1.8.