Derived category of moduli of pointed curves -- II
aa r X i v : . [ m a t h . AG ] F e b DERIVED CATEGORY OF MODULI OF POINTED CURVES - II
ANA-MARIA CASTRAVET AND JENIA TEVELEVA
BSTRACT . We show that the moduli space of stable rational curves with n marked points has a full exceptional collection equivariant under theaction of the symmetric group S n permuting the marked points. In par-ticular, its K-group with integer coefficients is a permutation S n -lattice.
1. I
NTRODUCTION
This is the second paper in the sequence devoted to derived category ofmoduli spaces of stable rational curves with marked points. We will provethe following theorem conjectured by Manin and Orlov:
Theorem 1.1.
The Grothendieck-Knudsen moduli space M ,n , of stable rationalcurves with n marked points, has a full exceptional collection invariant under theaction of the symmetric group S n permuting the marked points. In particular, theK-group of M ,n with integer coefficients is a permutation S n -lattice. An ordered collection of objects E , . . . , E r in the bounded derived cate-gory D b ( X ) of a smooth projective variety X over C is exceptional if R Hom( E i , E i ) = C , for all iR Hom( E i , E j ) = 0 , for all i > j. An exceptional collection in D b ( X ) is called full if the smallest full triangu-lated subcategory containing all E i is D b ( X ) . If a full, exceptional collectionexists, then the K-group of X with integer coefficients is freely generatedby the classes of the objects in the collection.The lines bundles O , O (1) , . . . , O ( n ) form a full exceptional collection in D b ( P n ) by a classical theorem of Beilinson [Be˘ı78]. In general, there aremany examples of full exceptional collections (see for ex. [BO02], [Huy06],[Kuz14]). The existence of full exceptional collections on M ,n is a straight-forward consequence of Kapranov’s blow-up description [Kap93] of M ,n as an iterated blow-up of P n , Orlov’s theorem on semi-orthogonal decom-positions on blow-ups [Orl92] and Beilinson’s theorem [Be˘ı78]. However,these collections are not S n -invariant.Understanding the derived category of M ,n was initiated in the workof Manin and Smirnov [MS13] (see also [Smi13, MS14]) and the work ofBallard, Favero and Katzarkov [BFK19]. Part of the motivation in [MS13]was to understand the relationship between derived categories and quan-tum cohomology, by analogy with Dubrovin’s conjecture for Fano varieties[Dub98, KS20] (itself motivated by homological mirror symmetry). Cohomology of M ,n , and in particular its S n -character, make an ap-pearance in the theory of modular operads [KSV95, GK98, Get95], as var-ious theories from physics provide a representation of these operads inthe category of differential graded vector spaces. In [Get95] Getzler givesrecursive formulas for the character of the S n -module H ∗ ( M ,n , Q ) , us-ing mixed Hodge theory. More recently, Bergstrom and Minabe [BM13]gave another recursive algorithm, using Hassett’s construction of mod-uli spaces of weighted stable curves [Has03]. In addition, Bergstrom andMinabe computed the length of the S n -module H ∗ ( M ,n , Q ) , improving,for genus , results of Faber and Pandharipande on the length of the S n -module H ∗ ( M g,n , Q ) [FP13]. New restrictions on the irreducible represen-tations that appear in the decomposition of the S n -module H ∗ ( M g,n , Q ) ap-pear in [Tos18], using work of Sam and Snowden on FS op -modules [SS17].Recent work on the S n -module given by the top weight cohomology of M g,n appears in [CFGP19]. All mentioned recursive formulas computingthe equivariant Poincar´e-Serre polynomials of M ,n are, however, not “ef-fective” in the sense that the sums involve ± signs. To the authors knowl-edge, the fact that the S n -module H ∗ ( M ,n , Q ) is a permutation representa-tion (i.e., it has a basis that is permuted, as a set, by the action of S n ) is a newresult. In addition, this allows for a straightforward decomposition into asum of irreducible S n -representations. Other work on S n -representationsgiven by (pieces of) the cohomology of special cases of Hassett spaces, suchas symmetric GIT quotients of ( P ) n , appear in [HK98]. The current workis a K-theoretic and categorical enhancement of all these results.We remark that, even ignoring the S n action, there has been a lot of workon the Chow ring and the Poincar´e polynomial of M ,n . For example, Keelgave a presentation of the Chow ring and recursive formulas for the Bettinumbers in [Kee92], with some further work by Fulton and MacPherson[FM94] and Manin [Man95]. The Chow rings of Hassett spaces of weightedstable rational curves has been calculated in [Cey09]. Example 1.2.
Let us give a simple example of a variety with an involutionsuch that the corresponding action of S in cohomology and K -theory isnot a permutation representation. Consider a del Pezzo surface S of de-gree , the double cover of P ramified along a smooth quartic curve, withthe usual involution, call it σ . For a generic S , Aut( S ) = h σ i ≃ S . Then S has
56 ( − -curves that come in pairs interchanged by σ (and mapped to bitangents). As σ preserves − K , it acts by − on the orthogonal com-plement ( − K ) ⊥ in Pic( S ) , the root lattice of type E . Therefore the actionof S on both the total cohomology H ∗ ( S, Z ) and the K-group K ( S ) is di-agonalizable with eigenvalues and eigenvalues − (signature (3 , ).It is not a permutation representation because the diagonalizable S actionon a lattice with signature ( a, b ) is a permutation representation if and onlyif a ≥ b . Philosophically, one can argue that the difference between S and M , , the del Pezzo surface of degree , is that S has moduli while M , (and M ,n ) is rigid and in some sense is characteristic-independent. So it isperhaps more reasonable to expect that the action of S n in K ( M ,n ) is thesimplest possible, i.e. a permutation representation. ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 3
To prove Thm. 1.1 we follow a strategy outlined in [CT17] and inspiredby the work in [BM13]. We quickly reduce the theorem to an analogousstatement about special Hassett spaces M p,q introduced below. The bulk ofthe paper is devoted to derived category of M p,q ’s. Notation 1.3.
Fix a vector of positive rational weights a = ( a , . . . , a n ) witheach a i ≤ and P a i > . Let M a denote the Hassett space of weightedpointed stable rational curves, i.e., pairs ( C, P a i p i ) with slc singularities,such that C is a nodal genus curve and the Q -line bundle ω C ( P a i p i ) isample. Note that M ,n = M ,..., . The polytope of weights has a chamberstructure with walls X i ∈ I a i = 1 for every subset I ⊂ { , . . . , n } . (1.1)Note that points { p i : i ∈ I } can become equal on C if and only if P i ∈ I a i ≤ .Moduli spaces within the interior of each chamber are isomorphic andcarry the same universal family. There exist birational reduction morphisms M a → M a ′ every time the weight vectors are such that a i ≥ a ′ i for every i . Notation 1.4.
Fix a vector of positive rational weights a = ( a , . . . , a n ) witheach a i ≤ and P a i = 2 . Let X a = ( P ) n // PGL be the GIT quotient of ( P ) n by the diagonal action of PGL with respect tothe fractional polarization O ( a , . . . , a n ) . The polytope of GIT weights canbe identified with the face of the polytope of Hassett weights and inheritsits chamber structure with the walls (1.1), which encodes variation of GIT(see [DH98]). Polarizations within the interior of each chamber have nostrictly semistable points and carry a universal P -bundle with n sections.More generally, there always exists a morphism M a := M a + ǫ,...,a n + ǫ → X a , < ǫ ≪ , which is an isomorphism if there are no strictly semistable points [Has03].In the presence of strictly semistable points, X a can acquire isolated sin-gularities (cones over P r × P s ) and M a is the blow-up of these singularities(“Kirwan resolution”). In any case, the universal family over these varieties M a “near the GIT face” has fibers with at most two irreducible components.In § Notation 1.5.
For p ≥ , q ≥ , we let X p,q , respectively M p,q , denote thespaces X a , respectively M a , above with weights a = a = . . . = a p , b = a p +1 = . . . = a p + q ≪ pa + qb = 2 . We call the points with the weight a heavy and the remaining points light .We denote by P (resp., Q ) a subset of heavy (resp., light) points. When q = 0 , we denote M n := M n, . Concretely, when p = 2 r + 1 , the space M p,q is given by: ANA-MARIA CASTRAVET AND JENIA TEVELEV • If q = 0 then p < a ≤ r (at most r of the points may coincide); • If q > then a = p − ǫ , b = pǫq , with < ǫ < r +1)( r +1) , i.e., at most r heavy points may coincide, and moreover, they may coincide withall the light points. This space is an iterated P -bundle over M p .When p = 2 r , the space M p,q is given by: • If q = 0 then r < a ≤ r − (at most ( r − of the points may coincide); • If q > then a = p − ǫ , b = pǫq , with < ǫ < r ( r +1) , i.e., r heavypoints may coincide, and they may further coincide with at most ⌊ q − ⌋ light points; moreover, at most ( r − of the heavy pointsmay coincide with each other and with all the light points.Note that X p,q ∼ = M p,q if and only if p or q is odd (as no partial sum of theweights equals when either p or q is odd). If p and q are both even, say p = 2 r , q = 2 s , then M p,q is a divisorial “Kirwan resolution” of singularitiesof X p,q . It has exceptional divisors P r + s − × P r + s − with normal bundle O ( − , − over each of the (cid:0) pr (cid:1)(cid:0) qs (cid:1) singular points of X p,q .We emphasize that all spaces M p,q are needed to prove Theorem 1.1 for n ≫ . The same argument also proves the following more general result. Theorem 1.6.
Let a be a Hassett weight such that a ≥ . . . ≥ a n and P a i > .Suppose further that either a = 1 or a j > /j for some j . Then M a has a full Γ a -invariant exceptional collection, where Γ a ⊆ S n is the stabilizer of the vector a . We note that full, exceptional (not Γ a -invariant) collections on some ofthe Hassett spaces M a have been constructed in [BFK19]. More generally,full, exceptional collections on many GIT quotients (without any require-ment of invariance) have been constructed in [HL15, BFK19].The vector bundles F l,E in the exceptional collections on M p,q are intro-duced in later sections. They are indexed by an integer l ≥ and a subset E ⊆ { , . . . , n } such that l + | E | is even. They have the following properties: • At a point [ C ] of M p,q which corresponds to an irreducible curve C ≃ P with n = p + q marked points, the fiber of F l,E is equal to F l,E | [ C ] = H (cid:18) C, ω ⊗ e − l C ( E ) (cid:19) , where we identify E with a subset of sections. Here e = | E | . • F l,E has rank l + 1 . For example, F , ∅ = O . • The group S p × S q , which acts naturally on M p,q by permuting heavyand light points separately, acts on { F l,E } via its action on E . Notation 1.7.
For every subset E ⊆ { , . . . , n } , we denote by E p (resp., E q )its intersection with P (resp., Q ) and their cardinalities by e p and e q .The result for M p,q is a combination of Theorems 1.8, 4.8, 1.11 and 1.16.We start with the following basic case: Theorem 1.8.
Let n = 2 r + 1 . The vector bundles { F l,E } form a full strong S n -equivariant exceptional collection in D b (M n ) under the following condition: l + min( e, n − e ) ≤ r − , where e = | E | , l + e even . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 5
The vector bundles are ordered by increasing e , and for a given e , arbitrarily. We prove exceptionality of this collection using the theory of windowsinto derived categories of GIT quotients [Tel00, HL15, BFK19].
Example 1.9.
Here we list these exceptional collections for small n : • M is a point. The collection contains object, F , ≃ O . • The reduction map M , → M is an isomorphism (although theuniversal families are different). The collection contains objects: – F , ≃ O . Line bundles F , can be identified with π ∗ i ( O (1)) forevery forgetful map π i : M , → M , ≃ P (conic bundle). – A rank vector bundle F , can be identified with the log tan-gent bundle of M , . The map given by its global sections H ( M , , F , ) = C gives a well-known embedding of M , (the del Pezzo surface of degree ) into the Grassmannian G (2 , . • The exceptional collection on M contains objects: F , F , F , F , F , F , . The reduction morphism M , → M is the blow-up of planes P ⊂ M intersecting transversally in points. • The exceptional collection on M contains objects: F , F , F , F , F , F , F , F , F , F , . As n grows, the reduction morphism M ,n → M n factors into moreand more steps. This process is analyzed in detail in Section 2.As it turns out, the theory of windows is not directly applicable for other M p,q ’s and we connect these cases to the basic case through various ad hocmethods. For example, if p is odd then we have a morphism M p,q → M p , aniterated ( q times) universal P -bundle of M p . By applying Orlov’s theoremon the derived category of a projective bundle, we immediately constructan equivariant exceptional collection in this case (see Theorem 4.8.)The case of even p is much more complicated. We introduce a new object: Notation 1.10.
For p = 2 r ≥ , q ≥ , R ⊆ P , | R | = r , denote by Z R thelocus in M p,q where the points from R come together. Let π R : U R → Z R bethe universal family over of M p,q restricted to Z R and let σ u be the sectionof π R that corresponds to the combined points of R .For l ≥ , E ⊆ P ∪ Q with e = | E | such that e + l is even and E p = R ,consider the following torsion sheaf on M p,q : T l,E = i R ∗ σ ∗ u (cid:18) ω e − l π R ( E ) (cid:19) , where i R : Z R ֒ → M p,q is the inclusion map. Theorem 1.11.
Let p = 2 r ≥ , q = 2 s + 1 ≥ . For subsets E p ⊆ P , E q ⊆ Q such that l + e is even, l ≥ consider the following collections: • The vector bundles F l,E on M p,q for l + min( e p , p + 1 − e p ) ≤ r − A ) ,l + min( e p + 1 , p − e p ) ≤ r − B ) , ANA-MARIA CASTRAVET AND JENIA TEVELEV • The torsion sheaves T l,E on M p,q for e p = r, l + min( e q , q − e q ) ≤ s − . There are full S p × S q invariant exceptional collections on M p,q obtained by com-bining the bundles from the group A (resp., B ) with the sheaves from group .These objects are arranged in blocks indexed by a subset E q . The order is as follows: • blocks are ordered by increasing e q , • arbitrarily if e q is the same (but the set E q is different).Within each block with the same E q we put the sheaves {T l,E } first • in arbitrary order if E p = E ′ p , • in order of decreasing l when E p = E ′ p ,followed by the bundles { F l,E }• in order of increasing e p , • and for a given e p , arbitrarily. Remark 1.12.
Note that we have an isomorphism of space M p, ∼ = M p +1 butthe collections from Thm. 1.11 and Thm. 1.8 are not the same. Remark 1.13.
Theorem 1.11 (and the next Theorem 1.16) remains true evenif r = 1 , i.e., if there are only p = 2 heavy points. In this case the col-lection contains no vector bundles F l,E and since Z R is the whole modulispace, the objects T l,E are line bundles. This collection is the same as theone constructed in [CT17, §
6] (“GIT version” of the Losev–Manin space).For the case of p , q even we need yet another type of object. Notation 1.14.
For more convenient bookkeeping, we retain q = 2 s + 1 oddand work with M p,q +1 instead. In particular, | Q | = q + 1 in the even case.The map M p,q +1 → X p,q +1 is a Kirwan resolution of singularities withexceptional divisors P r + s − × P r + s − . Let A ⊂ D b (M p,q +1 ) be a triangulatedsubcategory generated by torsion sheaves O P r + s − × P r + s − ( − a, − b ) where we have the following possibilities: • either ≤ a, b ≤ r + s − or • a = 0 and ≤ b ≤ r + s − or • b = 0 and ≤ a ≤ r + s − .Let B = ⊥ A = { T ∈ D b (M p,q +1 | Hom(
T, A ) = 0 for every A ∈ A} .We prove in Section 11 that A is an admissible ( S p × S q +1 ) invariant sub-category and thus B is an ( S p × S q +1 ) equivariant non-commutative resolu-tion of singularities of the GIT quotient X p,q +1 in the sense of [KL14]. Notethat X p,q +1 has many small resolutions related by flops obtained by con-tracting boundary divisors P r + s − × P r + s − onto one of the factors. How-ever, none of them is of course ( S p × S q +1 ) equivariant unlike the category B , which in some sense is the “minimal” equivariant resolution. Perhaps thebiggest technical issue contributing to the complexity of this case is that B is a non-commutative strongly crepant resolution only if r + s is odd.We prove in Prop. 6.1 that the vector bundles F l,E belong to B . The tor-sion objects have to be projected onto B : ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 7
Definition 1.15.
We define the objects in
B ⊂ D b (M p,q +1 ) by ˜ T l,E = ( T l,E ) B , where T → T B is a canonical functorial projection and the torsion sheaf T l,E is defined as in Notation 1.10 for E ⊆ P ∪ Q , l ≥ , e + l even. Theorem 1.16.
Let p = 2 r ≥ , q = 2 s + 1 ≥ . Consider the following objects: • The vector bundles F l,E on M p,q +1 (with l + e even) for l + min( e p , p + 1 − e p ) ≤ r − A ) ,l + min( e p + 1 , p − e p ) ≤ r − B ) , • The complexes ˜ T l,E on M p,q +1 (with l + e even) for e p = r, l + min( e q , q + 2 − e q ) ≤ s (group 2 B ) . Then M p,q +1 has two S p × S q +1 invariant full exceptional collections of • torsion sheaves O ( − a, − b ) in subcategory A ; • The bundles F l,E for pairs ( l, E ) in group A (alternatively B ), • The complexes ˜ T l,E for pairs ( l, E ) in group B .When combining group B with any of the groups A or B , the order is the sameas in Theorem 1.11. Remark 1.17.
The groups B and A are related by taking the complement E E c . From this point of view, it is natural to consider also { ˜ T l,E } for e p = r, l + min( e q + 1 , q + 1 − e q ) ≤ s (group 2 A ) , and combine it with groups A or B . The same proof as in Thm. 1.16shows that this collection is exceptional and of the expected length, but wedid not attempt to prove fullness.The same collection as in Thm. 1.16 works for the case of s = − : Theorem 1.18.
Let p = 2 r ≥ . We introduce the following collections of vectorbundles F l,E (with l + e even) on M p : l + min( e, p + 1 − e ) ≤ r − A ) ,l + min( e + 1 , p − e ) ≤ r − B ) . Then M p has two S p invariant full exceptional collections of • torsion sheaves O P r − × P r − ( − a, − b ) , where either ≤ a, b ≤ r − or a = 0 and ≤ b ≤ r/ − or b = 0 and ≤ a ≤ r/ − , followed by • vector bundles F l,E from collection A (alternatively, B ).The order is first by increasing e , and for a given e , arbitrarily. The category B isa strongly (resp. weakly) crepant non-commutative resolution of X p if and only if p ≡ (resp. p ≡ .) Example 1.19.
We list exceptional collections for small p (using group A ): • M ≃ M , ≃ P . In this case A is empty. The collection contains objects, F , ≃ O and F , ≃ O (1) . In fact F ,P is always the pull-back of the GIT polarization from the symmetric GIT quotient X p (see Corollary 6.2). ANA-MARIA CASTRAVET AND JENIA TEVELEV • The space X is the Segre cubic threefold in P with singularities. M ≃ M , is the blow-up of singularities of X with exceptionaldivisors P × P (this is the last case when M p ≃ M ,p ). The cate-gory A contains torsion sheaves O P × P ( − , − . The category B is aweakly crepant resolution of X with the following full exceptionalcollection of vector bundles: F , F , F , F , F , . Remark 1.20.
An interesting consequence of the main Theorem 1.6 is theexistence of a simple semi-orthogonal decomposition in the derived cate-gory of the Deligne–Mumford stack M ,n /S n and more generally, M a / Γ ,see Lemma 2.2. M ,n /S n is known as “symmetric M ,n ” and naturally ap-pears in moduli theory, for example in the study of ample line bundles on M g , see [GKM02]. Likewise, the coarse moduli space of X n /S n was exten-sively studied in the 19th century by Silvester and others in the frameworkof invariant theory of binary forms [Dol03]. Connection with [CT17]. In the first paper of this project [CT17] we provethe existence of a full S × S n -invariant collection on the Losev-Manin spaceLM n , i.e., the Hassett space M a with weights a = a = 1 , a = . . . = a n +2 = ǫ ≪ . The Losev-Manin spaces, together with the the spaces M p,q (the core of thepresent work) are the main cases on which the main theorem Thm. 1.6relies. The approach we take is different for the two types of spaces.In [CT17] we proved that for both LM n and M ,n it suffices to find full in-variant exceptional collections in the cuspidal block of the derived category(i.e., objects that push forward to by all the forgetful maps). In [CT17]we achieved this for the Losev-Manin spaces. For M ,n , we ran into dif-ficulties with the cuspidal block for n large and took a different approach.Equivariant description of D bcusp ( M ,n ) is an interesting open problem. Structure of paper/Comments.
In Section 2 we explain how to reduce themain theorem to the case of the Losev-Manin spaces LM n and the spaces M p,q using Hassett’s reduction maps and an invariant version of Orlov’sblow-up theorem. In Section 3 we introduce the vector bundles F l,E on thestack of n points on P and give some of their main properties.In Section 4 we recall Halpern-Leistner’s theory of windows and applyit to prove the exceptionality of the collections in Thm. 1.8 (the case of M p , p odd) and Thm. 4.8 (the case of M p,q , p odd, q > ). In Section 5 weextend the definition of the vector bundles F l,E to more general Hassettspaces (with non-empty boundary), while in Section 6 we give sufficientconditions for the vector bundles F l,E to be orthogonal to torsion sheavessupported on the boundary. In Section 7 we exhibit various inequalitiesinvolving the pairs ( l, E ) in Thm. 1.11 and Thm. 1.16. These will be crucialfor all of the remaining sections.The exceptionality of the “ F l,E -part” of the collections in Thm. 1.11 ( M p,q , p even, q odd) and Thm. 1.16 ( M p,q , p , q both even) is proven in Section 8and Section 9 by induction on q . Note that the case q odd and q even require ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 9 different arguments (hence, the two sections). This is where we introducewhat we call the alpha game (to go from M p,q − to M p,q when q is odd) andthe beta game (to go from M p,q to M p,q +1 when q is odd). Both “games”compare exceptionality between F l,E ’s on different Hassett spaces relatedby reduction maps.We finish the proof of the exceptionality of the collection on M p,q ( p even, q odd) of Thm. 1.11 (i.e., considering its “ T l,E part”) in Section 10 by reduc-ing it to a windows calculation on subvarieties Z R ⊆ M p,q (the supports ofthe torsion sheaves T l,E in the collection) and their intersections. Note that adirect windows argument on M p,q will not give exceptionality of most pairsin the collection, as the condition on weights is not satisfied. The exception-ality of the collection on M p,q +1 ( p even, q odd) in Thm. 1.16 is finished inSection 11 and requires yet new reduction maps in order to compare excep-tionality of different pairs, by reducing again to a window calculation onsubvarieties Z R ⊆ M p,q +1 and their intersections. It is here that we comparecommutative (not equivariant) and non-commutative (equivariant) smallresolutions of the singular GIT quotient X p,q +1 .Fullness of the exceptional collections on all the spaces M p,q is proved inthe remaining sections: Section 12 for the collections in Thm. 1.8 and Thm.4.8, Section 13 for the collections in Thm. 1.11 and in Section 14 for the col-lections in Thm. 1.16. Even in the most basic case of Thm. 1.8, where theexceptional collection is contained in the Halpern-Leistner’s ”window”, wewere unable to use directly his main theorem (see Thm. 4.4) to prove full-ness. However, we were inspired by its proof utilizing Koszul resolutionsof the unstable strata, although in our case we had to work on the universalfamily rather than on the moduli stack as in [HL15]. We use several con-structions based on the Koszul complex ( Koszul games ) in order to provefullness. One of them “replaces” the torsion sheaves T l,E (resp., complexes ˜ T l,E ) with the bundle F l,E for the the same pair ( l, E ) . We emphasize thatthe collections obtained by replacing T l,E (resp., ˜ T l,E ) with the correspond-ing F l,E ’s are not exceptional in general (although we prove that they arefull). We find this phenomenon very interesting and worthy of further ex-ploration even on toric varieties or whenever one has natural full but notexceptional collections of line or vector bundles. We remark that provingfullness in Section 14 for the collection on M p,q +1 ( q odd) involves yet againboth the alpha game and the beta game and relies on having proved alreadyfullness for the collection on M p,q and M p,q +2 (Section 13). Acknowledgements.
We thank Alexander Kuznetsov, Daniel Halpern-Leistner and Chunyi Li for helpful conversations related to this work. Wethank Matt Ballard, Alex Duncan and Patrick McFaddin for sending usa preliminary version of their work related to Thm. 1.8. We thank BenBakker, Matt Ballard, Arend Bayer, Emanuele Macr`ı, Georg Oberdieck, AlexPerry, Olivier Schiffmann and Maxim Smirnov for useful comments. Thefirst named author was supported by the NSF grant DMS-1701752. Thesecond named author was supported by the NSF grant DMS-1701704, aFulbright Scholarship and a Simons Fellowship. C ONTENTS
1. Introduction 12. Invariant exceptional collections. The proof of Theorem 1.6 103. Derived category of the moduli stack M n of n points on P M p,q ⊂ M n F l,E to some Hassett spaces 196. The bundles F l,E versus the sheaves O δ ( − a, − b ) ( l, E ) in groups and { F l,E } on M p,q − exceptional ⇒ { F l,E } on M p,q exceptional 419. { F l,E } on M p,q exceptional ⇒ { F l,E } on M p,q +1 exceptional 4610. Proof of Thm. 1.11 (exceptionality: P even, Q odd) 5211. Proof of Thm. 1.16 (exceptionality: P even, Q even) 5912. Proof of Thm. 1.8 (fullness: P odd) 7213. Proof of Thm. 1.11 (fullness: P even, Q odd) 7414. Proof of Thm. 1.16 (fullness: P even, ˜ Q even) 78References 872. I NVARIANT EXCEPTIONAL COLLECTIONS . T
HE PROOF OF T HEOREM Γ be a finite group acting on a smooth projective variety X and let {E • α } α ∈ I be an exceptional collection in D b ( X ) . There are two natural no-tions how a collection can be “permuted” by the action of Γ : Definition 2.1.
An exceptional collection is called Γ -invariant if, for every γ ∈ Γ and α ∈ I , there exists β ∈ I such that γ ∗ E • α ≃ E • β . An invariantcollection is called equivariant if every complex E • α is quasi-isomorphic toa complex in D b ( X/ Γ α ) = D b Γ α ( X ) , a bounded derived category of thecategory of Γ α -equivariant coherent sheaves, where Γ α ⊂ Γ is the stabilizerof the isomorphism class of E • α .Existence of a full Γ -invariant collection has two consequences: Lemma 2.2. If D b ( X ) admits a full Γ -invariant exceptional collection then(1) K ( X ) is a permutation Γ -lattice and(2) the bounded derived category of the quotient stack (orbifold) X/ Γ admits anatural semi-orthogonal decomposition with blocks isomorphic to representationcategories (if the collection is equivariant) and twisted representation categories (ifit is only invariant) of subgroups Γ α .Proof. Since K ( X ) is isomorphic to the Grothendieck K -group of D b ( X ) ,the first statement is clear from the definitions (a semi-orthogonal decom-position of D b ( X ) induces a direct sum decomposition of its K -group). Werefer to [Ela09, Theorem 2.3] for the precise formulation and proof of thesecond statement. (cid:3) Our collections will be not only Γ -invariant but in fact Γ -equivariant.For this we need a strengthened version of the equivariant Orlov blow-uplemma [CT17, Lemma 7.2]. ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 11
Let X be a smooth projective variety and let Y , . . . , Y n ⊂ X be smoothtransversal subvarieties of codimensions l , . . . , l n . Let Γ be a finite groupacting on X permuting Y , . . . , Y n . For every subset I ⊂ { , . . . , n } , we de-note by Y I the intersection ∩ i ∈ I Y i . In particular, Y ∅ = X . Let q : ˜ X → X be an iterated blow-up of (proper transforms of) Y , . . . , Y n . Since the in-tersection is transversal, blow-ups can be done in any order. More canon-ically, the iterated blow-up is isomorphic to the blow-up of the ideal sheaf I Y · . . . · I Y n . Let E i be the exceptional divisor over Y i for every i = 1 , . . . , n .For any subset I ⊂ { , . . . , n } , let E I = q − ( Y I ) = ∩ i ∈ I E i . In particular, E ∅ = ˜ X . Let i I : E I ֒ → ˜ X be the inclusion. The group Γ actson ˜ X and the morphism q is Γ -equivariant. Let Γ I ⊂ Γ be the normalizer of Y I for each subset I ⊂ { , . . . , n } (in particular, Γ ∅ = Γ ). Lemma 2.3.
Let { F βI } be a (full) Γ I -invariant (resp. equivariant) exceptional col-lection in D b ( Y I ) for every subset I ⊂ { , . . . , n } . Choosing representative of Γ -orbits on the set { Y I } , we can assume that if Y I = gY I ′ for some g ∈ Γ then { F βI } = g { F βI ′ } . Then there exists a (full) Γ -invariant (resp. equivariant) excep-tional collection in D b ( ˜ X ) with blocks B I,J = ( i I ) ∗ " ( Lq | E I ) ∗ ( F βI ) n X i =1 J i E i ! for every subset I ⊂ { , . . . , n } (including the empty set) and for every n -tuple ofintegers J such that J i = 0 if i I and ≤ J i ≤ l i − for i ∈ I .The blocks are ordered in any linear order which respects the following partialorder: B I ,J precedes B I ,J if n P i =1 J i E i ≥ n P i =1 J i E i (as effective divisors).Proof. Exceptionality, fullness and invariance of the collection was provedin [CT17, Lemma 7.2]. For equivariance, we use equivariant pull-back andpush-forward and endow line bundles O ( J E + . . . + J n E n ) with canon-ical linearizations with respect to the stabilizer of the divisor J E + . . . + J n E n . Here we use a canonical equivariant structure of the ideal sheaf ofany invariant subscheme. (cid:3) In the remainder of this section we discuss the proof of Theorem 1.6.All along we assume that the spaces M p,q have a full S p × S q -equivariantexceptional collection, which is proved in the subsequent sections. Proof of Theorem 1.6.
Let M a be a Hassett space as in Theorem 1.6. Let Γ a bethe stabilizer of the vector a in the symmetric group S n . Choose p ≤ n suchthat a = . . . = a p > a p +1 . We consider cases:(i) a = 1 , p ≥ ,(ii) a < , pa ≥ .(iii) a = 1 , p = 1 ,(iv) a j > /j for some j ,as well as the following general statement: (v) Consider Hassett weight vectors a = ( a , . . . , a n ) , a ′ = ( a ′ , . . . , a ′ n ) such that a i ≥ a ′ i for every i . Suppose Γ a ⊂ Γ a ′ . Then if M a ′ admitsa Γ a -equivariant exceptional collection then so does M a .We prove (i)–(v) simultaneously by induction on dimension of M a usingexistence of full S p × S q -equivariant exceptional collections on M p,q . Case (iv) . We can assume without loss of generaility that j is the largestindex with the property a j ≥ /j . The statement follows from (i) if a j = 1 .If a j < , we reduce to (ii) using (v) by taking the second weight vector a ′ = . . . = a ′ j = a j , a ′ i = a i for i > j . Case (iii) . Let A = a + . . . + a n > . Consider a ′ = 1 , a ′ i = a i A − ǫ for i ≥ for some fixed < ǫ < min( a n , A − . By (v), it suffices to show that M a ′ admits a Γ a -invariant exceptional collection. Note that M a ′ ∼ = P n − by[Has03, Section 6.2] since X i ≥ ,i = j a ′ i ≤ n − X i =2 a ′ i = A − a n A − ǫ < for all j ≥ . Note that P n − has an invariant full exceptional collection (the standardcollection O , O (1) , . . . , O ( n − is invariant under any group). For equiv-ariance, we need a bit more. All appearing groups are contained in thesymmetric group S n − which acts on P n − by permuting n − fixed pointsin general linear position. The homomorphism S n − → PGL n − can be fac-tored through GL n − , and therefore O (1) is a S n − -linearized line bundle.The corresponding action on k n − is the action on the irreducible ( n − -dimensional representation of S n − (cf. [KT09, Lemma 2.3]). Cases (i) and (ii) . If p = 2 (case (i)), we apply (v) to the weight vector a ′ = (1 , , ǫ, . . . , ǫ ) of the Losev–Manin space and use the main result of[CT17]. Let p > . In both (i) and (ii) we apply (v) to the weight vector a ′ = ( a, . . . , a, b . . . , b ) of M p,q (see Remark 1.5) with q = n − p . Concretely, • If q = 0 then a = p + ǫ and we choose a < a . • If q > then a = p − ǫ , b = pǫq . Then a < a and we choose b < a n . Case (v) . We connect the weights by a Γ a -invariant homotopy a ( t ) = t a ′ + (1 − t ) a . The Hassett chamber structure is semi-constant in the following sense: thereduction map M b ,...,b n → M b − ǫ,...,b n − ǫ is an isomorphism for < ǫ ≪ .It follows that our reduction map M a (0) = M a → M a ′ = M a (1) factorsinto the sequence of Γ a -equivariant reduction maps of the following form: f t : M a ( t − ǫ ) → M a ( t ) for < ǫ ≪ whenever there is a subset of indices J ⊂ { , . . . , n } such that X j ∈ J a j (0) > X j ∈ J a j ( t ) . (2.1)Arguing by induction on the number of wall crossings, it suffices to analyzea single reduction map f t . The following description of f t is from [Has03],see also [BM13]. Let J , . . . , J s be a full list of subsets satisfying (2.1). Thegroup Γ a permutes J i ’s. The reduction map f t blows up the loci M a ( t ) ( J i ) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 13 in M a ( t ) where the points in J i come together. Since P j ∈ J i a j ( t ) = 1 , theloci M a ( t ) ( J i ) intersect transversely: M a ( t ) ( J i ) ∩ M a ( t ) ( J j ) = ∅ if and only if J i ∩ J j = ∅ , in which case, the intersection is the locus where points in J i ,respectively J j , coincide. Clearly, the loci M a ( t ) ( J i ) and their intersectionsare themselves Hassett spaces of the form M a ′′ , with the vector a ′′ having atleast one weight (for marked points that correspond to combined pointsindexed by subsets J , . . . , J s ). These spaces have invariant exceptionalcollections by cases (i), (iii) in smaller dimension. The theorem follows byLemma 2.3. (cid:3)
3. D
ERIVED CATEGORY OF THE MODULI STACK M n OF n POINTS ON P Let M n be the moduli stack of n points on P , i.e. the quotient stack M n = [( P ) n / PGL ] . Equivalently, it is the stack of ´etale locally trivial P -bundles with n sec-tions. Concretely, an associated P -bundle of a PGL -torsor is the “univer-sal P -bundle”, i.e. a representable morphism π = π n +1 : M n +1 → M n , induced by the first projection ( P ) n +1 = ( P ) n × P → ( P ) n . The “univer-sal sections” are representable morphisms σ , . . . , σ n : M n → M n +1 , induced by big diagonals ( P ) n ≃ ∆ i,n +1 ֒ → ( P ) n +1 for i = 1 , . . . , n . Let Σ n (or Σ if n is clear from the context) be the sum of these sections.We identify the category of coherent sheaves on M n with the category of PGL -equivariant coherent sheaves on ( P ) n and likewise for their boundedderived categories. Definition 3.1.
For a vector i = ( i , . . . , i n ) with P i k even, the line bundle O ( P ) n ( i ) = O ( P ) n ( i , . . . , i n ) has a unique PGL - linearization and thus descends to M n . We call it O ( i ) . Remark 3.2.
It is clear from the definition that the ψ -class ψ i := σ ∗ i ω π = O (2 e i ) . Definition 3.3.
Fix a subset E ⊂ Σ n with e = | E | and an integer l ≥ suchthat e + l is even. Let ( i , . . . , i n ) be a sequence such that i j is if j ∈ E and otherwise. Consider the line bundle N l,E = O ( i , . . . , i n , l ) = ω ⊗ ( e − l ) π ( E ) on M n +1 . Let F l,E = π ∗ N l,E . Lemma 3.4. F l,E is a rank l +1 vector bundle on M n . As an SL -bundle on ( P ) n , F l,E ≃ SL O ( i , . . . , i n ) ⊗ V l , (3.1) where V l is an irreducible SL -module of dimension l + 1 . This has various immediate consequences:
Corollary 3.5. F ,E ≃ O ( i , . . . , i n ) . Corollary 3.6. F ∨ l,E ≃ π ∗ O ( − i , . . . , − i n , l ) ≃ SL O ( − i , . . . , − i n ) ⊗ V l . If n is even then F l,E ≃ F ∨ l,E c ⊗ F , Σ . Note that the formula Rπ ∗ (cid:18) ω ⊗ ( e − l ) π ( E ) (cid:19) allows to define F l,E ’s, at leastas -step complexes, on arbitrary families of pointed curves. The followingtheorem is our template for properties of F l,E ’s in the stable rational case. Theorem 3.7. D b ( M n ) has a S n -invariant semi-orthogonal decomposition with n blocks D bE ( M n ) indexed by subsets E ⊆ { , . . . , n } ordered by increasing e = | E | (different blocks with the same e are mutually perpendicular, i.e., R Hom between blocks is ). Each block has a full exceptional collection D bE ( M n ) = h F l,E : l + e even i of infinitely many mutually perpendicular vector bundles. The combined infiniteexceptional collection { F l,E } in D b ( M n ) is strong and S n -equivariant.The forgetful map π i : M n +1 → M n has the following properties: ( Rπ i ) ∗ F ∨ l,E = ( i ∈ EF ∨ l,E i E ( Lπ i ) ∗ F l,E = F l,E . Proof.
The line bundles L k = O ( i , . . . , i n ) , k = 1 , . . . , n , form a strong S n -equivariant exceptional collection in D b (( P ) n ) , where each j j = 0 or .Schur’s lemma and the fact that R Γ( P , O ( − imply that h F l,E i with l + e even is an exceptional collection in D b ( M n ) with required properties.It remains to prove fullness, i.e. that any complex F ∈ D b ( M n ) can be ob-tained by finitely many extensions starting with objects in the exceptionalcollection. This is a special case of [Ela09, Theorem 2.10]. Let’s give a simplead hoc argument. Viewing F as a bounded complex of coherent sheaves on ( P ) n , let k be the maximum index such that RHom( L k , F ) = 0 or k = 0 if RHom( L k , F ) = 0 for every k . We argue by induction on k . If k = 0 thenwe are done: F ≃ because O ( i , . . . , i n ) form a full exceptional collectionin D b (( P ) n ) . Otherwise, consider the left mutation in D b (( P ) n ) : F ′ → RHom( L k , F ) ⊗ L k → F → . Then
RHom( L k , F ′ ) = 0 . Moreover, viewing F as an SL -equivariant com-plex and using the unique SL -linearization of L k , we see that RHom( L k , F ) is isomorphic to a direct sum of irreducible SL -representations V l (up toshifts). Moreover − Id ∈ SL acts trivially on RHom( L k , F ) ⊗ L k , i.e. thelatter is a direct sum of vector bundles F l,E . Thus the above triangle is atriangle in D b ( M n ) and we are done by induction. (cid:3) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 15
Proposition 3.8. On M n , we have exact sequences → F l − ,E \ k → F l,E → Q kl,E → k ∈ E, where Q kl,E = O ( i , . . . , i k + l, . . . , i n ) , and → F ∨ l − ,E ∪{ k } → F ∨ l,E → S kl,E → k E, where S kl,E = O ( − i , . . . , − i k + l, . . . , − i n ) .Proof. Push-forward by π gives an exact sequence → O ( − β k − β n +1 ) → O → ∆ k,n +1 → on M n +1 tensored with O ( i , . . . , i n , l ) (resp. O ( − i , . . . , − i n , l ) ). (cid:3) Proposition 3.9. If E and E ′ are disjoint sets then F l,E ⊗ F l ′ ,E ′ = F l + l ′ ,E ∪ E ′ ⊕ F l + l ′ − ,E ∪ E ′ ⊕ . . . ⊕ F | l − l ′ | ,E ∪ E ′ . In particular, F l,E ⊗ F ,E ′ = F l,E ∪ E ′ . (Here we assume that all these bundles are defined, i.e., all parity conditions aresatisfied).Proof. Clebsch–Gordan formula. (cid:3)
Proposition 3.10.
More generally, define N l,E ≃ ω e − l π ( E ) for any l , positive ornegative, and define F l,E = Rπ ∗ N l,E . When l = − , we have F l,E = 0 . When l ≤ − , we have F l,E ≃ F − l − ,E [ − . Proof.
By Grothendieck–Verdier duality we have that Rα ∗ ( O ( − E, − l − ≃ F ∨ l,E [ − . But by Cor. 3.6 when − l − ≥ we have that F ∨− l − ,E ≃ Rα ∗ ( O ( − E, − l − . (cid:3)
4. W
INDOWS INTO DERIVED CATEGORIES OF SUBSTACKS M p,q ⊂ M n Proposition 4.1.
Let
U ⊆ M n be an open substack. We define vector bundles F l,E on U as restrictions of the corresponding vector bundles on M n . Then D b ( U ) is generated by the F l,E ’s (equivalently, by F ∨ l,E ’s ). All results of § U .Proof. We have U = [ U/ PGL ] for some open equivariant subset U ⊂ ( P ) n .It is enough to show that any equivariant coherent sheaf F on U can beobtained by a finite number of extensions starting with restrictions of vectorbundles F l,E . It is well-known (e.g. [Tho87]) that F is a restriction of anequivariant coherent sheaf on ( P ) n and we are done by Theorem 3.7. Forthe rest of the proposition, restrict isomorphisms of § U . (cid:3) These open substacks U include all Hassett spaces when the universalfamily is a P -bundle. More precisely, fix weights a = ( a , . . . , a n ) with P a i = 2 . The GIT quotient X a is a quotient of the semi-stable locus ( P ) nss for the fractional PGL -polarization given by a . The stack quotient M a = [( P ) nss / PGL ] , is an open substack of M n . If there are no strictly semistable points then PGL acts on ( P ) nss freely and M a = M a = X a .The most important case for us throughout the paper will be this: Definition 4.2.
For every partition P ` Q = { , . . . , n } with p = | P | ≥ , q = | Q | , we define an open substack M p,q ⊂ M n of P bundles such that atmost p/ sections indexed by P (heavy points) are allowed to coincide andif p = 2 r then r heavy points are allowed to coincide with at most q/ pointsindexed by Q (light points). The corresponding Hassett space is M p,q andthe GIT quotient is X p,q (see Notation 1.5). We have M p,q = M p,q = X p,q unless both p and q are even. We use notation M n , M n and X n if q = 0 . Wealso use the following notation: for every subset E ⊂ { , . . . , n } , we denoteby E p (resp. E q ) its intersection with P (resp. Q ).We would like to show the following: Theorem 4.3.
Consider a collection { F l,E } of PGL -equivariant vector bundleson ( P ) n . Order them by increasing e = | E | (and arbitrarily when e is the same).Choose an integer w K for every subset K such that P i ∈ K a i > . Suppose w K ≤ − l + ( e − e ∞ ) and l + ( e − e ∞ ) < w k + 2 | K | − for all bundles in the collection and all subsets as above, where e ∞ = | E ∩ K | and e = | E ∩ K c | . Then { F l,E } is a strong exceptional collection in D b ( M a ) . We will use the following theorem of Halpern–Leistner:
Theorem 4.4. [HL15]
Let [ X/G ] be the stack quotient of a smooth projectivevariety by a reductive group G and let [ X ss /G ] the open substack corresponding tothe semistable locus X ss with respect to a choice of polarization and linearization.For a choice of a Kempf–Ness (KN) stratification of the unstable locus X us withdata Z i , S i , λ i , σ i : Z i ֒ → S i , define the integers η i = weight λ i det (cid:0) N ∗ S i | X (cid:1) > . For each KN stratum S i , choose an integer w i ∈ Z . Define the full subcate-gory G w consisting of objects F ∈ D b [ X/G ] with the property that H ∗ ( σ ∗ i F ) hasweights in [ w i , w i + η i ) for all i . Then the restriction functor i ∗ : G w → D b [ X ss /G ] is an equivalence of categories. Remark 4.5.
There are two sign conventions for weights used in the liter-ature. Above we follow [HL15] where the ample polarization of the GITquotient has negative weights on the unstable locus, see (4.1) However,starting with §
10 we take the opposite weight as in [Tel00].
ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 17
In our case X = ( P ) n and G = PGL . Up to conjugation, a one parame-ter subgroup λ : G m → PGL has the form λ ( t ) = (cid:20) t t − (cid:21) . For every subset K ⊆ { , . . . , n } , consider the λ -invariant point z K = ( p , . . . , p n ) , with p i = (cid:26) if i / ∈ K ∞ if ∈ K (cid:27) , where and ∞ = [1 : 0] . We haveweight λ O (1) | p = (cid:26) if p = 0 − if p = ∞ Let k = | K | . If L = O ( a , . . . , a n ) , it follows thatweight λ L | z K = X i ∈ K c a i − X i ∈ K a i . (4.1)Here we follow the convention of [HL15], where the ample polarization ofthe GIT quotient has negative weights on the unstable locus.Let ∆ K be a diagonal in ( P ) n consisting of points ( p , . . . , p n ) such thatfor all i ∈ K the points p i are equal. Let S K ⊂ ∆ K be a locally closed diago-nal (the complement to the union of other diagonals). The KN stratificationof X us is given by the diagonals S K such that P i ∈ K a i > and by inclusions σ K : Z K = { z K } ֒ → S K . The destabilizing 1-PS is λ .Next we compute det (cid:0) N ∗ ∆ K | ( P ) n (cid:1) . We may assume without loss of gen-erality that n ∈ K . As ∆ K is a complete intersection in X of big diagonals ∆ jn , for all j ∈ K \ { n } , it follows that N ∗ ∆ K | ( P ) n ∼ = M j ∈ K \{ n } O ( − ∆ jn ) | ∆ K , det (cid:0) N ∗ ∆ K | ( P ) n (cid:1) ∼ = O (cid:0) − X j ∈ K \{ n } ∆ jn (cid:1) | ∆ K ∼ = O (0 , . . . , , − | K | − , via the identification ∆ K = ( P ) n − k × P given by the n -th marking. Itfollows that the weights of det (cid:0) N ∗ ∆ K | ( P ) n (cid:1) | z K depend only on | K | and η K := weight λ det (cid:0) N ∗ ∆ K | ( P ) n (cid:1) | z K = 2( | K | − . Lemma 4.6.
For subsets
E, K ⊆ { , . . . , n } with e = | E | , k = | K | , we denote e ∞ = | E ∩ K | , e = | E ∩ K c | . Note that e ∞ + e = e . The weights of F l,E | z K are l + ( e − e ∞ ) , ( l −
2) + ( e − e ∞ ) , ( l −
4) + ( e − e ∞ ) , . . . , − l + ( e − e ∞ ) . Proof.
For this calculation we can view F l,E as an SL - (rather than a PGL -)equivariant bundle. Then F l,E = O ( E ) ⊗ V l , V l = Sym l V , where V is a trivial rank vector bundle with the standard SL -action. Theweights of the rank l + 1 vector bundle V l (at any λ -fixed point) are l, l − , l − , . . . , − l and the formula follows. (cid:3) Proof of Theorem 4.3.
Follows from discussion above. (cid:3)
Proof of Theorem 1.8 (exceptionality).
We show that the vector bundles F l,E satisfy the conditions in Theorem 4.3. It follows from Lemma 4.6 that themaximum weight of F l,E over all subsets K with | K | = k is l + e if e ≤ n − k and l + 2 n − k − e if e > n − k . Equivalently, the maximum weight of F l,E is min { l + e, l + 2 n − k − e } . Similarly, the minimum weight of F l,E is − ( l + e ) if e ≤ k and − ( l + 2 k − e ) if e > k , or equivalently, the minimum weight of F l,E is − min { l + e, l + 2 k − e } . The conditions in Theorem 4.3 for existence of a window are equivalent torequiring that for any pairs ( l, E ) , ( l ′ , E ′ ) , if we let e = | E | and e ′ = | E ′ | min { l + e, l + 2 k − e } + min { l ′ + e ′ , l ′ + 2( n − k ) − e ′ } < η k = 2( k − , for all k ≥ r + 1 . We now prove that this is the case for the list of pairs ( l, E ) in Theorem 1.8. We consider three cases. Case I: e, e ′ ≤ r . By assumption l + e, l ′ + e ′ ≤ r − . Thenmin { l + e, l + 2 k − e } + min { l ′ + e ′ , l ′ + 2( n − k ) − e ′ } ≤≤ ( l + e ) + ( l ′ + e ′ ) ≤ r − < k − , for all k ≥ r + 1 . Case II: e, e ′ ≥ r + 1 . By assumption, l + n − e, l ′ + n − e ′ ≤ r − . Thensimilarly to Case I, we have:min { l + e, l + 2 k − e } + min { l ′ + e ′ , l ′ + 2( n − k ) − e ′ } ≤≤ ( l + 2 k − e ) + ( l ′ + 2( n − k ) − e ′ ) ≤ r − < k − for all k ≥ r + 1 . Case III: e ≤ r and e ′ ≥ r + 1 (or the opposite). By assumption l + e ≤ r − , l ′ + n − e ′ ≤ r − . It follows thatmin { l + e, l + 2 k − e } + min { l ′ + e ′ , l ′ + 2( n − k ) − e ′ } ≤≤ ( l + e ) + ( l ′ + 2( n − k ) − e ′ ) ≤ r − < k − for all k ≥ r + 1 . We finish by applying Theorem 3.7. (cid:3)
Corollary 4.7.
Let p = 2 r . A collection { F l,E } from Theorem 1.8 for n = 2 r + 1 is a strong S p -equivariant exceptional collection in D b (M p, ) .Proof. We have M p, ≃ M , { r ,..., r ,ǫ } ≃ M p +1 . Indeed, the stability conditionis the same: no r + 1 points are allowed to collide. (cid:3) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 19
Theorem 4.8.
Let p = 2 r + 1 , q > . The vector bundles F l,E form a full strong ( S p × S q ) -equivariant exceptional collection on M p,q for subsets E p ⊆ P , E q ⊆ Q such that l + e is even and l + min( e p , p − e p ) ≤ r − . The order is first by increasing e q , arbitrarily if e q is the same, but the set E q isdifferent. If E q = E ′ q , the order is by increasing e p , and for a given e p , arbitrarily.Proof. We have a morphism f : M p,q → M p , an iterated ( q times) universal P -bundle of M p . It is induced by the pro-jection ( P ) p + q → ( P ) p which maps ( P ) p + qss to ( P ) pss . In particular, wecan identify the pull-back f ∗ { F l,E p } of the collection of Theorem 1.8 with acollection { F l,E p } on M p,q when E p is a subset of P .For every j ∈ Q , let L j = F , { ,...,p,j } be a choice of an S p -equivariant rel-ative O π j (1) for the j -th copy of the universal P -bundle. For every subset J ⊂ Q , let L J = N j ∈ J L j . From Orlov’s theorem on the derived categoryof a projective bundle [Orl92], we have a semi-orthogonal decompositionof D b (M p,q ) into q blocks equivalent to D b (M p ) via functors D b (M p ) → D b (M p,q ) , E Lf ∗ ( E ) ⊗ L J for every subset J ⊂ Q . In blocks with | J | = 2 s even, we introduce anexceptional collection { F l,E p } ⊗ F ⊗− s , { ,...,p } ⊗ L J = { F l,E p ∪ J } . In blocks with | J | = 2 s + 1 odd, we use an exceptional collection { F l,E cp } ⊗ F ⊗− s , { ,...,p } ⊗ L J ≃ { F l,E p ∪ J } , where E cp = P \ E p . (cid:3)
5. E
XTENDING VECTOR BUNDLES F l,E TO SOME H ASSETT SPACES
The goal of this section is to construct an analogue of F l,E for certainHassett spaces. Let M := M A such that all A -stable curves have at mosttwo components. We assume in addition that no partial sum P i ∈ I a i with | I | ≥ equals . This includes all spaces M p,q when p and q are both even. Definition 5.1.
Let α : W → M be the universal family with sections σ , . . . , σ n : M → W .We will construct vector bundles F l,E on both M and W starting withDefinition 5.7 after we discuss geometry of these spaces.We recall several facts which will be used extensively in what follows.Recall that we have tautological line bundles ψ i = σ ∗ i ω α , δ ij = σ ∗ i ( σ j ) .(1) Assume M is a Hassett space whose universal family is a P -bundle.Then in Pic(M) we have ψ i + ψ j = − δ ij [CT17][Lemma 5.1].(2) If A = ( a , . . . , a n ) is such that a = 1 , P k =1 a k > and for all j ≥ we have P k = j, a k ≤ , then M ∼ = P n − [Has03, Section 6.2]. Furthermore, δ ij = O (1) ( i = j ) , ψ = O (1) , ψ j = O ( −
1) ( j = 1) . (3) If M = M A , M ′ = M B , A = ( a i ) , B = ( b i ) are Hasett spaces related bya reduction map p : M → M ′ (i.e., a i ≥ b i for all i ) then by [CT17][Lemma5.6] we have: p ∗ ψ i = ψ i − X i ∈ I, | I |≥ , P i ∈ I a i > , P i ∈ I b i ≤ δ I , (5.1) p ∗ δ ij = δ ij + X i,j ∈ I, | I |≥ , P i ∈ I a i > , P i ∈ I b i ≤ δ I . (5.2) Lemma 5.2. W is a Hassett space with markings , . . . , n and an extra point x .Every boundary divisor δ T := δ T,T c ⊆ M is isomorphic to P m − × P n − m − forsome m and its preimage in W is the union of δ T ∪{ x } and δ T c ∪{ x } , where δ T ∪{ x } = Bl p P m − × P n − m − , and the restriction map α | δ T ∪{ x } : δ T ∪{ x } = Bl P m − × P n − m − → δ T = P m − × P n − m − is the product map π × Id , where π : Bl p P m − → P m − is the canonical P -bundle. The description for δ T c ∪{ x } is similar.Proof. Since no partial sum P i ∈ I a i with | I | ≥ equals , the space W ∼ =M { ,...,n,x } with weights ( a , . . . , a n , ǫ ) . In particular, W is smooth.The boundary divisor δ T := δ T,T c ⊆ M corresponds to A - stable curveswith two components and markings from T , T c , respectively. Since δ T =M ′ × M ′′ , where M ′ and M ′′ are Hassett spaces parametrizing only irre-ducible curves, we have for all j ∈ T that X i ∈ T \{ j } a i ≤ < X i ∈ T a i . By Lemma 5.4, M ′ = P m − and similarly, M ′′ = P n − m − , with m = | T | .Consider now δ T ∪{ x } = δ T c = δ T ∪{ x } ,T c ⊆ W, T c = N \ T, N = { , . . . , n } . the corresponding boundary component in W . We have δ T ∪{ x } = U ′ × M ′′ , where π : U ′ → M ′ is the universal family over M ′ and U ′ can be identifiedwith the Hassett space with weights { a i } i ∈ T ∪ { ǫ, } , with the weight ǫ ,resp. , corresponding to the marking x , resp., the attaching point, call it u .It follows from Lemma 5.4 that δ T ∪{ x } = Bl p P m − × P n − m − , and the restriction map α | δ T ∪{ x } is as claimed. (cid:3) Lemma 5.3.
Using the identification δ T ∪{ x } = Bl p P m − × P n − m − , and denoting H = O P r − (1) , and by ∆ the exceptional divisor on Bl p P m − , we have(i) ( ω α ) | δ T ∪{ x } = ( − H ) ⊠ O . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 21 (ii) O ( σ i ) | δ T ∪{ x } = ( O if i / ∈ TH ⊠ O if i ∈ T. (iii) ( δ T ′ ∪{ x } ) | δ T ∪{ x } = ∆ ⊠ O if T ′ = T c − H ⊠ O ( −
1) if T ′ = T O if T ′ = T, T c . Furthermore, ( π × Id ) ∗ O ( − a, − b ) = π ∗ O ( − a ) ⊠ O ( − b ) = ( − aH + a ∆) ⊠ O ( − b ) . Proof.
Denote for simplicity δ = δ T ∪{ x } ⊆ W , δ = δ T ⊆ M . Since ω α = K U − α ∗ K M , we have by Lemma 5.4 and adjunction that K U | δ = K δ − δ | δ = (( − m + 1) H + ( m − ⊠ O ( − n + m + 2) ,K M | δ = K δ − δ | δ = (( − m + 2) H ⊠ O ( − n + m + 2) , and therefore ω α | δ = ( − H ) ⊠ O .Part (ii) follows from σ i = δ ix and Lemma 5.4(iii).To prove (iii), note that two boundary divisors δ T ∪{ x } , δ T ′ ∪{ x } intersectif and only if T = T ′ or T ′ = T c . Furthermore, δ T ∩{ x } ∩ δ T ′ ∪{ x } has class ∆ ⊠ O . For any boundary divisor δ on any Hassett space, we have: δ | δ = ( − ψ u ) ⊠ ( − ψ u ) . The last statement follows immediately from Lemma 5.4(iv). (cid:3)
Lemma 5.4.
Let N = { , . . . , n } . Assume a = 1 and assume that the universalfamily α : W → M is a P -bundle, i.e., all A -stable curves are irreducible, Then:(i) M ∼ = P n − and W ∼ = Bl p P n − . If H := α ∗ O (1) and ∆ denotes theexceptional divisor on W , then we have in Pic( W ) : ψ = H, δ N \{ } , { ,x } = ∆ , (here we identify W with the Hassett space with markings from N ∪ { x } ).(ii) The map α : W = Bl p P n − → M = P n − is induced by the linear system | H − ∆ | . In particular, if F be a fiber of α , then ψ · F = 1 .(iii) In Pic( W ) we have that δ jx = H , for all j in N \ { } .Proof. Since there are no reducible A -curves, it follows that for all j = 1 ,we have P k =1 ,j a k ≤ and therefore by [Has03, Section 6.2], we have M ∼ = P n − . The only reducible curves parametrized by W are those given by theboundary component δ := δ x . Let ˜ M be the Hassett space with the weights (1 , η, . . . , η, ǫ ) , with η = n − (i.e., such that all but one of the markings ( N \ { } ) ∪ { x } may coincide). Then ˜ M ∼ = P n − and the canonical reductionmap φ : W → ˜ M contracts δ to a point p ∈ ˜ M . Hence, W = Bl p ˜ P n − withexceptional divisor ∆ = δ . By (5.1, we haave φ ∗ ψ = ψ . This proves (i).Clearly, the map α restricted to ∆ = δ is an isomorphism. The onlymorphism π : Bl p P n − → P n − which is an isomorphism on the exceptionaldivisor ∆ is the one given by the linear system | H − ∆ | . We have then from(i) that H · F = 1 for any fiber F of π . This proves (ii). The curve F is obtained by moving the marking x along a P with fixedmarkings from N . Hence, δ jx · F = 1 . Furthermore, δ jx · ∆ = 0 . It followsthat δ jx = H in Pic( W ) . This proves (iii). (cid:3) Lemma 5.5.
Let π : Bl p P m − → P m − . If a ≥ we have Rπ ∗ (cid:0) O ( a ∆) (cid:1) = O ( − a ) ⊕ . . . ⊕ O ( − ⊕ O , while when a < we have Rπ ∗ (cid:0) O ( − ∆) (cid:1) = 0 , and when a ≥ we have Rπ ∗ (cid:0) O ( − a ∆) (cid:1) is generated by O (1) , . . . , O ( a − .More generally, Rπ ∗ (cid:0) O ( aH − b ∆)) is either if a = b − , or it is generated by:(i) O ( b ) , O ( b + 1) , . . . , O ( a ) if a ≥ b ;(ii) O ( a + 1) , O ( a + 2) , . . . , O ( b − if a ≤ b − ;In particular, we have that Rπ ∗ (cid:0) O ( aH − b ∆)) is generated by O ( u ) with min { b, a + 1 } ≤ u ≤ max { a, b − } . Proof.
The lemma follows by applying Rπ ∗ ( − ) to the exact sequences → O (( i − → O ( i ∆) → O ∆ ( − i ) → , along with the projection formula and π ∗ O (1) = O ( H − ∆) . (cid:3) Lemma 5.6. On Bl p P m − , the divisor − uH + v ∆ is acyclic if < u ≤ ( m − , ≤ v ≤ ( m − . Proof.
This is clear. (cid:3)
Definition 5.7 ( The vector bundles F l,E on M ). For every subset E ⊆ N := { , . . . , n } with e = | E | and integers l ≥ such that e + l is even, we let F l,E = Rα ∗ (cid:0) N l,E (cid:1) , where N l,E = ω e − l α ( E ) ⊗ O ( − X T α T,E,l δ T ∪{ x } ) , and the sum is over all T ⊆ N such that δ T ⊆ M is a boundary component,and where α T,E,l = ( | E ∩ T | ≥ e − l e − l − | E ∩ T | if | E ∩ T | < e − l . (5.3) Lemma 5.8.
Assume M is Hassett space such that all A -stable curves have atmost two components. Let a , { α T } be integers. The complex Rα ∗ (cid:0) ω aα ( E ) ⊗ O ( − X T α T δ T ∪{ x } ) (cid:1) is a vector bundle of rank e − a + 1 if | E ∩ T c | − a ≥ α T − α T c ≥ a − | E ∩ T | (5.4) (in which case, note that we have actually Rα ∗ = α ∗ ). In particular, the complex F l,E in Def. 5.7 is a vector bundle of rank l + 1 . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 23
Proof.
Note that (5.4) implies that e ≥ a . For simplicity, we denote L := ω aα ( E ) ⊗ O ( − X T α T δ T ∪{ x } ) . If C is a generic (i.e., irreducible) fiber of α : W → M , then we have deg (cid:0) L | C (cid:1) = deg (cid:0) ω aα ( E ) (cid:1) | C = − a + e ≥ . Let now C be a reducible fiber of α . By our assumption, the curve C isthe fiber above a point of δ T ⊆ M , for a partition T ⊔ T c of N . The curve C has components C and C , with C (resp., C ) having fixed markings from T (resp., T c ). The curve C i ⊆ W is given by the point x moving along C i .Therefore, we have C ⊆ δ T ∪{ x } = δ T c , C ⊆ δ T c ∪{ x } = δ T ,C · δ T = 1 , C · δ T c = 1 , and C i intersects all boundary other than δ T , δ T c trivially. Furthermore, onehas δ T c · C = − from ( δ T + δ T c ) · C = α − ( δ T ) · C = 0 . Similarly δ T · C = − . We have that (cid:0) ω α (cid:1) | C i = O ( − and deg( O ( σ j )) | C i =1 if j ∈ T and otherwise. It follows that deg( L | C ) = − a + | E ∩ T | + α T − α T c , deg( L | C ) = − a + | E ∩ T c | + α T c − α T . There is an exact sequence → L → L | C ⊕ L | C → O k ( u ) → . If deg( L | C i ) ≥ for i = 1 , , then it follows that H ( L | C i ) = 0 for i = 1 , and the induced map H ( L | C ⊕ L | C ) → k ( u ) is surjective. Therefore,H ( L ) = 0 , h ( L ) = e − a + 1 and Rα ∗ L = α ∗ L is a vector bundle of rank e − a + 1 .Assume now that a = e − l and α T is as in Def. 5.7. Case | E ∩ T | ≥ e − l , | E ∩ T c | ≥ e − l . In this case α T = α T c = 0 and deg( L | C ) = | E ∩ T | − e − l ≥ , deg( L | C ) = | E ∩ T c | − e − l ≥ . Case | E ∩ T | ≥ e − l , | E ∩ T c | < e − l . In this case α T = 0 , α T c = e − l − | E ∩ T c | , deg( L | C ) = l ≥ , deg( L | C ) = 0 . The case when | E ∩ T c | ≥ e − l , | E ∩ T | < e − l is similar. Note that since l ≥ , we cannot have | E ∩ T c | < e − l , | E ∩ T | < e − l . (cid:3) Lemma 5.9.
Let p = 2 r ≥ , q = 2 s + 1 ≥ − and consider the Hassett spaces M p,q +1 . Let Σ be the set of all indices ( | Σ | = p + q + 1 ). When l is even, the vectorbundles F l, Σ satisfy the property F l, Σ = F l, ∅ ⊗ F , Σ Proof.
Denote for simplicity
M := M p,q +1 . By definition, F l, Σ = Rα ∗ ( N l, Σ ) ,where N l,E = ω e − l α (Σ) is a line bundle on the universal family α : W → M .We claim that the line bundle ω r + s +1 α (Σ) on W is a pull-back from M . In-deed, this line bundle has degree on the generic fiber and restricts triviallyon each component of the reducible fibers of α . Therefore, it follows that ω r + s +1 α (Σ) = α ∗ (cid:0) α ∗ ( ω r + s +1 α (Σ)) (cid:1) = α ∗ (cid:0) F , Σ (cid:1) . It follows that F l, Σ = Rα ∗ (cid:0) ω − l α ⊗ α ∗ F , Σ (cid:1) = Rα ∗ (cid:0) ω − l α (cid:1) ⊗ F , Σ = F l, ∅ ⊗ F , Σ . (cid:3) The vector bundles F l,E on W . Assume p = 2 r ≥ and q = 2 s + 1 ≥ . Consider the Hassett space M p,q with markings P ⊔ Q , with | P | = p , | Q | = q . Let y ∈ Q and let M p,q − bethe Hassett space with markings from P ⊔ ( Q \ { y } ) . Let α : W → M p,q − be the universal family. Then W is a Hassett space with boundary δ T ∪{ y } = δ T ∪{ y } ,T c = Bl p P r + s − × P r + s − ,T ⊔ T c = P ⊔ ( Q \ { y } ) , | T ∩ P | = r, | T ∩ Q | = s. Notation 5.10.
There is a birational morphism f : W → M p,q that contractseach boundary δ T ∪{ y } using first projection f | δ T ∪{ y } ,Tc : Bl p P r + s − × P r + s − → P r + s − . This is because if ∆ denotes that the boundary divisor on Bl p P r + s − , then ∆ × P r + s − = δ T ∪{ y } ∩ δ T c ∪{ y } is contracted to a point in M p,q .Let π : U → W be the universal family, with x the new marking on U .The boundary in U has two types: δ T ∪{ y } ,T c ∪{ x } = Bl p P r + s − × Bl p P r + s − , with the restriction map of the form π | δ T ∪{ y } ,Tc ∪{ x } = ( Id, q ) : Bl p P r + s − × Bl p P r + s − → Bl p P r + s − × P r + s − = δ T ∪{ y } ,T c , (see Lemma 5.12). The second type is δ T ∪{ y,x } ,T c = M T ∪{ y,x,u } × M T c ∪{ u } = Bl , , P r + s × P r + s − , where u is the attaching point (weight ) and Bl , , P r + s denotes the blow-up of P r + s at two distinct points p , p and the proper transform of the linethrough them. The restriction map has the form π | δ T ∪{ y,x } = (˜ q, Id ) : Bl , , P r + s × P r + s − → Bl p P r + s − × P r + s − , (see Lemma 5.12). Notation 5.11. (i) On Bl p P r + s − we denote by H the hyperplane classand by ∆ the exceptional divisor.(ii) On Bl , , P r + s we denote by H the hyperplane class, and by E , E , E the corresponding exceptional divisors. ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 25
Lemma 5.12.
Let π : U → W be the universal family. Then:(1) Let δ := δ T ∪{ x } ,T c ∪{ y } = Bl p P r + s − × Bl p P r + s − . The restriction π | δ isgiven by the pair ( q, Id ) , where q : Bl p P r + s − → P r + s − , q ∗ O (1) = H − ∆ , ∆ = δ ux . Moreover, ω π | δ = ( − H ) ⊠ O , δ yx | δ = 0 ,δ jx | δ = ( H ⊠ O , if j ∈ T, O if otherwise . (2) On Bl , , P r + s = M T ∪{ y,x,u } , we have: E = δ T ∪{ x } , E = δ T ∪{ y } , E = δ T . Let δ := δ T ∪{ y,x } = Bl , , P r + s × P r + s − . The restriction π | δ is given by the pair (˜ q, Id ) , where ˜ q : Bl , , P r + s → Bl p P r + s − , ˜ q ∗ H = H − E , ˜ q ∗ ∆ = E + E , and ω π | δ = ( − H + E ) ⊠ O ,δ jx | δ = ( H − E ) ⊠ O , j ∈ T,δ jy | δ = ( H − E ) ⊠ O , j ∈ T,δ yx | δ = H ⊠ O . Proof.
For (1), the statements about the map q and δ jx | δ follow from Lemma5.4. Denoting δ = π ( δ ) , we have by adjunction that K W | δ = O ( − ( r + s − ⊠ ( − ( r + s − H + ( r + s − ,K U | δ = ( − ( r + s − H + ( r + s − ⊠ ( − ( r + s − H + ( r + s − . Using ω π = K U − π ∗ K W , we have that ω π | δ = ( − H ) ⊠ O .We now prove (2). Denote for simplicity M := M T ∪{ y,x,u } , M W := M T ∪{ y,u } , and let N ′ , resp., N ′′ be the Hassett space with markings T ∪ { y, x, u } suchthat the weights in T (resp., T , T ∪ { x } , T ∪ { y } ) are allowed to coincide.Note that N ′′ ∼ = P r + s , since all but one of the markings T ∪ { y, x } maycoincide. There are reduction maps π ′ : M → N ′ , π ′′ : N ′ → N ′′ . The map π ′′ contracts δ T ∪{ x } and δ T ∪{ y } to points, which we denote p := π ′′ ( δ T ∪{ x } ) , p := π ′′ ( δ T ∪{ y } ) , while the composition π ′′ ◦ π ′ contracts δ T to the line through p and p .Hence, M is isomorphic to the blow-up Bl , , P r + s and we have E = δ T ∪{ x } , E = δ T ∪{ y } , E = δ T . The map ˜ q : M = Bl , , P r + s → M W = Bl p P r + s − forgets the x mark-ing; hence, ˜ q is an isomorphism on E = ∆ T,y . Since the only fibrations Bl , , P r + s → Bl p P r + s − are given by the linear systems | H − E | , | H − E | ,it follows that ˜ q ∗ H = H − E . As ∆ = δ T ⊆ M W , it follows that ˜ q ∗ ∆ = δ T + δ T ∪{ x } = E + E . The restriction δ jx | δ is δ jx ⊠ O if j ∈ T and trivial otherwise. If j ∈ T , thepull-backs of δ jx via the reduction maps π ′ , π ′′ are given by π ′′∗ δ jx = δ jx + δ T ∪{ x } , ( π ′ ◦ π ′′ ) ∗ δ jx = δ jx + δ T ∪{ x } , As δ jx = O (1) on N ′′ = P r + s , it follows that on M we have δ jx = H − E . Bysymmetry, when j ∈ T c we also have δ jy = H − E on M . Similarly, since ( π ′ ◦ π ′′ ) ∗ δ yx = δ yx and δ yx = O (1) on N ′′ = P r + s , we have δ yx | δ = H ⊠ O .Denoting δ = π ( δ ) , by adjunction, we have K W | δ = ( − ( r + s − H + ( r + s − ⊠ O ( − ( r + s − ,K U | δ = ( − ( r + s ) H + ( r + s − E + E ) + ( r + s − E ) ⊠ O ( − ( r + s − . Using ω π = K U − π ∗ K W , it follows that ω π | δ = ( − H + E ) ⊠ O . (cid:3) Remark 5.13.
The same proof in Lemma 5.5 shows that if a ≥ we have R ˜ q ∗ (cid:0) O ( aE ) (cid:1) = O ( − aH ) ⊕ . . . ⊕ O ( − H ) ⊕ O . Definition 5.14.
For each E ⊆ P ∪ Q and l ≥ we define on WF l,E = Rπ ∗ (cid:0) N l,E (cid:1) ,N l,E = ω e − l π ( E ) ⊗ O ( − X T α T δ T ∪{ x } − X T α T ∪{ y } δ T ∪{ y,x } ) , where for S = T or S = T ∪ { y } , we define α S = ( | E ∩ S | ≥ e − l e − l − | E ∩ S | if | E ∩ S | < e − l , (5.5)(hence, either α T = α T ∪{ y } = 0 or α T ∪{ y } = α T − | E ∩ { y }| ≥ ). Lemma 5.15. F l,E is a vector bundle of rank l + 1 on W .Proof. Denote N := N l,E for simplicity. If C is an irreducible fiber of π , wehave deg( N | C ) = l . Consider now a reducible fiber C with two components C and C , with markings from T ∪ { y } (resp., T c ) on C (resp. C ). Then C · δ T ∪{ y,x } = − , C · δ T ∪{ y } = 1 ,C · δ T ∪{ y,x } = 1 , C · δ T ∪{ y } = − , while all other intersections with boundary are . We have: deg (cid:0) N | C (cid:1) = | E ∩ { y }| + | E ∩ T | − e − l α T ∪{ y } − α T c , deg (cid:0) N | C (cid:1) = | E ∩ T c | − e − l − α T ∪{ y } + α T c . Case 1) | E ∩ T | ≥ e − l , | E ∩ T c | ≥ e − l . Then α T ∪{ y } = α T c = 0 and deg (cid:0) N | C (cid:1) = | E ∩{ y }| + | E ∩ T |− e − l ≥ , deg (cid:0) N | C (cid:1) = | E ∩ T c |− e − l ≥ . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 27
Case 2) | E ∩ T | ≥ e − l , | E ∩ T c | < e − l . Then α T ∪{ y } = 0 , α T c = e − l − | E ∩ T c | , deg (cid:0) N | C (cid:1) = l ≥ , deg (cid:0) N | C (cid:1) = 0 . Case 3) | E ∩ T | < e − l , | E ∩ T c | ≥ e − l . Then α T ∪{ y } = e − l − | E ∩ T | − | E ∩ { y }| , α T c = 0 , deg (cid:0) N | C (cid:1) = 0 , deg (cid:0) N | C (cid:1) = l. It follows in all cases that h ( N | C ) = 0 , h ( N | C ) = l + 1 .Consider now a reducible fiber C with three components C , C , C , withmarkings from T , { y } , T c respectively. Then C · δ T ∪{ x } = − , C · δ T c ∪{ y,x } = 1 ,C · δ T ∪{ y,x } = C · δ T c ∪{ y,x } = − , C · δ T ∪{ x } = C · δ T c ∪{ x } = 1 ,C · δ T c ∪{ x } = − , C · δ T ∪{ y,x } = 1 , while intersections with other boundary are . We have: deg (cid:0) N | C (cid:1) = | E ∩ T | − e − l α T − α T c ∪{ y } , deg (cid:0) N | C (cid:1) = α T ∪{ y } + α T c ∪{ y } − α T − α T c + | E ∩ { y }| , deg (cid:0) N | C (cid:1) = | E ∩ T c | − e − l α T c − α T ∪{ y } . Case 1) | E ∩ T | ≥ e − l , | E ∩ T c | ≥ e − l . Then α T = α T ∪{ y } = α T c = α T c ∪{ y } = 0 , deg (cid:0) N | C (cid:1) = | E ∩ T | − e − l ≥ , deg (cid:0) N | C (cid:1) = | E ∩ T c | − e − l ≥ , deg (cid:0) N | C (cid:1) = | E ∩ { y }| ≥ . Case 2) | E ∩ T | ≥ e − l , | E ∩ T c | < e − l . Then α T = α T ∪{ y } = 0 , α T c = e − l − | E ∩ T c | , α T c ∪{ y } = α T c − | E ∩ { y }| , deg (cid:0) N | C (cid:1) = l ≥ , deg (cid:0) N | C (cid:1) = 0 , deg (cid:0) N | C (cid:1) = 0 . Case 3) | E ∩ T | < e − l , | E ∩ T c | ≥ e − l . Then α T c = α T c ∪{ y } = 0 , α T = e − l − | E ∩ T | , α T ∪{ y } = α T − | E ∩ { y }| , deg (cid:0) N | C (cid:1) = 0 , deg (cid:0) N | C (cid:1) = l, deg (cid:0) N | C (cid:1) = 0 . It follows in all cases that h ( N | C ) = 0 , h ( N | C ) = l + 1 .Note that if e + l is even and l ≥ , we cannot have | E ∩ T | < e − l , | E ∩ T c | < e − l , as otherwise we would have e − | E ∩ { y }| = | E ∩ T | + | E ∩ T c | ≤ ( e − l −
1) + ( e − l −
1) = e − l − , which is a contradiction. (cid:3) Lemma 5.16.
Let α : W → M p,q − be the universal family, with y the newmarking on W . Let l ≥ , E ⊆ P ∪ Q , with e + l even.(i) If y / ∈ E , then F l,E = α ∗ F l,E .(ii) Assume y ∈ E . For all l ≥ , e + l even, there is an exact sequence → F l − ,E \{ y } → F l,E → Q yl,E → , of vector bundles on W , with Q yl,E := σ ∗ y N l,E , where σ y is the section of π : U → W corresponding to the y marking. Furthermore, Rα ∗ ( F ∨ l,E ) = 0 . Proof of Lemma 5.16.
There is a commutative diagram U v −−−−→ V φ −−−−→ W π y ρ y α = α x y W Id −−−−→ W α = α y −−−−→ M p,q − where α : W → M p,q − is the universal family and α = α x , resp., α = α y in-dicates that the marking x (resp., y ) is the marking that is getting dropped.The right square is Cartesian. Let g = φ ◦ v . Note that v is a small map thatcontracts the codimension loci π − ( δ T ∩ δ T c ) = P r + s − × P × P r + s − → P r + s − × pt × P r + s − . Claim 5.17.
We have(i) Rv ∗ v ∗ O V ∼ = Rv ∗ O U ∼ = O V .(ii) In Pic( U ) we have v ∗ ω ρ = ω π , while on W we have ψ y = ω α y .(iii) g ∗ δ T ∪{ x } = δ T ∪{ x } + δ T ∪{ y,x } .Proof. We note that v is birational and its image has rational singularities,which are in fact locally isomorphic to the product of the affine cone xy = zt and a smooth variety (see e.g. [Kee92, page 548]). Part (i) follows. Parts (iii)is immediate since g is the map that forgets the marking y .By definition ψ y = σ ∗ y ω π . Since v ∗ ω ρ = ω π , it follows that if s y = v ◦ σ y ,then ψ y = s ∗ y ω ρ = s ∗ y φ ∗ ω α = ω α , since φ ◦ s y = Id W . This proves (ii). (cid:3) We have F l,E = Rπ ∗ L , where L := N l,E = e − l ω π + O ( E ) − X | E ∩ T | < e − l (cid:18) e − l − | E ∩ T | (cid:19) δ T ∪{ x } −− X | E ∩ T | < e − l (cid:18) e − l − | E ∩ T | − | E ∩ { y }| (cid:19) δ T ∪{ y,x } . We now compute α ∗ F l ′ ,E ′ , for y / ∈ E ′ . Using (i), we have α ∗ F l ′ ,E ′ = α ∗ y Rα x ∗ N l ′ ,E ′ = Rρ ∗ φ ∗ N l ′ ,E ′ = Rπ ∗ L , where L := v ∗ φ ∗ N l ′ ,E ′ = g ∗ N l ′ ,E ′ , with N l ′ ,E ′ is the usual line bundle on W : N l ′ ,E ′ = e ′ − l ′ ω π + O ( E ′ ) − X | E ′ ∩ T | < e ′− l ′ (cid:18) e ′ − l ′ − | E ′ ∩ T | (cid:19) δ T ∪{ x } . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 29
Since v has no exceptional divisors, it follows that L = e ′ − l ′ ω π + O ( E ′ ) − X | E ′ ∩ T | < e ′− l ′ (cid:18) e ′ − l ′ − | E ′ ∩ T | (cid:19) (cid:0) δ T ∪{ x } + δ T ∪{ y,x } (cid:1) . Case y / ∈ E, l ′ = l, E ′ = E . We clearly have L = L , and this proves thatwhen y / ∈ E , we have F l,E = α ∗ F l,E .Case y ∈ E, l ′ = l − , E ′ = E \ { y } . As e ′ − l ′ = e − l , | E ∩ { y }| = 1 , wehave L = L + O ( y ) + X | E ∩ T | < e − l δ T ∪{ y,x } , → L → L ( − y ) → M | E ∩ T | < e − l (cid:0) L ( − y ) (cid:1) | δ T ∪{ y,x } → , → L ( − y ) → L → (cid:0) L (cid:1) | σ y → . Assume T is such that | E ∩ T | < e − l . Then by Lemma 5.12 we have (cid:0) L ( − y ) (cid:1) | δ T ∪{ y,x } = ( − H ) ⊠ O (cid:18) e − l − | E ∩ T | − (cid:19) , on δ T ∪{ y,x } = Bl , , P r + s × P r + s − . Since on Bl , , P r + s we have R ˜ q ∗ ( − E ) =0 and ˜ q ∗ ( H ) = H − E (Lemma 5.12), it follows that R ˜ q ∗ ( − H ) = 0 . Hence, Rπ ∗ (cid:0) L ( − y ) (cid:1) | δ T ∪{ y,x } = 0 , Rπ ∗ L ∼ = Rπ ∗ L ( − y ) = F l − ,E \{ y } , (and here Rπ ∗ ( − ) = π ∗ ( − ) ). Applying Rπ ∗ ( − ) to the above two exact se-quences, it follows that there is an exact sequence on W → F l − ,E \{ y } → F l,E → Q yl,E → , where Q yl,E = σ ∗ y N l,E . Finally, we have Rα ∗ (cid:0) F ∨ l,E (cid:1) = Rα y ∗ ◦ Rπ ∗ (cid:0) ω π − L (cid:1) = Rα x ∗ ◦ Rg ∗ (cid:0) ω π − L (cid:1) . Hence, it suffices to prove that Rg ∗ (cid:0) ω π − L (cid:1) = 0 . Since ω π = g ∗ ω α , L is apull-back by g , and L = L + σ y + Σ , where Σ = X | E ∩ T | < e − l δ T ∪{ y,x } , it suffices to prove that Rg ∗ O ( − σ y − Σ) = 0 . Consider the exact sequence → O ( − σ y − Σ) → O ( − Σ) → O ( − Σ) | σ y → . It suffices to prove that g ∗ ( − ) induces an isomorphism when applied tothe restriction map O ( − Σ) → O ( − Σ) | σ y and all higher push forwards by g of O ( − Σ) and O ( − Σ) | σ y are . Since g ◦ σ y = Id and σ y = δ yx , we have R i g ∗ O ( − Σ) | σ y = 0 for all i > and g ∗ O ( − Σ) | σ y = O W ( − Σ ′ ) , where Σ ′ = g (Σ) = X | E ∩ T | < e − l δ T ∪{ x } . Because of the condition | E ∩ T | < e − l , the divisors in Σ are disjoint. Simi-larly, the divisors in Σ ′ are disjoint. Using the exact sequence → O ( − δ T ∪{ y,x } ) → O U → O δ T ∪{ y,x } → it suffices to prove that Rg ∗ O U = O W and Rg ∗ O δ T ∪{ y,x } ∼ = O δ T ∪{ x } . Since g = φ ◦ v , the first statement follows from Rv ∗ O U ∼ = O V (part (i)) and Rφ ∗ O V ∼ = O W , as φ is the pull-back of α y : W → M p,q − , the universalfamily over M p,q − . The second statement follows as the map g restrictedto δ T ∪{ y,x } is the map π | δ = (˜ q, Id ) from Lemma 5.12, where ˜ q : M T ∪{ y,x,u } = Bl , , → M T ∪{ y,u } = Bl p P r + s − with is the universal family, in this case a flat family of rational, at worstnodal curves over a smooth base. (cid:3)
6. T
HE BUNDLES F l,E VERSUS THE SHEAVES O δ ( − a, − b ) Proposition 6.1.
Assume P (resp., Q ) is the set of heavy (resp., light) indices and | P | = p = 2 r ≥ , | Q | = q + 1 = 2 s + 2 ≥ , Assume l ≥ , E ⊆ P ∪ Q with e = | E | such that e + l is even. Let δ T = δ T,T c ⊆ M p,q +1 be a boundary divisor such that (cid:12)(cid:12)(cid:12)(cid:12) | E ∩ T | − e − l (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) | E ∩ T c | − e − l (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ≤ r + s . Using the identification δ T = P × P r + s − , then RHom M p,q +1 ( F ∨ l,E , O δ T ( − a, − b )) = 0 (with a on the T -component) whenever one of the following happens: • ≤ a, b ≤ r + s − • a = r + s, µ < b < r + s • b = r + s, µ < a < r + s .Similarly, RHom M p,q +1 ( F l,E , O δ T ( − a, − b )) = 0 whenever one of the following happens: • ≤ a, b ≤ r + s − • a = 0 , < b < r + s − µ • b = 0 , < a < r + s − µ .Proof. By Serre duality, if E is a vector bundle on M p,q +1 , then RHom M p,q +1 ( E ∨ , O δ T ( − a, − b )) = RHom M p,q +1 ( E, K δ T ⊗ O δ T ( a, b )) ∨ . So the second statement is equivalent to the first.We now prove the first statement. If α : W → M p,q +1 is the universalfamily, let β be the restriction β := α | α − ( δ T ) : α − ( δ T ) = δ T ∪ δ T c → δ T , where we denote δ T := δ T,T c ∪{ y } ( y the new marking on W ). Then R Hom M p,q +1 (cid:0) F ∨ l,E , O δ T ( − a, − b ) (cid:1) = R Γ M p,q +1 (cid:0) F l,E | δ T ⊗ O δ T ( − a, − b ) (cid:1) . We emphasize that we allow here q + 1 = 0 (i.e., s = − ). ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 31
Since F l,E = Rα ∗ ( N l,E ) , it follows by cohomology and base change that F l,E | δ T = (cid:0) Rα ∗ ( N l,E ) (cid:1) δ T = Rβ ∗ (cid:0) N l,E | δ T ∪ δ Tc (cid:1) , and therefore by the projection formula R Γ M p,q +1 (cid:0) F l,E | δ T ⊗O δ T ( − a, − b ) (cid:1) = R Γ M p,q +1 (cid:0) Rβ ∗ (cid:0) N l,E | δ T ∪ δ Tc ⊗ β ∗ O ( − a, − b ) (cid:1)(cid:1) . Clearly, we have to show that the following line bundle on δ T ∪ δ T c ˜ N := (cid:0) N l,E | δ T ∪ δ Tc ⊗ β ∗ O ( − a, − b ) (cid:1)(cid:1) has no cohomology. Consider the following exact sequence: → O δ Tc ( − δ T ) → O δ T ∪ δ Tc → O δ T → . Tensoring with ˜ N , we obtain an exact sequence: → ˜ N ′′ → ˜ N → ˜ N ′ → , where ˜ N ′ := (cid:0) N l,E ⊗ β ∗ O ( − a, − b ) (cid:1) | δ T , ˜ N ′′ := (cid:0) N l,E ⊗ β ∗ O ( − a, − b ) ⊗ O ( − δ T ) (cid:1) | δ Tc ,N l,E = ω e − l π ( E )( − X α T,E,l δ T ∪{ y } ,T c ) . We now use Lemma 5.3 to compute ˜ N ′ and ˜ N ′′ . For simplicity, denote α T = α T,E,l and α T c = α T c ,E,l .Using the identification δ T = δ T,T c ∪{ y } = P r + s − × Bl P r + s , we have ˜ N ′ := O ( − a + α T c ) ⊠ (cid:0) ( | E ∩ T c | − e − l α T c − b ) H + ( b − α T )∆ (cid:1) . Similarly, using the identification δ T c = δ T c ,T ∪{ y } = P r + s − × Bl P r + s , wehave ˜ N ′′ := O ( − b + α T ) ⊠ (cid:0) ( | E ∩ T | − e − l α T − a ) H + ( a − α T c − (cid:1) . Here H , resp. ∆ , denotes O P r + s (1) , resp., the exceptional divisor on Bl P r + s .We prove that both ˜ N ′ , ˜ N ′′ are acyclic. To further simplify notations, let α := α T , β := α T c u := | E ∩ T | − e − l , v := | E ∩ T c | − e − l . Note that either u ≥ , α = 0 or u < , u + α = 0 . Similarly, either v ≥ , β = 0 or v < , v + β = 0 . Furthermore, u + v = l ≥ , hence, we must have u ≥ or v ≥ . Note that by assumption we have | u | , | v | ≤ µ. (6.1) Case u, v ≥ : then α = β = 0 and we have ˜ N ′ := O ( − a ) ⊠ (cid:0) ( v − b ) H + b ∆ (cid:1) . ˜ N ′′ := O ( − b ) ⊠ (cid:0) ( u − a ) H + ( a − (cid:1) . Clearly, if < a ≤ ( r + s − , < b ≤ ( r + s − then O ( − a ) , O ( − b ) areacyclic, hence so are ˜ N ′ , ˜ N ′′ . Assume that a = r + s , µ < b ≤ r + s − . Then O ( − b ) and therefore ˜ N ′′ is acyclic. By (6.1), we have − ( r + s − ≤ − b ≤ v − b < ≤ µ − b < . It follows that ˜ N ′ is acyclic by Lemma 5.6. Similarly, if b = r + s , µ < a ≤ r + s − , then O ( − a ) and ( u − a ) H + ( a − are acyclic, as we have − ( r + s − ≤ − a ≤ u − a < µ − a < . Case u ≥ , v < : then α = 0 , β + v = 0 and we have ˜ N ′ := O ( − a − v ) ⊠ (cid:0) − bH + b ∆ (cid:1) . ˜ N ′′ := O ( − b ) ⊠ (cid:0) ( u − a ) H + ( a + v − (cid:1) . If < b ≤ r + s − then O ( − b ) , − bH + b ∆ are acyclic, and the result follows.Assume now b = r + s , µ < a ≤ r + s − . By (6.1) ≤ µ + v < a + v < a ≤ r + s − − ( r + s − ≤ − a ≤ u − a ≤ µ − a < , hence, O ( − a − v ) and ( u − a ) H + ( a + v − are acyclic. Case v ≥ , u < : then β = 0 , α + u = 0 and we have ˜ N ′ := O ( − a ) ⊠ (cid:0) ( v − b ) H + ( b + u )∆ (cid:1) . ˜ N ′′ := O ( − b − u ) ⊠ (cid:0) − aH + ( a − (cid:1) . If < a ≤ r + s − then O ( − a ) , − aH + ( a − are acyclic, and the resultfollows. Assume now a = r + s , note that − aH + ( a − is still acyclic.Furthermore, O ( − b − u ) and ( v − b ) H + ( b + u )∆ are acyclic, as by (6.1) ≤ µ + u < b + u < b ≤ r + s − , − ( r + s − ≤ − b ≤ v − b ≤ µ − b < . (cid:3) Corollary 6.2.
The line bundle F , Σ on M p for p ≥ even is the pull-back of theGIT polarization via the morphism φ : M p → X p . When p = 4 , F , Σ ≃ O P (1) .Proof. The line bundle O (1 , . . . , on ( P ) p descends to X p by the Kempfdescent criterion giving a polarization L of X p . Note that, away from thesingularities, this agrees with our definition of F , Σ . It follows that F , Σ ≃ φ ∗ L ( X a T,T c δ T,T c ) . When p ≥ , it remains to show that a T,T c = 0 for every partition of P intotwo subsets with r elements each. For every a = 1 , . . . r − , by Prop. 6.1,we have M p ( F l,E , O δ T,Tc ( − a, − a )) == M T,T c R Γ( P r − × P r − , O δ T ( a T,T c − a, a T,T c − a )) It follows that all a T,T c = 0 . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 33
By definition, F , Σ = Rπ ∗ ( ω ⊗ π (Σ)) , which equals F , Σ = ψ ∗ (cid:0) ω ⊗ π (Σ) (cid:1) = − ψ + p X i =2 δ i . If p = 4 , then M , and M are isomorphic, with the same universal family.Therefore, ψ classes and boundary classes are the same. It follows that ψ i = O (1) = δ ij , and therefore F , Σ = O (1) . (cid:3) An analogue of Prop. 6.1 on the Hassett spaces W . Assume now | P | = p = 2 r ≥ , | Q | = q = 2 s + 1 ≥ . Let y ∈ Q and consider the universal family α : W → M p,q − = M P ∪ Q \{ y } . Proposition 6.3.
Let E ⊆ P ∪ Q with e = | E | , l ≥ such that e + l is even. Let δ T ∪{ y } ,T c ⊆ W be a boundary divisor ( T ⊔ T c = P ∪ ( Q \ { y } ) ) with the propertythat there exist m ( T c ) := max { , | E ∩ T c | − e − l } ≤ µ ′ , (6.2) m ( T ) := max { , | E ∩ T | − e − l } ≤ µ − | E ∩ { y }| , (6.3) µ ′ , µ ≤ r + s . Using the identification δ T ∪{ y } = δ T ∪{ y } ,T c = Bl p P r + s − × P r + s − , we have RHom W ( F l,E , ( − aH ) ⊠ O ( − b )) = 0 whenever one of the following happens: • ≤ a ≤ r + s − , ≤ b ≤ r + s − • a = 0 , < b < r + s − − µ ′ • b = 0 , < a < r + s − µ .Proof. Denote m ( T c ) := max { , e − l − | E ∩ T c |} ,m ( T ) := max { , e − l − | E ∩ T |} An immediate consequence of (6.2) is that when e − l − | E ∩ T c | ≥ then m ( T c ) = e − l − | E ∩ T c | = | E ∩ T | + | E ∩ { y }| − e + l | E ∩ { y }| + | E ∩ T | − e − l − l (use that | E ∩ T c | + | E ∩ T | + | E ∩ { y }| = e ). In particular, m ( T c ) ≤ µ − l. (6.4)Similarly, because of (6.3), when e − l − | E ∩ T | ≥ then we have m ( T ) = e − l − | E ∩ T | ≤ µ ′ + | E ∩ { y }| − l. (6.5) The proof is similar to that of Lemma 6.1. Let π : U → W be the universalfamily (with x the new index on U )). Denote for simplicity δ := δ T ∪{ y } ⊆ W, δ = δ T ∪{ y,x } ⊆ U , δ = δ T ∪{ y } ⊆ U ,β := π | π − ( δ ) : π − ( δ ) = δ ∪ δ → δ,N := N l,E , F := F l,E = Rπ ∗ N l,E . Using Groothendieck-Verdier duality, it suffices to prove that letting j : π − ( δ ) ֒ → U the inclusion map, then ˜ N := j ∗ ( N ∨ ⊗ ω π ) ⊗ β ∗ ( − aH ⊠ O ( − b )) (as a line bundle on π − ( δ ) ) is acyclic. We consider the exact sequence: → O δ ( − δ ) → O δ ∪ δ → O δ → . Tensoring with ˜ N , we obtain an exact sequence: → ˜ N ′′ → ˜ N → ˜ N ′ → , where ˜ N ′ := (cid:0) N ∨ ⊗ ω π (cid:1) | δ ⊗ β ∗ ( − aH ⊠ O ( − b )) , ˜ N ′′ := (cid:0) N ∨ ⊗ ω π ( − δ ) (cid:1) | δ ⊗ β ∗ ( − aH ⊠ O ( − b )) . We prove that both ˜ N ′ and ˜ N ′′ are acyclic. Using the identification δ = δ T ∪{ y } ,T c ∪{ x } = Bl P r + s − × Bl P r + s − , (the first copy of Bl P r + s − corresponds to T ∪ { y } ) by Lemma 5.12 we have ˜ N ′′ = M ′′ ⊠ M ′′ ,M ′′ = (cid:0) − α T c − a (cid:1) H + α T c ∪{ y } ∆ .M ′′ = (cid:0) e − l − | E ∩ T c | − α T c − b − (cid:1) H + (cid:0) α T ∪{ y } + b − (cid:1) ∆ . Similarly, using the identification δ = δ T ∪{ y,x } ,T c = Bl , , P r + s × P r + s − , by Lemma 5.12 we have ˜ N ′ = M ′ ⊠ M ′ , where M ′ = (cid:0) e − l − | E ∩ T | − | E ∩ { y }| − α T ∪{ y } − a − (cid:1) H ++ (cid:0) | E ∩ T | − e − l α T + 1 (cid:1) E + (cid:0) α T c + a (cid:1) E + α T c ∪{ y } E ,M ′ = O ( − α T ∪{ y } − b ) . Case 1) | E ∩ T | ≥ e − l , | E ∩ T c | ≥ e − l . Then α T = α T ∪{ y } = α T c = α T c ∪{ y } = 0 ,M ′′ = − aH,M ′′ = − d ′′ H + ( b − , where d ′′ = | E ∩ T c | − e − l b + 1 ,M ′ = − d ′ H + β E + β E , where d ′ = | E ∩ T | + | E ∩ { y }| − e − l a + 1 , β = | E ∩ T | − e − l , β = aM ′ = O ( − b ) . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 35 If a > then M ′′ is acyclic. If b > , then M ′ is acyclic. Furthermore, if < b ≤ r + s − − µ ′ , then M ′′ is acyclic since d ′′ = m ( T c ) + b + 1 ≤ µ ′ + b + 1 < r + s. If < a < r + s − µ then M ′ is acyclic since (6.3) implies that d ′ = m ( T ) + | E ∩ { y }| + a + 1 ≤ µ + a + 1 < r + s + 1 , and β = a ≤ r + s − , β = m ( T ) + 1 ≤ µ + 1 < r + s .Case 2) | E ∩ T | ≥ e − l , | E ∩ T c | < e − l . Then α T = α T ∪{ y } = 0 , α T c = m ( T c ) = e − l − | E ∩ T c | > ,α T c ∪{ y } = α T c − | E ∩ { y }| = m ( T c ) − | E ∩ { y }| ≥ ,M ′′ = ( − α T c − a ) H + ( α T c − | E ∩ { y }| )∆ ,M ′′ = ( − b − H + ( b − ,M ′ = − d ′ H + β E + β E + β E , where d ′ = | E ∩ T | + | E ∩ { y }| − e − l a + 1 , β = | E ∩ T | − e − l ,β = α T c + a, β = α T c − | E ∩ { y }| .M ′ = O ( − b ) . If b > then M ′′ and M ′ are both acyclic. Assume b = 0 , < a < r + s − µ .Then M ′′ is acyclic since by (6.4) we have ≤ α T c − | E ∩ { y }| = m ( T c ) − | E ∩ { y }| ≤ µ − l ≤ µ ≤ r + s − , < α T c + a = m ( T c ) + a ≤ µ + a < r + s. Furthermore, M ′ is acyclic since (6.3) implies that d ′ = m ( T ) + | E ∩ { y }| + a + 1 ≤ µ + a + 1 ≤ r + s + 1 . Furthermore, β = m ( T ) + 1 ≤ µ + 1 ≤ µ + a < r + s, β ≤ µ ≤ r + s − , and by (6.4) we have β = m ( T c ) + a ≤ µ + a < r + s .Case 3) | E ∩ T | < e − l , | E ∩ T c | ≥ e − l . Then α T c = α T c ∪{ y } = 0 , α T = m ( T ) = e − l − | E ∩ T | > ,α T ∪{ y } = α T − | E ∩ { y }| = m ( T ) − | E ∩ { y }| ≥ ,M ′′ = ( − a ) H,M ′′ = − d ′′ H + β ∆ , where d ′′ = | E ∩ T c | − e − l b + 1 , β = α T + b − | E ∩ { y }| − ,M ′ = ( − a − H + E + aE , M ′ = O ( − α T − b + | E ∩ { y }| ) . If a > then M ′′ and M ′ are acyclic. Assume a = 0 , < b < r + s − − µ ′ .Note that M ′ is still acyclic. Then M ′′ is acyclic since by (6.5) < β = m ( T ) − | E ∩ { y }| − b ≤ ( µ ′ − l ) + b − ≤ µ ′ + b − < r + s − ,d ′′ = m ( T c ) + b + 1 ≤ µ ′ + b + 1 < r + s. This finishes the proof. (cid:3)
7. I
NEQUALITIES FOR PAIRS ( l, E ) IN GROUPS AND Let ˜ Q = Q ∪ { z } , i.e., we single out a light index z . Let M p,q +1 = M P ∪ ˜ Q with | P | = p = 2 r ≥ heavy points and | ˜ Q | = q + 1 = 2 s + 2 ≥ lightpoints. Notation 7.1.
For l ∈ Z , E ⊆ P ∪ ˜ Q we define the score of a pair ( l, E ) as S ′ ( l, E ) = l + min { e p , p − e p } + min { e q , q + 1 − e q } . (We reserve the notation S ( l, E ) for the score of a pair ( l, E ) with E ⊆ P ∪ Q ,see Section 14.) Lemma 7.2.
Let l ≥ , E ⊆ P ∪ ˜ Q . Then max P ∪ ˜ Q = T ⊔ T c (cid:26) | E ∩ T | − e − l (cid:27) = S ′ ( l, E )2 , (7.1) max P ∪ ˜ Q = T ⊔ T c (cid:26) e − l − | E ∩ T | (cid:27) = S ′ ( l, E )2 − l. (7.2) Here we assume that T = T p ` T q is a subset of markings split into heavy andlight markings such that | T p | = r and | T q | = s . Equality in (7.1) is attained iff ( T p ⊆ E p or E p ⊆ T p ) or ( T q ⊆ E q or E q ⊆ T q ) while equality in (7.2) is attained iffProof. Indeed, | E ∩ T | is maximized when | E ∩ T | = 2 (min( r, e p ) + min( s + 1 , e q )) = min( p, e p ) + min( q + 1 , e q ) and so | E ∩ T | − ( e − l ) = min( p − e p , e p ) + min( q + 1 − e q , e q ) + l = S ′ ( l, E ) . Similarly, −| E ∩ T | is maximized when | E ∩ T c | − ( e − l ) = S ′ ( l, E ) and so ( e − l ) − | E ∩ T | = ( e − l ) + 2 | E ∩ T c | − e = S ′ ( l, E ) − l, which proves the lemma. (cid:3) Lemma 7.3.
Let p = 2 r , q = 2 s + 1 ≥ . Let ( l, E ) be a pair from the groups A , B , A or B of Theorem 1.16 for M p,q +1 . Then S ′ ( l, E ) ≤ r + s, i.e., (7.3) max P ∪ ˜ Q = T ⊔ T c (cid:26) | E ∩ T | − e − l (cid:27) ≤ r + s , where T = T p ` T q is a subset of markings split into heavy and light markingssuch that | T p | = r , | T q | = s . We call T critical for ( l, E ) if equality holds in (7.3)The inequality (7.3) is strict unless r + s is even and we have one of the followingcases, where we also list critical subsets: • l + e p = r − , e q = s + 1 , Ep ⊆ T p , E q = T q (group A ); • e p = r , l + q + 1 − e q = s , E p = T p , T q ⊆ E q (group A ); ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 37 • l + ( p − e p ) = r − , e q = s + 1 , T p ⊆ E p , E q = T q (group B ); • e p = r , l + e q = s , E p = T p , E q ⊆ T q (group B ).In addition, we have: | E ∩ T c | − e − l ≥ − r + s . (7.4) The inequality is strict unless r + s is even and we have one of the following cases: • e p = r − , e q = s + 1 , l = 0 , E ⊆ T (group A ); • e p = r , e q = s + 2 , l = 0 , T ⊆ E (group A ); • e p = r + 1 , e q = s + 1 , l = 0 , T ⊆ E (group B ); • e p = r , e q = s , l = 0 , E ⊆ T (group B ).This is the same list as for (7.3) but with l = 0 . In particular, T is critical.Proof of Lemma 7.3. If ( l, E ) is in group A then ( l, E c ) is in group B and S ′ ( l, E ) = S ′ ( l, E c ) . Since the conclusion of the lemma is symmetric under this operation, wecan assume without loss of generality that ( l, E ) is in group A or A .We first prove (7.3) and classify critical sets using Lemma 7.2.If e p ≤ r and e q ≤ s + 1 then S ′ ( l, E ) = l + e p + e q = l + e and l + e ≤ r + s for both groups A and A . The inequality is strict unlesswe have the first exception on the list.If e p > r and e q ≤ s + 1 (possible only in case 1A)) then S ′ ( l, E ) = l + ( p − e p ) + e q . Since l + ( p + 1 − e p ) ≤ r − for group A , we have S ′ ( l, E ) < r + s .If e p ≤ r and e q > s + 1 then S ′ ( l, E ) = l + e p + ( q + 1 − e q ) . which is ≤ r + s for both groups A and A . The inequality is strict unlesswe have the second exception on the list.Finally, if e p > r and e q > s + 1 (possible only in case 1A)) then S ′ ( l, E ) = l + ( p − e p ) + ( q + 1 − e q ) . which is < r + s for A .The second statement from the first, since | E ∩ T c | − e − l ≥ e − l − | E ∩ T | ≥ − r + s with the first inequality becoming an equality if and only if l = 0 . (cid:3) Corollary 7.4. If ( l, E ) is in group A or B on M p,q +1 , then for I ⊆ ˜ Q , | I | = s + 1 , we have | E q ∩ I | − e q − l ≤ s , with equality if and only if either e q = l + s + 2 , I ⊆ E q (group A ) or l + e q = s , E q ⊆ I (group B ). In addition, we have | E q ∩ I c | − e q − l ≥ − s . Proof.
Since | E p | = r , both inequalities follow from Lemma 7.3 for a set T with heavy indices T p = E p and light indices T q = I . (cid:3) Lemma 7.5.
Let p = 2 r , q = 2 s + 1 ≥ . Let ( l, E ) be a pair from groups A , B or of Theorem 1.11 for M p,q . Let y ∈ Q and T = T p ` T q ⊆ P ∪ ( Q \ { y } ) be a subset of markings split into heavy and light markings such that | T p | = r and | T q | = s . Then | E ∩ T | − e − l ≤ r + s − . (7.5) We call T critical if equality holds in (7.5). The inequality is strict if y ∈ E or if r + s is even. When r + s is odd equality holds precisely when y / ∈ E and we arein one of the following cases: • l + e p = r − , e q = s , E p ⊆ T p , E q = T q (group A ); • l + ( p − e p ) = r − , e q = s , T p ⊆ E p , E q = T q (group B ); • e p = r , l + e q = s − , E p = T p , E q ⊆ T q (group );In addition, we have | E ∩ T c | − e − l ≥ − r + s . (7.6) The inequality is strict, unless r + s is even, l = 0 , y ∈ E , and we have one thefollowing cases: • l = 0 , e p = r − , e q = s + 1 , E p ⊆ T p , E q = T q ∪ { y } (group A ); • l = 0 , e p = r + 1 , e q = s + 1 , T p ⊆ E p , E q = T q ∪ { y } (group B ). • l = 0 , e p = r , e q = s + 2 , E p = T p , T q ⊆ E q , y ∈ E q (group ).Furthermore, all the conditions on ( E, l ) in Lemma 6.3 hold for all T if we take µ ′ = r + s − , µ = r + s . Proof.
We need to show that | E ∩ T | ≤ r + s − e − l, | E ∩ T | ≥ − r − s + e − l. Letting e ′ q = | E ∩ ( Q \ { y } ) | , we have e = e p + e q = e p + e ′ q + | E ∩ { y }| , | E ∩ T | = 2 min( e p , r ) + 2 min( e ′ q , s ) , | E ∩ T c | = 2 max(0 , e p − r ) + 2 max(0 , e ′ q − s ) . Group 1A, e p < r . Assume ( l, E ) is in group A with l + e p ≤ ( r − .Then | E ∩ T | = 2 e p + 2 min( e ′ q , s ) ≤ e p + e ′ q + s ≤≤ ( e p + l ) + ( e p + e ′ q − l ) + s ≤ ( r −
1) + ( e − l ) + s, with equality if and only if l + e p = r − , e ′ q = s, y / ∈ E, E p ⊆ T p , E q = T q . Furthermore, in this case we have | E ∩ T c | = 2 max(0 , e ′ q − s ) ≥ e ′ q − s ≥ ( e ′ q − s ) − ( r − − e p − l ) == − r − s + e p + e ′ q + 1 + l ≥ − r − s + e + l ≥ − r − s + e − l, ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 39 with equality if and only if e p = r − , l = 0 , e ′ q = s, y ∈ E, E p ⊆ T p , E q = T q ∪ { y } . Group 1A, e p > r . Assume ( l, E ) is in group A with l + ( p + 1 − e p ) ≤ ( r − , i.e., r ≤ e p − l − . By exchanging E p with P \ E p will reduce thiscase to the Case 1A), e p < r . Since l + ( p − e p ) ≤ r − , this corresponds tothe l + e p ≤ r − in that case; hence, all inequalities are in fact strict. Wealso verify this directly as follows: | E ∩ T c | = 2 r + 2 min( e ′ q , s ) ≤ r + ( e ′ q + s ) ≤ r + ( e p − l − e ′ q + s ) == r + s + ( e p + e ′ q ) − l − ≤ r + s + e − l − . In particular, strict inequality holds in (7.5).Furthermore, since e p − r ≥ l + 2 , we have | E ∩ T c | = 2( e p − r ) + 2 max(0 , e ′ q − s ) ≥ ( e p − r ) + ( l + 2) + ( e ′ q − s ) == − r − s + ( e p + e ′ q ) + l + 2 ≥ − r − s + e + l + 1 ≥ − r − s + 1 + e − l. In particular, strict inequality holds in (7.6).Group 1B, e p > r . Assume ( l, E ) is in group B with l + ( p − e p ) ≤ ( r − .As in Case 1A, e p > r , we have r ≤ e p − l − and | E ∩ T | = 2 r + 2 min( e ′ q , s ) ≤ r + ( e ′ q + s ) ≤≤ r + ( e p − l −
1) + ( e ′ q + s ) ≤ r + s − e p + e ′ q ) − l ≤ r + s − e − l, with equality if and only if l + ( p − e p ) = r − , e ′ q = s, y / ∈ E, T p ⊆ E p , T q = E q . Furthermore, since e p − r ≥ l + 1 , we have | E ∩ T c | = 2( e p − r ) + 2 max(0 , e ′ q − s ) ≥ ( e p − r ) + ( l + 1) + ( e ′ q − s ) == − r − s + (1 + e p + e ′ q ) + l ≥ − r − s + e + l ≥ − r − s + e − l, with equality if and only if l = 0 , e p = r + 1 , e ′ q = s, y ∈ E, T p ⊆ E p , E q = T q ∪ { y } . Group 1B, e p < r . Assume ( l, E ) is in group B with l + e p + 1 ≤ ( r − .Using that e p ≤ r − − l , we have | E ∩ T | = 2 e p +2 min( e ′ q , s ) ≤ e p +( r − − l )+( e ′ q + s ) ≤ r + s + e − l − . In particular, strict inequality holds in (7.5). Furthermore, we have | E ∩ T c | = 2 max(0 , e ′ q − s ) ≥ ( e p − r + l + 2) + ( e ′ q − s ) == ( e p + e ′ q + 1) − r − s + l + 1 ≥ − r − s + e − l + 1 . In particular, strict inequality holds in (7.6).Group 2. When ( l, E ) is in group we have e p = r and | E ∩ T | = 2 r + 2 min( e ′ q , s ) , | E ∩ T | = 2 max(0 , e ′ q − s ) . Group 2, e q < s . Assume ( l, E ) is in group with l + e q ≤ ( s − . Itfollows that e ′ q = e q − | E ∩ { y }| < s and | E ∩ T | = 2 r + 2 e ′ q ≤ r + e ′ q + ( s − − l − | E ∩ { y }| ) = = r + s − r + e ′ q −| E ∩{ y }| ) − l ≤ r + s − e − | E ∩{ y }|− l ≤ r + s − e − l, with equality if and only if y / ∈ E, l + e q = s − , E p = T p , E q ⊆ T q . Furthermore, since e ′ q < s , we have | E ∩ T c | = 0 ≥ − ( s − − e q − l ) = − r − s + 1 + e p + e q + l == − r − s + 1 + e − l. In particular, strict inequality holds in (7.5).Group 2, e q > s . Assume ( l, E ) is in group with l + ( q − e q ) ≤ ( s − .In particular, e ′ q ≥ s and e q − l ≥ s + 2 implies that | E ∩ T | = 2 r + 2 s ≤ r + s + ( e q − l −
2) = r + s + e − l − In particular, strict inequality holds in (7.5). Furthermore, we have | E ∩ T c | = 2( e ′ q − s ) = 2 e q − | E ∩ { y }| − s which is ≥ − r − s + e + l = − s + e q + l since e q ≥ l + s + 2 and e p = r . Itfollows that equality holds in (7.6) if and only if y ∈ E, l = 0 , e q = s + 2 , E p = T p , T q ⊆ E q . (cid:3) Corollary 7.6.
Assume p = 2 r , q = 2 s + 1 . For ( l, E ) in group A or B on M p,q +1 or M p,q , with P = R ⊔ R ′ a partition of P with | R | = | R ′ | = r , then | E p ∩ R | − e p − l ≤ r − , or equivalently, l + | E p ∩ R | − | E p ∩ R ′ | ≤ r − . (7.7) Proof.
Note that if ( l, E ) is in group A (resp., B ) on M p,q +1 , then ( l, E p ) isin group A (resp., B ) on M p . Similarly if ( l, E ) is in group A (resp., B )on M p,q , then ( l, E p ) is in group A (resp., B ) on M p, . The result followsby applying Lemma 7.3 in the case when p = 2 r , s = − , and Lemma 7.5 inthe case when p = 2 r , s = 0 for the partition T = R , T c = R ′ . (cid:3) We will also need the following:
Lemma 7.7.
Let q = | Q | = 2 s + 1 , l ≥ , E ⊆ Q a set with e = | E | satisfying l + min { e, q − e } ≤ s − . Let I ⊆ Q be any subset. Then l + | E ∩ I c | − | E ∩ I | ≤ ( s − e ≤ s | I c | − s − e ≥ s + 1 . (7.8) Proof. If e ≤ s then the inequality follows from l + e ≤ s − . If e ≥ s + 1 ,then the inequality follows from l − e ≤ − s − . (cid:3) The same way we obtain the following:
Lemma 7.8.
Let q = | Q | = 2 s + 2 , l ≥ , E ⊆ Q a set with e = | E | , and let I ⊆ Q be any subset.If l + min { e + 1 , q + 1 − e } ≤ s , then l + | E ∩ I c | − | E ∩ I | ≤ ( s − e ≤ s | I c | − s − e ≥ s + 1 . (7.9) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 41 If l + min { e, q + 2 − e } ≤ s , then l + | E ∩ I c | − | E ∩ I | ≤ ( s if e ≤ s + 12 | I c | − s − e ≥ s + 2 . (7.10)8. { F l,E } ON M p,q − EXCEPTIONAL ⇒ { F l,E } ON M p,q EXCEPTIONAL
We will compare exceptional collections on various Hassett spaces re-lated by morphisms. We start with an abstract lemma, which encapsulatesour typical situation. If C = hA , Bi is a semi-orthogonal decomposition,then any object T ∈ C can be included in an exact triangle T B → T → T A , T A ∈ A , T B ∈ B . Furthermore, T B and T A are unique and functorial, called projection func-tors of T onto B and A . They are adjoint to inclusion functors. Lemma 8.1.
Assume S is a triangulated category endowed with a contravariantequivalence ∨ : S → S (“duality”) and two semi-orthogonal decompositions S = hA , Bi = hC , Di such that D ∨ = D . Then functors D → B , T T B , and B → D , T (cid:0) ( T ∨ ) D (cid:1) ∨ , are adjoint, i.e. for all T ′ ∈ B , T ∈ D , we haveR Hom B (cid:0) T ′ , T B (cid:1) = R Hom D (cid:0)(cid:0) ( T ′∨ ) D (cid:1) ∨ , T (cid:1) . Proof.
Indeed,R
Hom B (cid:0) T ′ , T B (cid:1) = R Hom S (cid:0) T ′ , T ) = R Hom S (cid:0) T ∨ , T ′∨ ) = R Hom D (cid:0) T ∨ , (cid:0) ( T ′∨ ) D (cid:1)(cid:1) = R Hom D (cid:0)(cid:0) ( T ′∨ ) D (cid:1) ∨ , T (cid:1) , which proves the lemma. (cid:3) We will use this lemma in the following situation:
Corollary 8.2.
Suppose we have objects T ′ , T ∈ B and ˜ T ′ , ˜ T ∈ D . Suppose ( T ′∨ ) D = ( ˜ T ′ ) ∨ and that T and ( ˜ T ) B are related by quotients Q , . . . , Q s suchthat RHom B ( T ′ , Q i ) = 0 for every i . Then RHom B ( T ′ , T ) = RHom D ( ˜ T ′ , ˜ T ) . Here we use the following definition:
Definition 8.3.
Let Q be a set of objects in a triangulated category T . Wesay that objects G and G ′ in T are related by quotients in Q if there are objects Q i in Q for ≤ i ≤ k and a sequence of triangles for some p i , q i ∈ Z : G i − → G i [ p i ] → Q i [ q i ] → G i − [1] ,G = G ′ , G k = G ′ . Throughout this section we assume p = 2 r ≥ , q = 2 s + 1 ≥ (inparticular, we have dim M p,q − ≥ ). Let y ∈ Q and consider the universalfamily α : W → M p,q − = M P ∪ Q \{ y } , and the birational morphism from Notation 5.10, f : W → M p,q . Proposition 8.4.
Let l ≥ , and let E ⊆ P ∪ Q be such that e + l even. For any T ⊆ P ∪ ( Q \ { y } ) with | T ∩ P | = r , | T ∩ Q | = s , we let m ( T ) := max (cid:26) , | E ∩ T | − e − l (cid:27) , (i) The vector bundles F l,E and f ∗ F l,E on W are related by exact sequences → G i − → G i → Q i → , ≤ i ≤ kG = F l,E , G k = f ∗ F l,E , with Q i direct sums of sheaves supported on boundary divisors δ T ∪{ y } ,T c = Bl P r + s − × P r + s − and having the form − jH ⊠ O ( u ) , < j ≤ m ( T c ) , ≤ u < m ( T c ) . (ii) The dual bundles F ∨ l,E and f ∗ F ∨ l,E on W are related by triangles Q ∨ i → G ∨ i → G ∨ i − → Q ∨ i [1] G ∨ = F ∨ l,E , G ∨ k = f ∗ F ∨ l,E , with Q ∨ i direct sums of sheaves (placed in cohomological degree ) sup-ported on boundary divisors δ T ∪{ y } ,T c as above, having the form ( j − H ⊠ O ( − u − , < j ≤ m ( T c ) , ≤ u < m ( T c ) . Proof.
We claim there are two types of quotients ( − jH ) ⊠ O ( u ) that appear:(Type I) < j ≤ m ( T c ) , ≤ u < m ( T c ) , if | E ∩ T | > e − l ;(Type II) < j ≤ m ( T c ) , ≤ u < ˜ m ( T ) , if | E ∩ T | + | E ∩ { y }| < e − l , where ˜ m ( T ) := e − l − | E ∩ T | − | E ∩ { y }| ≤ m ( T c ) . To prove the claim, recall that F l,E = Rπ ∗ ( N l,E ) (see Definition 5.14). Let N := N l,E = ω e − l π ( E ) (cid:0) − X T α T δ T ∪{ x } − X T α T ∪{ y } δ T ∪{ y,x } (cid:1) , where x is the new marking on the universal family π : U → W and either α T = α T ∪{ y } = 0 or α T = e − l − | E ∩ T | , α T ∪{ y } = α T − | E ∩ { y }| ≥ . If π ′ : U ′ → M p,q is the universal family, there is a commutative diagramwith a cartesian second square: U v −−−−→ V q −−−−→ U ′ π y ρ y π ′ y W Id −−−−→ W f −−−−→ M p,q and we have f ∗ F l,E = Rπ ∗ ( N ) , where we denote N = v ∗ ω e − l ρ ( E ) = ω e − l π ( E ) + X T (cid:0) | E ∩ T | − e − l (cid:1) δ T ∪{ x } , ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 43
Note that among the two types of boundary divisors δ T ∪{ x } , δ T ∪{ y,x } of U ,only the divisors δ T ∪{ x } are v -exceptional, and the restriction v | δ T ∪{ x } is thesecond projection δ T ∪{ x } ,T c ∪{ y } = Bl P r + s − × Bl P r + s − → Bl P r + s − (the markings from T ∪ { x } coincide in the image). We have: N = N + X T (cid:0) | E ∩ T |− e − l α T (cid:1) δ T ∪{ x } + X T α T ∪{ y } δ T ∪{ y,x } = N +Σ +Σ , Σ = X | E ∩ T | > e − l (cid:0) | E ∩ T | − e − l (cid:1) δ T ∪{ x } , Σ = X | E ∩ T | < e − l (cid:0) e − l − | E ∩ T | − | E ∩ { y }| (cid:1) δ T ∪{ y,x } . Any two distinct boundary divisors appearing in Σ do not intersect. Sim-ilarly, any two distinct boundary divisors appearing in Σ do not intersect,but each δ T ∪{ y,x } that appears in Σ intersects exactly one term from Σ ,namely, δ T c ∪{ x } . We add successively to N first terms from Σ , then Σ ,and get two types of quotients.Type I. The first type of quotient has direct summands of the form: Q ′ = (cid:0) N + iδ T ∪{ x } (cid:1) | δ T ∪{ x } = (cid:0) ω e − l π ( E ) + iδ T ∪{ x } (cid:1) | δ T ∪{ x } == (cid:0) αH (cid:1) ⊠ (cid:0) − iH (cid:1) , α := | E ∩ T | − e − l − i, supported on δ T ∪{ x } ,T c ∪{ y } = Bl P r + s − × Bl P r + s − , < i ≤ | E ∩ T | − e − l m ( T ) , (for various T with this property). Clearly, ≤ α < m ( T ) , < i ≤ m ( T ) . Hence, by Lemma 5.5, Rq ∗ ( αH )) = O ⊕ O (1) ⊕ . . . ⊕ O ( α ) ; hence, Rπ ∗ Q ′ isa direct sum of sheaves of the form O ( u ) ⊠ ( − iH ) on δ T,T c ∪{ y } = P r + s − × Bl P r + s − , where ≤ u ≤ α < m ( T ) , < i ≤ m ( T ) . Type II. The second type of quotient Q ′ has direct summands supported on δ T ∪{ y,x } = Bl , , P r + s × P r + s − , < i ≤ e − l −| E ∩ T |−| E ∩{ y }| = ˜ m ( T ) , (for various T with this property) and having the form Q ′ = (cid:0) N + ( | E ∩ T c | − e − l δ T c ∪{ x } + iδ T ∪{ y,x } (cid:1) | δ T ∪{ y,x } == (cid:0) ω e − l π ( E ) + ( | E ∩ T | − e − l δ T ∪{ x } ++( | E ∩ T | + | E ∩{ y }|− e − l i ) δ T ∪{ y,x } +( | E ∩ T c |− e − l δ T c ∪{ x } (cid:1) | δ T ∪{ y,x } = (cid:0) − iH + βE (cid:1) ⊠ O ( u ) , where β := m ( T c ) = | E ∩ T c | − e − l e + l − | E ∩ T | ≥ ˜ m ( T ) , u = e − l − | E ∩ T | − | E ∩ { y }| − i = ˜ m ( T ) − i < ˜ m ( T ) In particular, β ≥ i and ˜ m ( T ) > u ≥ . By Lemma 5.12 and Rmk. 5.13, Rπ ∗ (cid:0) − iH + βE (cid:1) = Rπ ∗ (cid:0) π ∗ ( − iH ) ⊗ ( β − i ) E (cid:1) == O ( − iH ) ⊕ O ( − ( i + 1) H ) ⊕ . . . ⊕ O ( − βH ) . The result follows. Part (ii) follows by dualizing the exact triangles in (i). (cid:3)
Corollary 8.5. On M p,q we have F ∨ l,E ∼ = Rf ∗ F ∨ l,E as long as m ( E, l ) := max (cid:18) , max P ∪ ( Q \{ y } )= T ⊔ T c (cid:26) | E ∩ T | − e − l (cid:27)(cid:19) ≤ r + s − , for example if m ( E, l ) ≤ r + s and r + s ≥ . If r + s ≥ , then r + s ≤ r + s − . When r + s = 3 , if m ( E, l ) ≤ r + s = ,then we still have m ( E, l ) ≤ r + s − . Proof.
Since Rf ∗ (( j − H ⊠ O ( − u − if ≤ u < r + s − , the resultfollows from Prop. 8.4. (cid:3) Remark 8.6.
Prop. 8.4 shows that when l = 0 and E = P ∪ ( Q \ { y } ) , on W we have f ∗ F l,E = F l,E ( m ( T ) = m ( T c ) = 0 ). Note that the same istrivially true when l = 0 , E = ∅ . In particular, Cor. 8.5 (i.e., F ∨ l,E ∼ = Rf ∗ F ∨ l,E )is still true in these cases. This is particularly relevant when M p,q − = M (i.e., r = 2 , s = 0 ), when { F , ∅ , F , Σ } form an exceptional collection on M = P (here Σ = P , | P | = 4 ). This is because F , ∅ = O , F , Σ = O (1) (Corollary 6.2). Corollary 8.7.
In the notation of Prop. 8.4, suppose that F l,E is one of the bundleson M p,q from group A (resp., B ). Then the sheaves Q i belong to the full triangu-lated subcategory of D b ( W ) generated by the sheaves ( − aH ) ⊠ O ( − b ) with either ≤ a ≤ r + s − , ≤ b ≤ r + s − , or b = 0 , ≤ a < r + s .Proof. By Lemma 7.5, we have m ( T c ) ≤ r + s − < r + s . The result follows by Prop. 8.4. (cid:3)
Corollary 8.8. (Alpha game) Assume p = 2 r ≥ , q = 2 s + 1 ≥ . Let F l,E and F l ′ ,E ′ be bundles on M p,q from group A (resp., B ). Then R Hom W ( F l ′ ,E ′ , F l,E ) = R Hom M p,q ( F l ′ ,E ′ , F l,E ) . Proof.
By Cor. 8.7 and Prop. 6.3 (with µ ′ = µ = r + s ), R Hom W ( F l ′ ,E ′ , F l,E ) = R Hom W ( F l ′ ,E ′ , f ∗ F l,E ) . On the other hand, R Hom W ( F l ′ ,E ′ , f ∗ F l,E ) = R Hom W ( f ∗ F ∨ l,E , F ∨ l ′ ,E ′ ) == R Hom M p,q ( F ∨ l,E , Rf ∗ F ∨ l ′ ,E ′ ) = R Hom M p,q ( F ∨ l,E , F ∨ l ′ ,E ′ ) by Corollary 8.5. The latter group is equal to R Hom M p,q ( F l ′ ,E ′ , F l,E ) . (cid:3) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 45
Corollary 8.9.
Assume p = 2 r ≥ , q = 2 s + 1 ≥ . The bundles { F l,E } on M p,q from group A (resp., B ) of Theorem 1.11 form an exceptional collection.Proof. Let F l,E , F l ′ ,E ′ be bundles from one of the two collections (group A or B ) on M p,q . By Cor. 8.8, for any y ∈ Q , we have R Hom M p,q ( F l ′ ,E ′ , F l,E ) = R Hom W ( F l ′ ,E ′ , F l,E ) . Case 1: E ′ q \ E q = ∅ . Choose y ∈ E ′ q \ E q . By Lemma 5.16, we have F ∨ l,E = α ∗ F ∨ l,E , Rα ∗ F ∨ l ′ ,E ′ = 0 , and therefore, R Hom W ( F l ′ ,E ′ , F l,E ) = R Hom W ( F ∨ l,E , F ∨ l ′ ,E ′ ) = R Hom W ( α ∗ F ∨ l,E , F ∨ l ′ ,E ′ ) = R Hom M p,q − ( F ∨ l,E , Rα ∗ F ∨ l ′ ,E ′ ) = 0 . Case 2: E q = E ′ q = Q . Choose y ∈ Q \ E q . Since the range in group A (resp., B ) on M p,q is precisely the range in group A (resp., B ) on M p,q − ,we are done by Lemma 5.16, since F l,E = α ∗ F l,E , F l ′ ,E ′ = α ∗ F l ′ ,E ′ and R Hom W ( α ∗ F l ′ ,E ′ , α ∗ F l,E ) = R Hom M p,q − ( F l ′ ,E ′ , F l,E ) . The latter is equal to or C (if ( l, E ) = ( l ′ , E ′ ) ) by Thm. 1.16 for M p,q − . Thecase of M ( p = 4 , q = 1 ) is covered by Remark. 8.6.Case 3: E q = E ′ q = Q , q > . Let z ∈ Q . By Prop. 3.9, F l,E = F l,E p ∪{ z } ⊗ F ,E q \{ z } ,R Hom M p,q ( F l ′ ,E ′ , F l,E ) = R Hom M p,q ( F l ′ ,E ′ p ∪{ z } , F l,E p ∪{ z } ) . Note that F l,E p ∪{ z } is still in group A (resp., group B ) on M p,q , and sinceby assumption Q = { z } we are in Case 2.Case 4: E q = E ′ q = Q , q = 1 . Note that M p, = M p +1 and since e ′ p ≥ e p , wehave e ′ ≥ e . If bundles are in group 1B on M p, , they are in the exceptionalcollection of Theorem 1.8, and we are done.Finally, suppose the bundles are in group 1A on M p, and E q = E ′ q = Q , q = 1 . Consider the line bundle L = O ( − P − z ) on M p +1 (i.e. we take -1 inthe position of every heavy point and − in the position of z . Then usingthe projection formula and Cor. 3.6 we have L ⊗ F l,E = O ( − P − z ) ⊗ Rπ ∗ O ( E, l ) = Rπ ∗ ( − ( P \ E p ) − z, l ) = F ∨ l, ( P \ E p ) ∪{ z } . Therefore, R Hom M p, ( F l ′ ,E ′ , F l,E ) = R Hom M p, ( F l ′ ,E ′ ⊗ L, F l,E ⊗ L ) == R Hom M p, ( F ∨ l ′ , ( P \ E ′ p ) ∪{ z } , F ∨ l, ( P \ E p ) ∪{ z } ) = R Hom M p, ( F l, ( P \ E p ) ∪{ z } , F l ′ , ( P \ E ′ p ) ∪{ z } ) . These bundles are in group 1B on M p, and are in the correct order: p − e p ≥ p − e ′ p . (cid:3) { F l,E } ON M p,q EXCEPTIONAL ⇒ { F l,E } ON M p,q +1 EXCEPTIONAL
Throughout this section we will assume P (resp., ˜ Q ) is the set of heavy(resp., light) indices and, unless otherwise noted, we have | P | = p = 2 r ≥ , | ˜ Q | = q + 1 = 2 s + 2 ≥ ,r + s ≥ i.e. , dim M p,q +1 ≥ . We emphasize that we allow here q +1 = 0 (i.e., s = − ). In this case we willprove unconditionally that vector bundles { F l,E } on M p are exceptional.The main result of this section is Corollary 9.9. Notation 9.1.
For every boundary divisor δ T,T c of M p,q +1 , we make a choiceof either T or T c and call it T . The other subset will be called T c . We willspecialize to concrete recipes later. For example, one way to make a choice,is to fix an index z , and take T such that z ∈ T . Notation 9.2.
We have a commutative diagram, where birational morphisms β and g depend on the choices of T ’s: W g −−−−→ U = M p + q +2 α y π y M p,q +1 β −−−−→ M p + q +1 The morphism g contracts W , the universal family over M p,q +1 to a P -bundle over β (M p,q +1 ) . Specifically, reducible fibers over δ T,T c are unionsof two P , with one component marked by T and another by T c . The mor-phism g collapses the first component: sections marked by T become iden-tified. Note that β (M p,q +1 ) (resp. g ( W ) ) is an open substack of M p + q +1 (resp. M p + q +2 ). They are proper algebraic spaces but not necessarily schemes.The morphism β contracts the boundary divisors δ T,T c ≃ P r + s − × P r + s − of M p,q +1 (with markings from T , resp., T c , on the first component, resp.,the second component) to P r + s − via the second projection. The boundarydivisors in W have the form δ T ∪{ y } and δ T c ∪{ y } , where y is the extra mark-ing on W . The morphism g contracts the boundary divisor δ T c ∪{ y } via thesecond projection δ T c ∪{ y } = P r + s − × Bl P r + s → Bl P r + s and the boundary divisor δ T ∪{ y } via the morphism δ T ∪{ y } = Bl P r + s × P r + s − → P r + s − ֒ → Bl P r + s , given by the second projection followed by embedding into Bl P r + s as theexceptional divisor (the locus where the marking corresponding to T and y coincide). Proposition 9.3.
Let l ≥ , and let E ⊆ Σ := P ∪ ˜ Q be such that e − l even. Let m := max (cid:18) , max T ⊔ T c =Σ (cid:26) | E ∩ T | − e − l (cid:27)(cid:19) . Then max (cid:18) , max T ⊔ T c =Σ (cid:26) −| E ∩ T c | + e − l (cid:27)(cid:19) ≤ m . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 47 (i) The bundles F l,E and β ∗ F l,E on M p,q +1 are related by exact sequences → G i − → G i → Q i → , ≤ i ≤ kG = F l,E , G k = β ∗ F l,E , with Q i direct sums of sheaves supported on boundary divisors δ T,T c = P r + s − × P r + s − ⊆ M p,q +1 having the form O ( u , − j ) , < j ≤ m , ≤ u < m , (where the component O ( u ) is the one corresponding to T ).(ii) The dual vector bundles F ∨ l,E and β ∗ F ∨ l,E on M p,q +1 are related by triangles Q ∨ i → G ∨ i → G ∨ i − → Q ∨ i [1] ,G ∨ = F ∨ l,E , G ∨ k = β ∗ F ∨ l,E , with Q ∨ i direct sums of sheaves (placed in cohomological degree ) sup-ported on boundary divisors δ T,T c as above and having the form O ( − u − , j − , < j ≤ m , ≤ u < m . Proof.
We prove that there are three types of quotients that appear:(Type I) if | E ∩ T | , | E ∩ T c | > e − l ;(Type II) if | E ∩ T | > e − l , | E ∩ T c | ≤ e − l ;(Type III) if | E ∩ T c | < e − l .Recall that F l,E = Rα ∗ ( N l,E ) and denote for simplicity: N := N l,E = ω e − l α ( E ) − X | E ∩ T | < e − l (cid:18) e − l − | E ∩ T | (cid:19) δ T ∪{ y } − X | E ∩ T c | < e − l (cid:18) e − l − | E ∩ T c | (cid:19) δ T c ∪{ y } . Furthermore, β ∗ F l,E = Rα ∗ ( N ) , where we denote N = g ∗ ω e − l π ( E ) = ω e − l α ( E ) + X T (cid:18) | E ∩ T | − e − l (cid:19) δ T ∪{ y } . We have N = N + S + S , where S = X | E ∩ T | > e − l (cid:18) | E ∩ T | − e − l (cid:19) δ T ∪{ y } ,S = X | E ∩ T c | < e − l (cid:18) e − l − | E ∩ T c | (cid:19) δ T c ∪{ y } . We break S into S ′ + S ′′ , where S ′ , resp. S ′′ , contains the terms with | E ∩ T c | > e − l , resp., | E ∩ T c | ≤ e − l . We add successively to N first termsfrom S ′ , then S ′′ , followed by S , to get three types of quotients on W . Type I. The first type of quotient has direct summands of the form: Q ′ = (cid:0) N + iδ T ∪{ y } (cid:1) | δ T ∪{ y } = (cid:0) ω e − l α ( E ) + iδ T ∪{ y } (cid:1) | δ T ∪{ y } == (cid:0) uH ) ⊠ O ( − i ) , where u := | E ∩ T | − e − l − i, where δ T ∪{ y } is of the type appearing in S ′ : δ T ∪{ y } = Bl P r + s × P r + s − , < i ≤ | E ∩ T | − e − l , | E ∩ T c | > e − l (for various T with this property). Clearly, ≤ u < m and < i ≤ m .Since Rα ∗ ( uH ) = O ⊕ O (1) ⊕ . . . ⊕ O ( u ) , the result follows.Type II. The next type of quotient has direct summands of the form: Q ′′ = (cid:0) N + iδ T ∪{ y } (cid:1) | δ T ∪{ y } == (cid:18) ω e − l α ( E ) + (cid:18) | E ∩ T c | − e − l (cid:19) δ T c ∪{ y } + iδ T ∪{ y } (cid:19) | δ T ∪{ y } == (cid:0) uH − v ∆) ⊠ O ( − i ) , where u := | E ∩ T |− e − l − i, v = e − l −| E ∩ T c | supported on δ T ∪{ y } of the type appearing in S ′′ : δ T ∪{ y } = Bl P r + s × P r + s − , < i ≤ | E ∩ T | − e − l , | E ∩ T c | < e − l , (for various T with this property). Clearly, we have ≤ u < m , < v ≤ m , < i ≤ m . By Lemma 5.5, if u ≥ v , Rπ ∗ ( uH − v ∆) = O ( v ) ⊕ O ( v + 1) ⊕ . . . ⊕ O ( u ) , whileif u < v , we have nevertheless that Rπ ∗ ( uH − v ∆) is either or generatedby O ( u + 1) , . . . , O ( v − and the result follows.Type III. The last type of quotient Q has direct summands of the form Q = (cid:0) N + S + iδ T c ∪{ y } (cid:1) | δ Tc ∪{ y } == (cid:18) ω e − l α ( E ) + (cid:18) | E ∩ T c | − e − l i (cid:19) δ T c ∪{ y } + (cid:18) | E ∩ T | − e − l (cid:19) δ T ∪{ y } (cid:19) | δ Tc ∪{ y } == O ( u ) ⊠ (cid:0) − iH + β ∆ (cid:1) , where β = | E ∩ T | − e − l , u = e − l − | E ∩ T c | − i, supported on δ T c ∪{ y } (of the type appearing in S ): δ T c ∪{ y } = P r + s − × Bl P r + s , < i ≤ e − l − | E ∩ T c | ≤ β (for various T with this property). Clearly, < i ≤ m , < β ≤ m , ≤ u < m . Since β ≥ i , it follows by Lemma 5.5 that Rα ∗ ( − iH + β ∆) = O ( − i ) ⊕ O ( − i − ⊕ . . . ⊕ O ( − β ) , and (i) follows. Part (ii) follows from (i) by dualizing the triangles. (cid:3) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 49
Corollary 9.4.
Suppose F l,E is one of the bundles from groups A or B on M p,q +1 . Then sheaves Q i belong to subcategory A (see Notation 1.14) with thefollowing possible exceptions when r + s is even and for the following subsets T : • l + e p = r − , e q = s + 1 , E p ⊆ T p , T q = E q (group A ); • e p = r + 1 + l , e q = s + 1 , T p ⊆ E p , T q = E q (group B );Proof. See Lemma 7.3. (cid:3)
Corollary 9.5.
In the notations of Prop. 9.3, if m ≤ r + s − (for example, if m ≤ r + s ), then on β (M p,q +1 ) we have: F ∨ l,E ≃ Rβ ∗ F ∨ l,E . Proof.
Since Rβ ∗ O ( − u − , j −
1) = 0 for ≤ u < r + s − , we have Rβ ∗ F ∨ l,E ≃ Rβ ∗ β ∗ F ∨ l,E ≃ F ∨ l,E , which proves the corollary. (cid:3) Corollary 9.6. (Beta game) Let F l,E and F l ′ ,E ′ be bundles from group A (resp.,group B ) on M p,q +1 . Suppose | E q | ≤ | E ′ q | . Then R Hom M p,q +1 ( F l ′ ,E ′ , F l,E ) = R Hom β (M p,q +1 ) ( F l ′ ,E ′ , F l,E ) unless r + s is even and we have one of the following exceptions. We also listcritical subsets T that violate the conditions.( A ) l + e p = l ′ + e ′ p = r − , e q = e ′ q = s + 1 , E p ⊆ T p , E q = T q andeither E ′ p ⊆ T p , E ′ q = T q or E ′ p ⊆ T cp , E ′ q = T cq ;( B ) e p = r + 1 + l , e ′ p = r + 1 + l ′ , e q = e ′ q = s + 1 , T p ⊆ E p , T q = E q and either T p ⊆ E ′ p , T q = E ′ q or T cp ⊆ E ′ p , T cq = E ′ q .Here T = T p ` T q is a subset of markings split into heavy and light markingssuch that T p = r and T q = s + 1 .Proof. Indeed, R Hom M p,q +1 ( F l ′ ,E ′ , β ∗ F l,E ) = R Hom M p,q +1 ( β ∗ F ∨ l,E , F ∨ l ′ ,E ′ ) == R Hom β (M p,q +1 ) ( F ∨ l,E , Rβ ∗ F ∨ l ′ ,E ′ ) = R Hom β (M p,q +1 ) ( F ∨ l,E , F ∨ l ′ ,E ′ ) by Corollary 9.5 (since by Lemma 7.3, we have m ≤ r + s ). The latter groupis equal to R Hom β (M p,q +1 ) ( F l ′ ,E ′ , F l,E ) . It remains to show that R Hom M p,q +1 ( F l ′ ,E ′ , F l,E ) = R Hom M p,q +1 ( F l ′ ,E ′ , β ∗ F l,E ) , which is true by Corollary 9.4 and Proposition 6.1, unless F l,E is listed inCorollary 9.4 with some T = T , in which case we also need that R Hom M p,q +1 ( F l ′ ,E ′ , O δ T ( − a, − b )) = 0 , where a = 0 , b = r + s . By Prop. 6.1 (with µ = r + s − ), this holds unless (cid:12)(cid:12)(cid:12) | E ′ ∩ T | − e ′ − l ′ (cid:12)(cid:12)(cid:12) or (cid:12)(cid:12)(cid:12) | E ′ ∩ T c | − e ′ − l ′ (cid:12)(cid:12)(cid:12) equals r + s . This means that ( l ′ , E ′ ) should be listed in Lemma 7.3, with a critical subset either T or T c . (cid:3) Notation 9.7.
Let p = 2 r ≥ , q = 2 s + 1 ≥ − . Consider the Hassett space M p,q +1 with markings from P ∪ ˜ Q with P the set of p heavy points and ˜ Q = Q ∪ { z } the set of q + 1 light points. We will distinguish between T and T c using the following recipe: for every boundary divisor δ T,T c of M p,q +1 ,choose T to be a subset containing index z and let T c be its complement.If s ≥ , then β (M p,q +1 ) = U p,q , where π : U p,q → M p,q is the universal P -bundle and the composition f = π ◦ β : M p,q +1 → M p,q is the forgetful morphism that forgets the marking z .If s = − , then β (M p,q +1 ) = M P \{ z } , { z } = M p − , , and if π : M p − , → M p − is the universal family, then f = π ◦ β is again theforgetful morphism that forgets the marking z . Therefore, we let U p, − := M p − , , M p, − := M p − . Note that U p,q is a GIT quotient. Lemma 9.8.
Assume s ≥ . Consider the following collections of vector bundles F l,E on U p,q (see Definition 3.3), where we assume that l + e is even: l + min( e p , p + 1 − e p ) ≤ r − A ) ,l + min( e p + 1 , p − e p ) ≤ r − B ) , The vector bundles from group A (resp., B ) form an exceptional collectionon U p,q . The order is as follows: we put all subsets E containing z last and beforethem all subsets not containing z . Within each of these two blocks, we order firstby increasing e q , arbitrarily if e q is the same but E q ’s are different, if E q ’s are thesame, in order of increasing e p , and for a given e p , arbitrarily.Proof. It follows from Corollary 3.6 that we have(i) If z / ∈ E , then F l,E ≃ π ∗ F l,E ;(ii) If z ∈ E , letting E c = Σ \ E , we have that F l,E ≃ F ∨ l,E c ⊗ F , Σ ≃ π ∗ F ∨ l,E c ⊗ F , Σ . In other words, the part of the collection with z / ∈ E is identical to the onein Theorem 1.11, while the part with z ∈ E corresponds to vector bundles F l,E = F ∨ l,E c ⊗ F , Σ with the collection { F ∨ l,E c } being the dual collection tothe one in Theorem 1.11.By Orlov’s theorem on the derived category of a projective bundle, D b ( U p,q ) = h D b (M p,q ) , D b (M p,q ) ⊗ F , Σ i . Now use the collection F l,E of Theorem 1.11 for the block D b (M p,q ) and thedual collection F ∨ l,E (tensored with F , Σ ) for the block D b (M p,q ) ⊗ F , Σ . (cid:3) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 51
We now compare the vector bundles F l,E on U p,q and M p,q . Note that therange for the pairs ( l, E ) for collections A and B on M p,q +1 in Thm. 1.16is precisely the range for these collections on U p,q in Lemma 9.8. Corollary 9.9.
The bundles F l,E from Thm. 1.16 form an exceptional collection.Proof. Consider first the case s ≥ . Let F l,E , F l ′ ,E ′ be bundles from oneof the two collections (either A or B ) on M p,q +1 . If E q = E ′ q , choose z ∈ E ′ q \ E q . By Cor. 9.6, we have R Hom M p,q +1 ( F l ′ ,E ′ , F l,E ) = R Hom U p,q ( F l ′ ,E ′ , F l,E ) , since the only exceptions happen when E q = E ′ q or E q and E ′ q are disjointand have s + 1 elements, but then z must be in both E q and E ′ q (use that z ∈ T ). By Lemma 9.8, we have R Hom U p,q ( F l ′ ,E ′ , F l,E ) = 0 .Suppose now that E q = E ′ q . If e q < q + 1 , let z be any light index notin E q , while if e q = q + 1 , let z be any light index. Then Cor. 9.6 applies: R Hom M p,q +1 ( F l ′ ,E ′ , F l,E ) = R Hom U p,q ( F l ′ ,E ′ , F l,E ) , since the only exception for E q = E ′ q is when e q = s + 1 and E q contains z .The latter group is equal to or C when ( l, E ) = ( l ′ , E ′ ) , by Lemma 9.8.Consider now the case s = − . In this case E = E p , E ′ = E ′ p aresubsets of P . We first prove the statement for group A . Note that if ( l, E ) is in group A on M p , it is also in the collection on M p +1 . Considerthe set-up in Section 8: let α : W → M p be the universal family and let f : W → M p, = M p +1 be the birational morphism that contracts theboundary divisors δ T ∪{ x } ,T c by identifying the points in T c . Using that F l,E = α ∗ F l,E (Lemma 5.16), R Hom M p ( F l ′ ,E ′ , F l,E ) = R Hom W ( α ∗ F l ′ ,E ′ , α ∗ F l,E ) = RHom W ( F l ′ ,E ′ , F l,E ) , which by Cor. 8.8, equals R Hom M p, ( F l ′ ,E ′ , F l,E ) . As M p +1 = M p, , by Thm.1.8, we have that R Hom M p, ( F l ′ ,E ′ , F l,E ) is if e ′ ≥ e , ( l, E ) = ( l ′ , E ′ ) , and C when ( l, E ) = ( l ′ , E ′ ) .We now prove the statement for group B . Note, the previous proofworks only if none of ( l, E ) , ( l ′ , E ′ ) satisfy l + ( p − e ) = r − . We thereforeproceed with a different proof for group B ..As in the case q > , by Cor. 9.6, if z / ∈ E we have R Hom M p ( F l ′ ,E ′ , F l,E ) = R Hom M p − , ( F l ′ ,E ′ , F l,E ) , since the only exceptions for group B happen when z ∈ E . If E ′ \ E = ∅ ,we pick z ∈ E ′ \ E . Since F l,E = π ∗ F l,E and Rπ ∗ ( F ∨ l ′ ,E ′ ) = 0 on M p − , , R Hom M p − , ( F ∨ l,E , F ∨ l ′ ,E ′ ) = R Hom M p − , ( π ∗ F ∨ l,E , F ∨ l ′ ,E ′ ) == R Hom M p − ( F ∨ l,E , Rπ ∗ ( F ∨ l ′ ,E ′ )) = 0 . If E ′ ⊆ E , since e ′ p ≥ e p , we must have E ′ = E . If E = Σ = P , let z ∈ Σ \ E . Since F l,E = π ∗ F l,E , F l ′ ,E ′ = π ∗ F l ′ ,E ′ on π : M p − , → M p − , R Hom M p − , ( F ∨ l,E , F ∨ l ′ ,E ′ ) = R Hom M p − ( F ∨ l,E , F ∨ l ′ ,E ′ ) . As ( l, E ) is in group B , we have l + min { e + 1 , p − e } ≤ r − , i.e., l + min { e, ( p − − e } ≤ r − , which is the range of pairs ( l, E ) in Thm.1.8 with E ⊆ P \ { z } . The result follows in this case from Thm. 1.8.If E = E ′ = Σ , by Cor. 9.6, we still have R Hom M p ( F l ′ ,E ′ , F l,E ) = R Hom M p − , ( F l ′ ,E ′ , F l,E ) , unless l = l ′ = r − (and r is odd since l + e is even). Assume this is notthe case. We have R Hom M p − , ( F l ′ ,E ′ , F l,E ) = R Hom M p − , ( F ∨ l ′ , ∅ , F ∨ l, ∅ ) = R Hom M p − ( F l, ∅ , F l ′ , ∅ ) (use F l,E = F ∨ l,E c ⊗ F , Σ ) and we are done by Thm. 1.8. It remains to provethat the vector bundle F r − , Σ ( r odd) is exceptional on M p . By Lemma 5.9,we have that F r − , Σ = F r − , ∅ ⊗ F , Σ . Hence, it suffices to prove that F r − , ∅ is exceptional on M p . As in the case of group A , we have R Hom M p ( F r − , ∅ , F r − , ∅ ) = R Hom M p, ( F r − , ∅ , F r − , ∅ ) , which equals C by Thm. 1.8. (cid:3)
10. P
ROOF OF T HM . 1.11 ( EXCEPTIONALITY : P EVEN , Q ODD ) Notation 10.1.
Throughout this section we assume p = 2 r , q = 2 s + 1 . Let P = { , . . . , p } , R = { , . . . , r } , R ′ = P \ R. Recall that Z R denotes the locus in M p,q where points in R come together: Z R = δ r ∩ δ r ∩ . . . ∩ δ r − ,r ∼ = M { u }∪ R ′ ∪ Q , where u is the marking corresponding to the points in R . Note that u canonly coincide with at most s light points and with none of the points in R ′ .Let Y = Z R ∩ Z R ′ . We identify Y = M { u,v }∪ Q , where u (resp., v ) is themarking corresponding to R (resp., R ′ ).Theorem 1.11 follows from Corollary 8.9 and the following theorem,which takes care of torsion objects: Theorem 10.2.
Assume p = 2 r , q = 2 s + 1 . In the notations of Thm. 1.11:(1) If ( l, E ) is in group A (resp., B ) and ( l ′ , E ′ ) is in group , we haveR Hom M p,q ( F l,E , T l ′ ,E ′ ) = 0 if e q ≥ e ′ q . (2) If ( l, E ) is in group and ( l ′ , E ′ ) is in group A (resp., B ), we haveR Hom M p,q ( T l,E , F l ′ ,E ′ ) = 0 if e q > e ′ q . (3) If ( l, E ) , ( l ′ , E ′ ) are both in group and E p = E ′ p = R , we haveR Hom M p,q ( T l,E , T l ′ ,E ′ ) = 0 , if e q ≥ e ′ q , E q = E ′ q , or if E q = E ′ q , l < l ′ , and R Hom M p,q ( T l,E , T l,E ) = C .(4) If ( l, E ) , ( l ′ , E ′ ) are both in group and E p = R , E ′ p = R ′ , we haveR Hom M p,q ( T l,E , T l ′ ,E ′ ) = 0 if e q ≥ e ′ q . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 53
Lemma 10.3.
We have the following: ψ j | Z R ≃ ψ u if j ∈ R − ψ u if j ∈ R ′ − ψ u − δ ju if j ∈ Q ,δ ij | Z R ≃ − ψ u if i, j ∈ Rψ u if i, j ∈ R ′ i ∈ R, j ∈ R ′ δ ij if { i, j } ∩ Q = ∅ , (where if j ∈ R , i ∈ Q , we identify δ ij = δ iu ). Furthermore, K | Z R ≃ K Z R + ( r − ψ u , where K = K M p,q (resp., K Z R ) is the canonical class of M p,q (resp., Z R ).Proof. It follows from the definition that the restriction of ψ j to Z R is theclass ψ j on the corresponding Hassett space for all j . Note that the univer-sal family over Z R is still a P -bundle. It follows from [CT17, Lemma 5.1]that ψ i + ψ j = − δ ij . Since δ uj = ∅ if j ∈ R ′ , the result follows. The lastequality follows from adjunction since c ( N Z R | M p,q ) = r X j =2 ( δ j ) | Z R = − ( r − ψ u , which is also a useful formula. (cid:3) We reduce Theorem 10.2 to a calculation of R Γ on Z R or Y : Lemma 10.4.
In order to prove Theorem 10.2, it suffices to prove the following:(1) If ( l, E ) is in group A (resp., B ) and ( l ′ , E ′ ) is in group , we haveR Γ (cid:0) Z R , ( F l,E | Z R ) ∨ ⊗ T l ′ ,E ′ ) = 0 if e q ≥ e ′ q , R := E ′ p (2) If ( l, E ) is in group and ( l ′ , E ′ ) is in group A (resp., B ), we haveR Γ (cid:0) Z R , T ∨ l,E ⊗ F l ′ ,E ′ | Z R ⊗ c ( N Z R | M p,q ) (cid:1) = 0 if e q > e ′ q , R := E p (3) If ( l, E ) , ( l ′ , E ′ ) are both in group and E p = E ′ p = R , we haveR Γ (cid:0) Z R , T ∨ l,E ⊗ T l ′ ,E ′ ⊗ ( X j ∈ J δ j (cid:1) ) = 0 , for all J ⊆ R \ { } if e q ≥ e ′ q , E q = E ′ q , or if E q = E ′ q , l < l ′ , andR Γ (cid:0) Z R , T ∨ l,E ⊗ T l,E ⊗ ( X j ∈ J δ j (cid:1) ) = 0 , for all ∅ 6 = J ⊆ R \ { } .(4) If ( l, E ) , ( l ′ , E ′ ) are both in group and E p = R , E ′ p = R ′ , we haveR Γ Y, (cid:18) l + l ′ (cid:19) ψ u + 12 X j ∈ E q ψ j − X j ∈ E ′ q ψ j = 0 if e q ≥ e ′ q . Proof. (1) R Hom M p,q ( F l,E , T l ′ ,E ′ ) = R Hom Z R ( F l,E | Z R , T l ′ ,E ′ ) == R Γ (cid:0) Z R , ( F l,E | Z R ) ∨ ⊗ T l ′ ,E ′ ) . (2) Using that i ! F l ′ ,E ′ = i ∗ F l ′ ,E ′ ⊗ c ( N Z R | M p,q ) , where i : Z R ֒ → M p,q isthe canonical embedding, we have (up to a shift) that R Hom M p,q ( T l,E , F l ′ ,E ′ ) = R Hom Z R ( T l,E , F l ′ ,E ′ | Z R ⊗ c ( N Z R | M p,q )) == R Γ (cid:0) Z R , T ∨ l,E ⊗ F l ′ ,E ′ | Z R ⊗ c ( N Z R | M p,q ) (cid:1) . ([Huy06, Cor. 3.38]).(3) Follows from tensoring the Koszul resolution of Z R with a line bun-dle L such that i R ∗ ( L | Z R ) = T l,E and applying R Hom( − , T l ′ ,E ′ ) .(4) Consider the canonical embeddings i : Z R ֒ → M p,q , i ′ : Z R ′ ֒ → M p,q , j : Y ֒ → Z R , j ′ : Y ֒ → Z R ′ . Since for any line bundles L on Z R and L ′ on Z R ′ we have R Hom( i ∗ L, i ′∗ L ′ ) = R Hom( i ′∗ i ∗ L, L ′ ) == R Hom( j ∗ j ′∗ L, L ′ ) = R Hom( j ′∗ L, j ! L ′ ) it follows that R Hom M p,q ( T l,E , T l ′ ,E ′ ) = R Hom Y ( T l,E | Y , T l ′ ,E ′ | Y ⊗ c ( N Y | Z R ′ )) == R Γ (cid:0) Y, ( T l,E | Y ) ∨ ⊗ T l ′ ,E ′ | Y ⊗ ( X j ∈ R \{ } δ j ) | Y (cid:1) == R Γ (cid:0) Y, ( l + l ′ ψ u + 12 X j ∈ E q ψ j − X j ∈ E ′ q ψ j ) using Lemma 10.3. (cid:3) We prove this vanishing by a windows calculation on Z R and Y . Notation 10.5.
Let X = ( P ) r + q +1 = P u × ( P ) r × ( P ) q , corresponding tothe partition { u } ⊔ R ′ ⊔ Q of the markings on Z R = M { u }∪ R ′ ∪ Q . Then Z R isa GIT quotient of X by PGL .For a ∈ Z , A ⊆ R ′ , B ⊆ Q , consider on X the line bundle O ( a, A, B ) = pr ∗ O ( a ) ⊗ pr ∗ R ′ O ( A ) ⊗ pr ∗ Q O ( B ) where as usual O ( A ) = O ( i , . . . , i r ) with i j = 1 if j ∈ A and otherwise,and similarly, O ( B ) = O ( i , . . . , i q ) with i j = 1 if j ∈ B and otherwise.We also denote O ( − A ) the dual of O ( A ) .We have the following correspondence between vector bundles on Z R and P GL -linearized vector bundles on X = ( P ) r + q +1 : F l,E | Z R = O ( | E p ∩ R | , E p ∩ R ′ , E q ) ⊗ V l , T l,E = O ( r + l, , E q ) , if E p = R,K Z R = O ( − , − , . . . , − . Likewise, let X ′ = ( P ) q +2 = P u × P v × ( P ) q , corresponding to the partition { u } ⊔ { v } ⊔ Q of the markings on M { u,v }∪ Q .Then Y is the GIT quotient of X ′ by PGL . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 55
Instead of Thm. 4.4 we are going to use the following related resultproved for vector bundles by Teleman [Tel00].
Theorem 10.6. [HL15, Th. 3.29]
In the notations of Thm. 4.4, for any object F ∈ D b [ X/G ] such that H ∗ ( σ ∗ i F ) has weights < η i , we haveR Γ [ X/G ] ( F ) = R Γ [ X ss /G ] ( F ) . The calculation of the Kempf-Ness stratification and the correspondingweights is similar to the one in Section 4. However, starting from this pointon, we will follow a convention of [Tel00] and take an opposite weight to(4.1): weight λ O X ( a , . . . , a k ) | z K = X i ∈ K a i − X i ∈ K c a i . (10.1)We find this convention more natural since the ample polarization of theGIT quotient has positive weights on the unstable locus. The condition ofTheorem 10.6 becomes H ∗ ( σ ∗ i F ) has weights > − η i . Remark 10.7. (The devil’s trick) The line bundle D := O ( r, R ′ , on X = ( P ) r + q +1 is trivial on Z R , since for any j ∈ R ′ , we have δ uj = ∅ and therefore, ψ u + ψ j = 0 in Z R = M { u }∪ R ′ ∪ Q . Likewise, the line bundle D := O (1 , , on X ′ descends to the trivial linebundle on Y . Instead of proving vanishing in Lemma 10.4 directly, we willprove vanishing on X (resp., X ′ ) after tensoring with a high multiple of D since it’s this tensor product that will satisfy conditions of Theorem 10.6.This is a useful observation for any GIT quotient X//G such that the unsta-ble locus contains a divisorial component.
Proof of Thm.10.2.
We prove the vanishings in Lemma 10.4. First we checkthat the
PGL -invariant vanishing holds on X = ( P ) r + q (cases (1), (2)and (3)) and on X ′ = ( P ) q (case (4)) after tensoring a vector bundle withthe devil line bundle D N , N ≫ . Later on we check the weight conditionin Thm. 10.6.For (1), assuming the weight condition, R Γ (cid:0) Z R , ( F l,E | Z R ) ∨ ⊗T l ′ ,E ′ ⊗ D N ) == R Γ (cid:0) X, O ( r + l ′ − | E p ∩ R | + N r, − E p ∩ R ′ + N R ′ , E ′ q − E q ) ⊗ V l (cid:1) PGL , which is clearly if E q * E ′ q . Since here we assume e q ≥ e ′ q , we have that E q ⊆ E ′ q if and only if E q = E ′ q . Assume E q = E ′ q . In this case, we need toconsider V r + l ′ −| E p ∩ R | + Nr ⊗ V N − y ⊗ . . . ⊗ V N − y r ⊗ V l , where y i = 1 if the corresponding index in E p ∩ R ′ and otherwise. Weclaim that the PGL -invariant part is by the Clebsch-Gordan formula. Itsuffices to check that r + l ′ − | E p ∩ R | + N r > N r − | E p ∩ R ′ | + l. This follows if l + | E p ∩ R | − | E p ∩ R ′ | < r. Since ( l, E ) is in group A or B , this follows by Cor. 7.6.For (2), assuming the weight condition, R Γ (cid:0) Z R , T ∨ l,E ⊗ F l ′ ,E ′ | Z R ⊗ c ( N Z R | M p,q ) ⊗ D N (cid:1) == R Γ (cid:0) X, O ( r − l + | E ′ p ∩ R | − N r, E ′ p ∩ R ′ + N R ′ , E ′ q − E q ) ⊗ V l ′ (cid:1) PGL . (use c ( N Z R | M p,q ) = O (2 r − , , ). This is since E q * E ′ q .For (3), assuming the weight condition, R Γ (cid:0) Z R , T ∨ l,E ⊗T l ′ ,E ′ ⊗ X j ∈ J δ j ⊗ D N (cid:1) = R Γ X (cid:0) O (2 | J |− l + l ′ + N r, + N R ′ , E ′ q − E q ) (cid:1) PGL for all J ⊆ R \ { } . This is if e q ≥ e ′ q , E q * E ′ q , while if E q = E ′ q the PGL -invariant part is when l < l ′ or when l = l ′ , | J | > by the Clebsch-Gordanformula since in these cases we have | J | − l + l ′ + N r > N r.
Assume now we are in situation (4). The question is equivalent to the van-ishing for N ≫ of the PGL -invariant part of R Γ (cid:0) O ( − l − l ′ − N, N, E ′ q − E q ) (cid:1) . This is clear if e q ≥ e ′ q , E q * E ′ q . If E q = E ′ q then this follows from theClebsch-Gordan formula since N > N − l − l ′ − . We now check that for each stratum, each of the cases (1)-(4) fall underthe assumption on weights of Thm. 10.6. Up to symmetry, the unstable lociin X have the following form:(1) The locus K I , for I ⊆ Q , | I | ≥ s + 1 , where u and the indices in I come together. In this case, η = 2 | I | , (2) The locus K J,I , for J ⊆ R ′ , I ⊆ Q , J = ∅ , | I | ≥ , where u and theindices in J and I come together. In this case, η = 2 | I | + 2 | J | , (3) The locus L I , for I ⊆ Q , | I | ≥ s + 1 , where the indices in R ′ and I come together. In this case, η = 2 | I | + 2 r − . The devil line bundle has the property thatweight λ O ( r, R ′ , | z { u }∪ J ∪ I = r + | J | − | R ′ \ J | = 2 | J | > while its weight for the other strata is . Therefore, the condition in Thm.10.6 for the stratum K J,I can be achieved by tensoring with a high enoughmultiple of this line bundle. We only need to consider the remaining strata.
Strata K I . Assume | I | ≥ s + 1 . In Case (1) we need to verify that theweights of O ( r + l ′ − | E p ∩ R | , − E p ∩ R ′ , E ′ q − E q ) ⊗ V l ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 57 are > − | I | . Since the weights of V l are between − l and l , it suffices toprove that r + l ′ − | E p ∩ R | + | E p ∩ R ′ | + | E ′ q ∩ I |− | E q ∩ I |− | E ′ q ∩ I c | + | E q ∩ I c |− l > − | I | . By (7.7) for the pair ( l, E p ) , it suffices to prove that l ′ + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | ≥ − | I | . Since the left hand side equals l ′ + ( e q − e ′ q ) − | E q ∩ I | + 2 | E ′ q ∩ I | and we assume e q ≥ e ′ q , the result follows from − | E q ∩ I | ≥ − | I | .In Case (2), we need to verify that the weights of O ( r − l + | E ′ p ∩ R | − , E ′ p ∩ R ′ , E ′ q − E q ) ⊗ V l ′ are > − | I | . Again, it suffices to prove that r − l + | E ′ p ∩ R |− −| E ′ p ∩ R ′ | + | E ′ q ∩ I |−| E q ∩ I |−| E ′ q ∩ I c | + | E q ∩ I c |− l ′ > − | I | . Using (7.7) for the pair ( l ′ , E ′ p ) , it suffices to prove that − l + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | ≥ − | I | + 2 . The left hand side is greater than − l − | E q ∩ I | + | E q ∩ I c | − | I c | , and this is ≥ − | I | + 2 by Lemma 7.7 applied to the pair ( l, E q ) .In Case (3) we need to verify that the weight of O (2 | J | − l + l ′ , , E ′ q − E q ) is > − | I | for all ≤ | J | ≤ r − . Equivalently, we need to prove that − l + l ′ + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | > − | I | . The left hand side is greater or equal than − l − | E q ∩ I | + | E q ∩ I c | − | I c | and this is > − | I | by Lemma 7.7 applied to the pair ( l, E q ) . Strata L I . In Case (1), we need to verify that − r − l ′ + | E p ∩ R |−| E p ∩ R ′ | + | E ′ q ∩ I |−| E q ∩ I |−| E ′ q ∩ I c | + | E q ∩ I c |− l > − | I |− r +2 . Using (7.7) for the pair ( l, E p ) , it suffices to prove that − l ′ + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | ≥ − | I | + 2 . The left hand side is greater than − l ′ + | E ′ q ∩ I | − | E ′ q ∩ I c | − | I | , hence, it suffices to show that − l ′ + | E ′ q ∩ I | − | E ′ q ∩ I c | ≥ −| I | + 2 , but this follows from Lemma 7.7 applied to the pair ( l ′ , E ′ q ) .In Case (2), we need to verify that − ( r − l + | E ′ p ∩ R |− | E ′ p ∩ R ′ | + | E ′ q ∩ I |−| E q ∩ I |−| E ′ q ∩ I c | + | E q ∩ I c |− l ′ > − | I |− r +2 . Using (7.7) for the pair ( l ′ , E ′ p ) , it suffices to prove that l + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | ≥ − | I | . The left hand side is clearly greater than −| E q ∩ I | − | E ′ q ∩ I c | ≥ −| I | − | I c | = − s − , and the inequality follows since | I | ≥ s + 1 .In Case (3), we need to verify that for all ≤ | J | ≤ r − we have − | J | + l − l ′ + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | > − | I | − r + 2 , or equivalently, l − l ′ + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | > − | I | . The left hand side is clearly greater than − l ′ + | E ′ q ∩ I | − | E ′ q ∩ I c | − | I | , which is > − | I | by Lemma 7.7 applied to the pair ( l ′ , E ′ q ) and | I | ≥ s + 1 .We now consider Case (4). Up to symmetry, the unstable loci in X ′ = ( P ) q +2 = P u × P v × ( P ) q have the following form:(1) The locus K ′ I , for I ⊆ Q , | I | ≥ s + 1 , where u and the indices in I come together. In this case, η = 2 | I | .(2) The locus K ′′ I , for I ⊆ Q , | I | ≥ s + 1 , where v and the indices in I come together. In this case, η = 2 | I | .(3) The locus K ′′′ I , for I ⊆ Q , | I | ≥ , where u , v and the indices in I come together. In this case, η = 2 | I | + 2 .The devil line bundle O (1 , , has the property thatweight λ O (1 , , | z { u,v }∪ I = 2 > but its weight on other strata is . Therefore we only have to consider thestrata K ′ I , K ′′ I .Consider the stratum K ′ I . We need to verify that the weight of O (0 , l + l ′ + 2 , E ′ q − E q ) is > − | I | . Equivalently, we need to prove that − l − l ′ − | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | > − | I | . This follows from Lemma 7.7 applied to the pairs ( l, E q ) and ( l ′ , E ′ q ) . Forthe stratum K ′′ I , one needs to verify that l + l ′ + 2 + | E ′ q ∩ I | − | E q ∩ I | − | E ′ q ∩ I c | + | E q ∩ I c | > − | I | , which is clearly satisfied as the left hand side is greater than −| E q ∩ I | − | E ′ q ∩ I c | ≥ −| I | − | I c | > − | I | . This completes the proof. (cid:3)
ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 59
11. P
ROOF OF T HM . 1.16 ( EXCEPTIONALITY : P EVEN , Q EVEN )We are going to apply Lemma 8.1 for S = D b (M p,q +1 ) , the usual dualityfunctor ∨ and the s.o.d. hA , Bi of Notation 1.14. Lemma 11.1.
The subcategory A is admissible and ( S p × S q +1 ) equivariant. Thesubcategory B is an ( S p × S q +1 ) equivariant non-commutative resolution of sin-gularities of the GIT quotient X p,q +1 in the sense of [KL14] .Proof. Easily follows from [Kuz08] where arbitrary cones over Segre vari-eties are considered. Alternatively, it is easy to check that sheaves that gen-erate A form an exceptional collection (or see [CT17][Lemma 6.19]), whichimplies admissibility. (cid:3) Notation 11.2.
Let p = 2 r ≥ , q = 2 s + 1 ≥ , | P | = p , | Q | = q + 1 , R = { , . . . , r } ⊆ P, R ′ = P \ R. Let M R ′ be the Hassett space obtained by contracting all the boundary divi-sors δ T,T c ≃ P r + s − × P r + s − ⊆ M p,q +1 with T p = R ′ onto the second factor.We let β R ′ : M p,q +1 → M R ′ be the corresponding contraction. For E ⊆ P ∪ Q , l ≥ , e p = r , let T l,E bethe torsion sheaf T l,E on M R ′ defined as in (1.10).The second s.o.d. we consider is given by D = Lβ ∗ R ′ D b (M R ′ ) , C = D ⊥ . i.e., C is the subcategory generated by the torsion sheaves O δ T,Tc ( − a, − b ) , with T p = R ′ and < a ≤ r + s − , ≤ b ≤ r + s − .Theorem 1.16 follows from Prop. 11.3, which reduces analysis of torsionobjects to the calculations on the Hassett space M R , which in turn is donein Thm. 11.10. The fact that vector bundles { F l,E } in group A (resp., B )form an exceptional collection was proved in Sections 9 and 8.Throughout this section we assume p = 2 r ≥ , q = 2 s + 1 , s ≥ . Notethat in the case of M p,q +1 = M p ( s = − ) the collection of Theorem 1.16consists only of the bundles { F l,E } , hence, we are done in that case. Proposition 11.3.
We haveR
Hom M p,q +1 ( G ′ , G ) = R Hom M R ′ ( G ′ , G ) , for any of the following cases:(1) G = ˜ T l,E , G ′ = F l ′ ,E ′ , G = T l,E , G ′ = F l ′ ,E ′ if E p = R ;(2) G = F l,E , G ′ = ˜ T l ′ ,E ′ , G = F l,E , G ′ = T l ′ ,E ′ if E ′ p = R ′ ;(3) G = ˜ T l,E , G ′ = ˜ T l ′ ,E ′ , G = T l,E , G ′ = T l ′ ,E ′ if E p = E ′ p = R ;(4) G = ˜ T l,E , G ′ = ˜ T l ′ ,E ′ , G = T l,E , G ′ = T l ′ ,E ′ if E p = R , E ′ p = R ′ .where all F l,E are for pairs ( l, E ) in group A or B , and all ˜ T l,E , T l,E are forpairs ( l, E ) in group A or B . Proof.
To see (1), (3) and (4), we apply Lemma 8.1 with G = ˜ T l,E . Since inthese cases E p = R , by Prop. 11.8(iii), (cid:0) Lβ ∗ R ′ T l,E (cid:1) B = ˜ T l,E , and thereforeR Hom M p,q +1 ( G ′ , ˜ T l,E ) = R Hom M R ′ ((( G ′∨ ) D ) ∨ , T l,E ) , where we identify objects in D with the corresponding elements in D b (M R ′ ) .For simplicity, denote β = β R ′ . The identity (( G ′∨ ) D ) ∨ = Lβ ∗ G ′ is im-plied by R β ∗ (cid:0) G ′∨ (cid:1) = G ′∨ , since for any object T , T D = Lβ ∗ Rβ ∗ T . For each G ′ in cases (1), (3) and (4), we have that R β ∗ (cid:0) G ′∨ (cid:1) = G ′∨ by Lemma 11.9(2)(for G ′ = F l ′ ,E ′ ) and Prop. 11.8((ii) and (iv)) (for G ′ = T l ′ ,E ′ ).In case (2), we distinguish two cases. Assume first that ( l, E ) is not oneof the exceptions in Lemma 11.9(1), i.e., none of the following hold: • E p ⊆ R ′ , e p + l = r − , e q = s + 1 (case A ), • R ′ ⊆ E p , e p = r + 1 + l , e q = s + 1 (case B ).In this case Lemma 11.9(1) givesR Hom M p,q +1 ( G ′ , F l,E ) = R Hom M R ′ ((( G ′∨ ) D ) ∨ , F l,E ) . As by Prop. 11.8(ii), Rβ R ′ ∗ (cid:0) G ′∨ (cid:1) = G ′∨ for G ′ = T l ′ ,E ′ , the result follows.Assume now that ( l, E ) is one of the exceptions in Lemma 11.9(1). Usingagain Prop. 11.8(ii) and Lemma 11.9(1), we haveR Hom M p,q +1 ( ˜ T l ′ ,E ′ , F l,E ) = R Hom M R ′ ( T l ′ ,E ′ , F l,E ) provided that for ≤ u < r + s we can prove the following claim: Claim 11.4. If ( l ′ , E ′ ) is in group A or B and E ′ p = R ′ , then for all ≤ u < r + s and boundary δ T,T c with T p = R ′ , we have:R Hom M p,q +1 (cid:0) ˜ T l ′ ,E ′ , O δ T,Tc ( u, − r + s (cid:1) = 0 . We will prove the claim after writing an explicit resolution of ˜ T l ′ ,E ′ . (cid:3) Notation 11.5.
Let R = { , . . . , r } . A sheaf T l,E with E p = R is isomorphicin D b (M p,q +1 ) to the Koszul resolution of the stratum Z R = T j ∈ R \{ } δ j tensored with the line bundle L ′ : → L R \{ } → . . . → M J ⊆ R \{ } , | J | = r − L J → . . . → M j ∈ R \{ } L j → L ∅ → , where for every subset J ⊆ R \ { } , with ≤ | J | ≤ r − , we let L J = L ′ (cid:0) − X j ∈ J δ j (cid:1) , L ′ = e q − r − l ψ + X k ∈ E q δ k . Likewise, we are going to show that ( T l,E ) B is isomorphic to the follow-ing complex, which from now on we will call ˜ T l,E : → ˜ L R \{ } → . . . → M J ⊆ R \{ } , | J | = r − ˜ L J → . . . → M j ∈ R \{ } ˜ L j → ˜ L ∅ → , where for every subset J ⊆ R \ { } , with ≤ | J | ≤ r − , we let ˜ L J = L J ⊗ O (cid:0) − X T p = R α J,T,E,l δ T,T c (cid:1) , with ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 61 α J,T,l,E = ( e q − r − l − | E q ∩ T q | + | J | if e q − r − l − | E q ∩ T q | + | J | > otherwise . Lemma 11.6.
Let ( l, E ) be in group A or B . Then ˜ T l,E = ( ˜ L J ) as in Notation11.5 has a structure of a complex in D b (M p.,q +1 ) isomorphic to ( T l,E ) B (see theproof for description of differentials). In particular, we have an exact triangle ˜ T l,E → T l,E → Q, with Q generated by sheaves in A supported on δ T,T c , T p = R .Proof. Let A R ⊂ D b (M p,q ) be an admissible subcategory generated by sheavesin A supported on δ T,T c with T p = R . Let B R = ⊥ A R . Since L J = ˜ L J + X T p = R α J,T,l,E δ T,T c , the canonical morphisms ˜ L J → L J have the cokernels generated by sheavessupported on boundary δ T,T c , with T p = R . We claim that these cokernelsare in A R . Indeed, quotients relating ˜ L J and L J have the form Q = (cid:0) ˜ L J + iδ (cid:1) | δ = O ( − i ) ⊠ O ( α J,T,E,l − i ) , < i ≤ α J,T,E,l , where α J,T,E,l = e q − r − l − | E q ∩ T | + | J | ≤ e q − l − | E q ∩ T | + r − , and this is < r + s by Cor. 7.4. It follows that ( ˜ L J ) B R = ( L J ) B R . Next we claim that ˜ L J ∈ B R , in other words that R Hom (cid:0) ˜ L J , O δ ( − a, − b ) (cid:1) = 0 for any O δ ( − a, − b ) as in Thm. 1.16 with T p = R . We have ( ˜ L J ) ∨ ⊗ O ( − a, − b ) == O (cid:18) e q − r − l − | E q ∩ T | + | J | − α J,T,l,E − a (cid:19) ⊠ O ( − α J,T,l,E − b ) , Consider the case when α J,T,l,E = 0 . Clearly, the sheaf is acyclic when b = 0 . Assume b = 0 , < a < r + s . It suffices to prove that < | E q ∩ T | − e q − r − l − | J | + a ≤ r + s − . Since α J,T,l,E = 0 , we have that | E q ∩ T | − e q − r − l − | J | ≥ . To prove thesecond inequality, as a < r + s , it suffices to prove that | E q ∩ T | − e q − r − l ≤ r + s . This follows from Cor. 7.4.Consider now the case when α J,T,l,E > . In this case ( ˜ L J ) ∨ ⊗ O ( − a, − b ) = O ( − a ) ⊠ O ( − α J,T,l,E − b ) . Clearly, this is acyclic when a > . Assume a = 0 , < b < r + s . Therefore itsuffices to prove that for all J ⊆ R \ { } we have α J,T,l,E = e q − r − l − | E q ∩ T | + | J | ≤ r + s . or equivalently that e q − l − | E q ∩ T | ≤ s . This holds by Cor. 7.4 and this finishes the proof.Now we use the following simple lemma applied to the functor D b (M p,q +1 ) → B R ֒ → D b (M p,q +1 ) : Lemma 11.7.
Let F : D b ( A ) → D b ( B ) be an exact functor. Let L • ∈ D b ( A ) be a complex L → . . . → L r . Suppose F ( L ) , . . . , F ( L r ) ∈ B . Then F ( L • ) isisomorphic to a complex F ( L ) → . . . → F ( L r ) obtained by applying the functor F to differentials of L • .Proof. Induction on r using naive truncations. (cid:3) It follows from the lemma that ˜ T l,E = ( ˜ L J ) is a complex isomorphic to ( T l,E ) B R and its differentials are obtained by applying the functor T → T B R to differentials in T l,E = ( L J ) . More concretely, for j ∈ J , we have L J \{ j } = L J + δ j and the differentials in T l,E are built from the maps σ : L J → L J \{ j } givenby multiplication with a canonical section of O ( δ j ) . Likewise, ˜ L J \{ j } = ˜ L J + δ j + X Tp = R,αJ,T,l,E> δ T,T c , since α J \{ j } ,T,l,E = ( α J,T,l,E − if α J,T,l,E > α J,T,l,E = 0 otherwiseWe claim that differentials in ˜ T l,E are built from the maps ˜ σ : ˜ L J → ˜ L J \{ j } given by multiplication with a canonical section of O ( δ j + P Tp = R,αJ,T,l,E> δ T,T c ) (note that these maps are in B R since it is a full subcategory). Indeed, con-sider the diagram ˜ L J ˜ σ −−−−→ ˜ L J \{ j } y y L J σ −−−−→ L J \{ j } where the left, resp., right, vertical maps are the canonical inclusions, givenby multiplication with a non-zero section of X α J,T,l,E > α J,T,l,E δ T,T c ( resp. , X α J,T,l,E > ( α J,T,l,E − δ T,T c ) . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 63
Clearly, this is a commutative diagram. Applying the functor T → T B R toit gives a diagram ˜ L J ˜ σ −−−−→ ˜ L J \{ j } = y y = ˜ L J σ B R −−−−→ ˜ L J \{ j } Which proves the claim. In particular, we have an exact triangle of a s.o.d. ˜ T l,E → T l,E → ( T l,E ) A R . (11.1)We claim that ˜ T l,E ∈ B . Since T l,E is supported on Z R and Z R does notintersect any of the boundary δ = δ T,T c in M p,q +1 with T p = R, R ′ , we haveR Hom (cid:0) T l,E , O δ ( − a, − b ) (cid:1) = 0 , for all a, b . Hence, R Hom (cid:0) ˜ T l,E , O δ ( − a, − b ) (cid:1) = 0 by (11.1) since supports ofobjects in A R are also disjoint from δ . (cid:3) Proof of Claim 11.4.
Let R ′ = { , . . . , r } . By Lemma 11.6, it suffices to provethat for all subsets J ⊆ R ′ \ { } we have R Γ (cid:0) ˜ L ∨ J ⊗ O ( u, − r + s ) (cid:1) = 0 . For δ = δ T,T c , with T p = R ′ , letting α := α J,T,E ′ ,l ′ , we have ( ˜ L J ) | δ = O (cid:0) − e ′ q − r − l ′ | E ′ q ∩ T q | − | J | + α, α (cid:1) == ( O (0 , α ) if α > O ( − e ′ q − r − l ′ + | E ′ q ∩ T q | − | J | ,
0) if α = 0 . If α = 0 , the second component of ˜ L ∨ J ⊗ O ( u, − r + s ) is O ( − r + s ) , which isacyclic on P r + s − , and the result follows. Assume now α > . We have toprove that ˜ L ∨ J ⊗ O ( u, − r + s O ( u, − α − r + s is acyclic, or equivalently, that α + r + s ≤ r + s − , i.e., that α = e ′ q − r − l ′ − | E ′ q ∩ T q | + | J | ≤ r + s − . As | J | ≤ r − , it suffices to prove that e ′ q − l ′ − | E ′ q ∩ T q | ≤ s . This follows from Cor. 7.4 applied to the pair ( E ′ , l ′ ) . (cid:3) Proposition 11.8.
Let ( l, E ) be in group A or B on M p,q +1 , with E p = R .Then(i) (cid:0) Lβ ∗ R T l,E (cid:1) B = ˜ T l,E , except when r + s is even and one of the following holds: – e q = l + s + 2 and T q ⊆ E q (case A ), – e q + l = s , E q ⊆ T q (case B ), in which case there is an exact triangle ˜ T l,E → (cid:0) Lβ ∗ R T l,E (cid:1) B → Q B , where Q is generated by the sheaves O δ T,Tc (0 , − r + s ) with T q ⊆ E q .(ii) Rβ R ∗ (cid:0) ˜ T ∨ l,E (cid:1) = T ∨ l,E , (iii) (cid:0) Lβ ∗ R ′ T l,E (cid:1) B = ˜ T l,E ,(iv) Rβ R ′ ∗ (cid:0) ˜ T ∨ l,E (cid:1) = T ∨ l,E .Proof. We note that T l,E isomorphic in D b (M R ) to the Koszul resolution ofthe stratum Z R = T j ∈ R \{ } δ j tensored with the line bundle L ′ : → L R \{ } → . . . → M J ⊆ R \{ } , | J | = r − L J → . . . → M j ∈ R \{ } L j → L ∅ → , where for every subset J ⊆ R \ { } , with ≤ | J | ≤ r − , we let L J = L ′ (cid:0) − X j ∈ J δ j (cid:1) , L ′ = e q − r − l ψ + X k ∈ E q δ k , where all tautological classes are on M R .Let ˜ T l,E = ( ˜ L J ) as in Notn. 11.5. To show (i), we prove that there is anexact triangle ˜ T l,E → (cid:0) Lβ ∗ R T l,E (cid:1) B → Q B (11.2)with Q ∈ A (and so in fact Q = 0 ), except when r + s is even and either e q = l + s + 2 (case A ) or e q + l = s (case B ), in which case Q is generatedby the sheaves O δ T,Tc (0 , − r + s ) .For J ⊆ R \ { } , ≤ | J | ≤ r − , we have when E p = R that β R ∗ L J = L J + X T p = R α ′ J,T,E,l δ T,T c = ˜ L J + X Tp = R,α ′ J,T,l,E> α ′ J,T,l,E δ T,T c . where we denote for simplicity α ′ J,T,E,l = | E q ∩ T q | − e q − r − l − | J | . (Using Notn. 1.14, if α J,T,E,l > we have α J,T,E,l = − α ′ J,T,E,l .)Let Q J = Coker( ˜ L J → β ∗ R L J ) . Then Q J is generated by the (push-forwards from δ T,T c to M p,q +1 of the) successive quotients Q iJ := (cid:0) ˜ L J + iδ T,T c (cid:1) | δ T,Tc = O T ( α ′ J,T,l,E − i ) ⊠ O T c ( − i ) , (11.3)for all < i ≤ α ′ J,T,l,E . Then Q iJ ∈ A if and only if i < r + s . We have i ≤ α ′ J,T,l,E = | E q ∩ T q | − e q − r − l − | J | ≤ | E q ∩ T q | − e q − l r ≤ r + s . (11.4)from Cor. 7.4, with equality exactly when J = ∅ , and either e q = l + s + 2 , T q ⊆ E q (case A ), or e q + l = s , E q ⊆ T q (case B ), in which case Q iJ = O (0 , − r + s , ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 65 with the component corresponding to markings from T . We concludearguing as in the proof of Lemma 11.6.We now prove (ii). It suffices to prove that Rβ R ∗ ( Q iJ ) ∨ = 0 for all quo-tients (11.3). Up to shift, ( Q iJ ) ∨ = (cid:0) ˜ L J + iδ T,T c (cid:1) ∨ ⊗ O ( δ T,T c ) | δ T,Tc = O T ( − α ′ J,T,l,E + i − ⊠ O T c ( i − As β R contracts the T -component, it suffices to prove that < α ′ J,T,l,E − i + 1 ≤ r + s − , for all < i ≤ | α J,T,l,E | . The inequality follows if α ′ J,T,l,E ≤ r + s − , whichfollows from (11.4) as r + s ≤ r + s − as r + s ≥ .We now consider the case of the map β R ′ . We have β ∗ R ′ L J = e q − r − l ψ + X k ∈ E q δ k − X j ∈ J δ j = ˜ L J + X T p = R,α
J,T,l,E > α J,T,l,E δ T,T c . As before, let Q J = Coker( ˜ L J → β ∗ R ′ L J ) . Then Q J is generated by the(push-forwards of the) successive quotients Q iJ := (cid:0) ˜ L J + iδ T,T c (cid:1) | δ T,Tc = O T ( − i ) ⊠ O T c ( α J,T,l,E − i ) , for all < i ≤ α J,T,l,E = e q − r − l − | E q ∩ T q | + | J | . Then Q iJ ∈ A if and only if i < r + s . As | J | ≤ r − , we have α J,T,l,E = e q − r − l − | E q ∩ T q | + | J | ≤ e q + r − l − | E q ∩ T q | − ≤ r + s − , (11.5)by Cor. 7.4. Therefore, Q ∈ A and this proves (iii). To prove (iv), it sufficesto prove that ( Q iJ ) ∨ = ( ˜ L J + iδ T,T c ) ∨ ⊗O ( δ T,T c ) | δ T,Tc = O T ( i − ⊠ O T c ( − ( α J,T,l,E − i ) − , pushes forward to by β R ′ . As β R ′ contracts the T c -component, it sufficesto prove that < α J,T,l,E − i + 1 ≤ r + s − , for all < i ≤ α J,T,l,E , or equivalently that α J,T,l,E ≤ r + s − , whichfollows from (11.5) as r + s − ≤ r + s − . (cid:3) There is a similar result for the bundles F l,E : Lemma 11.9.
For any ( l, E ) in group A or B , we have(1) (cid:0) Lβ ∗ R ′ F l,E (cid:1) B = F l,E , except when – E p ⊆ R ′ , e p + l = r − , e q = s + 1 (case A ); – R ′ ⊆ E p , e p = r + 1 + l , e q = s + 1 (case B ),in which case there is an exact triangle F l,E → (cid:0) Lβ ∗ R ′ F l,E (cid:1) B → Q B , where Q is generated by the sheaves O δ T,Tc ( u, − r + s , T p = R ′ , T q = E q , ≤ u < r + s . (2) Rβ R ′ ∗ F ∨ l,E = F ∨ l,E .Proof. Part (1) follows from Prop. 9.3 and Cor. 9.4. Part (2) is a particularcase of Cor. 9.5. (cid:3)
Theorem 11.10.
Assume p = 2 r ≥ , q = 2 s + 1 ≥ . In the notations of 11.2:(1) If E p = R , we haveR Hom M R ′ ( F l ′ ,E ′ , T l,E ) = 0 if e ′ q ≥ e q . (2) If E ′ p = R ′ , we haveR Hom M R ′ ( T l ′ ,E ′ , F l,E ) = 0 if e ′ q > e q . (3) If E p = E ′ p = R , we haveR Hom M R ′ ( T l ′ ,E ′ , T l,E ) = 0 , if e ′ q ≥ e q , E q = E ′ q or if E q = E ′ q and l > l ′ , andR Hom M R ′ ( T l,E , T l,E ) = C . (4) If E p = R , E ′ p = R ′ , we haveR Hom M R ′ ( T l ′ ,E ′ , T l,E ) = 0 if e ′ q ≥ e q . where all F l,E for pairs ( l, E ) in group A or B , and T l,E are for ( l, E ) in group A or B . For the remaining of this section we use the following:
Notation 11.11.
Set R = { , . . . , r } , R ′ = { r + 1 , . . . , p } and consider the loci Z R ֒ → M R ′ , Z R ′ ֒ → M R ′ , Y = Z R ∩ Z R ′ . We reduce Theorem 11.10 to a calculation of R Γ on loci Z R , Z R ′ and Y : Lemma 11.12.
In order to prove Theorem 11.10, it suffices to prove the following:(1) If ( l ′ , E ′ ) is in group A or B , and ( l, E ) is in group A or B , we haveR Γ (cid:0) Z R , ( F l ′ ,E ′ | Z R ) ∨ ⊗ T l,E ) = 0 if e ′ q ≥ e q , R := E p (2) If ( l, E ) is in group A or B and ( l ′ , E ′ ) is in group A or B , we haveR Γ (cid:0) Z R ′ , T ∨ l ′ ,E ′ ⊗ F l,E | Z R ′ ⊗ c ( N Z R ′ | M R ′ ) (cid:1) = 0 if e ′ q > e q , R ′ := E ′ p (3) If ( l, E ) , ( l ′ , E ′ ) are both in group A or B , then for R := E p = E ′ p , R Γ (cid:0) Z R , T ∨ l ′ ,E ′ ⊗ T l,E ⊗ ( X j ∈ J δ j (cid:1) ) = 0 , for all J ⊆ R \ { } if e ′ q ≥ e q , E q = E ′ q , or if E q = E ′ q , l > l ′ , andR Γ (cid:0) Z R , T ∨ l,E ⊗ T l,E ⊗ ( X j ∈ J δ j (cid:1) ) = 0 , for all ∅ 6 = J ⊆ R \ { } . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 67 (4) If ( l, E ) , ( l ′ , E ′ ) are both in group A or B , then for R := E p , R ′ := E ′ p , R Γ Y, − (cid:18) l + l ′ (cid:19) ψ u + 12 X j ∈ E ′ q ψ j − X j ∈ E q ψ j = 0 if e ′ q ≥ e q . Proof.
This is very similar to the proof of Lemma 10.4:(1) R Hom M R ′ ( F l ′ ,E ′ , T l,E ) = R Γ (cid:0) Z R , ( F l ′ ,E ′ | Z R ) ∨ ⊗ T l,E ) . (2) As i ! F l,E = i ∗ F l,E ⊗ c ( N Z R ′ | M R ′ ) , where i : Z R ′ ֒ → M R ′ is the canon-ical embedding, we have by [Huy06, Cor. 3.38] (up to a shift) that R Hom M R ′ ( T l ′ ,E ′ , F l,E ) = R Γ (cid:0) Z R ′ , T ∨ l ′ ,E ′ ⊗ F l,E | Z R ′ ⊗ c ( N Z R ′ | M R ′ ) (cid:1) . (3) Follows from tensoring the Koszul resolution of Z R with a line bun-dle L such that i R ∗ ( L | Z R ) = T l ′ ,E ′ and applying R Hom( − , T l,E ) .(4) As in the proof of Lemma 10.4, we have R Hom M R ′ ( T l ′ ,E ′ , T l,E ) = R Γ (cid:0) Y, ( T l ′ ,E ′ | Y ) ∨ ⊗ T l,E | Y ⊗ c ( N Y | Z R ) (cid:1) = R Γ (cid:0) Y, ( T l ′ ,E ′ | Y ) ∨ ⊗ T l,E | Y ⊗ ( X j ∈ R ′ \{ p } δ jp ) | Y (cid:1) == R Γ (cid:0) Y, − ( l + l ′ ψ u + 12 X j ∈ E ′ q ψ j − X j ∈ E q ψ j ) using an analogue of Lemma 10.3. (cid:3) We prove this vanishing by a windows calculation on Z R , Z R ′ and Y . Notation 11.13.
Since Z R and Z R ′ intersect only boundary δ T,T c with T p = R or R ′ , which get contracted in M R ′ , they are smooth GIT quotients by PGL of ( P ) r + q +1 . More precisely, let X = ( P ) r + q +1 = P u × ( P ) r × ( P ) q , corresponding to the partition { u } ⊔ R ′ ⊔ Q of the markings on Z R =M { u }∪ R ′ ∪ Q . Then Z R is a GIT quotient of X by PGL . Likewise, let X ′ = ( P ) r + q +1 = ( P ) r × P v × ( P ) q , corresponding to the partition R ⊔ { v } ⊔ Q of the markings on Z R ′ =M R ∪{ v }∪ Q . Then Z R ′ is a GIT quotient of X ′ by PGL . In addition, let X ′′ = ( P ) q +2 = P u × P v × ( P ) q , corresponding to the partition { u } ⊔ { v } ⊔ Q of the markings on M { u,v }∪ Q .Then Y is the GIT quotient of X ′′ by PGL .Vector bundles on Z R (resp., Z R ′ ) correspond to P GL -linearized vectorbundles on X = ( P ) r + q +1 (resp., X ′ ) as follows: F l,E | Z R = O X ( | E p ∩ R | , E p ∩ R ′ , E q ) ⊗ V l , T l,E = O X ( r + l, , E q ) , if E p = R,K Z R = O ( − , − , . . . , − . F l,E | Z R ′ = O X ′ ( E p ∩ R, | E p ∩ R ′ | , E q ) ⊗ V l ,δ j | Z R = O (2 , ,
0) ( j ∈ R \ { } ) , δ pj | Z R ′ = O (0 , ,
0) ( j ∈ R ′ \ { p } ) . Remark 11.14. (The devil’s trick reloaded.) Since for any j ∈ R ′ (resp., j ∈ R ), we have ψ u + ψ j = 0 in Z R ( ψ v + ψ j = 0 in Z R ′ ) as δ uj = ∅ (resp., δ vj = ∅ ). Therefore, as in Rmk. 10.7, the line bundles D := O ( r, R ′ , on X = ( P ) r + q +1 , D := O ( R, r, on X ′ = ( P ) r + q +1 , descend to trivial line bundles on Z R and Z R ′ . Likewise, the line bundle D := O (1 , , on X ′′ descends to the trivial line bundle on Y . Proof of Theorem 11.10.
We prove the vanishings in Lemma 10.4. As before,we will first prove the
PGL -invariant vanishing holds on on X (cases (1)and (3)), on X ′ (case (2)) and on X ′′ (case (4)) (resp., X ′ or X ′′ ) after ten-soring with the devil line bundle D N , N ≫ , since it’s this tensor productthat will satisfy conditions of Theorem 10.6. Later on we check the weightcondition in Thm. 10.6.For (1), assuming the weight condition, R Γ (cid:0) Z R , ( F l ′ ,E ′ | Z R ) ∨ ⊗ T l,E ⊗ D N ) = R Γ (cid:0) X, O ( r + l − | E ′ p ∩ R | + N r, − E ′ p ∩ R ′ + N R ′ , − E ′ q + E q ) ⊗ V l ′ (cid:1) PGL . which is clearly if E ′ q * E q . Since here we assume e ′ q ≥ e q , we have that E ′ q ⊆ E q if and only if E q = E ′ q . Assume E q = E ′ q . In this case, we need toconsider V r + l −| E ′ p ∩ R | + Nr ⊗ V N − y ⊗ . . . ⊗ V N − y r ⊗ V l ′ , where y i = 1 if the corresponding index in E ′ p ∩ R ′ and otherwise. Weclaim that the PGL -invariant part is by the Clebsch-Gordan formula. Itsuffices to check that r + l − | E ′ p ∩ R | + N r > N r − | E ′ p ∩ R ′ | + l ′ . This follows if l ′ + | E ′ p ∩ R | − | E ′ p ∩ R ′ | < r. Since ( l ′ , E ′ ) is in group A or B , this follows by Cor. 7.6.For (2), assuming the weight condition, R Γ (cid:0) Z R ′ , T ∨ l ′ ,E ′ ⊗ F l,E | Z R ′ ⊗ c ( N Z R ′ | M R ′ ) ⊗ D N (cid:1) == R Γ (cid:0) X ′ , O ( E p ∩ R + N R, | E p ∩ R ′ |− r − l ′ +2( r − N r, E q − E ′ q ) ⊗ V l (cid:1) PGL . This is if E ′ q * E q , which follows from e ′ q > e q .For (3), assuming the weight condition, R Γ (cid:0) Z R , T ∨ l ′ ,E ′ ⊗ T l,E ⊗ ( X j ∈ J δ j (cid:1) ⊗ D N ) == R Γ (cid:0) X, O (2 | J | + l − l ′ + N r, N R ′ , E q − E ′ q ) (cid:1) PGL , for all J ⊆ R \ { } . This is if E ′ q * E q . Assume now E q = E ′ q . In this case,we need to consider V | J | + l − l ′ + Nr ⊗ V ⊗ rN , ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 69 whose
PGL -invariant part is when l > l ′ or when l = l ′ , | J | > by theClebsch-Gordan formula since in these cases we have | J | + l − l ′ + N r > N r.
For (4), assuming the weight condition, R Γ (cid:0) Y, ( T l ′ ,E ′ | Y ) ∨ ⊗ T l,E | Y ⊗ c ( N Y | Z R ) ⊗ D N (cid:1) = R Γ (cid:0) X ′′ , O ( l + l ′ + 2 + N, N, E q − E ′ q ) (cid:1) PGL . This is if E ′ q * E q . If E q = E ′ q we have to consider V l + l ′ +2+ N ⊗ V N , whose PGL -invariant part is by the Clebsch-Gordan formula as l + l ′ +2 + N > N .We now check that for each stratum, each of the cases (1)-(4) fall underthe assumption on weights of Thm. 10.6. The unstable loci in X (resp., X ′ )corresponding to the loci Z R (resp., Z R ′ ) in M R ′ have the following form:– The locus K I , for I ⊆ Q , | I | ≥ s + 1 (resp., | I | ≥ s + 2 ), where u (resp., v )and the indices in I come together. In this case, η = 2 | I | . – The locus K J,I , for J ⊆ R ′ (resp., J ⊆ R ), I ⊆ Q , J = ∅ , | I | ≥ , where u (resp., v ) and the indices in J and I come together. In this case, η = 2 | I | + 2 | J | . – The locus L I , for I ⊆ Q , | I | ≥ s + 2 (resp., | I | ≥ s + 1 ), where the indicesin R ′ and I (resp., R and I ) come together. In this case, η = 2 | I | + 2 r − . The devil line bundle O ( r, R ′ , on X has the property thatweight λ O ( r, R ′ , | z { u }∪ J ∪ I = r + | J | − | R ′ \ J | = 2 | J | > while its weight for the other strata is (similarly for O ( R, r, on X ′ ).Therefore, the condition in Thm. 10.6 for the stratum K J,I can be achievedby tensoring with a high enough multiple of this line bundle. We only needto consider the remaining strata.Consider first cases (1) and (3) involving Z R ⊆ M R ′ . For case (1) we needto verify that the weights of the following vector bundle on X are > − η : O ( r + l − | E ′ p ∩ R | , − E ′ p ∩ R ′ , E q − E ′ q ) ⊗ V l ′ . For the stratum K I ( | I | ≥ s + 1 ), we need to prove that r + l − l ′ − | E ′ p ∩ R | + | E ′ p ∩ R ′ | + | E q ∩ I |− | E ′ q ∩ I |− | E q ∩ I c | + | E ′ q ∩ I c | > − | I | . By (7.7) for the pair ( l ′ , E ′ p ) , it suffices to prove that l + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | ≥ − | I | . Since the left hand side is greater than −| I c | − | I | = − q − − s − and | I | ≥ s + 1 , the result follows. For the stratum L I ( | I | ≥ s + 2 ), we need toprove that − r − l − l ′ + | E ′ p ∩ R |−| E ′ p ∩ R ′ | + | E q ∩ I |−| E ′ q ∩ I |−| E q ∩ I c | + | E ′ q ∩ I c | > − | I |− r +2 . By (7.7) for the pair ( l ′ , E ′ p ) , it suffices to prove that − l + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | ≥ − | I | + 2 . As −| E ′ q ∩ I | + | E ′ q ∩ I c | ≥ −| I | , it suffices to prove that − l + | E q ∩ I | − | E q ∩ I c | ≥ −| I | + 2 . This follows from Lemma 7.8 since | I | ≥ s + 2 .For case (3) we need to verify that the weights of the following line bun-dle on X are > − η : O (2 | J | + l − l ′ , , E q − E ′ q ) , for all ≤ | J | ≤ r − . For the stratum K I , we need to prove that l − l ′ + 2 | J | + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | > − | I | . Note that it suffices to prove that − l ′ − | E ′ q ∩ I | + | E ′ q ∩ I c | − | I c | > − | I | . This follows from Lemma 7.8 since | I | ≥ s + 1 for the locus K I in X . Forthe stratum L I ( | I | ≥ s + 2 ), we need to prove that − l + l ′ − | J | + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | > − | I | − r + 2 , or equivalently, − l + l ′ + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | > − | I | . As the left hand side is greater or equal than − l + | E q ∩ I | − | E q ∩ I c | − | I | ,it suffices to prove − l + | E q ∩ I | − | E q ∩ I c | > −| I | . This follows from Lemma 7.8 since | I | ≥ s + 2 .For case (2), involving Z R ′ ⊆ M R ′ , the relevant vector bundle on X ′ is O ( E p ∩ R, | E p ∩ R ′ | + r − l ′ − , E q − E ′ q ) ⊗ V l . For the stratum K I ( | I | ≥ s + 2 ), we need to prove that r − l − l ′ − | E p ∩ R ′ |−| E p ∩ R | + | E q ∩ I |−| E ′ q ∩ I |−| E q ∩ I c | + | E ′ q ∩ I c | > − | I | . By (7.7) for the pair ( l, E p ) , it suffices to prove that − l ′ − | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | > − | I | . As the left hand side is greater or equal than − l ′ − −| E ′ q ∩ I | + | E ′ q ∩ I c |−| I c | ,it suffices to prove that − l ′ − | E ′ q ∩ I | + | E ′ q ∩ I c | ≥ | I c | − | I | + 2 . This follows from Lemma 7.8 since | I | ≥ s + 2 .For the stratum L I ( | I | ≥ s + 1 ), we need to prove that − r + l ′ − l +2+ | E p ∩ R |−| E p ∩ R ′ | + | E q ∩ I |−| E ′ q ∩ I |−| E q ∩ I c | + | E ′ q ∩ I c | > − | I |− r +2 . By (7.7) for the pair ( l, E p ) , it suffices to prove that l ′ + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | ≥ − | I | . As the left hand side is greater or equal than −| I | − | I c | = − q − , theinequality follows since | I | ≥ s + 1 . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 71
We now consider Case (4). Up to symmetry, the unstable loci in X ′ = ( P ) q +3 = P u × P v × ( P ) q +1 corresponding to Y = Z R ∩ Z R ′ ⊆ M R ′ have the following form:(1) The locus K ′ I , for I ⊆ Q , | I | ≥ s + 1 , where u and the indices in I come together. In this case, η = 2 | I | .(2) The locus K ′′ I , for I ⊆ Q , | I | ≥ s + 2 , where v and the indices in I come together. In this case, η = 2 | I | .(3) The locus K ′′′ I , for I ⊆ Q , | I | ≥ , where u , v and the indices in I come together. In this case, η = 2 | I | + 2 .As before, the devil line bundle O (1 , , on X ′′ has the property thatweight λ O (1 , , | z { u,v }∪ I = 2 > . while its weight for the other strata is . Hence, as before, we only have toconsider the strata K ′ I , K ′′ I .We verify that the weights of O ( l + l ′ + 2 , , E q − E ′ q ) are > − | I | . For thestratum K ′ I ( | I | ≥ s + 1 ), we need to prove that l + l ′ + 2 + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | > − | I | . This is clear, as the left hand side is greater or equal than − | I | − | I c | =2 − ( q + 1) = − s and | I | ≥ s + 1 .For the stratum K ′′ I ( | I | ≥ s + 2 ), we need to prove that − l − l ′ + | E q ∩ I | − | E ′ q ∩ I | − | E q ∩ I c | + | E ′ q ∩ I c | > − | I | + 2 . We apply Lemma 7.8 to the pairs ( l, E q ) (for the set I c ) and ( l ′ , E ′ q ) (for theset I ). Recall that e ′ q ≥ e q . If in case A and e ′ q ≤ s (resp., case B and e ′ q ≤ s + 1 ), then Lemma 7.8 implies that − l + | E q ∩ I | − | E q ∩ I c | ≥ − s + 1 , ( resp., ≥ − s ) , − l ′ − | E ′ q ∩ I | + | E ′ q ∩ I c | ≥ − s + 1 , ( resp., ≥ − s ) , and the inequality follows by summing up, as − s > − | I | + 2 .If in case A and e ′ q > s ≥ e q (resp., case B and e ′ q > s + 1 ≥ e q ), thenLemma 7.8 implies that − l + | E q ∩ I | − | E q ∩ I c | ≥ − s + 1 , ( resp., ≥ − s ) , − l ′ − | E ′ q ∩ I | + | E ′ q ∩ I c | ≥ − | I | + s + 2 , ( resp., ≥ − | I | + s + 3) , and the inequality follows again by summing up.If in case A and e q > s (resp., case B and e q > s + 1 ), then Lemma 7.8implies that − l + | E q ∩ I | − | E q ∩ I c | ≥ − | I c | + s + 2 ( resp., ≥ | I c | + s + 3) , − l ′ − | E ′ q ∩ I | + | E ′ q ∩ I c | ≥ − | I | + s + 2 ( resp., ≥ | I c | + s + 3) , and the inequality follows again by summing up, since − | I | − | I c | + 2 s + 4 > − | I | + 2 , (as | I c | < s + 1 , since | I | ≥ s + 2 ). (cid:3)
12. P
ROOF OF T HM . 1.8 ( FULLNESS : P ODD )Recall that Cor. 4.7 and Thm. 4.8 follow from Thm. 1.8, as proved inSection 4. In this section we prove that exceptional collection of Thm. 1.8,call it C , is full. By Prop. 4.1, it suffices to prove the following lemma: Lemma 12.1.
Let p = 2 r + 1 . Every vector bundle F l,E (with l + e is even) on M p is generated by vector bundles in the collection C .Proof. We consider the following function, which we call the score : S ( l, E ) = l + min( e, p − e ) . We will prove an equivalent dual statement that every vector bundle F ∨ l,E on M p is generated by bundles F ∨ l,E in the range S ( l, E ) ≤ r − . We argueby induction on the score and for fixed score, by induction on l . Clearly, thestatement holds when S ( l, E ) ≤ r − by definition of the collection C ∨ .Let a ≥ r and assume that all the bundles F ∨ l,E with S ( l, E ) < a aregenerated by C ∨ . Let F ∨ l,E be a bundle such that S ( l, E ) = a . We considertwo cases: e = | E | ≤ r and e = | E | ≥ r + 1 .Assume e ≤ r . Let I ⊆ { , . . . , p } with I ∩ E = ∅ and | I | = r + 1 . Weconsider the Koszul complex for the diagonal U = ∆ I ∪{ x } ⊆ ( P ) p × P x , ← O U ← O ← M i ∈ I O ( − e i − e x ) ← M i,k ∈ I O ( − e i − e k − e x ) ← . . . ← O ( − X i ∈ I e i − ( r + 1) e x ) ← We tensor this resolution with O ( − X j ∈ E e j + le x ) and take derived push-forwards of its terms via the projection map π :( P ) p +1 → ( P ) p , which we then restrict to the semistable locus in ( P ) p .Since π ( U ) is in the unstable locus, we obtain the following objects: F ∨ l,E M i ∈ I F ∨ l − ,E ∪{ i } . . . M J ⊆ I, | J | = j F ∨ l − j,E ∪ J . . . M J ⊆ I, | J | = l F ∨ ,E ∪ J M J ⊆ I, | J | = l +2 F ∨ ,E ∪ J [ − . . . M F ∨ j − l − ,E ∪ J [ − . . . F ∨ r − l − ,E ∪ I [ − , where the terms F ∨ l − j,E ∪ J appear as R π ∗ as long as ≤ j ≤ l , the termappears when j = l + 1 (which happens if l ≤ r ; if l > r then the last termthat appears is F ∨ l − r − ,E ∪ I ), while the terms F ∨ j − l − ,E ∪ J appear as R π ∗ inthe range l + 2 ≤ j ≤ r + 1 by applying the Grothendieck duality.We claim that the first object ( F ∨ l,E ) is generated by the remaining objects,which are all generated by C ∨ by induction. It will follow that F ∨ l,E is gener-ated by C ∨ . For the first claim, we can apply the standard spectral sequence[Huy06, 2.66] to get an exact sequence: ← F ∨ l,E ← M i ∈ I F ∨ l − ,E ∪{ i } ← . . . ← M J ⊆ I, | J | = j F ∨ l − j,E ∪ J ← . . . ← M J ⊆ I, | J | = l F ∨ ,E ∪ J ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 73 ← M J ⊆ I, | J | = l +2 F ∨ ,E ∪ J ← . . . ← M J ⊆ I, | J | = j F ∨ j − l − ,E ∪ J ← . . . ← M F ∨ r − l − ,E ∪ I ← . The term in the middle disappears as it is bypassed by the differential d .But in fact we do not need to be so specific and can just use the followingsimple lemma which we are going to use repeatedly in the remainder ofthe paper: Lemma 12.2.
Let F : D ( A ) → T be an exact functor of triangulated categoriesand let → A → . . . → A n → be an exact sequence in A . Then any of theobjects F ( A ) , . . . , F ( A n ) belong to the triangulated subcategory of T generatedby the remaining objects.Proof. It suffices to prove that each A i is generated by the remaining onesin D ( A ) . This is clear by considering the short exact sequences arising from → A → . . . → A n → . (cid:3) We now prove that the remaining terms are generated by C ∨ by induc-tion. Consider the two types of terms:(a) The terms F ∨ ˜ l, ˜ E = F ∨ l − j,E ∪ J have scores S (˜ l, ˜ E ) = ( l − j )+min { e + j, p − e − j } ≤ ( l − j )+( e + j ) = l + e = S ( l, E ) = a, and we are done by induction, since ˜ l ≤ l , with equality if and only if (˜ l, ˜ E ) = ( l, E ) .(b) The terms F ∨ ˜ l, ˜ E = F ∨ j − l − ,E ∪ J ( l + 2 ≤ j ≤ r + 1 ) have scores S (˜ l, ˜ E ) = ( j − l −
2) + min { e + j, p − e − j } ≤ ( j − l −
2) + ( p − e − j ) == p − e − l − p − a − since S ( l, E ) = l + e = a . But p − a − < a since since a ≥ r . It follows thatthese terms are generated by C ∨ by induction.Assume e ≥ r + 1 . Let I ⊆ E , | I | = r + 1 and let E ′ = E \ I . As before,we tensor the above Koszul resolution of ∆ I ∪{ x } ⊆ ( P ) p × P with O − X k ∈ E ′ e k + ( r − − l ) e x ! , take derived push-forwards via the projection map π and apply Lemma 12.2: F ∨ r − − l,E ′ M i ∈ I F ∨ r − − l,E ′ ∪{ i } . . . M J ⊆ I, | J | = j F ∨ r − − l − j,E ′ ∪ J . . .. . . M J ⊆ I, | J | = r − − l F ∨ ,E ′ ∪ J M J ⊆ I, | J | = r +1 − l F ∨ ,E ′ ∪ J . . .. . . M J ⊆ I, | J | = j F ∨ l + j − r − ,E ′ ∪ J . . . F ∨ l,E , where the terms F ∨ r − − l − j,E ′ ∪ J appear as R π ∗ for ≤ j < r − l , the termappears when j = r − l , and the terms F ∨ l + j − r − ,E ′ ∪ J appear as R π ∗ for r + 1 − l ≤ j < r + 1 .Consider the two types of terms and we compare theirs scores with a = S ( l, E ) = l + p − e. (a) The terms F ∨ ˜ l, ˜ E = F ∨ r − − l − j,E ′ ∪ J have scores S (˜ l, ˜ E ) = ( r − − l − j )+min { e ′ + j, p − e ′ − j } ≤ ( r − − l − j )+( e ′ + j ) = e − l − , (where e ′ = | E ′ | = e − r − ). But since we assume a = l + p − e ≥ r , wehave l + p − e > e − l − and we are done by induction.(b) The terms F ∨ ˜ l, ˜ E = F ∨ l + j − r − ,E ′ ∪ J have scores S (˜ l, ˜ E ) = ( l + j − r −
1) + min { e ′ + j, p − e ′ − j } ≤ ( l + j − r −
1) + p − e ′ − j == l + p − e = S ( l, E ) , and we are done by induction, since ˜ l = l + j − r − ≤ l (as j ≤ r + 1 ) withequality if and only if j = r + 1 , i.e., J = I (that is (˜ l, ˜ E ) = ( l, E ) ). (cid:3)
13. P
ROOF OF T HM . 1.11 ( FULLNESS : P EVEN , Q ODD ) Theorem 13.1.
Assume p = 2 r ≥ , q = 2 s + 1 ≥ . The vector bundles { F l,E } for all l ≥ , E ⊆ P ∪ Q , e = | E | , with l + e even, are generated by any of thetwo exceptional collections in D b (M p,q ) given by Thm. 1.11. In particular, theseexceptional collections are full.Proof. We prove equivalently that the dual of the collection in Thm. 1.11(groups A and , resp. B and ), call it C , generates all the duals F ∨ l,E . Wewill do an induction based on the “score” of a pair ( l, E ) , defined as: S ( l, E ) = l + min { e p , p − e p } + min { e q , q − e q } . For equal scores, we do an induction on l .If S ( l, E ) ≤ r − , then l + min { e p + 1 , p + 1 − e p } ≤ r − , i.e., the pair ( l, E ) belongs to both groups A and B . This ensures the base case of theinduction.We now assume that we have a pair ( l, E ) (not in group A , resp. B )such that for any ( l ′ , E ′ ) with S ( l ′ , E ′ ) < S ( l, E ) , or if S ( l, E ) = S ( l ′ , E ′ ) and l ′ < l , then the bundle F ∨ l ′ ,E ′ is generated by C ∨ .We will use three constructions based on the Koszul complex. We willcall a set I ⊆ P ∪ Q stable if i p ≤ r − or if i p = r , i q ≤ s . Otherwise, I is called unstable (the usual notions of GIT stability giving M p,q ). Given asubset I ⊆ P ∪ Q , we consider the Koszul complex for U = ∆ I ∪{ x } ⊆ ( P ) p + q × P x ← O U ← O ← M k ∈ I O ( − e k − e x ) ← M k,j ∈ I O ( − e k − e j − e x ) ← . . . ← O ( − X k ∈ I e k − ie x ) , i = | I | . Koszul Game 1: Assume I is an unstable set and E ⊆ P ∪ Q is such that E ∩ I = ∅ . Assume e + l even. Tensoring the Koszul complex above with O ( − P k ∈ E e k + le x ) and pushing forward gives objects: F ∨ l,E . . . M J ⊆ I, | J | = j F ∨ l − j,E ∪ J . . . M J ⊆ I, | J | = l F ∨ ,E ∪ J ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 75 M J ⊆ I, | J | = l +2 F ∨ ,E ∪ J [ − . . . M J ⊆ I, | J | = j F ∨ j − l − ,E ∪ J [ − . . . F ∨ i − l − ,E ∪ I [ − , where the second part of the sequence appears only when i ≥ l + 2 , and if i ≤ l , the first part stops at F ∨ l − i,E ∪ I .Koszul Game 2: Assume I is an unstable set and E ′ ⊆ P ∪ Q is such that E ′ ∩ I = ∅ . We will apply this for E = E ′ ∪ I . In particular, we assume e + l even. Tensoring the Koszul complex above with O ( − E ′ , i − − l ) , pushingforward gives objects: F ∨ i − − l,E ′ . . . M J ⊆ I, | J | = j F ∨ i − − l − j,E ′ ∪ J . . . M J ⊆ I, | J | = i − − l F ∨ ,E ′ ∪ J M J ⊆ I, | J | = i − l F ∨ ,E ′ ∪ J [ − . . . M J ⊆ I, | J | = j F ∨ j − i + l,E ′ ∪ J [ − . . . F ∨ l,E ′ ∪ I [ − , where the first group of objects appears only when i ≥ l + 2 , while if i ≤ l ,the second part starts at F ∨ l − i,E ′ .Koszul Game 3: Assume I = E p = R , E ⊆ P ∪ Q , where R ⊆ P , | R | = r , e + l even. We tensor the Koszul complex above with O − X j ∈ E q e j + ( r − − l ) e x and push forward. Unlike in the previous games, the torsion object is in thesemistable locus. Note that identifying U ∼ = ∆ R = ( P ) u × ( P ) r + q +1 , where u is the marking corresponding to indices in R , we have O − X j ∈ E q e j + ( r − − l ) e x | U = O (cid:0) − X j ∈ E q e j + ( r − − l ) e u (cid:1) , which descends to Z R ⊆ M p,q as the line bundle − r − − l ψ u + 12 X j ∈ E q ψ j = − r − − l + e q ψ u − X j ∈ E q δ ju . Using Lemma 13.2, terms have the following derived push-forwards: T ∨ l,E [ r − F ∨ r − l − ,E q . . . M J ⊆ R, | J | = j F ∨ r − − l − j,E q ∪ J . . . M J ⊆ R, | J | = r − − l F ∨ ,E q ∪ J M J ⊆ R, | J | = r − l F ∨ ,E q ∪ J [ − . . . M J ⊆ R, | J | = j F ∨ l − r + j,E q ∪ J [ − . . .. . . F ∨ l,E q ∪ R [ −
1] = F ∨ l,E [ − , where the first part of the sequence of bundles F ∨ r − − l − j,E q ∪ J appears onlywhen r ≥ l + 2 , while if r ≤ l , then the second part of the sequence starts at F ∨ l − r,E q . Note that the score of every F ∨ ˜ l, ˜ E with (˜ l, ˜ E ) = ( l, E ) in the Koszulgame 3 is strictly lower than the score of F ∨ l,E :(a) For (˜ l, ˜ E ) = ( r − − l − j, E q ∪ J ) ( ≤ j ≤ r − − l ), the score ˜ S is ˜ S = ( r − − l − j ) + j + min { e q , q − e q } = r − − l + min { e q , q − e q } , and clearly, we have ˜ S < S := S ( l, E ) = l + r + min { e q , q − e q } .(b) For (˜ l, ˜ E ) = ( l − r + j, E q ∪ J ) ( ≤ j ≤ r ), the score ˜ S is ˜ S = ( l − r + j ) + j + min { e q , q − e q } = l − r + 2 j + min { e q , q − e q } , and clearly, we have again ˜ S < S if J = R , i.e., ˜ E = E .In particular, if our starting pair ( l, E ) is in group , since T l,E is in C ,the bundle F ∨ l,E is generated by C ∨ by induction, since all other terms inthe above Koszul resolution have lower score. There are four cases in theinduction argument.Case 1: e p < r . Since we assume that ( l, E ) is not in group A (resp. B ),we have l + e p ≥ r (resp. l + e p ≥ r − ). We play the Koszul game 1 witha set I = I p , i p = r + 1 (“minimal” unstable I disjoint from E ). We verifythat the score of every F ∨ ˜ l, ˜ E in the complex is less or equal than S := S ( l, E ) = l + e p + min { e q , q − e q } . (a) Let (˜ l, ˜ E ) = ( l − j, E ∪ J ) ( < j ≤ l ). Note that l = 0 cannot occur forthis type of F ∨ ˜ l, ˜ E . The score ˜ S is ˜ S = ( l − j ) + min { e p + j, p − e p − j } + min { e q , q − e q } == l + min { e p , p − e p − j } + min { e q , q − e q } ≤ l + e p + min { e q , q − e q } = S. As j > , we are done by induction on l .(b) For (˜ l, ˜ E ) = ( j − l − , E ∪ J ) ( l + 2 ≤ j ≤ i ), the score ˜ S is ˜ S = ( j − l −
2) + min { e p + j, p − e p − j } + min { e q , q − e q } . Since e p + j ≥ e p + l + 2 ≥ r + 1 , we have min { e p + j, p − e p − j } = p − ( e p + j ) and therefore, in case A , using l + e p ≥ r , we have ˜ S = p − e p − l − { e q , q − e q } < S. In case B , we still have to consider the situation l + e p = r − . But then ˜ l + ( p − ˜ e p ) = ( j − l −
2) + ( p − e p − j ) = r − , and therefore F ˜ l, ˜ E is in B .Case 2: e p = r , e q ≤ s . If ( E, l ) is in group , as remarked above, we canplay the Koszul game 3 to reduce the score. Hence, we may assume ( E, l ) is not in group , i.e., l + e q ≥ s .We now play the Koszul game 1 with a set I with i p = r , i q = s + 1 (“minimal” unstable I ). We verify that the the score of every F ∨ ˜ l, ˜ E in thecomplex is less or equal than S := S ( l, E ) = l + r + e q . a) Let (˜ l, ˜ E ) = ( l − j, E ∪ J ) ( < j ≤ l ). Note again that l = 0 cannotoccur for this type of F ∨ ˜ l, ˜ E . The score ˜ S is ˜ S = ( l − j ) + ( r − j p ) + min { e q + j q , q − e q − j q } ≤≤ ( l − j ) + ( r − j p ) + ( e q + j q ) = l + r + e q − j p ≤ S, As j > , we are done by induction on l . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 77 (b) For (˜ l, ˜ E ) = ( j − l − , E ∪ J ) ( l + 2 ≤ j ≤ i ), the score ˜ S is ˜ S = ( j − l −
2) + ( r − j p ) + min { e q + j q , q − e q − j q } ≤≤ ( j − l −
2) + ( r − j p ) + ( q − e q − j q ) < l + r + e q = S since we assume l + e q ≥ s .Case 3: e p ≥ r + 1 . We may assume ( E, l ) is not in group A (resp., B ).Then l + ( p + 1 − e p ) ≥ r (resp., l + ( p − e p ) ≥ r ), or equivalently, e p ≤ l + r + 1 (resp., e p ≤ l + r ). We now play the Koszul game 2 with the set I = I p ⊆ E p , i p = r + 1 ( i q = 0 ) and with E ′ p = E p \ I p , E ′ q = E q ( e ′ p = e p − ( r + 1) , e ′ q = e q – so “minimal” unstable I ).We verify that the score of every F ∨ ˜ l, ˜ E in the complex is less than or equalto S := S ( l, E ) = l + ( p − e p ) + min { e q , q − e q } . a) For (˜ l, ˜ E ) = ( i − l − − j, E ′ ∪ J ) ( ≤ j ≤ i − l − ), the score ˜ S is ˜ S = ( i − l − − j ) + min { e ′ p + j, p − e ′ p − j } + min { e q , q − e q } ≤≤ ( i − l − − j ) + ( e ′ p + j ) + min { e q , q − e q } = e p − l − { e q , q − e q } . In case B , since e p ≤ l + r , we have ˜ S < S . In case A , since e p ≤ l + r + 1 ,we have ˜ S < S , unless e p = l + r + 1 , when ˜ S = S . However, if e p = l + r + 1 ,we can move on as F ∨ ˜ l, ˜ E is in group A : ˜ l + min { e ′ p + j, p + 1 − e ′ p − j } ≤ ( i − l − − j ) + ( e ′ p + j ) = r − . b) (˜ l, ˜ E ) = ( j − i + l, E ′ ∪ J ) ( max { i − l, } ≤ j < i ), the score ˜ S is ˜ S = ( j − i + l ) + min { e ′ p + j, p − e ′ p − j } + min { e q , q − e q } ≤≤ ( j − i + l ) + ( p − e ′ p − j ) + min { e q , q − e q } = l + p − e p + min { e q , q − e q } = S (since e ′ p = e p − i ). If ˜ S = S , we are done by induction on l since j − i + l < l .Note again that in this case l > .Case 4: e p = r , e q ≥ s + 1 . If ( E, l ) is in group , as remarked above, we canplay the Koszul game 3 to reduce the score. Hence, we may assume ( E, l ) is not in group , i.e., l + ( q − e q ) ≥ s , or equivalently, e q ≤ l + s + 1 .We now play the Koszul game 2 with the set I with I p = E p , I q = E q , i.e., E ′ = ∅ (“maximal” unstable I with I ⊆ E ). In particular, we have i = e .We verify that the score of every F ∨ ˜ l, ˜ E in the complex is less or equal than S := S ( l, E ) = l + r + q − e q . a) For (˜ l, ˜ E ) = ( i − l − − j, J ) ( ≤ j ≤ i − l − ), the score ˜ S is ˜ S = ( i − l − − j ) + j p + min { j q , q − j q } ≤ ( i − l − − j q ) + j q = . = i − l − r + e q − l − < l + r + q − e q = S, since by assumption e q ≤ l + s + 1 .b) (˜ l, ˜ E ) = ( j − i + l, J ) ( max { i − l, } ≤ j < i = e ), the score ˜ S is ˜ S = ( j − i + l ) + j p + min { j q , q − j q } ≤ j q − i + l + 2 j p + ( q − j q ) == q − i + l + 2 j p = q − e q − r + l + 2 j p ≤ l + r + q − e q , with equality if and only if j q ≥ s + 1 , j p = r . Since j < i , we are done byinduction on l since j − i + l < l . Note again that in this case l > . (cid:3) Lemma 13.2.
Let M be one of M p,q or M p,q +1 . Let R ⊆ P , | R | = r and let i : Z R → M denotes the inclusion map. If ( l, E ) is a pair with E p = R , thederived dual of T l,E = i ∗ (cid:0) e q − r − l ψ u + P j ∈ E q δ ju (cid:1) is given by T ∨ l,E = i ∗ − e q − r + l ψ u − X j ∈ E q δ ju [1 − r ] . Proof. If L is a line bundle such that L | Z R = T l,E then T ∨ l,E = L ∨ ⊗ O ∨ Z R = L ∨ ⊗ det N Z R ⊗ O Z R [ − c ] [Huy06, 3.40], where c = r − is the codimension of Z R and (det N Z R ) | Z R = X j ∈ R \{ } ( δ j ) | Z R = − ( r − ψ u is the determinant of the normal bundle. (cid:3) Corollary 13.3. D b (M p,q ) is generated by the bundles F l,E with ( l, E ) in group A (resp., B ) and group .Proof. This follows immediately from the proof of Thm. 13.1. (cid:3)
We finish this section by analyzing fullness on U p,q , the universal familyover M p,q . Note that this is also a Hassett space as well as a GIT quotient. Notation 13.4.
We let ˜ Q = Q ∪ { z } , where z is an extra index.On U p,q we have bundles F l,E by Def. 3.3. We use the same groups A , B , A and B as in Thm. 1.16. Theorem 13.5. D b ( U p,q ) is generated by the bundles F l,E with ( l, E ) in one ofthe following groups: (i) A (resp., B ), (ii) A with z / ∈ E , (iii) B with z ∈ E .Proof. This is a consequence of Orlov’s Theorem for U p,q → M p,q (a P bun-dle) and is similar to the proof of Lemma 9.8. If z / ∈ E , the range A on U p,q is exactly the range of group on M p,q and the range A (resp., B ) on U p,q corresponds to A (resp., B ) on M p,q . Hence, by Cor. 13.3 the objectswith z / ∈ E generate D b (M p,q ) . If z ∈ E , the range B on U p,q is exactly therange of group on M p,q and the range A (resp., B ) on U p,q correspondsto B (resp., A ) on M p,q . Hence, the corresponding duals { F ∨ l,E c } generate D b (M p,q ) . As F l,E = F ∨ l,E c ⊗ F , Σ , it follows that the objects with z ∈ E gen-erate D b (M p,q ) ⊗ F , Σ . As in the proof of Lemma 9.8, the result now followsby Orlov’s theorem. (cid:3)
14. P
ROOF OF T HM . 1.16 ( FULLNESS : P EVEN , ˜ Q EVEN ) Notation 14.1.
Throughout this section p = 2 r , q = 2 s + 1 , ˜ Q = Q ∪ { z } . Asin Notn. 9.7, let β = β z : M p,q +1 → U p,q be the morphism that contracts the T component of any boundary δ T,T c if z ∈ T . Recall, when q + 1 = 0 , thisis the map M p → M p − , which lowers the weight of a heavy index (whichwe call z ). Let S ′ be the score of Notation 7.1. ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 79
Notation 14.2.
We let A to be the subcategory in D b (M p,q +1 ) generated bythe torsion sheaves O δ T,Tc ( − a, − b ) of Notn. 1.14. Let C be the collection ofTheorem 1.16 (exceptional by §
11) which consists of sheaves in A , vectorbundles F l,E in group A (resp., B ) combined with the complexes ˜ T l,E in group B . We let C ′ , resp., C ′ F , be the collection obtained from C byreplacing complexes ˜ T l,E in C with the torsion sheaves T l,E , resp., the vectorbundles F l,E . Note that these collections are not, in general, exceptional. Theorem 14.3.
Let l ≥ , E ⊆ P ∪ ˜ Q , S ′ ( l, E ) ≤ r + s . Then the bundle F l,E isgenerated by C ′ F . More precisely, F l,E can be generated by A together with { F l ′ ,E ′ } with ( l ′ , E ′ ) in group A (resp., B ) or B and such that S ′ ( l ′ , E ′ ) ≤ S ′ ( l, E ) . To prove Thm. 14.3, recall the notations of Section 8: let α : W → M p,q +1 be the universal family and f : W → M p,q +2 = M P, ˜ Q ∪{ y } the birationalmap that contracts the T c component of boundary divisors δ T ∪{ y } ,T c on W (where y is the new marking on W ). When q +1 = 0 , we have M p,q +2 = M p, and f : W → M p − , is the map considered in Section 12. Remark 14.4.
For l ≥ , E ⊆ ˜ Q , the pair ( l, E ) can be considered on both of M p,q +1 and M p,q +2 . Such a pair ( l, E ) is in group A (resp., B ) on M p,q +1 ifand only if ( l, E ) is in group A (resp., B ) on M p,q +2 . Furthermore, ( l, E ) is in group B on M p,q +1 if and only if ( l, E ) is in group on M p,q +2 . Thescore S ′ ( l, E ) on M p,q +1 relates to S ( l, E ) = l + min { e p , p − e p } + min { e q , q + 2 − e q } , (the score on M p,q +2 as introduced in the previous section) by S ′ ( l, E ) = ( S ( l, E ) if e q ≤ s + 1 S ( l, E ) − e q ≥ s + 2 . Lemma 14.5.
For all ( l, E ) on M p,q +2 in groups A , B , , we have S ( l, E ) ≤ r + s, with equality if and only if we are in one of the following cases:(1A) e p = r − − l , e q = s + 1 or s + 2 ;(1B) e p = r + 1 + l , e q = s + 1 or s + 2 ;(2) e p = r , e q = s − l or l + s + 3 . Lemma 14.6.
Let l ≥ and E ⊆ P ∪ ˜ Q . Then on M p,q +1 the bundle F l,E andthe complex Rα ∗ f ∗ F l,E are related by quotients of the form Q = O δ T,Tc ( − v, u ) , ≤ u ≤ m ( T c ) − , < v ≤ m ( T c ) − , (14.1) m ( T c ) := max (cid:18) , | E ∩ T c | − e − l (cid:19) ≤ S ′ ( l, E )2 . In particular, F l,E and R α ∗ f ∗ F l,E are related by quotients in A if the score S ′ ( l, E ) ≤ r + s .Proof. By Prop. 8.4, the bundles F l,E and f ∗ F l,E on W are related by quo-tients − jH ⊠ O ( u ) , < j ≤ m ( T c ) , ≤ u ≤ m ( T c ) − , supported on δ T ∪{ y } ,T c = Bl P r + s × P r + s − . By Lemma 5.5, Rπ ∗ ( − jH ) = 0 if j = 1 , while if j ≥ , it is generated by O ( − v ) with ≤ v ≤ j − .It follows that Rα ∗ (cid:0) − jH ⊠ O ( u ) (cid:1) is generated by quotients (14.1). The restfollows from Lemma 7.2. (cid:3) Lemma 14.7.
Let l ≥ , E ⊆ P ∪ ˜ Q . Assume S ′ ( l, E ) ≤ r + s . On M p,q +2 ,the bundle F l,E can be generated by { F l ′ ,E ′ } for ( l ′ , E ′ ) belonging to group A (resp., B ) and on M p,q +2 such that y / ∈ E ′ , and such that S ′ ( l ′ , E ′ ) ≤ S ′ ( l, E ) .Proof. We follow in the footsteps of the proof of Thm. 13.1 and do an in-duction on the score S ( l, E ) on M p,q +2 and for equal scores induction on l .(Recall, S ( l, E ) = S ′ ( l, E ) if e q ≤ s +1 and S ( l, E ) = S ′ ( l, E )+1 if e q ≥ s +2 ).We only point out the main arguments for each case.Case 1: e p < r . As in Case 1 of Thm. 13.1, we may assume ( E, l ) is notin group A (resp., B ). Using the Koszul game 1, we generate F l,E with F ˜ l, ˜ E with ˜ E of the form E ∪ J , with J a set of heavy indices (in particular, y / ∈ ˜ E ) and such that either S (˜ l, ˜ E ) < S ( l, E ) or S (˜ l, ˜ E ) = S ( l, E ) , ˜ l < l (towhich one has to add, when working with group B , the possibility that S (˜ l, ˜ E ) = S ( l, E ) and (˜ l, ˜ E ) is in group B ). Since ˜ E q = E q , we have thateither that S ′ (˜ l, ˜ E ) = S (˜ l, ˜ E ) , S ′ ( l, E ) = S ( l, E ) , or S ′ (˜ l, ˜ E ) = S (˜ l, ˜ E ) − , S ′ ( l, E ) = S ( l, E ) − . In particular, S ′ (˜ l, ˜ E ) ≤ S ′ ( l, E ) ≤ r + s , i.e., thehypotheses in the Lemma are satisfied for (˜ l, ˜ E ) .Case 2: e p = r , e q ≤ s + 1 . By our assumption that S ′ ( l, E ) ≤ r + s , we can-not have that e q = s + 1 . Hence, e q ≤ s . As in Case 2 of Thm. 13.1, we mayassume ( E, l ) is not in group on M p,q +2 . Using the Koszul game 1 for a set I disjoint from E with | I p | = r and | I q | = s + 2 , and in addition with y / ∈ I q (possible since e q ≤ s ), we generate F l,E with F ˜ l, ˜ E with ˜ E of the form E ∪ J ,with J ⊆ I (in particular, y / ∈ ˜ E ) and such that either S (˜ l, ˜ E ) < S ( l, E ) or S (˜ l, ˜ E ) = S ( l, E ) , ˜ l < l . Furthermore, we have S ′ (˜ l, ˜ E ) ≤ S (˜ l, ˜ E ) ≤ S ( l, E ) = S ′ ( l, E ) ≤ r + s, (as S ′ ( l, E ) = S ( l, E ) because e q ≤ s ).Case 3: e p ≥ r + 1 . As in Case 3 of Thm. 13.1, we may assume ( E, l ) is notin group A (resp., B ). Using the Koszul game 2, we generate F l,E with F ˜ l, ˜ E with ˜ E of the form E \ J ′ , with J ′ a set of heavy indices (in particular, y / ∈ ˜ E ) and such that either S (˜ l, ˜ E ) < S ( l, E ) , or S (˜ l, ˜ E ) = S ( l, E ) , ˜ l < l , or,when working with group A , S (˜ l, ˜ E ) = S ( l, E ) and (˜ l, ˜ E ) is in group A .Since ˜ E q = E q , we have again (as in Case 1) that S ′ (˜ l, ˜ E ) ≤ S ′ ( l, E ) ≤ r + s ,i.e., the hypotheses in the Lemma are satisfied for (˜ l, ˜ E ) .Case 4: e p = r , e q ≥ s + 2 . As in Case 4 of Thm. 13.1, we may assume ( E, l ) is not in group on M p,q +2 . Using the Koszul game 2, we generate F l,E with F ˜ l, ˜ E with ˜ E of the form J ⊆ E , (in particular, y / ∈ ˜ E ) and such thateither S (˜ l, ˜ E ) < S ( l, E ) or S (˜ l, ˜ E ) = S ( l, E ) , ˜ l < l . Note, if S (˜ l, ˜ E ) < S ( l, E ) then S ′ (˜ l, ˜ E ) ≤ S (˜ l, ˜ E ) < S ( l, E ) = S ′ ( l, E ) + 1 , (since e q ≥ s + 2 ). In particular, it follows that S ′ (˜ l, ˜ E ) ≤ S ′ ( l, E ) ≤ r + s . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 81
Assume now S (˜ l, ˜ E ) = S ( l, E ) , ˜ l < l . We still need to prove that in thiscase S ′ (˜ l, ˜ E ) ≤ S ′ ( l, E ) ≤ r + s . As in Case 4 of Thm. 13.1, the only way tohave S (˜ l, ˜ E ) = S ( l, E ) is when we are in case b) (˜ l, ˜ E ) = ( j − e + l, J ) ( j < e ,so ˜ l < l ) and j q ≥ s +2 . Since | ˜ E q | = j q ≥ s +2 , we have S ′ (˜ l, ˜ E ) = S (˜ l, ˜ E ) − .Hence, S ′ (˜ l, ˜ E ) = S (˜ l, ˜ E ) − S ( l, E ) − S ′ ( l, E ) ≤ r + s . (cid:3) Proof of Thm. 14.3.
Let l ≥ , E ⊆ P ∪ ˜ Q . Assume S ′ ( l, E ) ≤ r + s . ByLemma 14.7 and Rmk. 14.4, the bundle F l,E on M p,q +2 can be generated by { F l ′ ,E ′ } for ( l ′ , E ′ ) belonging to group A (resp., B ) and , with y / ∈ E ′ ,and such that S ′ ( l ′ , E ′ ) ≤ S ′ ( l, E ) . It follows that the object Rα ∗ f ∗ F l,E on M p,q +1 can be generated by Rα ∗ f ∗ { F l ′ ,E ′ } . It follows from Lemma 14.6 that F l,E can be generated by C ′ F . (cid:3) Definition 14.8.
Let l ∈ Z (positive or negative). Recall that F l,E on thestack M n , and therefore also on all GIT quotients without strictly semistablepoints, including U p,q , were defined in Proposition 3.10. It was proved therethat F l,E = 0 for l = − and F l,E ≃ F − l − ,E [ − for l ≤ − . We would liketo define analogous objects on M p,q +1 . Let α : W → M p,q +1 be the universalfamily. For any l ∈ Z (positive or negative) and E ⊆ P ∪ ˜ Q , let N ′ l,E = ω e − l α ( E ) , F ′ l,E = Rα ∗ ( N ′ l,E ) . Proposition 14.9.
Let l ≥ − , E ⊆ P ∪ ˜ Q . The bundle F l,E as defined in Def. 5.7and the object F ′ l,E are related by quotients of the form O δ ( − v, u ) , v > , u ≥ , u, v ≤ S ′ ( l, E )2 − l − . In particular, if l ≥ , S ′ ( l, E ) ≤ r + s , F ′ l,E , F l,E are related by quotients in A .Proof. By Def. 5.7, F l,E = Rα ∗ ( N l,E ) . We compare F ′ l,E with F l,E by com-paring on W the line bundles N ′ l,E and N l,E : N ′ l,E = N l,E + X T ⊔ T c = P ∪ ˜ Q, | E ∩ T | < e − l (cid:18) e − l − | E ∩ T | (cid:19) δ T ∪{ y } . Since for l ≥ − and e − l is even, we cannot have both | E ∩ T | < e − l , | E ∩ T c | < e − l . Hence, for every partition T ⊔ T c = P ∪ ˜ Q at most oneof δ T ∪{ y } ,T c , δ T c ∪{ y } ,T appears on the right hand side of the equality Thequotients on W relating N ′ l,E and N l,E have the form Q = (cid:0) N ′ l,E + ( − α T + i ) δ (cid:1) | δ = (cid:0) − iH (cid:1) ⊠ O ( α T − i ) ,δ = δ T ∪{ y } ,T c , < i ≤ α T := e − l − | E ∩ T | . Since by Lemma 5.5, Rπ ∗ ( − iH ) is if i = 1 and generated by O ( − v ) , with v = 1 , . . . , i − if i ≥ , it follows that the quotients relating F ′ l,E and F l,E ,if non-zero, have the form O T ( − v ) ⊠ O T c ( u ) , ≤ u ≤ α T − , < v ≤ α T − , and the result follows by Lemma 7.2. (cid:3) Corollary 14.10.
Consider a fixed boundary δ = δ T ,T c and let J ⊆ T , j = | J | (0 ≤ j ≤ r + s ) . Then F j − ,T c ∪ J and F ′ j − ,T c ∪ J are related by objects in A if j > , while for j = 0 , F ′− ,T c and O δ ( − r + s , are related by objects in A .Proof. When ( l, E ) = ( j − , T c ∪ J ) , | J | = j , we have e − l = r + s + 1 and inthe notations of the proof of Prop. 14.9, if T ⊔ T c = P ∪ ˜ Q then α T − r + s − | E ∩ T | ≤ r + s with equality if and only if E = T c ∪ J ⊆ T c , which happens if and only if j = 0 , i.e., J = ∅ , and T = T , i.e., one copy of O δ ( − r + s , appears. (cid:3) Corollary 14.11.
Consider a partition T ⊔ T c = P ∪ ˜ Q . The sheaf O δ (cid:18) − r + s , (cid:19) , δ = δ T,T c ⊆ M p,q +1 is generated by C ′ F .Proof. We prove that O δ ( − r + s , is generated by the bundles { F l,E } on M p,q +1 with score S ′ ( l, E ) = r + s and objects in A . In particular, by Thm.14.3, it is generated by the collection C ′ F .Consider the Koszul resolution of T j ∈ T δ iy = ∅ in W : ← O ← ⊕ i ∈ T O ( − δ iy ) ← . . . ← O ( − X i ∈ T δ iy ) ← . Dualizing and tensoring with ω r + s +1 α (cid:0) P i ∈ T c δ iy (cid:1) and using the notation N ′ l,E := ω e − l α ( E ) (see Def. 14.8,), we have the following long exact sequenceof line bundles on W : → N ′− ,T c → ⊕ i ∈ T N ′ ,T c ∪{ i } → . . . → ⊕ J ⊆ T, | J | = j N ′ j − ,T c ∪ J → . . .. . . → N ′ r + s,T c ∪ T → . By applying Rα ∗ (cid:0) − (cid:1) , we obtain the following objects on M p,q +1 : F ′− ,T c ⊕ i ∈ T F ′ ,T c ∪{ i } . . . ⊕ J ⊆ T, | J | = j F ′ j − ,T c ∪ J . . . F ′ r + s,T c ∪ T . All the vector bundles F j − ,T c ∪ J have score r + s and the statement followsby Thm. 14.3 and Cor. 14.10. (cid:3) Proposition 14.12.
Let l ∈ Z (positive or negative), E ⊆ P ∪ ˜ Q , z ∈ ˜ Q . Then F ′ l,E and β ∗ z F l,E are related by quotients of the form O δ ( − v, u ) , u ≥ , < v ≤ max (cid:26) S ′ ( l, E )2 − l − , S ′ ( l, E )2 (cid:27) . Proof.
Let N = N ′ l,E . Recall that F ′ l,E = Rα ∗ ( N ) . We have that β ∗ z F l,E = Rα ∗ ( N ) , where N = N + X z ∈ T (cid:0) | E ∩ T | − e − l (cid:1) δ T ∪{ x } . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 83
We let on WN := N + X z ∈ T, | E ∩ T |− e − l > (cid:0) | E ∩ T | − e − l (cid:1) δ T ∪{ y } == N + X z ∈ T, e − l −| E ∩ T | > (cid:0) e − l − | E ∩ T | (cid:1) δ T ∪{ y } . The line bundles N and N are related by quotients (cid:0) N + iδ (cid:1) | δ = (cid:0) | E ∩ T | − e − l − i (cid:1) H ⊠ O T c ( − i ) ,δ = δ T ∪{ y } ,T c ( z ∈ T ) , < i ≤ | E ∩ T | − e − l . It follows by Lemma 5.5 that F ′ l,E = Rα ∗ ( N ) and Rα ∗ ( N ) are related byquotients of the form Q = O T ( u ) ⊠ O T c ( − i ) , u ≥ , < i ≤ | E ∩ T | − e − l . Recall that | E ∩ T | − e − l ≤ S ′ ( l,E )2 .The line bundles N and N are related by quotients (cid:0) N + iδ (cid:1) | δ = (cid:18) N − (cid:18) e − l − | E ∩ T | (cid:19) δ + iδ (cid:19) | δ == ( − iH ) ⊠ O T c (cid:18) e − l − | E ∩ T | − i (cid:19) ,δ = δ T ∪{ y } ,T c ( z ∈ T ) , < i ≤ e − l − | E ∩ T | . Since by Lemma 5.5 that Rπ ∗ ( − iH ) is either or generated by O ( − v ) with v = 1 , . . . , i − , it follows that F ′ l,E = Rα ∗ ( N ) and Rα ∗ ( N ) are related byquotients of the form O T ( − v ) ⊠ O T c ( u ) , u ≥ , < v ≤ e − l − | E ∩ T | − . Since e − l − | E ∩ T | ≤ S ′ ( l,E )2 − l , the result follows. (cid:3) We will also need the following immediate corollary of Prop. 9.3(i) andLemma 7.2:
Corollary 14.13.
Let l ≥ , E ⊆ P ∪ ˜ Q , ˜ Q = Q ∪ { z } . Assume S ′ ( l, E ) ≤ r + s. Then F l,E and β ∗ z F l,E are related by quotients in A and sheaves of type O δ (cid:0) , − r + s (cid:1) .Furthermore, they are related by quotients in A if any of the following holds:(i) r + s is odd, or more generally, if S ′ ( l, E ) < r + s ;(ii) r + s is even, S ′ ( l, E ) = r + s , e q ≥ s + 1 and z / ∈ E ; Cor. 14.13, Thm. 14.3 and Corollary 14.11 imply the following:
Corollary 14.14.
Let l ≥ , E ⊆ P ∪ ˜ Q , S ′ ( l, E ) ≤ r + s . For any z ∈ ˜ Q , thebundle β ∗ z F l,E on M p,q +1 is generated by C ′ F . Lemma 14.15.
Assume G ∈ D b (M p,q +1 ) is such that R Hom(
C, G ) = 0 , for any C ∈ C ′ F . Then G = 0 .Proof. By Cor. 14.14, for any z ∈ ˜ Q , we have R Hom( β ∗ z F l,E , G ) = 0 if S ′ ( l, E ) ≤ r + s . It follows that R Hom( F l,E , Rβ z ∗ G ) = 0 whenever S ′ ( l, E ) ≤ r + s. Since by Thm. 13.5 the collection { F l,E } with S ′ ( l, E ) ≤ r + s containsas a subcollection a full collection of D b ( U p,q ) , it follows that Rβ z ∗ G = 0 for every z ∈ ˜ Q . In particular, G has support on the boundary. Since theboundary is a disconnected union of components, G is isomorphic to a di-rect sum of complexes supported on irreducible boundary divisors. Theresult follows from Lemma 14.16. (cid:3) Lemma 14.16.
Let X be a smooth variety and let X → X be a contraction of adivisor E ≃ P l × P l with normal bundle O ( − , − . Let f ± : X → X ± be twosmall resolutions of X (contracting E to P l in two directions). Let G ∈ D b ( X ) and suppose Rf −∗ ( G ) = Rf + ∗ ( G ) = R Hom( O E ( − a, − b ) , G ) = 0 for every a, b = 1 , . . . , l . Then G = 0 .Proof. Applying Orlov’s blow-up theorem to f + , we see that G belongsto a subcategory generated by exceptional collection O E ( − a, − b ) with forevery a = 1 , . . . , l + 1 and b = 1 , . . . , l , which has a s.o.d. hB , Ai with A generated by sheaves with a = 1 , . . . , l and B by sheaves with a = l + 1 .Since R Hom( A , G ) = 0 , we conclude that G ∈ B . Now we prove that G = 0 by proving, by induction on i , that G belongs to a subcategory B i generatedby exceptional collection {O E ( − ( l + 1) , − l ) , . . . , O E ( − ( l + 1) , − i ) } for every i > (and therefore G = 0 ). Applying Rf −∗ to the triangle X → Y → G → X [1] with Y ≃ O E ( − ( l + 1) , − i ) ⊗ K • and X ∈ B i +1 (and K • a complex ofvector spaces with trivial differentials) gives Rf −∗ X ≃ Rf −∗ O E ( − ( l + 1) , − i ) ⊗ K • = O P l ( − i )[ − l ] ⊗ K • . On the other hand, Rf −∗ X belongs to a triangulated subcategory gener-ated by O P l ( − l )[ − l ] , . . . , O P l ( − ( i + 1))[ − l ] . By shrinking X we can assumethat the restriction map Pic X − → Pic P l is surjective. Tensoring with anappropriate line bundle shows that O P l ⊗ K • belongs to a triangulated sub-category generated by O P l ( − l + i ) , . . . , O P l ( − . Computing R Γ shows that K • = 0 . It follows that Rf −∗ X = 0 and therefore G ≃ X ∈ B i +1 . (cid:3) In the remaining part of this section we prove the following Theorem(whose consequence is then the fullness part of Thm. 1.16).
Theorem 14.17.
The collection C ′ F is generated by the exceptional collection C ofThm. 1.16.Proof. By Lemma 11.6, T l,E and ˜ T l,E differ by quotients in A . Hence, itsuffices to prove that C ′ F is generated by C ′ , i.e., that we can “replace” T l,E with F l,E when ( l, E ) is in group B . ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 85
Claim 14.18.
Let l ≥ , E ⊆ P ∪ ˜ Q , ˜ Q = Q ∪ { z } . Assume that S ′ ( l, E ) ≤ r + s. Then the bundle F l,E is generated by the collection C ′ . We prove the statement by induction on the score S ′ ( l, E ) and for equalscore, by induction on l . If ( l, E ) is in group A (resp. B ), there is noth-ing to prove, so we assume this is not the case. Assume we are in the casewhen ( l, E ) is not in group B . By Thm. 14.3, the bundle F l,E is generatedby { F l ′ ,E ′ } with pairs ( l ′ , E ′ ) either in group A (resp. B ) or B and in ad-dition with S ′ ( l ′ , E ′ ) ≤ S ′ ( l, E ) , i.e., we are reduced to prove the statementfor ( l, E ) in group B . Assume ( l, E ) is in group B . By Prop. 14.9, it’ssuffices to prove that F ′ l,E is generated by C ′ . Claim 14.19. F ′ l,E and T l,E ( E p = R ), are related by the following objects: { F ′ ˜ l, ˜ E } for (˜ l, ˜ E ) = ( j + l − r, E q ∪ J ) , J ⊆ R, j = | J | , ≤ j < r. Proof.
Let W → M p,q +1 be as usual the universal family ( y new markingon W ). Consider the Koszul complex for U = ∆ R ∪{ y } ⊆ W : ← O U ← O ← M k ∈ R O ( − δ ky ) ← . . . ← M J ⊆ R,j = | J | O ( − X k ∈ J δ ky ) ← . . . ←← O ( − X k ∈ R δ ky ) ← We claim that after tensoring with ω − eq + r − l α ( − E q ) and applying Rα ∗ (cid:0) − (cid:1) ,we obtain objects ( T l,E ) ∨ ( F ′ l − r,E q ) ∨ M k ∈ R ( F ′ l − r,E q ∪{ k } ) ∨ . . .. . . M J ⊆ R,j = | J | ( F ′ l + j − r,E q ∪ J ) ∨ . . . ( F ′ l,E ) ∨ , i.e., that we have Rα ∗ (cid:0) ω − eq + r − l α ( − E q ) | U (cid:1) = ( T l,E ) ∨ , (14.2) Rα ∗ (cid:0) ω − eq + r − l α ( − E q − J ) (cid:1) = ( F ′ j + l − r,E q ∪ J ) ∨ for all J ⊆ R, (14.3)(where the last equality is up to a shift).To see (14.2), consider the universal family W R → Z R over Z R with thesection σ u corresponding to the indices in R . Then Rα ∗ (cid:0) ω − eq + r − l α ( − E q ) | U (cid:1) = σ ∗ u (cid:0) ω − eq + r − l α ( − E q ) (cid:1) == (cid:0) − e q + r − l (cid:1) ψ u − X k ∈ E q δ ku , and this equals ( T l,E ) ∨ by Claim 13.2 (up to a shift).To see (14.3), note that for any ( l ′ , E ′ ) , Grothendieck-Verdier duality gives (cid:0) F ′ l ′ ,E ′ (cid:1) ∨ = Rα ∗ (cid:0) ω − e ′− l ′ α ( − E ′ ) (cid:1) , (up to a shift). This is because for a morphism α : X → Y of relativedimension , with X , Y smooth varieties, if N is a line bundle on X , then Rα ∗ (cid:0) N ∨ ⊗ ω α [1] (cid:1) = (cid:0) Rα ∗ N (cid:1) ∨ . It follows that ( F ′ l,E ) ∨ and ( T l,E ) ∨ are related by { ( F ′ ˜ l, ˜ E ) ∨ } . By dualizing,the Claim follows. (cid:3) Claim 14.20.
The objects F ′ ˜ l, ˜ E for (˜ l, ˜ E ) = ( j + l − r, E q ∪ J ) , J ⊆ R, j = | J | , ≤ j < r, (where note that ˜ l = j + l − r could be negative) are generated by C ′ .Proof. The score S ′ (˜ l, ˜ E ) = ( j + l − r ) + j + min { e q , q + 1 − e q } = < l + r + min { e q , q + 1 − e q } = S ′ ( l, E ) . If ˜ l ≥ , then by Prop. 14.9 we have that F ′ ˜ l, ˜ E and F ˜ l, ˜ E are related byquotients in A , since S ′ (˜ l, ˜ E ) < S ′ ( l, E ) ≤ r + s .If ˜ l ≤ − , then we claim that F ′ ˜ l, ˜ E and β ∗ z F ˜ l, ˜ E are related by quotientsin A . When ˜ l ≤ − , we have max ( S ′ (˜ l, ˜ E )2 − ˜ l − , S ′ (˜ l, ˜ E )2 ) = S ′ (˜ l, ˜ E )2 − ˜ l − − l − r + min { e q , q + 1 − e q } ≤ r + s − , (since l ≥ ) and the result follows from Prop. 14.12.It follows that it suffices to prove that the following are generated by C ′ :(a) F ˜ l, ˜ E , when ˜ l ≥ and (˜ l, ˜ E ) = ( l, E ) ;(b) β ∗ z F ˜ l, ˜ E , when ˜ l ≤ − .In case (a), we proved that S ′ (˜ l, ˜ E ) < S ′ ( l, E ) . Hence, we are done byinduction. In case (b), since F − ,E = 0 on U p,q for any E , we may assume ˜ l ≤ − . By Prop. 3.10, on U p,q we have F ˜ l, ˜ E ∼ = F − ˜ l − , ˜ E [ − . We have S ′ ( − ˜ l − , ˜ E ) = ( − ˜ l −
2) + j + min { e q , q + 1 − e q } == − l − r + min { e q , q + 1 − e q } = S ′ ( l, E ) − l − ≤ r + s − , since l ≥ . Therefore, by Cor. 14.13(i), β ∗ z F − ˜ l − , ˜ E and F − ˜ l − , ˜ E are relatedby quotients in A . Since S ′ ( − ˜ l − , ˜ E ) = S ′ ( l, E ) − l − ≤ r + s − , byinduction, F − ˜ l − , ˜ E (and hence, β ∗ z F − ˜ l − , ˜ E ) are generated by C ′ . (cid:3) This finishes the proof of the Theorem. (cid:3)
Proof of fullness in Thm. 1.16.
It suffices to prove that if G ∈ D b (M p,q +1 ) is such that R Hom(
C, G ) = 0 for every C in the exceptional collection ofThm. 1.16 then G = 0 . By Thm. 14.17, we have R Hom(
C, G ) = 0 for every C ∈ C ′ F . The result follows from Lem. 14.15. (cid:3) ERIVED CATEGORY OF MODULI OF POINTED CURVES - II 87 R EFERENCES [Be˘ı78] A. A. Be˘ılinson,
Coherent sheaves on P n and problems in linear algebra , Funktsional.Anal. i Prilozhen. (1978), no. 3, 68–69.[BFK19] M. Ballard, D. Favero, and L. Katzarkov, Variation of geometric invariant theoryquotients and derived categories , J. Reine Angew. Math. (2019), 235–303.[BO02] A. Bondal and D. Orlov,
Derived categories of coherent sheaves , Proceedings of theInternational Congress of Mathematicians, Vol. II (Beijing, 2002), 2002, pp. 47–56.[BM13] J. Bergstr ¨om and S. Minabe,
On the cohomology of moduli spaces of (weighted) stablerational curves , Math. Z. (2013), no. 3-4, 1095–1108.[Cey09] ¨O. Ceyhan,
Chow groups of the moduli spaces of weighted pointed stable curves of genuszero , Adv. Math. (2009), no. 6, 1964–1978.[CFGP19] M. Chan, C. Faber, S. Galatius, and S. Payne,
The S n -equivariant top weight Eulercharacteristic of M g,n (2019), available at arXiv:1904.06367 .[CT17] A.-M. Castravet and J. Tevelev, Derived category of moduli of pointed curves - I (2017), available at arXiv:1708.06340 .[Dol03] I. Dolgachev,
Lectures on invariant theory (2003), xvi+220.[DH98] I. Dolgachev and Y. Hu,
Variation of Geometric Invariant Theory Quotients , Inst.Hautes ´Etudes Sci. Publ. Math. (1998), 5–56.[Dub98] B. Dubrovin, Geometry and analytic theory of Frobenius manifolds , Proceedingsof the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998,pp. 315–326.[Ela09] A. D. Elagin,
Semi-orthogonal decompositions for derived categories of equivariant co-herent sheaves , Izv. Ross. Akad. Nauk Ser. Mat. (2009), no. 5, 37–66.[FM94] W. Fulton and R. MacPherson, A compactification of configuration spaces , Ann. ofMath. (2) (1994), no. 1, 183–225.[FP13] C. Faber and R. Pandharipande,
Tautological and non-tautological cohomology of themoduli space of curves , Handbook of moduli. Vol. I, 2013, pp. 293–330.[Get95] E. Getzler,
Operads and moduli spaces of genus Riemann surfaces , The moduli spaceof curves (Texel Island, 1994), 1995, pp. 199–230.[GK98] E. Getzler and M. M. Kapranov,
Modular operads , Compositio Math. (1998),no. 1, 65–126, DOI 10.1023/A:1000245600345.[GKM02] A. Gibney, S. Keel, and I. Morrison,
Towards the ample cone of M g,n , J. Amer. Math.Soc. (2002), no. 2, 273–294.[Has03] B. Hassett, Moduli spaces of weighted pointed stable curves , Adv. Math. (2003),no. 2, 316–352.[HK98] J.-C. Hausmann and A. Knutson,
The cohomology ring of polygon spaces , Vol. 48,1998.[HL15] D. Halpern-Leistner,
The derived category of a GIT quotient , J. Amer. Math. Soc. (2015), no. 3, 871–912.[Huy06] D. Huybrechts, Fourier-Mukai transforms in algebraic geometry , Oxford Mathemat-ical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006.[Kap93] M. M. Kapranov,
Veronese curves and Grothendieck-Knudsen moduli space M ,n , J.Algebraic Geom. (1993), no. 2, 239–262.[Kee92] S. Keel, Intersection theory of moduli space of stable n -pointed curves of genus zero ,Trans. Amer. Math. Soc. (1992), no. 2, 545–574.[KT09] S. Keel and J. Tevelev, Equations for M ,n , International Journal of Mathematics (2009), no. 09, 1159-1184.[KL14] A. Kuznetsov and V. A. Lunts, Categorical Resolutions of Irrational Singularities ,International Mathematics Research Notices (2014), no. 13, 4536-4625.[KS20] A. Kuznetsov and M. Smirnov,
On residual categories for Grassmannians , Proc.Lond. Math. Soc. (3) (2020), no. 5, 617–641, DOI 10.1112/plms.12294.[KSV95] T. Kimura, J. Stasheff, and A. A. Voronov,
On operad structures of moduli spaces andstring theory , Comm. Math. Phys. (1995), no. 1, 1–25.[Kuz08] A. Kuznetsov,
Lefschetz decompositions and categorical resolutions of singularities ,Selecta Math. (N.S.) (2008), no. 4, 661–696. [Kuz14] , Semiorthogonal decompositions in algebraic geometry , Proceedings of the In-ternational Congress of Mathematicians—Seoul 2014. Vol. II, 2014, pp. 635–660.[Man95] Yu. I. Manin,
Generating functions in algebraic geometry and sums over trees , Themoduli space of curves (Texel Island, 1994), 1995, pp. 401–417.[MS13] Yu. I. Manin and M. N. Smirnov,
On the derived category of M ,n , Izv. Ross. Akad.Nauk Ser. Mat. (2013), no. 3, 93–108.[MS14] , Towards motivic quantum cohomology of M ,S , Proc. Edinb. Math. Soc. (2) (2014), no. 1, 201–230.[Orl92] D. Orlov, Projective bundles, monoidal transformations, and derived categories of co-herent sheaves , Izv. Ross. Akad. Nauk Ser. Mat. (1992), no. 4, 852–862.[Smi13] M. N. Smirnov, Gromov-Witten correspondences, derived categories, andFrobenius manifolds , Ph.D. thesis, Univ. of Bonn (2013), available at http://hss.ulb.uni-bonn.de/2013/3125/3125.pdf .[SS17] S. V. Sam and A. Snowden,
Gr¨obner methods for representations of combinatorialcategories , J. Amer. Math. Soc. (2017), no. 1, 159–203.[Tel00] C. Teleman, The quantization conjecture revisited , Ann. of Math. (2) (2000), 1–43.[Tho87] R. W. Thomason,
Equivariant resolution, linearization, and Hilbert’s fourteenth prob-lem over arbitrary base schemes , Adv. in Math. (1987), no. 1, 16–34.[Tos18] P. Tosteson, Stability in the homology of Deligne-Mumford compactifications , preprint(2018), available at https://arxiv.org/abs/1801.03894 .U NIVERSIT ´ E P ARIS -S ACLAY , UVSQ, CNRS, L
ABORATOIRE DE M ATH ´ EMATIQUES DE V ERSAILLES , 78000, V
ERSAILLES , F
RANCE
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U
NIVERSITY OF M ASSACHUSETTS A MHERST , 710 N
ORTH P LEASANT S TREET , A
MHERST , MA 01003
AND L ABORATORY OF A LGEBRAIC G EOMETRY AND ITS A PPLICATIONS , HSE, M
OSCOW , R
USSIA
E-mail address ::