Exceptional collections on certain Hassett spaces
aa r X i v : . [ m a t h . AG ] M a y EXCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES
ANA-MARIA CASTRAVET AND JENIA TEVELEVA
BSTRACT . We construct an S × S n invariant full exceptional collectionon Hassett spaces of weighted stable rational curves with n + 2 markingsand weights ( + η, + η, ǫ, . . . , ǫ ) , for < ǫ, η ≪ and can be identi-fied with symmetric GIT quotients of ( P ) n by the diagonal action of G m when n is odd, and their Kirwan desingularization when n is even. Theexistence of such an exceptional collection is one of the needed ingredi-ents in order to prove the existence of a full S n -invariant exceptional col-lection on M ,n . To prove exceptionality we use the method of windowsin derived categories. To prove fullness we use previous work on the ex-istence of invariant full exceptional collections on Losev-Manin spaces.
1. I
NTRODUCTION
A conjecture of Manin and Orlov states that Gorthendieck-Knudsen mod-uli space M ,n of stable, rational curves with n markings admits a full, ex-ceptional collection which is invariant (as a set) under the action of the sym-metric group S n premuting the markings. The conjecture has been provedby the authors in [CT20] by reducing it to the similar statement for severalHassett spaces, one of which is the space under consideration in this paper.While the proof presented in [CT20] for other needed Hassett spaces seemsvalid in this particular case as well, it was not discussed in [CT20] and weprefer to give a different and much simpler proof here.For a vector of rational weights a = ( a , . . . , a n ) with < a i ≤ and P a i > , the Hassett space M a is the moduli space of weighted pointedstable rational curves, i.e., pairs ( C, P a i p i ) with slc singularities, such that C is a genus , at worst nodal, curve and the Q -line bundle ω C ( P a i p i ) is ample. For example, M ,n = M ,..., . There exist birational reductionmorphisms 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 con-sidered in the work of Ballard, Favero and Katzarkov [BFK12], and ear-lier, for M ,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 de-notes the stabilizer of the set of weights a , we ask whether there exists afull, Γ a -invariant exceptional collection on M a . Theorem [CT20, Thm. 1.5]reduces the existence of such collections on M ,n , as well as many otherHassett spaces M a , to the following cases:(I) The Losev-Manin spaces M a , where a = (1 , , ǫ, . . . , ǫ ) , < ǫ ≪ . (II) The Hassett spaces M p,q , for p + q = n ( q ≥ , p ≥ ) having p heavy weights and q light weights with the following properties: a = . . . = a p = a + η, a p +1 = . . . = a n = ǫ ≪ ,pa + qǫ = 2 , < η ≪ . To reduce to the above cases, the authors were inspired by results ofBergstrom and Minabe [BM13, BM14] that used reduction maps betweenHassett spaces. The existence of a full, invariant, exceptional collection incase (I) was proved in [CT17]. The work in [CT20] proves the statementfor the spaces M p,q in (II) with p ≥ and is the most difficult part of theargument. The current paper treats the spaces M p,q in (II) with p = 2 . Weemphasize that this case is not explicitly proved in [CT20]. However, theproof for p > seems valid even when p = 2 . The proof for p > requiresa lot of different comparisons between different Hassett spaces. Here weprove that this can be avoided when p = 2 . More precisely, the main spaceunder consideration when p = 2 is the following: Notation 1.1.
Let Z N denote the Hassett space with markings N ∪ { , ∞} with weights of markings and ∞ equal to + η and the markings from N equal to ǫ , with < ǫ, η ≪ . We also write Z n := Z N for n = | N | when there is no ambiguity. When n is odd, the space Z n is isomorphic tothe symmetric GIT quotient Z n = ( P ) n // O (1 ,..., G m , with respect to thediagonal action of G m on ( P ) n , coming from G m acting on P by z · [ x, y ] =[ zx, z − y ] (see Lemma 3.4). When n is even, Z n is isomorphic to the Kirwandesingularization of the same GIT quotient (see Lemma 4.3).The group S × S n acts on Z n by permuting , ∞ , and the markingsfrom N respectively. In a similar fashion, the Losev-Manin space LM N (orLM n , for n = | N | ) of dimension ( n − is the Hassett space with weights (1 , , ǫ, . . . , ǫ ) , with markings from N ∪ { , ∞} with the weights of , ∞ equal to , while markings from N are equal to ǫ , with < ǫ ≪ . The spaceLM N is isomorphic to an iterated blow-up of P n − along points q , . . . , q n in linearly general position, and all linear subspaces spanned by { q i } . Inparticular, LM n is a toric variety. The action of S n permuting the mark-ings from N corresponds to a relabeling of the points { q i } , while the actionof S , permuting , ∞ , corresponds, at the level of P n − , to a Cremonatransformation with center at the points { q i } . There is a birational S × S N -equivariant morphism, reducing the weights of and ∞ : p : LM N → Z N .In particular, Z N is also a toric variety. Our main theorem is the following: Theorem 1.2.
The Hassett space Z n = M ( + η, + η,ǫ,...,ǫ ) has a full exceptionalcollection which is invariant under the action of ( S × S n ) . In particular, theK-group K ( Z n ) is a permutation ( S × S n ) -module. Thm. 1.2 is the immediate consequence of Thm. 1.5 (case of n odd) andThm. 1.7 (case of n even). We now describe the collections. Definition 1.3. If ( π : U → M , σ , . . . , σ n ) is the universal family over theHassett space M , one defines tautological classes ψ i := σ ∗ i ω π , δ ij = σ ∗ i σ j . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 3
Note that when n is odd, we have ψ + ψ ∞ = 0 on Z n . For other relations,including the case when n is even, see Section 2. Definition 1.4.
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: L E,p := − (cid:18) e − p (cid:19) ψ ∞ − X j ∈ E δ j ∞ . As sums of Q -line bundles, L E,p = p ψ ∞ + P j ∈ E ψ j = − p ψ + P j ∈ E ψ j .In particular, the action of S 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).
Theorem 1.5.
Let n = 2 s + 1 odd. The line bundles { L E,p } (Def. 1.4) form afull, ( S × S n ) invariant exceptional collection in D b ( Z n ) under the condition: | p | + min( e, n − e ) ≤ s, where e = | E | , p + e even . The line bundles are ordered by decreasing e , and for a fixed e , arbitrarily. The collection in Thm. 1.5 is the dual of the collection in [CT20, Thm.1.10] for p = 2 , with some of the constraints on the order removed. See alsoRmk. 3.7 for a more precise statement.Consider now the case when n = 2 s + 2 ≥ is even. In this case theuniversal 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 componentparametrizing nodal rational curve with two components, with markingsfrom T ∪ {∞} on one component and T c ∪ { } on the other. Moreover, δ T ∪{∞} = P s × P s and we have that Z n → ( P ) n // O (1 ,..., PGL is a Kirwanresolution of singularities with exceptional divisors δ T ∪{∞} . Definition 1.6.
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: L E,p := − (cid:18) e − p (cid:19) ψ ∞ − X j ∈ E δ j ∞ − X | E ∩ T |− e − p > (cid:18) | E ∩ T | − e − p (cid:19) δ T ∪{∞} . The line bundles L E,p are natural from the GIT point of view, see Def. 4.6and the discussion after. From this point of view, it is also clear that theaction of S exchanges L E,p with L E, − p . Theorem 1.7.
Assume n = 2 s + 2 is even, s ≥ . The following form a full, ( S × 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: – < a ≤ s , < b ≤ s , – a = 0 , < b < s +12 , – b = 0 , < a < s +12 . • The line bundles { L E,p } (Def. 1.6) under the following condition: | p | + min( e, n + 1 − e ) ≤ s + 1 , where e = | E | , p + e even . The order is as follows: all torsion sheaves precede the line bundles, the torsionsheaves are arranged in order of decreasing ( a + b ) , while the line bundles arearranged in order of decreasing e , and for a fixed e , arbitrarily. ANA-MARIA CASTRAVET AND JENIA TEVELEV
The torsion part of the collection in Thm. 1.7 is the same as the torsionpart of the collection in [CT20, Thm. 1.15] for p = 2 . However, the remain-ing parts are not the same, nor are they dual to each other, as in the caseof Thm. 1.5. There is a relationship between the dual collection { L ∨ E,p } andthe torsion free part of the collection in [CT20, Thm. 1.15] for p = 2 , but thisis more complicated - see Rmk. 4.23 for a precise statement.To prove that our collections are exceptional, we use the method of win-dows [HL15, BFK12]. We then use some of the main results of [CT17, Prop.1.8, Thm. 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 thefull exceptional collection on the Losev–Manin space. We emphasize thatwhile in [CT20] we prove exceptionality and fullness on spaces like Z N in-directly, by working on their contractions (small resolutions of the singularGIT quotient when n is even), in this paper we prove both exceptionalityand fullness directly, by using the method of windows (for n even on theKirwan resolution, the blow-up of the strictly semistable locus).As remarked in [CT17], we do not know any smooth projective toric va-rieties X with an action of a finite group Γ normalizing the torus actionwhich do not have a Γ -equivariant exceptional collection { E i } of maximalpossible length (equal to the topological Euler characteristic of X ). Fromthis point of view, the Losev-Manin spaces LM N and their birational con-tractions Z N provide evidence that this may be true in general. The exis-tence of such a collection implies that the K-group K ( X ) is a permutation Γ -module. In the Galois setting (when X is defined over a field which isnot algebraically closed and Γ is the absolute Galois group), an analogousstatement was conjectured by Merkurjev and Panin [MP97]. Of course onemay further wonder if { E i } is in fact full, which is related to (non)-existenceof phantom categories on X , another difficult open question.We refer to [CT15, CT13, CT12] for background information on the bira-tional geometry of M ,n , the Losev–Manin space and other related spaces. Organization of paper.
In Section 2 we discuss preliminaries on Hassettspaces and prove some general results on how tautological classes pullback under reduction morphisms. These results are of independent inter-est and have been already used in a crucial way in [CT20]. In Section 3,we discuss the GIT interpretation of the Hassett spaces Z n in the n oddcase and prove Thm. 1.5. In Section 4, we do the same for the n even caseand prove Thm. 1.7. Section 5 serves as an appendix, recalling results onLosev-Manin spaces from [CT17] and calculating the push forward to Z n of the full exceptional collection on the Losev-Manin space LM n . These re-sults are used in Sections 3 and 4 to prove fullness in Theorems 1.5 and 1.7.Throughout the paper, we do not distinguish between line bundles and thecorresponding divisor classes. Acknowledgements.
We are grateful to Alexander Kuznetsov for suggest-ing the problem about the derived categories of moduli spaces of pointedcurves in the equivariant setting. We thank Daniel Halpern–Leistner for hishelp with windows in derived categories.
XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 5
The first author was supported by NSF grants DMS-1529735 and DMS-1701752. The second author was supported by NSF grants DMS-1303415and DMS-1701704. Parts of this paper were written while the first authorwas visiting the Institut des Hautes ´Etudes Scientifiques in France and thesecond author was visiting the Fields Institute in Toronto, Canada.2. P
RELIMINARIES ON H ASSETT SPACES
We refer to [Has03] for background on the Hassett moduli spaces. Recallthat for a choice of weights a = ( a , . . . , a n ) , a i ∈ Q , < a i ≤ , X a i > , 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 . More-over, M a is a smooth projective variety of dimension ( n − . Note that thepolytope of weights has a chamber structure with walls P i ∈ I a i = 1 forevery subset I ⊆ { , . . . , n } . One obtains the Losev-Manin space LM N byconsidering weights on the set of markings { , ∞} ∪ N : (cid:0) , , n , . . . , n (cid:1) , n = | N | . Replacing the weights equal to n with some ǫ ∈ Q , for some < ǫ ≪ ,defines the same moduli problem, hence, gives isomorphic moduli spaces.Similarly, the moduli space Z N of Notn. 1.1 is the moduli space with setof markings { , ∞} ∪ N and weights (cid:0)
12 + η,
12 + η, n , . . . , n (cid:1) , η ∈ Q , < η ≪ . If a = ( a , . . . , a n ) and a ′ = ( b , . . . , 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 morphismwhose exceptional locus consists of boundary divisors δ I (parametrizingreducible curves with a node that disconnects the markings from I and I c )for every subset I ⊆ N such that P i ∈ I a i > , but P i ∈ I b i ≤ . For us aspecial role will be played by the reduction map p : LM N → Z N whichreduced the weights of { , ∞} from 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 thelocus where the map π is smooth, the identity σ i · ω π = − σ i holds on S .Therefore, − ψ i = π ∗ (cid:0) σ i (cid:1) = σ ∗ i σ i . Lemma 2.1.
Assume M is a Hassett space whose universal family π : U → M isa P -bundle. Then the identity − ω π = 2 σ i + π ∗ ( ψ i ) holds on U , and therefore, on M we have for all i = j : ψ i + ψ j = − δ ij . Hence, for all distinct i, j, k , we have ψ i = − δ ij − δ ik + δ jk .Proof. Indeed, − ω π − σ i restricts to the fibers of the P -bundle trivially, andtherefore should have form π ∗ ( L ) for some line bundle on M . Pulling backby σ i shows that L = ψ i . (cid:3) ANA-MARIA CASTRAVET AND JENIA TEVELEV
When n is odd, the universal family U → Z N is a P -bundle and thesections σ and σ ∞ are distinct. Lemma 2.1 has the following: Corollary 2.2.
The following identities hold on Z N when n is odd: ψ = − ψ ∞ = − δ i + δ i ∞ , ψ i = − δ i − δ i ∞ . (2.1) Lemma 2.3.
Let
M = M a , M ′ = M a ′ be Hassett spaces, with a = ( a i ) , a ′ = ( b i ) , a i ≥ b i for all i . Consider the corresponding reduction map p : M ′ → M . Let ( π : U → M , { σ i } ) , ( π ′ : U ′ → M ′ , { σ ′ i } ) be the universal families. Denote by ( ρ : V → M ′ , { s i } ) the pull-back of ( π ′ : U → M , { σ i } ) to M ′ . Then there exists acommutative diagram: U ′ v −−−−→ V q −−−−→ U y π ′ ρ y π y M ′ Id −−−−→ M ′ p −−−−→ M Furthermore, identifying U ′ with a Hassett space M ˜ a , where ˜ a = ( a , . . . , a n , (with an additional marking x with weight ) [Has03, 2.1.1] , we have: v ∗ ω ρ = ω π ′ − X | I |≥ , P i ∈ I a i > , P i ∈ I b i ≤ δ I ∪{ x } ,v ∗ s i = σ i + X i ∈ I, | I |≥ , P i ∈ I a i > , P i ∈ I b i ≤ δ I ∪{ x } ,p ∗ ψ i = ψ i − X i ∈ I, | I |≥ , P i ∈ I a i > , P i ∈ I b i ≤ δ I ,p ∗ δ ij = δ ij + X i,j ∈ I, | I |≥ , P i ∈ I a i > , P i ∈ I b i ≤ δ I . Proof.
The spaces U and U ′ are smooth [Has03, Prop. 5.3 and 5.4]. Theexistence of the commutative diagram follows from semi-stable reduction[Has03, Proof of Thm. 4.1]. The map v is obtained by applying the relativeMMP for the line bundle ω π ′ ( P b i σ ′ i ) . Concretely, the relative MMP resultsin a sequences of blow-downs, followed by a small crepant map: U ′ = S → S → . . . → S r = V , (all over M ′ ). The resulting map v : U ′ → V is a birational map whichcontracts divisors in U ′ to codimension loci in V (as the relative dimensiondrops from to ). Note that V is generically smooth along these loci. The v -exceptional divisors can be identified via U ′ ∼ = M ˜ a with boundary divisors δ I ∪{ x } ( I ⊆ N ), with the property that P i ∈ I a i > , P i ∈ I b i ≤ .For a flat family of nodal curves u : C → B with Gorenstein base B (inour case smooth) the relative dualizing sheaf ω u is a line bundle on C withfirst Chern class K C − u ∗ K B , where K C and K B denote the correspondingcanonical divisors. In particular: ω π ′ = K U ′ − π ′∗ K M ′ , ω ρ = K V − ρ ∗ K M ′ . Since the map v on an open set is the blow-up of codimension loci in V , it follows that K U ′ = v ∗ K V + P E, by the blow-up formula. Hence, XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 7 v ∗ ω ρ = ω π ′ − P E , where the sum runs over all prime divisors E whichare v -exceptional. This proves the first identity. For the second, we identifythe 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 | ≥ ), for δ I ∪{ x } v -exceptional if and only if i ∈ I . Moreover, inthis case, v ( δ I ∪{ x } ) is contained in s i (with codimension ) and s i is smooth(since M ′ is). The second identity follows. By Def. 1.3 and the diagram, p ∗ ψ i = p ∗ σ ∗ i ω π = s ∗ i q ∗ ω π = s ∗ i ω ρ = σ ′ i ∗ v ∗ ω ρ ,p ∗ δ ij = p ∗ σ ∗ i ( σ j ) = s ∗ i q ∗ ( σ j ) = s ∗ i s j = σ ′∗ i v ∗ s j . 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 otherwise. (cid:3) Corollary 2.4.
Let p : LM N → Z N be the reduction map. Let s := j n − k . Then p ∗ ψ = ψ − X I ⊆ N, ≤| I |≤ s δ I ∪{ } ,p ∗ ψ i = − X i ∈ I ⊆ N, ≤| I |≤ s (cid:0) δ I ∪{ } + δ I ∪{∞} (cid:1) ( i ∈ N ) ,p ∗ δ i = X i ∈ I ⊆ N, ≤| I |≤ s δ I ∪{ } ( i ∈ N ) ,p ∗ δ ij = δ ij + X i,j ∈ I ⊆ N, ≤| I |≤ s (cid:0) δ I ∪{ } + δ I ∪{∞} (cid:1) ( i, j ∈ N ) . Lemma 2.5.
On the Losev-Manin space LM N , we have ψ i = 0 for all i ∈ N .Proof. Apply Lemma 2.3 to a reduction map p : M ,N ∪{ , ∞} → LM N : p ∗ ψ i = ψ i − X i ∈ I, | I |≥ , , ∞∈ I c δ I . The right hand side of the equality is [KT09, Lemma 3.4]. Therefore, p ∗ ψ i = 0 . As p ∗ O = O , by the projection formula, we have ψ i = 0 . (cid:3) Proof of Cor. 2.4.
Follows from Lemma 2.3 and Lemma 2.5. In the notationsof the Lemma, the universal family U ′ over M ′ = LM N can be identifiedwith M ˜ a , where ˜ a = (1 , , ǫ, . . . , ǫ, , with an additional marking x with weight. But U ′ can also be identified with LM N ∪{ x } = M (1 , ,ǫ,...ǫ ) (here x has weight ǫ ). Via this identification, boundary divisors δ J correspond toboundary divisors δ J , for any J ⊆ N ∪{ , ∞ , x } . The v -exceptional divisorsappearing in the sum are δ I ∪{ x, } , δ I ∪{ x, ∞} , I ⊆ N , | I | ≤ j n − k . (cid:3) When n = | N | is even, the Hassett space Z N = M ( + η, + η, n ,..., n ) of Notn.1.1 is closely related to the following Hassett spaces: Z ′ N = M ( + ǫ, , n ,..., n ) , Z ′′ N = M ( , + ǫ, n ,..., n ) , with weights assigned to ( ∞ , , p , . . . , p n ) . There exist p ′ : Z N → Z ′ N , p ′ : Z N → Z ′′ N , reduction maps that contract the boundary divisors using ANA-MARIA CASTRAVET AND JENIA TEVELEV the two different projections. The universal families over Z ′ N and Z ′′ N are P -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 thetautological classes on the Hassett space Z N : ψ = δ i ∞ − δ i + X i ∈ T, | T | = n δ T ∪{∞} , ψ ∞ = δ i − δ i ∞ + X i/ ∈ T, | T | = n δ T ∪{∞} ,ψ + ψ ∞ = X | T | = n δ T ∪{∞} . Proof.
The second relation follows from the first using the S 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 denoteby ψ ′ i , δ ′ ij (resp., ψ i , δ ij ) the tautological classes on Z ′ N (resp., on Z N ). Theuniversal family C ′ → Z ′ N is a P -bundle. By Lemma 2.1, we have ψ ′∞ = δ ′ i − δ ′ i ∞ (since δ ′ ∞ = 0 ). The relation follows, as by Lemma 2.3, we have p ′∗ ψ ′∞ = ψ ∞ − X | T | = n δ T ∪{∞} , p ′∗ δ ′ i ∞ = δ i ∞ + X i ∈ T, | T | = n δ T ∪{∞} , p ′∗ δ ′ i = δ i . (cid:3)
3. P
ROOF OF T HEOREM ( P ) nss // G m . For n odd,we first show that the Hassett space Z N introduced in (1.1) can be identifiedwith symmetric GIT quotients ( P ) nss // G m . We use the method of windowsfrom [HL15] to prove exceptionality of the collections in Thm. 1.5. We thenprove that the collection is full, by using the full exceptional collection onthe Losev-Manin spaces LM N (see Section 5).3.1. Generalities on GIT quotients ( P ) nss // G m . Assume n is an arbitrarypositive integer. Let G m = Spec k [ z, z − ] act on A by z · ( x, y ) = ( zx, z − y ) .Let P G m := G m / {± } . Note that P G m acts on P faithfully. Let ∈ P bethe point with homogeneous coordinates [0 : 1] and let ∞ = [1 : 0] .We use concepts of “linearized vector bundles” and “equivariant vectorbundles” interchangeably. For (complexes of) coherent sheaves, we prefer“equivariant”. We endow the line bundle O P ( − with a G m -linearizationinduced by the above action of G m on its total space V O P ( − ⊂ P × A .Consider the diagonal action of G m on ( P ) n . For ¯ j = ( j , . . . , j n ) in Z n ,we denote O (¯ j ) the line bundle O ( j , . . . , j n ) on ( P ) n with G m -linearizationgiven by the tensor product of linearizations above. We denote O ⊗ z k thetrivial 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 + . . . + j n + k is even.There is an action of S × S n on ( P ) n which normalizes the G m action.Namely, S n permutes the factors of ( P ) n and S acts on P by z z − . Thisaction permutes linearized line bundles O (¯ j ) ⊗ z k as follows: S n permutescomponents of ¯ j and S flips k
7→ − k . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 9
Notation 3.1.
Consider the GIT quotient Σ n := ( P ) nss // L G m , L = O (1 , . . . , , with respect to the ample line bundle L (with its canonical G m -linearizationdescribed above). Here ( P ) nss denotes the semi-stable locus with respect tothis linearization. Let φ : ( P ) nss → Σ n denote the canonical morphism.As GIT quotients X// L G are by definition Proj (cid:0) R ( X, L ) G (cid:1) , where R ( X, L ) G is the invariant part of the section ring R ( X, L ) , we may replace L with anypositive multiple. As the action of P G m on ( P ) n is induced from the actionof G m , Σ n is isomorphic to the GIT quotient ( P ) nss // P G m (with respect toany even multiple of L ). The action of S × S n on ( P ) n descends to Σ n .By the Hilbert-Mumford criterion, a point ( z i ) in ( P ) n is semi-stable(resp., stable) if ≤ n (resp., < n ) of the z i equal or equal ∞ .3.2. The space Z N as a GIT quotient when n is odd. When n is odd, thereare no strictly semistable points and the action of P G m on ( P ) nss is free. Inparticular, Σ n is smooth and by Kempf’s descent, any P G m -linearized linebundle on ( P ) nss descends to a line bundle on Σ n . Furthermore, Σ n canbe identified with the quotient stack [( P ) nss /P G m ] and its derived category D b (Σ n ) with the equivariant derived category D bP G m (( P ) nss ) .Consider the trivial P -bundle on ( P ) n with the following sections: ρ : ( P ) n × P = Proj(Sym( O ⊕ O )) → ( P ) n ,s ( z ) = ( z, , s ∞ ( z ) = ( z, ∞ ) , s i ( z ) = ( z, pr i ( z )) , where pr i : ( P ) n → P is the i -th projection. The sections s , resp., s ∞ areinduced by the projection p : O ⊕ O → O , resp., p : O ⊕ O → O , while thesection s i is induced by the map O ⊕ O → pr ∗ i O (1) given by the sections x i = pr ∗ i x, y i = pr ∗ i y of pr ∗ O (1) that define and ∞ on the i -th copy of P . Notation 3.2.
Let ∆ i = pr − i ( { } ) ⊆ ( P ) n and ∆ i ∞ = pr − i ( {∞} ) ⊆ ( P ) n .Note that ∆ i is the zero locus of the section x i , or the locus in ( P ) n where s i = s . Similarly, let ∆ i ∞ the zero locus of the section y i .We now endow all the above vector bundles with G m -linearizations. Let L = O ⊗ z, L ∞ = O ⊗ z − , L i = pr ∗ i O (1) ⊗ , E = L ⊕ L ∞ . The maps L → L i , L ∞ → L i (given by the sections x i , y i ) are G m -equivariant,hence, induce G m -equivariant surjective maps E → L i . The projectionmaps E → L and E → L ∞ are clearly G m -equivariant. While none of E , L , L ∞ , L i are P G m -linearized vector bundles, tensoring with O (1 , . . . , solves this problem, and we obtain a non-trivial P -bundle π : P ( E ) → Σ n with disjoint sections σ , σ ∞ and additional sections σ , . . . , σ n .Denote δ i the locus in Σ n where σ i = σ . This is the zero locus of thesection giving the map L ∞ → L i on Σ n , i.e., the section whose pull-back to ( P ) 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) ⊗ defining ∆ i , ∆ i ∞ descend to globalsections of the corresponding line bundle on Σ n and define δ i , δ i ∞ . Lemma 3.3.
Assume n is odd. We have the following dictionary between linebundles on the GIT quotient Σ n and P G m -linearized line bundles on ( P ) n : O ( δ i ) = pr ∗ i O (1) ⊗ z, O ( δ i ∞ ) = pr ∗ i O (1) ⊗ z − ψ = O ⊗ z − , ψ ∞ = O ⊗ z , ψ i = pr ∗ i O ( − ⊗ . Proof.
The first two formulas follows from the previous discussion: O ( δ i ) corresponds to the P G m -linearized line bundle L i ⊗ L ∨∞ . The remainingformulas follow from Lemma 3.4 the identities (2.1). (cid:3) Lemma 3.4. If n = | N | is odd, the Hassett space Z N (see Notn. 1.1) is isomorphicto the GIT quotient Σ n = ( P ) nss // O (1 ,..., G m . Proof.
The trivial P -bundle ρ : ( P ) nss × P → ( P ) nss with sections s , s ∞ , s i is the pull-back of the P -bundle π : P ( E ) → Σ n and sections σ , σ ∞ , σ i . Since the former is a family of A -stable rational curves, where A = ( + η, + η, n , . . . , n ) , we have an induced morphism f : Σ n → Z N .Clearly, every A -stable pointed rational curve is represented in the familyover ( P ) nss (hence, Σ n ). Furthermore, two elements of this family are iso-morphic 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 aresmooth, f must be an isomorphism. Alternatively, there is an induced mor-phism F : ( P ) nss → Z N which is G m -equivariant (with G m acting triviallyon 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. (cid:3) Exceptionality.
When n is odd, Σ n is a smooth polarized projectivetoric variety for the torus G n − m and its polytope is a cross-section of the n -dimensional cube (the polytope of ( P ) n with respect to L ) by the hy-perplane normal to and bisecting the big diagonal. In particular, the topo-logical Euler characteristic e (Σ n ) is equal to the number of edges of thehypercube intersecting that hyperplane: e (Σ n ) = n (cid:18) n − n − (cid:19) = n (cid:18) n (cid:19) + ( n − (cid:18) n (cid:19) + ( n − (cid:18) n (cid:19) + . . . . By Lemma 3.3, the line bundles { L E,p } in Thm. 1.5 correspond to restric-tions to ( P ) nss of P G m linearized line bundles on ( P ) n L E,p = O ( − E ) ⊗ z p , (3.1)where O ( − E ) = O (¯ j ) , with ¯ j is a vector of ’s and ( − ’s, with − ’s corre-sponding to the indices in E ⊆ N . (Here we abuse notations and we denoteby L E,p both the line bundle on ( P ) n and the corresponding one on Σ n .)The collection is ( S × S n ) -equivariant and consists of e (Σ n ) line bundles. Proof of Thm. 1.5 - exceptionality,
Let G := P G m . We use the method of win-dows [HL15]. We describe the Kempf–Ness stratification [HL15, Section2.1] of the unstable locus ( P ) nus with respect to L . The G -fixed points are Z I = { ( x i ) | x i = 0 for i I, x i = ∞ for i ∈ I } for every subset I ⊆ { , . . . , n } . Let σ I : Z I ֒ → ( P ) n be the inclusionmap. The stratification comes from an ordering of the pairs ( λ, Z ) , where XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 11 λ : G m → G is a -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 µ ( λ, Z ) = − weight λ L| Z | λ | , is decreasing. Here | λ | is a euclidian norm on Hom( G m , G ) ⊗ Z R . We refer[HL15, Section 2.1] for the details. As µ ( λ, Z ) = µ ( λ k , Z ) for any integer k > , it follows that, in our situation, one only has to consider only pairs ( λ, Z I ) and ( λ ′ , Z I ) , for the two -PS λ ( z ) = z and λ ′ ( z ) = z − . Recall that weight λ O ( − | ∞ = +1 , weight λ O ( − | = − , weight λ ( O ⊗ z p ) | q = p for all points q ∈ P . It follows that weight λ ′ O ( − | ∞ = − , weight λ ′ O ( − | = +1 and weight λ L| Z I = | I c | − | I | , weight λ ′ L| Z I = −| I c | + | I | . The unstable locus is the union of the following Kempf–Ness strata: S I = { ( x i ) | x i = ∞ if i ∈ I, x i = ∞ if i / ∈ I } ∼ = A | I c | for | I | > n/ ,S ′ I = { ( x i ) | x i = 0 if i I, x i = 0 if i ∈ I } ≃ A | I | for | I | < n/ . The destabilizing -PS for S I (resp. for S ′ I ) is λ (resp. λ ′ ). The -PS λ (resp., λ ′ ) acts on the conormal bundle N ∨ S I | ( P ) n (resp., N ∨ S ′ I | ( P ) n ) restricted to Z I with positive weights and their sum η I (resp., η ′ I ) can be computed as η I = 2 | I | , resp. η ′ I = 2 | I c | . To see this, note that the sum of λ -weights of (cid:0) N ∨ S I | ( P ) n (cid:1) | Z I equals weight λ (cid:0) det N ∨ S I | ( P ) n (cid:1) | Z I = weight λ (cid:0) det T S I (cid:1) | Z I − weight λ (cid:0) det T ( P ) n (cid:1) | Z I . Note that S I can be identified with A | I c | and the point Z I ∈ S I with thepoint ∈ A | I c | . The action of G on A | I c | is via z · ( x j ) = ( z x j ) . It follows that weight λ T S I | Z I = 2 | I c | . Similarly, the tangent space (cid:0) det T ( P ) n (cid:1) | Z I can beidentified with the tangent space of T A n , with the action of G on ( x j ) ∈ A n being z · x j = z x j if j ∈ I c and z · x j = z − x j if j ∈ I . It follows that weight λ (cid:0) T ( P ) n (cid:1) | Z I = 2 | I c | − | I | . Hence, η I = 2 | I | . Similarly, η ′ I = 2 | I c | .For the Kempf-Ness strata S I and S ′ I we make a choice of “weights” w I = w ′ I = − s, where n = 2 s + 1 . By the main result of [HL15, Thm. 2.10], D bG (( P ) nss ) is equivalent tothe window G w in the equivariant derived category D bG (( P ) n ) , namely afull subcategory of all complexes of equivariant sheaves F • such that allweights (with respect to corresponding destabilizing -PS) of the cohomol-ogy sheaves of the complex σ ∗ I F • lie in the segment [ w I , w I + η I ) or [ w ′ I , w ′ I + η ′ I ) , respectively. We prove that the window G w contains all linearized line bundles L E,p = O ( − E ) ⊗ z p from Thm. 1.5. Recall that n = 2 s + 1 . Since the collectionis S invariant and S flips the strata S I and S ′ I , it suffices to check thewindow conditions for S I . The λ -weight of O ( − E ) ⊗ z p restricted to Z I equals | I ∩ E | − | I c ∩ E | + p . It is straightforward to check that the maximum of this quantity over all E is equal to s + 2 | I | − n + 1 when s is odd, or s +2 | I | − n − when s is even, and the minimum to − s , hence the claim. Sinceour collection of linearized line bundles is clearly an exceptional collectionon D bG (( P ) n ) , it follows it is an exceptional collection in D bG ( Z n ) . (cid:3) Fullness.
We will prove the following general statement.
Theorem 3.5.
The collection in Thm. 1.5 generates all line bundles L E,p := O ( − E ) ⊗ z p , for all E ⊆ N , e = | E | , p ∈ Z with e + p even.Proof of Thm. 1.5 - fullness. By Thm. 3.5, the collection in Thm. 1.5 generatesall the objects Rp ∗ ( π ∗ I ˆ G ) from Cor. 5.11. Fullness then follows by Cor. 5.5.Alternatively, it is easy to see that line bundles L E,p generate derived cate-gory of the stack [( P ) n /P G m ] and we can finish as in [CT20, Prop. 4.1]. (cid:3) Proof Thm. 3.5.
For simplicity, denote by C the collection in the theorem. Weintroduce 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 statementby induction on the score s ( E, p ) , and for equal score, by induction on | p | .Let ( E, p ) be any pair as in Thm. 3.5. If s ( E, p ) ≤ s , there is nothing toprove. Assume s ( E, p ) > s . Using S -symmetry, we may assume w.l.o.g.that p ≥ . We will use the two types of P G m -equivariant Koszul resolu-tions 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 L E,p = O ( − E ) ⊗ z p , gives an exact sequence → L E ∪ I,p − s − → . . . → M J ⊆ I, | J | = j L E ∪ J,p − j → . . . → L E,p → . We prove that each term L E ∪ J,p − j is generated by C for all j > . Note that s ( E, p ) = | p | + e = p + e . If p − j ≥ , then s ( E ∪ J, p − j ) ≤ ( p − j ) + ( e + j ) = p + e = s ( E, p ) , but as p − j < p , we are done by induction on | p | . If p − j < then s ( E ∪ J, p − j ) ≤ ( j − p ) + n − ( e + j ) = n − e − p < e + p = s ( E, p ) 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 Lemma3.6 for the set I , followed by tensoring with L E ′ ,p − s − = O ( E ′ ) ⊗ z p − s − ,where E ′ = E \ I , gives an exact sequence → L E,p → . . . → M J ⊆ I, | J | = j L E ′ ∪ J,j + p − s − → . . . → L E ′ ,p − s − → . We prove that each term L E ∪ J,j + p − s − is generated by C for all J = I (when ( E ′ ∪ J, j + p − s −
1) = (
E, p ) ). Note that s ( E, p ) = p + n − e . We let e ′ := | E ′ | = e − s − . If j + p − s − ≥ , then s ( E ′ ∪ J, j + p − s − ≤ ( j + p − s −
1) + ( n − e ′ − j ) = p + n − e = s ( E, p ) . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 13 As p + j − s − ≤ p with equality if and only if J = I , we are done byinduction on | p | . If j + p − s − < , then s ( E ′ ∪ J, j + p − s − ≤ − ( j + p − s − e ′ + j ) = e − p < s ( E, p ) = p + n − e, since we assume s ( E, p ) > s , which gives e − p ≤ s . (cid:3) Lemma 3.6.
Let n = 2 s + 1 , I ⊆ N , | I | = s + 1 . There are two types of P G m -equivariant resolutions:(1) The restriction to ( P ) nss of the Koszul complex of the intersection of thedivisors ∆ i (Notation 3.2) for i ∈ I , which takes the form → O ( − I ) ⊗ z − ( s +1) → . . . → M J ⊆ I, | J | = j O ( − J ) ⊗ z − j → . . . → O ⊗ → (2) The restriction to ( P ) nss of the Koszul complex of the intersection of thedivisors ∆ i ∞ (Notation 3.2) for i ∈ I , which takes the form → O ( − I ) ⊗ z ( s +1) → . . . → M J ⊆ I, | J | = j O ( − J ) ⊗ z j → . . . → O ⊗ → Proof.
Let G = P G m . Denote for simplicity D i = ∆ i , for all i ∈ N .The divisors D , . . . , D n intersect with simple normal crossings. Let Y I := ∩ i ∈ I D i ⊆ ( P ) n . Consider the Koszul resolution of Y I : . . . → ⊕ i We explain the connection with case p = 2 , q = n = 2 s + 1 of[CT20, Thm. 1.10]. The collection there is the following:(i) The line bundles F ,E := − P j ∈ E ψ j ( e = | E | is even) in the so-calledgroup (group A and group B of the theorem coincide in this case).(ii) The line bundles in the so-called group : T l, { u }∪ E := σ ∗ u (cid:0) ω e +1 − l π ( E ∪{ u } ) (cid:1) = e − l − ψ u + X j ∈ E δ ju = − l + 12 ψ u − X j ∈ E ψ j where e = | E | , u ∈ { , ∞} , l ≥ , l + | E ∩ { u }| even (i.e., l + e odd), with l + min { e, n − e } ≤ s − . This collection is the dual of the one in Thm. 1.5. The elements in group with l = p − , u = ∞ recover the dual of the collection in Thm. 1.5 when p > . Similarly, elements in group with l = − p − , u = 0 recover thedual of the collection in Thm. 1.5 when p < . The elements of group recover the dual of the collection in Thm. 1.5 when p = 0 .4. P ROOF OF T HEOREM Z N (see (1.1)) when n = | N | is even with the Kirwan resolution ofthe symmetric GIT quotient Σ n . We use the method of windows [HL15] toprove the exceptionality part of Thm. 1.7. We prove fullness using previousresults on Losev-Manin spaces LM N (see Section 5).4.1. The space Z N as a GIT quotient, n even. Assume n = 2 s + 2 . Thereare (cid:0) ns +1 (cid:1) strictly semistable points { p T } ∈ ( P ) nss one for each subset T ⊆ N , | T | = s + 1 . More precisely, the point p T is obtained by taking ∞ forspots in T and for spots in T c . Instead of the GIT quotient Σ n , which issingular at the images of these points, we consider its Kirwan resolution ˜Σ n constructed as follows.Let W = W n be the blow-up of ( P ) n at the points { p T } and let { E T } be the corresponding exceptional divisors. The action of G m lifts to W . Todescribe this action locally around a point p T , assume for simplicity T = { s + 2 , . . . , n } around the point p T . Consider the affine chart A n = ( P \ {∞} ) s +1 × ( P \ { } ) s +1 In the new coordinates, we have p T = 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 − . Then W is locally the blow-up Bl A n , with equations x i t j = x j t i , x i u j = y j t i , y i u j = y j u i . The action of G m on W is given by z · (cid:0) ( x i , y i ) , [ t i , u i ] (cid:1) = (cid:0) ( z x i , z − y i ) , [ z t i , z − u i ] (cid:1) . The fixed locus of the action of G m on E T consists of the subspaces Z + T = { u = . . . = u s +1 = 0 } = P s ⊆ P n − = E T ,Z − T = { t = . . . = t s +1 = 0 } = P s ⊆ P n − = E T . As Bl A n is the total space V ( O E T ( − of the line bundle O E T ( − 1) = O E T ( E T ) and the action of G m on Bl A n coincides with the canonical actionof G m on V ( O E T ( − coming from the action of G on E T = P n − given by z · [ t , . . . , t s +1 , u , . . . , u s +1 ] = [ z t , . . . , z t s +1 , z − u , . . . , z − u s +1 ] . It follows that O E T ( E T ) (and hence, O ( E T ) ) has a canonical G m -linearization.With respect to this linearization, we have:weight λ O E T ( − | q = weight λ O ( E T ) | q = +2 , q ∈ Z + T , λ ( z ) = z, (4.1)weight λ O E T ( − | q = weight λ O ( E T ) | q = − , q ∈ Z − T , λ ( z ) = z. and similarly,weight λ ′ O E T ( − | q = weight λ ′ O ( E T ) | q = − , q ∈ Z + T , λ ′ ( z ) = z − . weight λ ′ O E T ( − | q = weight λ ′ O ( E T ) | q = +2 , q ∈ Z − T , λ ′ ( z ) = z − . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 15 We denote by O (¯ j )( P α T E T ) the line bundle π ∗ O ( j , . . . , j n )( P α T E T ) on W n (where j i , α T integers and π : W n → ( P ) n is the blow-up map), withthe G m -linearization given by the tensor product of the canonical lineariza-tions above. As before, for every equivariant coherent sheaf F , we denoteby F ⊗ z k the tensor product with O ⊗ z k . For a subset E ⊆ N , we denote O ( − E ) := π ∗ O (¯ j ) with j i = − if i ∈ E and j i = 0 otherwise. Note that the action of S exchanges O ( − E ) ⊗ z p with O ( − E ) ⊗ z − p and E T with E T c (Lemma 4.4).Consider the GIT quotient with respect to a (fractional) polarization L = O (1 , . . . , (cid:16) − ǫ X E T (cid:17) , where < ǫ ≪ , ǫ ∈ Q , and the sum is over all exceptional divisors (withthe canonical polarization described above): ˜Σ n = ( W n ) ss // L G m . Lemma 4.1. The G m -linearized line bundle O (¯ j )( P α T E T ) ⊗ z p descends to theGIT quotient ˜Σ n if and only if for all subsets I ⊆ N with | I | 6 = s + 1 − X i ∈ I j i + X i ∈ I c j i + p is evenand for all subsets I ⊆ N with | I | = s + 1 , we have − X i ∈ I j i + X i ∈ I c j i + p ± α I is divisible by . Proof. By Kempf’s descent, a G -linearized line bundle L descends to theGIT quotient if and only if the stabilizer of any point in the semistable locusacts trivially on the total space of L , or equivalently, weight λ L | q = 0 , for anysemistable point q and any -PS λ : G m → G . By definition, weight λ L | q = weight λ L | p , where p is the fixed point lim t → λ ( t ) · q .For any point q in ( P ) 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 → λ ( t ) · q is the pointwith coordinates z i = ∞ for i ∈ I and z i = 0 for i ∈ I c , and hence:weight λ (cid:0) O (¯ j )( X α T E T ) ⊗ z p (cid:1) | q = − X i ∈ I j i + X i ∈ I c j i + p. (4.2)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 for i ∈ I , λ ′ ( z ) = z − :weight λ ′ (cid:0) O (¯ j )( X α T E T ) ⊗ z p (cid:1) | q = X i ∈ I j i − X i ∈ I c j i + p. Note, q is semistable iff | I | > s + 1 . The stabilizer of q is {± } in both cases.If q ∈ E T \ ( Z + T ⊔ Z − T ) then lim t → λ ( t ) · q ∈ Z − T , lim t → λ ′ ( t ) · q ∈ Z + T andusing (4.1) we obtainweight λ (cid:0) O (¯ j )( X α T E T ) ⊗ z p (cid:1) | q = − X i ∈ I j i + X i ∈ I c j i + p − α T , weight λ ′ (cid:0) O (¯ j )( X α T E T ) ⊗ z p (cid:1) | q = X i ∈ I j i − X i ∈ I c j i + p − α T . A point q ∈ E T \ ( Z + T ⊔ Z − T ) has stabilizer {± , ± i } . The conclusion follows. (cid:3) Corollary 4.2. For E ⊆ N , p ∈ Z , the line bundle O ( − E )( P α T E T ) ⊗ z p descends to the GIT quotient ˜Σ n if and only if for all subsets I ⊆ N with | I | 6 = s +1 | I ∩ E | − | I c ∩ E | + p is evenand for all subsets I ⊆ N with | I | = s + 1 , we have | I ∩ E | − | I c ∩ E | + p − α I is divisible by . Lemma 4.3. If n = | N | is even, the Hassett space Z N = M , ( + η, + η, n ,..., n ) isisomorphic to the GIT quotient ˜Σ n = ( W n ) ss // O (1 ,..., − ǫ P E T ) G m . Proof. The trivial P -bundle ( P ) n × P → ( P ) n has sections s , s ∞ , s i ,which we pull back to W ss × P → W ss which we denote still by s , s ∞ , s i .The family is not A -stable at the points p T , where s i = s ∞ for all i ∈ T and s i = s for all i ∈ T c (markings in T are identified with ∞ , and markingsin T c with ). Here A = ( + η, + η, n , . . . , n ) . Let C ′ be the blow-up of W × P along the codimension loci E T × { } = s ( E T ) , E T × {∞} = s ∞ ( E T ) . Denote by ˜ E T and ˜ E ∞ T be the corresponding exceptional divisors in C ′ . Theresulting family π ′ : C ′ → W has fibers above points p ∈ E T a chain of P ’s of the form C ∪ ˜ F ∪ C ∞ , where ˜ F is the proper transform of the fiberof W × P → W and ˜ F meets each of C (the fiber of ˜ E T → E T at p ) and C ∞ (the fiber of ˜ E ∞ T → E T at p ). The proper transforms of s i for i ∈ T (resp., i ∈ T c ) intersect C ∞ (resp., C ) at distinct points. The dualizingsheaf ω π ′ is relatively nef, with degree on ˜ F . It follows that ω π ′ inducesa morphism C ′ → C over W ss which contracts the component ˜ F in eachof the above fibers, resulting in an A -stable family. Therefore, we have aninduced 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 quotient ˜Σ n is a categoricalquotient, there is an induced morphism f : ˜Σ n → Z N . Two elements of thefamily C → W ss are isomorphic if and only if they belong to the same orbitunder 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 categoricalquotient [Dol03, p. 94]). It follows that f is an isomorphism. (cid:3) Lemma 4.4. Assume n = 2 s + 2 is even. We have the following dictionarybetween tautological line bundles on the Hassett space Z N (idenitified with theGIT quotient ˜Σ n ) and G m -linearized line bundles on W n : O ( δ i ) = pr ∗ i O (1)( − X i/ ∈ T E T ) ⊗ z, O ( δ i ∞ ) = pr ∗ i O (1)( − X i ∈ T E T ) ⊗ z − ψ = O ( X E T ) ⊗ z − , ψ ∞ = O ( X E T ) ⊗ z , ψ i = pr ∗ i O ( − X E T ) ⊗ , O ( δ T ∪{∞} ) = O (2 E T ) ⊗ | T | = s + 1) . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 17 Proof. Denote δ T = δ T ∪{∞} . We start with the proof of O ( δ T ) = O (2 E T ) ⊗ .Consider the affine chart A n = ( P \ {∞} ) s +1 × ( P \ { } ) s +1 around the point p T (markings in T = { s + 2 , . . . , n } are identified with ∞ ,and markings in T c with ). We have coordinates x , . . . , x s +1 , y , . . . , y s +1 .The GIT quotient map ( P ) nss → Σ is locally at p T given by f : A n → Y = f ( A n ) ⊆ A ( s +1) , f (( x i ) , ( y j )) = ( x i y j ) ij . 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 ofthe rational map (which we still call F ) F : Bl A n Bl Y ⊆ Bl A ( s +1) . 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 A n ⊆ A n × P n − and coordinates ( z ij , [ w ij ]) on Bl A ( s +1) (with z ij w kl = z kl w ij ). Consider the affine charts U = { t = 0 } ⊆ Bl A n and V j = { w j =0 } ⊆ Bl A r . The map F | U is the rational map F : U = A nx ,t ,...,t r ,u ,...,u r V j = A r z j , ( w kl ) kl =1 j ,z j = x u j , w kl = t k u l u j . The exceptional divisor ˜ E in Bl A ( s +1) has local equation z j = 0 in V j ,while the exceptional divisor E T of Bl A n has equation x = 0 in U . Itfollows that F ∗ O ( ˜ E ) = O (2 E T ) . In particular, as δ T = Bl Y ∩ ˜ E , it followsthat F ∗ O ( δ T ) = O (2 E T ) . It follows that O ( δ T ) = O (2 E T ) ⊗ z k , for someinteger k (the same for all T , by the S n -symmetry). On the other hand, bythe S -symmetry, O ( δ T c ) = O (2 E T c ) ⊗ z − k . Hence, we must have k = 0 .We now prove that O ( δ i ) = pr ∗ i O (1)( − P i/ ∈ T E T ) ⊗ z . (Note that allother relations will then follow by S -symmetry and Lemma 2.6.) Clearly, F ∗ O ( δ i ) is the line bundle O ( ˜∆ i ) | W ss , where ˜∆ i is the proper transformin W of the diagonal ∆ i in ( P ) n defined by x i = 0 , where z i = [ x i , y i ] nowdenote coordinates on ( P ) n . As ˜∆ i = ∆ i − P i/ ∈ T E T (markings in T c areidentified with ), it follows that O ( δ i ) = pr ∗ i O (1)( − X i/ ∈ T E T ) ⊗ z k , for some integer k . The pull-back of the canonical section of the effectivedivisor δ i (which is x i ) must be an invariant section. The section x i of O P (1) becomes the constant section in the open chart U : x i = 0 . Con-sidering a point q = ( q , . . . , q n ) in U , with q i = ∞ and q j ∈ P general for j = i , it follows that for the -PS λ ( z ) = z we have weight λ pr ∗ i O (1) | q = − ,weight λ O ⊗ z k | q = k , hence, the constant section becomes z − k under theaction of λ and we must have k = 1 for the section to be invariant. (cid:3) Lemma 4.5. Let δ T := δ T ∪{∞} = P s × P s . We have δ i ∞| δ T = ( O (1 , if i ∈ T O if i / ∈ T , δ i | δ T = ( O (0 , if i / ∈ T O if i ∈ T, ,ψ ∞| δ T = O ( − , , ψ | δ T = O (0 , − , δ T | δ T = O ( − , − . Proof. By symmetry, it suffices to compute δ i ∞| δ T and ψ ∞| δ T . Clearly, theintersection δ i ∞ ∩ δ T = ∅ if i / ∈ T . We identify δ T = M ′ × M ′′ = P s × P s ,where M ′ , resp., M ′′ are Hassett spaces with weights ( + η, n , . . . , n , ,with the attaching point x having weight . We identify M ′ = P s via theisomorphism | ψ x | : M ′ → P s . We have δ i ∞| δ T = δ i ∞ ⊗ O , ψ ∞| δ T = ψ ∞ ⊗ O .By Lemma 2.1, on M ′ we have ψ ∞ + ψ x = 0 since δ x ∞ = 0 , and δ i ∞ = − ψ ∞ = O (1) if i ∈ T . The identity δ T | δ T = O ( − , − follows now fromthe previous ones by restricting to δ T any of the identities in Lemma 2.6. (cid:3) Exceptionality. Note that W n is a polarized toric variety with the poly-tope ∆ obtained by truncating the n -dimensional cube at vertices lying onthe hyperplane H normal to and bisecting the big diagonal. Then ˜Σ n is asmooth polarized projective toric variety for the torus G n − m and its poly-tope is ∆ ∩ H . In particular, the topological Euler characteristic e ( ˜Σ n ) isequal to the number of edges ∆ intersecting H : e ( ˜ Z n ) = ( s + 1) (cid:18) ns + 1 (cid:19) = s (cid:18) ns + 1 (cid:19) + ( n − (cid:18) ns + 1 (cid:19) ( n = 2 s + 2) . Note that ( s + 1) (cid:0) ns +1 (cid:1) = n (cid:0) n (cid:1) + ( n − (cid:0) n (cid:1) + ( n − (cid:0) n (cid:1) + . . . + 2 (cid:0) ns (cid:1) . Definition 4.6. For E ⊆ N , e = | E | , p ∈ Z such that p + e is even, let L E,p := O ( − E )( X T ⊆ N, | T | = s +1 α T,E,p E T ) ⊗ z p where α T,E,p := −| x T,E,p | , x T,E,p := | E ∩ T | − e − p (4.3)i.e., the descent to ˜Σ n of the restriction to ( W n ) ss of the above G m -linearizedline bundle on W n . By Lemma 4.4 we recover Def. 1.6: L E,p = − (cid:18) e − p (cid:19) ψ ∞ − X i ∈ E δ i ∞ − X x T,E,p > x T,E,p δ T ∪{∞} . (4.4)We write x T if there is no ambiguity. Note that x T,E,p = − x T c ,E, − p . Lemma 4.7. The action of S on Z N exchanges L E,p with L E, − p .Proof. The statement follows immediately from (4.4) and Lemma 2.6. (cid:3) Proof of Thm. 1.5 - exceptionality. Lemma 4.9 implies that the torsion sheaves O δ ( − a, − b ) form an exceptional collection. Let now δ := δ T ∪{∞} . To provethat {O δ ( − a, − b ) , L E,p } form an exceptional pair, i.e., that L ∨| δ ⊗ O ( − a, − b ) is acylic, note that by Lemma 4.5 and (4.6) we have, letting α T := α T,E,p : L ∨| δ = ( O (0 , α T ) if p + | E ∩ T | − | E ∩ T c | ≥ O ( α T , if p + | E ∩ T | − | E ∩ T c | ≤ , . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 19 Clearly, if a, b > then L ∨| δ ⊗ O ( − a, − b ) is acylic. Consider now the casewhen one of a, b is . Using the S -symmetry, we may assume a = 0 . Let < b < s +12 . Since by (4.5) we have −⌊ s +12 ⌋ ≤ α T ≤ , the result follows.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 -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 − . The G -fixed locus in W = W n consists of the points Z I = { ( x i ) | x i = ∞ for i ∈ I, x i = 0 for i I, } ∈ ( P ) n \ { p T } for every subset I ⊆ N with | I | 6 = s + 1 and the loci Z + T ⊔ Z − T ⊆ E T , for eachsubset T ⊆ N , | T | = s + 1 . The pairs ( λ, Z ) to be considered are therefore ( λ, Z I ) , ( λ ′ , Z I ) ( I ⊆ N, | I | 6 = s + 1) , ( λ, Z + T ) , ( λ ′ , Z + T ) , ( λ, Z − T ) , ( λ ′ , Z − T ) ( T ⊆ N, | T | = s + 1) . Recall that our polarization is L = O (1 , . . . , − ǫ P E T ) and for any subset I ⊆ N with | I | 6 = s + 1 we have weight λ L| Z I = | I c | − | I | , weight λ ′ L| Z I = −| I c | + | I | , while for all subsets T ⊆ N with | T | = s + 1 we have: weight λ L| q = − ǫ, weight λ ′ L| q = +2 ǫ ( q ∈ Z + T ) , weight λ L| q = +2 ǫ, weight λ ′ L| q = − ǫ ( q ∈ Z − T ) . As in the n odd case, we define for any subset I ⊆ N affine subsets: S I = { ( x i ) | x i = ∞ if i ∈ I, x i = ∞ if i / ∈ I } ∼ = A | I c | S ′ I = { ( x i ) | x i = 0 if i I, x i = 0 if i ∈ I } ∼ = A | I | . 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) ,S + T = Bl p T S T = Bl A | T c | , S − T = Bl p T S ′ T = Bl A | T | ( for | T | = s + 1) . The destabilizing -PS for S I (resp., for S ′ I ) is λ (resp. λ ′ ). The -PS λ (resp., λ ′ ) acts on the restriction to Z I of the conormal bundle N ∨ S I | ( P ) n (resp., N ∨ S ′ I | ( P ) n ) with positive weights. Their sum η I (resp., η ′ I ) is: η I = 2 | I | , resp., η ′ I = 2 | I c | . When | T | = s + 1 , the destabilizing -PS for S + T (resp. for S − T ) is λ (resp. λ ′ ). The -PS λ (resp., λ ′ ) acts on N ∨ S + T | W (resp., N ∨ S − T | W ) restrictedto q ∈ Z + T (resp., Z − T ), with positive weights. Their sum η + T (resp., η − T ) is: η + T = 4 | T | = 2 n, resp., η − T = 4 | T c | = 2 n. To see this, let q ∈ Z + T . The sum of λ -weights of (cid:0) N ∨ S + T | W (cid:1) | q equals weight λ (cid:0) det N ∨ S + T | W (cid:1) | q = weight λ (cid:0) det T S + T (cid:1) | q − weight λ (cid:0) det T W (cid:1) | q . We use the local coordinates introduced in 4.1 (assume again w.l.o.g that T = { s + 2 , . . . , n } ). We may assume also that the point q = [ t , . . . , t s +1 , . . . , ∈ Z + T ⊆ E T = P n − has t = 1 . Then local coordinates on an open set U = A n ⊆ W around q are given by x , t , . . . , t s +1 , u , . . . , u s +1 , with the blow-up map A n → A n : ( x , t , . . . , t s +1 , u , . . . , u s +1 ) ( x , x t , . . . , x t s +1 , x u , . . . , x u s +1 ) . Then S + T ∩ U ⊆ U has equations u = . . . = u s +1 (the proper transform of S T : y = . . . = y s +1 ). The action of G on W induces an action on U : z · x = z x , z · t i = t i ( i = 2 , . . . , s + 1) , z · t i = z − t i ( i = 1 , . . . , s + 1) . It follows that weight λ (cid:0) det T W (cid:1) | q = − − s, weight λ (cid:0) det T S + T (cid:1) | q = 2 . Hence, η T = 4 s + 4 = 2 n . Similarly, η ′ T = 2 n : for q ∈ Z − T and coordinates y , t , . . . , t s +1 , u , . . . , u s +1 on the chart u = 1 , the action of G given by: z · y = z − y , z · t i = z t i ( i = 1 , . . . , s + 1) , z · u i = u i ( i = 2 , . . . , s + 1) . Letting m := j n k = j s +12 k , we make a choice of windows G w : [ w I , w I + η I ) , [ w ′ I , w ′ I + η ′ I ) , [ w + T , w + T + η + T ) , [ w − T , w − T + η − T ) ,w I = w ′ I = − ( s + 1) , w + T = w − T = − m = − n if s is odd ,w I = w ′ I = − s, w + T = w − T = − m = − n + 2 if s is even . We prove that G w contains the G -linearized line bundles that descend tothe L E,p in Thm. 1.7. Since the collection is S invariant and S flips thestrata S I and S ′ I , it suffices to check the window conditions for the strata S I , S + T . For I ⊆ N , | I | > s + 1 , at the point Z I ∈ S I we have by (4.2) weight λ (cid:0) L E,p (cid:1) | Z I = | E ∩ I | − | E ∩ I c | + p, which lies in [ w I , w I + η I ) by Lemma 4.10.For T ⊆ N with | T | = s + 1 , we have by (4.1) and (4.2) that weight λ (cid:0) L E,p (cid:1) | q ∈ Z + T = | E ∩ T | − | E ∩ T c | + p − | x T | == ( if | E ∩ T | − | E ∩ T c | + p ≥ − | x T | if | E ∩ T | − | E ∩ T c | + p ≤ . , which by (4.5) lies in [ w + T , w + T + η + T ) . Hence, all { L E,p } in Thm. 1.7 arecontained in the window G w .We now check exceptionality. Consider two line bundles as in Thm. 1.7: L E,p = O ( − E )( X | T | = r α T E T ) ⊗ z p , L E ′ ,p ′ = O ( − E ′ )( X | T | = r α ′ T E T ) ⊗ z p ′ . 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, Thm. 2.10], wehave that R Hom( L E ′ ,p ′ , L E,p ) equals the weight ( p ′ − p ) part (with respectto the canonical action of G ) of R Hom W ( L E ′ ,p ′ , L E,p ) = R Γ( O ( E ′ − E ) ⊗ O ( X T ( α ′ T − α T ) E T )) . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 21 Hence, letting M := O ( E ′ − E ) ⊗ O (cid:0) X β T ≤ ( − β T ) E T (cid:1) , where β T := α T − α ′ T , we need to understand the weight ( p ′ − p ) part of R Γ (cid:0) M ⊗ O (cid:0) X | T | = r,β T > ( − β T ) E T (cid:1)(cid:1) . Note that M is a pull-back from ( P ) n ; hence, by the projection formula, R Γ( M ) = R Γ( O ( E ′ − E )) (which is if E * E ′ ).Consider a simplified situation. For a line bundle M on W , G := E T , β := β T > consider the exact sequences: → M ( − ( i + 1) G ) → M ( − iG ) → M ( − iG ) | G → , ( i = 0 , . . . β − . To prove that the weight ( p ′ − p ) of R Γ( M ( − βG )) is , it suffices to prove R Γ( M ) , R Γ( M ( − iG ) | G ) ( i = 0 , , . . . , β − , have no weight ( p ′ − p ) part. Put an arbitrary order on the subsets T with β T > ( T , T , . . . ). Applying the above observation successively, first for M , E T , then inductively for M ( − β T − . . . − β i T i ) , E T i +1 , it suffices toprove that for all T , the following spaces R Γ( M ) , R Γ( M ( − iE T ) | E T ) ( i = 0 , , . . . , β − have no weight ( p ′ − p ) part.We start with R Γ( M ) . If E = E ′ , then R Γ ∗ ( M ) = 0 . If E = E ′ , then M = O and the action of G on R Γ( M ) is trivial. Hence, unless p = p ′ (i.e., L E,p = L E ′ ,p ), R Γ( M ) has no weight ( p ′ − p ) part.We now continue with R Γ( M ( − iE T ) | E T ) . By the projection formula, R Γ( M ( − iE T ) | E T ) = M | p T ⊗ R Γ( O ( − iE T ) | E T ) , where M | p T is the fiber of M at p T (we denote M both the line bundle on ( P ) n and its pull back to W ). By (4.2), the action of G on M | p T has weight (cid:0) | E ∩ T | − | E ∩ T c | (cid:1) − (cid:0) | E ′ ∩ T | − | E ′ ∩ T c | (cid:1) . Consider coordinates t i , u i on E = P n − , such that t i (resp., u i ) have weight (resp., weight − ). There is a canonical identification R Γ( O ( − iE T ) | E T ) = C { Y t a k k Y u b k k | a k , b k ∈ Z ≥ , X a k + X b k = i } , with the weight of Q t a k k Q u b k k equal to P a k − P b k . As P a k − P b k ranges through all even numbers between − i and i , it follows that thepossible weights of elements in R Γ( M ( − iE T ) | E T ) are (cid:0) | E ∩ T | − | E ∩ T c | (cid:1) − (cid:0) | E ′ ∩ T | − | E ′ ∩ T c | (cid:1) + 2 j, for all the values of j between − i and i .Assume now that for some ≤ i ≤ β T − α T − α ′ T − , − i ≤ j ≤ i , (cid:0) | E ∩ T | − | E ∩ T c | (cid:1) − (cid:0) | E ′ ∩ T | − | E ′ ∩ T c | (cid:1) + 2 j = p ′ − p. Using the definition of α T , α T ′ , it follows that ± α T ± α ′ T = − j . Claim 4.8. None of ± α T ± α ′ T lies in the interval [ − ( α T − α ′ T − , ( α T − α ′ T − . Proof. By symmetry, it is enough to prove that none of ± α T ± α ′ T lies in theinterval [0 , ( α T − α ′ T − . As α T , α ′ T ≤ and α T > α ′ T . Hence, it remainsto prove that − α T − α ′ T , α T − α ′ T do not lie in the interval [0 , ( α T − α ′ T − .But clearly, − α T − α ′ T > α T − α ′ T − and α T − α ′ T > α T − α ′ T − . (cid:3) This finishes the proof that the collection in Thm. 1.7 is exceptional. (cid:3) Lemma 4.9. Let ≤ a, b ≤ s . Let δ be a divisor in a Hassett space M such that δ = P s × P s and with normal bundle O ( − , − . Assume that the restrictionmap Pic(M) → Pic( δ ) is surjective. Then {O δ ( − a, − b ) , O δ ( − a ′ , − b ′ ) } is not anexceptional collection if and only if one of the following happens: • a ′ ≥ a , b ′ ≥ b , • a ′ = 0 , a = s , b ′ > b , • b ′ = 0 , b = s , a ′ > a , • a ′ = b ′ = 0 , a = b = s .When a = a ′ , b = b ′ , we have R Hom( O δ ( − a ′ , − b ′ ) , O δ ( − a, − b )) = C .Proof. As any line bundle on δ is the restriction of a line bundle on M , wehave that R Hom( O δ ( − a ′ , − b ′ ) , O δ ( − a, − b )) = R Hom( O δ , O δ ( a ′ − a, b ′ − b )) . Applying R Hom( − , O δ ( a ′ − a, b ′ − b )) to the canonical sequence → O ( − δ ) → O → O δ → , it follows that there is a long exact sequence on M . . . → Ext i ( O δ , O δ ( a ′ − a, b ′ − b )) →→ H i ( O δ ( a ′ − a, b ′ − b )) → H i ( O δ ( a ′ − a − , b ′ − b − → . . . It is clear now that if any of the conditions in the Lemma hold, then R Hom( O δ ( − a ′ , − b ′ ) , O δ ( − a, − b )) = 0 . Assume now that none of the conditions holds. Then either a ′ < a or b ′ < b .Assume a ′ < a . Since a ′ − a ≥ − a ≥ − s , O δ ( a ′ − a, b ′ − b ) is acyclic. But inthis case O δ ( a ′ − a − , b ′ − b − is not acyclic if and only if a ′ = 0 , a = s andeither b ′ − b > or b ′ − b ≤ − s (in which case, we must have b ′ = 0 , b = s ).This gives precisely two of the listed cases. The case b ′ < b is similar. (cid:3) Lemma 4.10. Let n = 2 s + 2 . For a fixed set I ⊆ N with | I | > s + 1 , we have max ( E,p ) (cid:0) | E ∩ I | − | E ∩ I c | + p (cid:1) = ( | I | − ( s + 3) if s is odd | I | − ( s + 2) if s is even , min ( E,p ) (cid:0) | E ∩ I | − | E ∩ I c | + p (cid:1) = ( − ( s + 1) if s is odd − s if s is even , where the maximum and the minimum are taken over all the pairs ( E, p ) corre-sponding to each line bundle L E,p in Thm. 1.7. Similarly, for T ⊆ N , | T | = s + 1max ( E,p ) (cid:0) p + | E ∩ T | − | E ∩ T c | (cid:1) = 2 m, min ( E,p ) (cid:0) p + | E ∩ T | − | E ∩ T c | (cid:1) = − m, where m := j n k = j s + 12 k . In particular, when ( E, p ) are as in Thm. 1.7, the coefficients α T,E,p in (4.3) satisfy − m ≤ α T,E,p = −| x T,E,p | ≤ (4.5) XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 23 The proof is straightforward and we omit it.4.3. Fullness. Let C be the collection in Thm. 1.7. We denote by A ⊂ C thecollection of torsion sheaves in Thm. 1.7. We prove more generally: Theorem 4.11. The collection C in Thm. 1.7 generates all line bundles { L E,p } (see Def. 1.6 and Def. 4.6) for all E ⊆ N , e = | E | , p ∈ Z with e + p even.Proof of Thm. 1.7 - fullness. By Thm. 4.11, the collection C generates all theobjects Rp ∗ ( π ∗ I ˆ G ) from Cor. 5.11. Fullness then follows by Cor. 5.5. (cid:3) To prove Thm. 4.11 we do an induction on the score S ( E, p ) : S ( E, p ) := | p | + min { e, n − e } , (4.6)written as S ( E, p ) = 2 j s k + 2 q, q ∈ Z . (4.7) Remark 4.12. As S ( E, p ) is even, the range of ( E, p ) in Thm. 1.7 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 − if e ≥ s + 2 .Using notation (4.7), ( E, p ) is not in the range of Thm. 1.7 if q ≥ when s is even or s is odd and e ≥ s + 2 , and if q ≥ when s is odd and e ≤ s + 1 .To prove Thm. 4.11 we introduce three other types of line bundles. Notation 4.13. Let n = 2 s + 2 , E ⊆ N , e = | E | and p ∈ Z . On Z N let R E,p = − (cid:18) e − p (cid:19) ψ ∞ − X i ∈ E δ i ∞ , Q E,p = − (cid:18) e + p (cid:19) ψ − X i ∈ E δ i ,V E,p := R E,p + X x T,E,p < | x T,E,p | δ T ∪{∞} = Q E,p + X x T,E,p > | x T,E,p | δ T ∪{∞} , (4.8)where the last equality follows from (4.9) and (4.10).We recall for the reader’s convenience that using Notn. 4.3 we have L E,p = − (cid:18) e − p (cid:19) ψ ∞ − X i ∈ E δ i ∞ − X x T,E,p > x T,E,p δ T ∪{∞} . Therefore, R E,p = L E,p + X x T,E,p > | x T,E,p | δ T ∪{∞} (4.9)and by using Lemma 2.6, we have also Q E,p = L E,p + X x T,E,p < | x T,E,p | δ T ∪{∞} . (4.10)We remark that using Lemma 4.4, we have: R E,p = O ( − E )( X x T E T ) ⊗ z p , Q E,p = O ( − E )( − X x T E T ) ⊗ z p ,L E,p = O ( − E )( − X | x T | E T ) ⊗ z p , V E,p = O ( − E )( X | x T | E T ) ⊗ z p . Remark 4.14. It is clear by the definition that by the S symmetry (i.e.,exchanging with ∞ ) the line bundle R E,p is exchanged with Q E, − p . Theline bundles R E,p , Q E,p will be crucial for the proof of Thm. 4.11. We notethat the line bundles V E,p are used only in the proof of Cor. 4.18. For every divisor δ T := δ T ∪{∞} , we have by Lemma 4.5 that R E,p | δ T = O ( − x T,E,p , , Q E,p | δ T = O (0 , x T,E,p ) . (4.11)From here on, the notation O ( − a, − b ) indicates that O ( − a ) (resp., O ( − b ) )corresponds to the component marked by ∞ (resp., marked by ). Definition 4.15. We say that line bundles L and L ′ are related by quotients Q i if there are exact sequences → L i − → L i → Q i → i = 1 , . . . , t ) ,L = L, L t = L ′ . Note that when L ′ = L + P β T δ T with β T ≥ for all T , the quotients Q i are direct sums of torsion sheaves of type O δ T ( − a, − b ) . Lemma 4.16. Let E ⊆ N , e = | E | , p ∈ Z , such that e + p even. Then:(i) L E,p and R E,p are related by quotients which are direct sums of type O δ T ( − x T + i, i ) , ≤ i < | x T | = x T ( x T > (ii) L E,p and Q E,p are related by quotients which are direct sums of type O δ T ( i, x T + i ) , ≤ i < | x T | = − x T ( x T < (iii) R E,p and V E,p are related by quotients which are direct sums of type O δ T ( − x T − i, − i ) , < i ≤ | x T | = − x T ( x T < (iv) Q E,p and V E,p are related by quotients which are direct sums of type O δ T ( − i, x T − i ) , < i ≤ | x T | = x T ( x T > , where we denote for simplicity δ T := δ T ∪{∞} and x T := x T,E,p . In particular, allpairs are related by quotients of type O ( − a, ∗ ) , O ( ∗ , − a ) , with < a ≤ S ( E, p )2 . Proof. This follows immediately from (4.11), (4.9), (4.10) and (4.8). The laststatement follows by Lemma 4.17. (cid:3) Lemma 4.17. Let n = 2 s + 2 , E ⊆ N , e = | E | , p ∈ Z , e + p even. Then for all T | x T,E,p | ≤ S ( E, p )2 , (4.12) where S ( E, p ) is the score of the pair ( E, p ) (Notn. 4.6). Furthermore, | x T,E,p | = x T,E,p = S ( E, p )2 if and only if T ⊆ E, p ≥ The proof is straightforward and we omit it. Note, (4.5) is a particular case. Corollary 4.18. Let e = s + 1 , p ≥ , and ( E, p ) such that S ( E, p ) = 2 j s k + 2 q, with p = 2 q − , q ≥ if s is even, and p = 2 q − , q ≥ if s is odd. Assume thefollowing objects are generated by C :(i) All torsion sheaves O δ T ( − a, for all < a < j s k + q and all T ,(ii) The line bundles R E,p , Q E,p . XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 25 Then O δ T ( − ( j s k + q ) , with T = E is generated by C . Here δ T := δ T ∪{∞} . As C is invariant under the action of S , it follows from Cor. 4.18 that asimilar statement holds when replacing O δ T ( − a, 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 byassumption, using Lemma 4.16(iii), it suffices to prove that when x T < , O ( − x T − i, − i ) is generated by A , for all < i ≤ | x T | , i.e., | x T | < s +12 . Sincethe assumptions on q imply that p > , we have that | x T | = − x T = e − p − | E ∩ T | ≤ e − p s + 1 − p < s + 12 , and the claim follows. By Lemma 4.16(iv), the quotients relating Q E,p and V E,p have the form O δ T ( − i, x T − i ) for < i ≤ x T . By Lemma 4.17, wehave that x T ≤ S ( E,p )2 = j s k + q , with equality if an only if T ⊆ E . Since e = s + 1 , we must have T = E . It follows that all but one quotient,namely O δ T ( − ( j s k + q ) , for T = E (when i = x T = S ( E,p )2 ) are already byassumption 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. (cid:3) Corollary 4.19. Let q ∈ Z , q > . Assume that R E,p , Q E,p are generated by C whenever S ( E, p ) = 2 j s k + 2 q ′ , with < q ′ ≤ q , and e = s + 1 . Then for all T , δ T := δ T ∪{∞} , the following torsion sheaves are generated by C : O δ T ( − a, , O δ T (0 , − a ) when < a ≤ j s k + q Proof. By the S symmetry, it suffices to prove the statement for O δ T ( − a, .For any q > , taking E ⊆ N with e = s +1 and p = 2 q − when s is even, or p = 2 q − when s is odd, gives a pair ( E, p ) with S ( E, p ) = 2 j s k + 2 q . If s iseven, or if s is odd and q ≥ , the assumptions of Cor. 4.18 are satisfied. Byinduction on q > , O δ T ( − a, is generated by C when T = E , a = j s k + q .The only case left is when s is odd and q = 1 ( p = 0 ). By assumption R E, , Q E, are generated by C if e = s + 1 ( S ( E, 0) = s + 1 ). We have to prove that O δ T ( − s +12 , is generated by C . Taking E = T , p = 0 , we have that the pair ( E, is in the range of Thm. 1.7. Hence, L E, is in C . By Lemma 4.16 andLemma 4.17 L E, and R E, are related by quotients which are direct sumsof sheaves in A , with only one quotient which is O δ T ( − s +12 , for T = E (the only possibility to have x T = S ( E, = s +12 is when T = E ). Note thatthis quotient appears exactly once. The statement follows. (cid:3) Lemma 4.20. (Koszul resolutions) Let p ∈ Z , E ⊆ N .(K1) If e ≤ s + 1 , letting I ⊆ N \ E , | I | = s + 1 , there is a long exact sequence: → Q E ∪ I,p − s − → . . . → M J ⊆ I, | J | = j Q E ∪ J,p − j → . . . → Q E,p → . (K2) If e ≥ s + 1 , letting I ⊆ E , | I | = s + 1 , there is a long exact sequence: → R E,p → . . . → M J ⊆ I, | J | = j R E \ J,p − j → . . . → R E \ I,p − s − → . Proof. We have T i ∈ I δ i ∞ = ∅ and the boundary divisors { δ i ∞ } i ∈ I intersecttransversely (the divisors intersect properly and the intersection is smooth,being a Hassett space). It follows that there is a long exact sequence → O ( − X i ∈ I δ i ∞ ) → M j ∈ I O ( − X i ∈ I \{ j } δ i ∞ ) → M j,k ∈ I O ( − X i ∈ I \{ j,k } δ i ∞ ) → . . .. . . → M i ∈ I O ( − δ i ∞ ) → O → . Tensoring this long exact sequence by − P i ∈ E \ I δ i ∞ − e − p ψ ∞ , gives thesecond long exact sequence in the lemma. The first long exact sequenceis obtained in a similar way by considering the Koszul resolution of theintersection of the the boundary divisors { δ i } i ∈ I . (cid:3) Lemma 4.21. Assume p ≥ and E ⊆ N such that S ( E, p ) = 2 j s k + 2 q, and the pair ( E, p ) is such that q ≥ if s is even and q ≥ if s is odd. In thenotations of Lemma 4.20, we have:(1) If e ≤ s + 1 then Q E ∪ J,p − j in Lemma 4.20(K1) satisfies S ( E ∪ J, p − j ) ≤ S ( E, p ) . If equality holds, then | p − j | < p if j = 0 (2) If e ≥ s + 1 then R E \ J,p − j in Lemma 4.20(K2) satisfies S ( E \ J, p − j ) ≤ S ( E, p ) . If equality holds, then | p − j | < p if j = 0 .Proof. We prove (1). We have S ( E, p ) = p + e . If p − j ≥ , then S ( E ∪ J, p − j ) ≤ ( p − j ) + e + j = p + e = S ( E, p ) , and clearly | p − j | = p − j < p if j = 0 . If p − j < , we prove that theinequality on slopes is strict. We have S ( E ∪ J, p − j ) ≤ ( j − p ) + ( n − e − j ) = n − p − e < e + p = S ( E, p ) , since S ( E, p ) = e + p = 2 j s k + 2 q > s + 1 .We prove (2). We have S ( E, p ) = p + ( n − e ) . If p − j ≥ , then S ( E \ J, p − j ) ≤ ( p − j ) + ( n − e + j ) = p + ( n − e ) = S ( E, p ) , and clearly | p − j | = p − j < p if j = 0 . If p − j < , we prove that theinequality on slopes is strict. We have S ( E \ J, p − j ) ≤ ( j − p ) + ( e − j ) = e − p < p + n − e = S ( E, p ) , since e − p < s + 1 , as S ( E, p ) = p + n − e = 2 j s k + 2 q > s + 1 . (cid:3) Proof of Thm. 4.11. Case s even. For any ( E, p ) write the score S ( E, p ) as S ( E, p ) = s + 2 q. (4.13)Note that if q ≤ then L E,p is already in C (Rmk. 4.12). Moreover, if q ≤ ,by Lemma 4.16 R E,p and Q E,p are related by quotients which are directsums of torsion sheaves of the form O ( − a, ∗ ) or O ( ∗ , − a ) , with < a ≤| x T | . As | x T | ≤ S ( E,p )2 ≤ s < s +12 , such quotients are in A .We prove by induction on q ≥ , and for equal q , by induction on | p | ,that R E,p , Q E,p with S ( E, p ) = s + 2 q are generated by C . By Cor. 4.19, it XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 27 follows that all O δ T ( − a, − b ) are generated by C . Then Lemma 4.16 impliesthen that all line bundles L E,p are generated by C .We now prove the inductive statement. For q ≤ , we already provedthat R E,p , Q E,p are generated by C . Assume q ≥ . Take a pair ( E, p ) withscore S ( E, p ) = s + 2 q . Using the S symmetry, we may assume p ≥ . Forany ( E ′ , p ′ ) with strictly smaller score than s + 2 q , or equal score and strictlysmaller | 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 . UsingLemma 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 Cor.4.19 and the induction assumption, O δ T ( − a, , O δ T (0 , − a ) if < a ≤ s + q are generated by C . By Lemma 4.16 we have that L E,p is related to eachof Q E,p , R E,p by quotients which are direct sums of O δ T ( − a, ∗ ) , O δ T ( ∗ , − a ) with < a ≤ S ( E,p )2 = s + q . Since for any e = s + 1 , one of Q E,p , R E,p isgenerated by C , it follows that L E,p is generated by C . Case s odd. For any ( E, p ) write the score S ( E, p ) as S ( E, p ) = ( s − 1) + 2 q. (4.14)We prove by induction on q ≥ , and for equal q , by induction on | p | , thatthe line bundles R E,p and Q E,p with S ( E, p ) = ( s − 1) + 2 q are generatedby C . This proves the theorem, as Cor. 4.19 gives that all torsion sheavessupported on boundary are generated by C . The inductive argument wedid for s even goes through verbatim if q ≥ (the assumption is used inLemma 4.21). Hence, we only need to prove that R E,p , Q E,p are generatedby C for q = 0 and q = 1 . We may assume w.l.o.g. that p ≥ .Assume q = 0 . Fix a pair ( E, p ) with S ( E, p ) = s − . Then ( E, p ) is inthe range of Thm. 1.7 and L E,p is in C . As in the previous case, by Lemma4.16, the line bundles R E,p , Q E,p are related to L E,p by quotients generatedby 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 . Claim 4.22. O δ T ( − s +12 , , O δ T (0 , − s +12 ) are generated by C .Proof. By Cor. 4.19, it suffices to prove that R E, , Q E, are generated by C for some E with e = | E | = s + 1 . Take such an E . By Rmk. 4.14, R E, and Q E, are exchanged by the action of S . Hence, by symmetry, it sufficesto prove that R E, is generated by C . Consider the resolution in Lemma4.20(ii) for ( E, , with I = E . The terms that appear, other than R E, , are R E \ J, − j , with J ⊆ E , j > . For all j ≥ , S ( E \ J, − j ) = s + 1 and all ( E \ J, − j ) are in the range of Thm. 1.7. Hence, L E \ J, − j are generated by C .We claim that if j > , the quotients relating R E \ J, − j to L E \ J, − j are gen-erated by A . By Lemma 4.16 the quotients relating R E \ J, − j to L E \ J, − j are O δ T ( − x T + i, i ) , ≤ i < x T = x T,E \ J, − j where x T = | ( E \ J ) ∩ T | − s + 12 ≤ | E \ J | − s + 12 ≤ s − s + 12 = s − . The claim follows. It follows that for j > , R E \ J, − j is generated by C .Using the resolution, it follows that R E, is generated by C . (cid:3) Assume that e ≤ s + 1 . Then ( E, p ) is in the range of Thm. 1.7 and L E,p is in C . Since R E,p , Q E,p are related to L E,p by quotients O δ T ( − a, ∗ ) , O δ T ( ∗ , − a ) with < a ≤ S ( E,p )2 = s +12 , it follows by Claim 4.22 that R E,p , Q E,p are generated by C .Assume now that e > s + 1 . Then ( E, p ) is not in the range of Thm. 1.7.Note that it suffices to prove that R E,p is generated by C , since by Lemma4.16 R E,p , L E,p are related by quotients which are direct sums of O δ T ( − a, ∗ ) with < a ≤ S ( E,p )2 = s +12 (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 . (cid:3) Remark 4.23. For n = 2 s + 2 ≥ , the exceptional collection on Z n given in[CT20, Thm. 1.15] consists of:(i) The same torsion sheaves O δ T ( − a, − b ) as in Thm. 1.7.(ii) The line bundles in the so-called group (group A and group B ofthat theorem coincide in this case): for all E ⊆ N , with e = | E | even, F ,E = e ψ ∞ + X j ∈ E δ j ∞ − X e −| E ∩ T | > (cid:0) e − | E ∩ T | (cid:1) δ T ∪{∞} . (4.15)The line bundles F ,E are defined in [CT20] as Rπ ∗ ( N ,E ) , for certainline bundles N ,E on the universal family over Z n . One checks directly(or see the proof of [CT20, Lemma 5.8]) that N ,E restrict trivially to everycomponent of any fiber of the universal family π : U → Z n . Hence, N ,E = π ∗ F ,E , F ,E = σ ∗ u N ,E , for any marking u . In particular, for u ∈ { , ∞} , we obtain formula (4.15).(iii) The objects in the so-called group B , which in this case are linebundles (corresponding only to the J = ∅ term in [CT20, Notn.11.5]): ˜ T l, { u }∪ E := e − l − ψ u + X j ∈ E δ ju −− X e − l − −| E ∩ T | > (cid:18) e − l − − | E ∩ T | (cid:19) δ T ∪{ u } where u ∈ { , ∞} , E ⊆ N , e = | E | , l ∈ Z , l ≥ such that | E ∩ { u }| + l iseven (i.e., e + l is odd), subject to the condition l + min { e, n + 1 − e } ≤ s ( group B ) . The formula generalizing both expressions in (ii) and (iii) is e − p ψ u + X j ∈ E δ ju − X e − p −| E ∩ T | > (cid:18) e − p − | E ∩ T | (cid:19) δ T ∪{ u } which, when u = ∞ , is exactly the line bundle V ∨ E,p (the dual of V E,p - see(4.13). Hence, the group B with l = p − , u = ∞ recovers all the { V ∨ E,p } when p > . Similarly, the group B with l = − p − , u = 0 recovers all the { V ∨ E,p } when p < . The elements of group recover all the { V ∨ E,p } when p = 0 . A similar proof as in this section will prove that the collection in XCEPTIONAL COLLECTIONS ON CERTAIN HASSETT SPACES 29 [CT20, Thm. 1.15] - the torsion sheaves (i) and the line bundles { V ∨ E,p } , for ( E, p ) as in Thm. 1.7- is a full exceptional collection.5. P USHFORWARD OF THE EXCEPTIONAL COLLECTION ON THE L OSEV -M ANIN SPACE LM N TO Z N We refer to [CT17] for background on Losev-Manin spaces. Recall thatthe Losev-Manin moduli space LM N is the Hassett space with markings N ∪ { , ∞} and weights (1 , , n , . . . , n ) , where n = | N | . The space LM N parametrizes nodal linear chains of P ’s marked by N ∪ { , ∞} with is onthe left tail and ∞ is on the right tail of the chain. Both ψ and ψ ∞ inducebirational morphisms LM N → P n − (Kapranov models) which realize LM N as an iterated blow-up of P n − in n points (standard basis vectors) followedby blowing up (cid:0) n (cid:1) proper transforms of lines connecting points, etc. Inparticular, LM N is a toric variety of dimension n − . Its toric orbits (or theirclosures, the boundary strata as a moduli spaces) are given by partitions N = N ⊔ . . . ⊔ N k , | N i | > for all i , which correspond to boundary strata Z N ,...,N k = δ N ∪{ } ∩ δ N ∪ N ∪{ } ∩ . . . ∩ δ N ∪ ... ∪ N k − ∪{ } which parametrizes (degenerations of) linear chains of P ’s with pointsmarked by, respectively, N ∪ { } , N ,. . . , N k − , N k ∪ {∞} . We can identify Z N ,...,N k ≃ LM N × . . . × LM N k , where the left node of every P is marked by and the right node by ∞ .There are forgetful maps π K : LM N → LM N \ K , for all K ⊆ N , ≤ | K | ≤ n − , given by forgetting points marked by K and stabilizing. Definition 5.1. [CT17, Def. 1.4] The cuspidal block D bcusp ( LM N ) consists ofobjects E ∈ D b ( LM N ) such that for all i ∈ N we have Rπ i ∗ E = 0 . Proposition 5.2. [CT17, Prop. 1.8] There is a semi-orthogonal decomposition D b ( LM N ) = h D bcusp ( LM N ) , { π ∗ K D bcusp ( LM N \ K ) } K ⊂ N , Oi where subsets K with ≤ | K | ≤ n − are ordered by increasing cardinality. Definition 5.3. [CT17, Def. 1.9] Let G N = { G ∨ , . . . , G ∨ n − } be the set offollowing line bundles on LM N : G a = aψ − ( a − X k ∈ N δ k − ( a − X k,l ∈ N δ kl − . . . − X J ⊂ N, | J | = a − δ J ∪{ } for every a = 1 , . . . , n − . Let ˆ G be the collection of sheaves of the form T = ( i Z ) ∗ L , L = G ∨ a ⊠ . . . ⊠ G ∨ a t for all massive strata Z = Z N ,...,N t , i.e., such that N i ≥ for every i and forall ≤ a i ≤ | N i | − . Here i Z : Z ֒ → LM N is the inclusion map. If t = 1 weget line bundles G N and for t ≥ these sheaves are torsion sheaves. Theorem 5.4. [CT17, Thm. 1.10] ˆ G is a full exceptional collection in D bcusp ( LM N ) ,which is invariant under the group S × S N . Clearly, by Thm. 5.4, Prop. 5.2 and adjointness, we have the following Corollary 5.5. If E ∈ D b ( Z N ) is such that R Hom( E, F ) = 0 for all F of theform Rp ∗ ( π K ∗ ˆ G ) , for all K ⊆ N , including K = ∅ , then E = 0 . We now proceed to calculate the objects in the collection Rp ∗ ( π K ∗ ˆ G ) . Proposition 5.6.