TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 7, Pages 2695–2729 S 0002-9947(99)02230-8 Article electronically published on February 5, 1999

THE QUANTUM COHOMOLOGY RING OF FLAG VARIETIES

IONUT¸ CIOCAN-FONTANINE

Abstract. We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263–277). We also give a geometric proof of the quantum Monk formula.

Introduction The quantum cohomology ring of a projective manifold X is a deformation of the usual cohomology ring which has been introduced by physicists (see [W]). The quantum multiplication of cohomology classes on X is obtained by adding to the usual cup-product the so-called instanton corrections. These can be in turn inter- preted as intersection numbers on a sequence of moduli spaces of (holomorphic) maps P1 X. To make this interpretation rigorous according to mathematical standards,→ one encounters severe problems, mainly because these moduli spaces are not compact and they may have the wrong dimension. Recently, substantial efforts have been made to put the theory on firm math- ematical footing, and the first construction of quantum cohomology rings, using methods of symplectic topology, has been given by Ruan and Tian [RT] for a large class of manifolds (semi-positive symplectic manifolds), which includes projective Fano and Calabi-Yau varieties. Meanwhile, an algebro-geometric axiomatic approach has been started by Kont- sevich and Manin in [KM], and the program outlined there has just been completed, using Kontsevich’s notion of stable maps, in the works of Behrend, Behrend and Manin and Li and Tian (see [Beh], [BM], [LT2]). An account for some of the results along these lines can be found in [FP]. Another algebro-geometric approach for the case of homogeneous spaces can be found in [LT1]. There are actually two different deformations of the cohomology ring of X that can be considered; they are called the big quantum cohomology ring and the small quantum cohomology ring in [FP]. The big quantum ring is an algebra over a power series ring and contains most of the interesting data from the point of view of enumerative geometry. The “parameter” space for the deformation is the vector space H∗(X; C) itself. The small quantum ring is obtained by deforming the coho- mology classes over H1,1(X) only. For Fano varieties this gives an algebra over a

Received by the editors April 2, 1997. 1991 Mathematics Subject Classification. Primary 14M15; Secondary 14N10. Key words and phrases. Quantum cohomology, flag varieties, hyperquot schemes, degeneracy loci.

c 1999 American Mathematical Society

2695

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polynomial ring, while for Calabi-Yau threefolds, which is the case of main interest for physicists, the two rings are essentially the same. In this paper we shall be concerned only with the small quantum cohomology ring, and from now on we drop the adjective “small” when referring to it. Computations of quantum cohomology rings have been worked out for several examples: Batyrev [Bat] for toric varieties, Crauder-Miranda [CM] for some Del Pezzo surfaces, Siebert-Tian [ST] for Grassmannians (see also [Be2]), and Qin-Ruan [QR] for some projective bundles over Pn. In [GK], based on the conjectures of conformal field theory, Givental and Kim conjectured a presentation of the quantum cohomology ring of complete flag vari- eties. Their computation relied on the assumed existence of an equivariant version of quantum cohomology, satisfying some functorial properties. Specifically, let F (n) be the variety of complete flags in Cn. The classical cohomology ring with Z-coefficients has a presentation given by Z[x1,...,xn]/I, where x1,...,xn are the Chern classes of some natural line bundles on F (n)and th I=(R1(n),R2(n),...,Rn(n)), with Ri(n)thei elementary symmetric function in x1,...,xn. By a general result of Siebert and Tian [ST], the quantum cohomology ring of F (n) will then have a presentation given by

(*) Z[x1,...,xn,q1,...,qn 1]/Iq, −

where the new variables qi are the deformation parameter, and Iq is an ideal gener- ated by deformations of the generators of I. The Givental-Kim conjecture identifies the generators of Iq. The present paper is the result of a project whose goals were on the one hand to give an algebro-geometric proof of this conjecture, by building on the ideas in [Be2] (see also [Be1] and [BDW]), and on the other hand to develop an analogue in the case of complete flag varieties of the “quantum Schubert calculus,” introduced by A. Bertram [Be2] for Grassmannians. In analogy with the classical case, this consists of solving the

Quantum Giambelli problem. Find polynomial representatives for the cosets corre- sponding to the Schubert classes in the presentation (*).

Quantum Pieri problem. Express the (quantum) product of a “special” Schubert class and a general one in the basis of Schubert classes.

Here special Schubert classes are the Chern classes of the tautological vector bundles on F (n). The main ideas of this approach were presented in the note [C-F1], together with a proof for the Givental-Kim conjecture. Another result obtained there is the solution of the quantum Giambelli problem for the special Schubert classes. Using this special case and the (geometrically obvious) nonnegativity of the struc- ture constants for the basis of the quantum cohomology ring consisting of Schubert classes, Fomin, Gelfand, and Postnikov [FGP] were able to give a combinatorial construction of the quantum Schubert polynomials, thus solving the quantum Gi- ambelli problem completely! They also stated and proved the quantum Monk for- mula, which is a very important case of the quantum Pieri problem. In view of the above, this is a certain identity involving the quantum Schubert polynomials. The proof they give is again combinatorial.

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The starting point of our study is the observation that the 3-point genus 0 Gromov-Witten invariants of F (n), which are the structure constants for the quan- tum cohomology ring, can be realized as intersection numbers on a certain com- pactification of the scheme Hom(P1,F(n)). This compactification has been named the hyperquot scheme in [C-F1] and is a natural generalization of Grothendieck’s Quot scheme. It was discovered first (although we were not aware of it until very recently) several years ago by G. Laumon [Lau1], [Lau2], in a completely different context. In this paper we complete, clarify and simplify the geometric approach of [C-F1]. We provide an argument which establishes simultaneously (see Theorem 5.6) both the validity of the Givental-Kim presentation and the special case of the quantum Giambelli formula mentioned above. Although the idea is certainly the same, the proof here is significantly simpler. The second main point of the paper is to give a direct geometric proof of the quantum Monk formula. Both proofs are based on an analysis of the boundary of the compactification, which is much more precise than the one we sketched in [C-F1]. We mention that since [C-F1] appeared, several different proofs (and general- izations) of the Givental-Kim conjecture have been found. First, the equivariant approach of Givental and Kim has been recently completed in [G] and [Ki2], giving in particular an independent proof of the presentation conjectured in [GK]. Yet another proof, which works for all homogeneous spaces of type G/B, was given in [Ki3]. Finally, in a May 1996 talk at the University of Washington, D. Peterson announced that he obtained the analogue of the quantum Monk formula for all G/B, and he also computed the quantum cohomology ring for all G/P ’s. As far as we know, this work has not been written down yet in preprint form.

Acknowledgements. This paper is a rewriting of my Ph.D. thesis [C-F2]. My special gratitude goes to my advisor, Aaron Bertram, for suggesting the problem, and for his continuous help and encouragement. Thanks are also due to William Fulton, whose many comments and suggestions led to substantial improvements of earlier versions of the paper.

0. Flag varieties and their cohomology Let V be a complex n-dimensional vector space and define F = F (V )tobethe variety parametrizing complete flags of subspaces:

U : 0 = U0 U1 U2 Un 1 Un =V. • { } ⊂ ⊂ ⊂···⊂ − ⊂ Throughout the paper we will let VX = V X be the trivial rank n vector bundle determined by V on a scheme X. There⊗O is a universal flag of subbundles

E1 E2 En 1 En =VF ⊂ ⊂···⊂ − ⊂ and a universal sequence of quotient bundles

VF = Ln  Ln 1  L1 0, − ··· where Li = VF /En i for i =1,...,n 1. − − Fix a complete flag of subspaces V := V1 Vn 1 Vn = V.LetSnbe • ⊂ ··· ⊂ − ⊂ the symmetric group on n letters. For w Sn,let ∈ rw(q, p) = card i i q, w(i) p . { | ≤ ≤ }

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We denote by `(w) the length of the permutation w (i.e., the number of inversions in w). The Schubert variety associated to w is defined by

Ωw =Ωw(V )= U F rankU (Vp F Lq) rw(q, p), 1 p, q n . • { • ∈ | • ⊗O → ≤ ≤ ≤ } Ωw is an irreducible subvariety of F ,ofcodimension`(w). We will identify as needed each of these subvarieties with the elements determined by their fundamen- tal classes in H (F ; Z) or, via Poincar´e duality, in H∗(F ; Z). This should not lead to confusion. ∗ Let w Sn be the permutation of longest length, given by 0 ∈ w (i)=n i+1,i=1,...,n. 0 − The following results are classical.

Theorem 0.1 (Ehresmann, [E]). Ωw w Sn is an additive basis for H∗(F ; Z). { } ∈ Moreover, its Poincar´e dual basis is given by Ωw w w S . { 0· } ∈ n Theorem 0.2 (Borel, [Bo]). Let xi = c1(ker(Li Li 1)),i=1,...,n.Then → − H∗(F;Z)∼=Z[x1,x2,...,xn]/(R1(n),R2(n),...,Rn(n)), th where Ri(n) is the i elementary symmetric function in x1,x2,...,xn.

To express Ωw in H∗(F ; Z) in terms of the above presentation, we need the Schubert polynomials of Lascoux and Sch¨utzenberger [LS1], [LS2]. Define operators ∂i,i=1,...,n 1, on Z[x1,...,xn]by − P(x1,...,xn) P(x1,...,xi 1,xi+1,xi,xi+2,...,xn) ∂iP= − − . xi xi − +1 n(n 1) For any w Sn ,writew=w si si , with k = − `(w), where ∈ 0· 1 · ··· · k 2 − si =(i, i + 1) is the transposition interchanging i and i + 1. The polynomial Sw(x) Z[x1,...,xn] defined by ∈ n 1 n 2 Sw(x)=∂ik ∂i1(x1− x2− ...xn 1) ◦···◦ − is the Schubert polynomial associated to w. With this definition, we have a Giambelli type formula, due to Bernstein, Gelfand and Gelfand [BGG] and Demazure [D].

Theorem 0.3. Ωw = Sw(x1,...,xn) in H∗(F ; Z).

1. The moduli space of maps and the hyperquot scheme

Let Yi := Ωw0 si be the Poincar´e dual of Ωsi . By Theorem 0.1, these classes form · 1 a basis of H2(F ; Z). Let f : P F be a holomorphic map. Since F (Ωsi ) are nef 1 n 1 → O line bundles, f [P ]= i− diYi, with di 0. For fixed d =(d1,d2,...,dn 1), let ∗ =1 ≥ − 1 P Hd := Homd(P ,F) be the moduli space of such maps. The usual way to put a structure of quasipro- jective scheme on this space is to embed it in the Hilbert scheme of P1 F ,by identifying a map with its graph. Since F is a homogeneous space, its tangent× bun- 1 1 dle TF is generated by sections, and therefore so is f ∗TF . Hence H (P ,f∗TF)=0 1 for every f : P F . Standard deformation theory implies then that Hd is smooth, 0→1 of dimension h (P ,f∗TF). It is well known (see e.g. [Mo]) that the canonical class

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of F is given by KF = 2(Ωs1 + +Ωsn 1). From Riemann-Roch it follows that − ··· − Hd has dimension n 1 − 1 n(n 1) dimF + f [P ] ( KF )= − +2 di. ∗ · − 2 i=1 X The moduli space of maps comes with a universal “evaluation” morphism 1 ev : P H F, × d → given by ev(t, [f]) = f(t), which can be used to pull-back Schubert varieties to Hd. 1 More precisely, for t P , w Sn, define a subscheme of Hd by ∈ ∈ 1 Ωw(t)=ev− (Ωw) ( t H ). ∩ { }× d Set-theoretically, this pull-back can be described as

Ωw(t)= [f] H f(t) Ωw . { ∈ d | ∈ } The definition of the quantum cohomology ring, which will be made precise in Defi- nition 4.4, involves counting the number of points in zero-dimensional intersections

of such subvarieties of Hd, under suitable genericity conditions. Since Hd is only quasiprojective, to make the counting well-defined we will realize these numbers as

intersection numbers on a certain compactification of Hd. This compactification is a “flagged” version of Grothendieck’s Quot scheme, and we will discuss first how this scheme appears naturally in our context. 1 1 n 1 Amapf :P F with f [P ]= i=1− diYi induces (by pull-back of the universal sequence of→ quotients on∗ F ) a flag of quotient bundles P ( ) V 1  Qn 1  Q1, † P − ··· 1 with rankQi = i,deg(Qi)=di, and conversely, every such sequence on P comes from a map as above. Hence one can also think of Hd as being the moduli space of flagged vector bundle quotients of the trivial rank n bundle on P1, with given ranks and degrees, and a compactification is obtained by considering the flat limits for these quotients. The pull-backs of Schubert varieties defined above can also be described as de- generacy loci by

Ωw(t)= [f] rank (Vp ev∗Lq) rw(q, p), 1 p, q n 1 ( t H ), { | [f] ⊗O→ ≤ ≤ ≤ − }∩ { }× d and one may try to extend them across the boundary of the compactification by extending ev∗Lq and using the same degeneracy locus definition. However, the natural extensions of ev∗Lq are NOT locally free, so this will not work in this form. Fortunately, a slight modification of the above idea will do the job. From the sequence ( ), by setting Si := Q∗, one gets a sequence of subbundles † i S1 Sn 1 V∗1, ⊂···⊂ − ⊂ P with rankSi = i,deg(Si)= di, and which encodes the map f. We let now the − quotients V ∗1 /Si degenerate in flat families to obtain a compactification of H . P d It turns out that on this compactification (ev∗Lq)∗ extend to vector bundles q, S and we will use them in Section 3 to define and study extensions Ωw(t)ofthe subvarieties Ωw(t) t Hd. We now turn to⊂{ the}× precise definition of the compactification.

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Let X be a smooth projective variety over an algebraically closed field k and be a rank r vector bundle on X. For any scheme S over k,letπ:X S XbeE the projection. Fix 2 n r and consider the functor × → ≤ ≤ n(X, ): Schemes over k Sets F E { }→{ } given by equivalence classes of flagged quotient sheaves n(X, )(S)= , F E π∗  Qn 1  Q1,which are flat over S  E − ···  ϕ and for a morphism S T , −→ n ( X, )(ϕ) = pull-back by ϕ. F E Here π∗  Qn 1   Q1 is equivalent to π∗  Qn0 1   Q10 if there E − ··· E − ··· are isomorphisms θi : Qi Qi0 such that all the squares commute. ( 2 is the usual Quot functor.) → F Let P =(P1(m),...,Pn 1(m)) be numerical polynomials and define the subfunc- − tor n,P (X, ) by requiring that χ(Xs,Qis(m)) = Pi(m) for all s S. Extending theF constructionE of the Quot scheme we have the following ∈ Theorem 1.1. For fixed P (m), (X, ) is represented by a projective scheme. Fn,P E We will denote this scheme by (X, )andrefertoitasthehyperquot HQn,P E scheme associated to X, , n and P . In general this scheme may be very compli- cated, but in the case of interestE for our purposes it is well-behaved. More precisely, let Pi(m)=(m+1)i+dn i, which is the Hilbert polynomial of a vector bundle of − 1 rank i and degree dn i on P .ThenPis determined by d =(d1,...,dn 1)only. − 1 − Denote the hyperquot scheme associated to P , V ∗1 and d by . P HQd n(n 1) n 1 − Theorem 1.2. d is smooth, irreducible, of dimension 2− +2 i=1 di,and contains H as anHQ open, dense subscheme. d P These two theorems were discovered by G. Laumon ([Lau1]) and rediscovered, independently, in [C-F1] and [Ki1] in the context of quantum cohomology. For completeness, we have included the proofs from [C-F2] in an appendix to this paper.

2. The structure of the boundary

As a fine moduli space, d comes equipped with a universal sequence of quo- tients HQ d d d V ∗1  n 1   2  1 0 P ×HQd T − ··· T T 1 d d on P d.Let i := ker V ∗1  n i . ×HQ S { P ×HQd T − } d For i =1,...,n 1, i is a vector bundle of rank i and relative degree di − S d − (this follows from flatness and the fact that i is locally free on each fibre of the 1 S projection P d d). However, the inclusions in the induced sequence ×HQ →HQ d d d 0 , 1 , 2 , , n 1 , V∗1 →S →S →··· →S − → P ×HQd are injective as maps of sheaves only!

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1 Obviously, Hd is the largest subscheme of d with the property that on P Hd all the inclusions are injections of vector bundles.HQ In this section we will describe× in detail the loci where these maps degenerate. We start with a simple lemma, describing the possible ranks for the fibres of the quotient sheaves. For a sheaf on a scheme X, we will denote the rank of the fibre F (x):= k(x)atapointx Xby rankx( ). F F⊗ ∈ F Lemma 2.1. Assume we are given a sequence of vector bundles and sheaf inclu- sions on P1:

S1 , , Sn 1 , V∗1, →··· → − → P

1 with rank Si = i and deg Si = di.DenotebyTn ithe quotient V ∗1 /Si.Lett P − − P ∈ be a point, and define nonnegative integers ei by rankt(Tn i)=n i+ei.Then − − (2.1.1) ei min(i, di), for 1 i n 1, ≤ ≤ ≤ − (2.1.2) ei ei 1 1, for 2 i n 1. − − ≤ ≤ ≤ −

Proof. Since there are sheaf surjections Tn (i 1)  Tn i, one has rankt(Tn (i 1)) − − − − − ≥ rankt(Tn i), which gives (2.1.2). − To prove (2.1.1), note first that ei i,sinceTn i is a quotient of V ∗1 .Next, ≤ − P consider the dual map VP1 Si∗. By assumption, its cokernel Ni has rank ei at t. Look at the surjection →

S∗ Ni(t) 0. i → →

If we let Mi denote the kernel, then deg Mi = di ei.SinceV1 S∗factors − P → i through Mi, the degree of Mi is nonnegative, and we get ei di. ≤

Construction 2.2. Let e =(e1,...,en 1) be a multiindex satisfying the condi- − 1 tions (2.1.1) and (2.1.2) of the lemma. Consider on P d e the universal sequence ×HQ −

d e d e d e 0 , − , − , , − , V∗1 . 1 2 n 1 P d e →S →S →··· →S − → ×HQ − For each 1 i n 1, let ≤ ≤ − 1 πi : Xi P d e −→ ×HQ −

d e be the Grassmann bundle of ei-dimensional quotients of i − .LetXebe the 1 S fibre product of the Xi’s over P d e, with natural projection π : Xe 1 ×HQ − −→ P d e. In what follows, we will use the same notation for the bundles living ×HQ − on Xi and their pull-backs to Xe via the various natural projections. Let

d e 0 K i π ∗ − Q i 0 −→ −→ i S i −→ −→

be the universal sequence on Xi. Ki and Qi are vector bundles of ranks i ei and − ei, respectively. On Xe we have then the following diagram:

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0 00 0 ··· ···

    K1 Ki Ki+1 Kn 1 y ··· y y ··· y−

d e d e d e d e π∗ 1− π ∗ i − π ∗ i +1− π ∗ n − 1 V X∗ Sy −→ · · · −→ Sy −→ Sy −→ · · · −→ Sy− −→ e

    Q1 Qi Qi+1 Qn 1 y ··· y y ··· y−

    0 00 0 y ··· y y ··· y Let e be the locally-closed subscheme of Xe determined by the closed conditions U (2.2.1) rank(Ki Q i )=0, for i =1,...,n 2, −→ +1 − and the open conditions

(2.2.2) rank(Ki V ∗ )=i ei, for i =1,...,n 1. −→ X e − − Theorem 2.3. (i) e is smooth, irreducible, and is of dimension U n 1 n 1 n(n 1) − − 1+ − +2 di ei(1 + ei ei 1), 2 − − − i=1 i=1 X X 1 where we set, once and for all, e0 =0. The projection π : e P d e is U → 1 ×HQ − smooth and proper, has irreducible fibres, and its image contains P Hd e. × − (ii) There exist morphisms he : e d with the following properties: d U −→ H Q (a) If rank(t,x) n i = n i + ei for every 1 i n 1 at a point (t, x) 1 T − − ≤ ≤ − ∈ P d,thenx he( e). In particular d Hd is covered by the union of ×HQ ∈ U HQ \ he( e). U 1 1 (b) The restriction of he to π− (P Hd e) is an isomorphism onto its image. × − Proof. (i) We will make use of the following observation. Let E be a vector bundle of rank r on a smooth variety Y , with a subbundle S of rank s r k.Letp:Gˇk(E) Ybe the Grassmann bundle of k-dimensional ≤ − → quotients of E, with universal quotient bundle p∗E Q, and let Ω(S)bethe ˇ → subscheme of Gk(E) determined by the vanishing of the map p∗S Q.Then Ω(S) can be identified with the Grassmann bundle of k-dimensional→ quotients of E/S over Y . In particular it is smooth, of pure codimension ks in Gˇk(E), and p :Ω(S) Y is smooth, with irreducible fibres. We now→ prove by induction on i the following (more general)

Claim 2.4. Let Ui X1 P1 P1 Xi+1 be the locally closed subscheme determined by the conditions⊂ × ×HQ ···× ×HQ

rank(Kj V ∗)=j, j =1,...,i+1, → and

rank(Kj Qj )=0,j=1,...,i. → +1

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i Then Ui is smooth, of pure codimension j=1 ej+1(j ej) in X1 P1 P1 1 − × ×HQ···× ×HQ Xi+1. The projection %i : Ui P d e is smooth, with irreducible fibres, and 1 → ×HQP− its image contains P Hd e. × − 1 Proof. In the case i =0,U0 is either P d e,ife1 = 1, or the open subscheme ×HQ − 1 d e of P d e where the map 1 − V ∗ is nondegenerate, if e1 =0.Inboth cases the×HQ claim− is clearly true. S → ˇ Assume that 2.4 is true for Ui 1.Letp:Ge Ui 1 be the Grassmann bundle − i+1 → − over Ui 1, obtained by restricting the projection − X1 1 1 Xi+1 X1 1 1 Xi. ×P ×HQ ···×P ×HQ → ×P ×HQ ···×P ×HQ ˇ Ui is the intersection of the Schubert variety in Gei+1 determined by the vanishing of the map p∗Ki Qi and the open subscheme of Gˇe given by rank(Ki → +1 i+1 +1 → V ∗)=i+1 ei . It follows from the above observation and the induction − +1 hypothesis that Ui has the claimed codimension, is smooth, and %i = %i 1 p is smooth, with irreducible fibres. This completes the proof of 2.4. − ◦

Because Un 2 = e, it follows from 2.4 that e has pure codimension n 2 − U U i=1− ei+1(i ei)inXe. A simple computation shows that the dimension of e is given by the− formula in the theorem. U P (ii) Consider the map 1 1 1 ψ : P e P P d e, ×U −→ × ×HQ − 1 1 ˜ 1 ψ=(id, π). Let ∆ P P be the diagonal and let ∆ P e be the preimage of ⊂ × 1 ⊂ ×U ∆ d e via ψ.Letp:P e e be the projection. For every 1 i n 1, ×HQ − ×U →U ≤ ≤ − define ˜e to be the kernel of the natural map Si d e ψ∗ − p ∗ Q i ˜ . Si −→ | ∆ Since ∆˜ is a divisor intersecting each fibre of the projection p in exactly one point, ˜ ˜e and p∗Qi ∆˜ is locally free of rank ei on ∆, it follows that i is a vector bundle of | 1 S˜e rank i and relative degree di on P e. By [H, III.9.9] i is therefore flat over −d e ×U S ˜e e. Via its inclusion in ψ∗ − , this new bundle comes with a sheaf injection , U Si Si → V ∗1 . Moreover, by the construction of e, there are induced sheaf injections P ×Ue U ˜e ˜e ˜e i , i+1,i=1,...,n 2, and n 1 , V ∗1 , S → S − S − → P ×Ue making the following diagram commutative: ˜e ˜ e ˜ e 1 2 ... n 1 V ∗1 S −−−−→ S −−−−→ −−−−→ S − −−−−→ P ×Ue

d e d e d e ψ∗ 1 − ψ ∗ 2 − ... ψ ∗ n − 1 V ∗1 Sy −−−−→ Sy −−−−→ −−−−→ Sy − −−−−→ P ×Ue ˜e In fact, (2.2.1) is precisely the condition needed for the map , V ∗1 to factor i P e ˜e S → ×U through i+1. Since S represents the hyperquot functor, there exists a map HQd he : e , U −→ H Q d d e such that (id, he)∗ = ˜ , for every i =1,...,n 1. Si Si −

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To prove (a), let x d be a point, represented by a sequence of vector bundles, together with sheaf∈HQ injections

(*) S1 , , Sn 1 , V∗1, →··· → − → P such that rank(Si)=i,deg(Si)= di. Assume that rankt(Tn i)=n i+ei, 1 − − − where t and Tn i = V ∗1 /Si.Togetapointin e whose image under he is x, P P we need∈ to construct− a sequence of bundles U

E1 , , En 1 , V∗1, →··· → − → P together with rank ei quotients of the fibres at t

Ei(t) Qi(t) 0, → → satisfying the following conditions: (2.3.1) rank(Ei)=iand deg(Ei)= (di ei); − − (2.3.2) if Ki(t):=ker(Ei(t) Qi(t)), the induced map of vector spaces Ki(t) → → V ∗1 (t) is injective; P (2.3.3) Si =ker(Ei Qi(t)) and all the squares → Si S i −−−−→ +1

  Ei E i +1 y −−−−→ y commute (in particular, this implies that Ki(t) Qi+1(t) vanishes). Consider the dual sequence →

VP1 Sn∗ 1 S1∗, → − →···→ and let Ni be the cokernel of the map VP1 Si∗.LetMibe the kernel of the → ei induced quotient Si∗  Ni(t). By assumption, Ni(t) = C , hence Mi is a vector 1 ∼ bundle of rank i and degree di ei on P . Since there are natural induced maps − Ni Ni 1, we have the following commutative diagram: → − 0 ... 0 ... 0

   V 1 M n 1 ... Mi ... M1 P −−−−→ y− −−−−→ −−−−→ y −−−−→ −−−−→ y

   V P 1 S n∗ 1 ... Si∗ ... S1∗ −−−−→ y− −−−−→ −−−−→ y −−−−→ −−−−→ y

   N n 1 ( t ) ... N i( t ) ... N 1( t ) −y −−−−→ −−−−→ y −−−−→ −−−−→ y

   0 ... 0 ... 0 y y y Set Ei := Mi∗ and Qi(t):=coker(Si Ei). It is straightforward to check that the required conditions are satisfied. Indeed,→ (2.3.1) and (2.3.3) are obvious from the definition of the bundles Ei. Moreover, by construction the images of Ki(t) and Si(t)inV∗ coincide. By assumption, the latter image has rank i ei, hence (2.3.2) holds as well. −

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1 Notice that if for every 1 i n 1 the subscheme of P where the sheaf Tn i ≤ ≤ − 1 − has rank n i + ei is the reduced point t P , and the rank is n i everywhere else, then the− sequence ∈ −

V 1 Mn 1 M1 P → − →···→ is uniquely determined, and all the maps are surjective. Therefore he is 1-to-1 on 1 1 π− (P Hd e), which gives (b) set-theoretically. By globalizing this observation × − in exactly the same way as in [Be2, Thm. 3.1], one constructs an inverse to he on 1 1 the image of π− (P Hd e), so we obtain (b). × −

Remark 2.5. Denote by ei the multiindex (0,...,0,1,0,...,0) (with 1 in the ith

spot). It follows from Construction 2.2 and the proof of Theorem 2.3 that π : ei 1 is an open immersion. The images D := h ( ) are birationalU → to P d ei i ei ei 1 ×HQ − U P d e . In particular, each Di has codimension 1 in d. ×HQ − i HQ Example 2.6. Let n =3andd=(1,1). In this case the boundary consists of the two codimension 1 strata, D1 and D2, and a codimension 2 stratum E := h(1,1)( (1,1)). U 1 Since (0,0) = H(0,0) = F , it follows that (1,1) = X(1,1) is the P -bundle HQ ∼ ∼ U ˇ (0,0) π 1 P( ) P F ,themaph(1,1) : (1,1) E is an isomorphism, K1 =0,and S2 → × U → K2 is the kernel of the natural surjection (0,0) π∗ E(1) 0. S2 →O →

Remark 2.7. A somewhat different stratification of d isgivenin[Lau2]andis studied further in the recent paper [Kuz]. HQ

3. Schubert classes on and their restrictions to the boundary HQd 1 Recall from Section 1 the universal evaluation morphism ev : P Hd F and × → the subschemes of Hd: 1 1 Ωw(t)=ev− (Ωw) ( t Hd),tP,w Sn. ∩ { }× ∈ ∈ 1 To extend these Ωw(t) to the hyperquot scheme, consider on P the dual ×HQd universal sequence V 1 and the fixed flag 0 = V V P d n∗ 1 1∗ 0 1 ×HQ →S − →···→S ⊂ ⊂ Vn 1 Vn = V. We define Ωw(t) as the restriction to t d of the ··· ⊂ − ⊂ { }×HQ appropriate degeneracy locus on P1 : ×HQd Ωw(t):= rank(Vp ∗) rw(q, p), 1 p, q n t . { ⊗O→Sq ≤ ≤ ≤ }∩{ }×HQd Obviously, Ωw(t) Ωw(t). Via the identification t = , the sub- ⊂ { }×HQd ∼HQd scheme Ωw(t) determines a class in the Chow ring CH∗( d). We will prove in the next section some “general position” results showing thatHQ these classes have the expected codimension and are independent of the choices made in the definition (see Theorem 4.1 and its corollaries). In order to do this, we need to understand first the restriction of Ωw(t) to the boundary. We keep the notation introduced in Section 2. Let e =(e1,...,en 1)bea − multiindex satisfying the conditions (2.1.1) and (2.1.2) of Lemma 2.1, let π : e 1 U → P d e be the projection, and let he : e d be the map defined in Theorem×HQ 2.3.− U →HQ

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Lemma 3.1. 1 1 1 ˜ e he− (Ωw(t)) = π− (P Ωw(t)) Ωw(t), × ∪ ˜ e 1 Ωw(t) being the degeneracy locus inside π− ( t d e) given by { }×HQ − (*) rank (Vp K∗) rw(q, p),p,q=1,...,n. ⊗O→ q ≤ Proof. We reconsider the construction in the proof of Theorem 2.3, (ii), keeping d ˜e the same notation. Since (id, he)∗ = for every q =1,...,n 1, it follows that Sq Sq − 1 e ∗ h− (Ωw(t)) = y e rank Vp ˜ rw(q, p), for all p, q , e { ∈U | (t,y) ⊗O→ Sq ≤ }   e ∗ where Vp ˜ is obtained as the composition ⊗O→ Sq   d e ∗ e ∗ Vp V ψ∗ − ˜ , ⊗O→ ⊗O→ Sq −→ S q     1 1 1 d e ∗ ˜ e ∗ with ψ =(id, π):P e P P d e.Themap ψ∗ q− q is ×U → × ×HQ − S −→ S ˜ ˜ an isomorphism outside ∆, while its restriction to ∆ factors through Kq∗,andthe  lemma follows.

e We would like now to estimate the codimension of Ω˜ (t)in e. The problem w U is that since rank(Kq∗)=q eq, some of the rank conditions in (∗) may become irrelevant. To deal with this− we need some facts about degeneracy loci associated to permutations. The following definitions and results are taken from [F1]. Let w Sn be a permutation. The following subset of 1,...,n 1,...,n is called the∈essential set associated to w: { }×{ } 1 1 ss(w)= (q, p) w(q) >p,w(q+1) p, w− (p) >q,w− (p+1) q . E { | ≤ ≤ } Assume that on an arbitrary scheme X we are given a morphism of filtered vector bundles h E1 En 1 En Fn Fn 1  F1 ⊂···⊂ − ⊂ → − ··· with rank(Ei) = rank(Fi)=i. We consider the subscheme Dw = Dw(h) defined by

rank(Ep Fq) rw(q, p), 1 p, q n. → ≤ ≤ ≤ This subscheme can be given however using a more economical set of rank condi- tions.

Proposition 3.2 (Fulton, [F1, Prop. 4.2]). The scheme Dw is defined by

rank(Ep Fq) rw(q, p), → ≤ for all p, q 1,...,n such that (q, p) ss(w). ∈{ } ∈E Assume now that we are given integers m and n together with sequences

1 m <

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and

rj,i = rw(nj ,mi), for all 1 j k, 1 i l. ≤ ≤ ≤ ≤ Fulton proves that such w, if it exists, is unique up to the embeddings of Sh in Sh+1. Let V be our fixed n-dimensional complex vector space and let F (n1,...,nk)be the partial flag variety parametrizing sequences of quotients of V with ranks given by the ni.Let

V Qn Qn ⊗O k ··· 1 be the universal quotient sequence on F (n1,...,nk). For any collection of nonnegative integers r =(rj,i), 1 j k,1 i n,let ≤ ≤ ≤ ≤ D F(n,...,nk) be the degeneracy locus defined by the conditions r⊂ 1 rank(Vi Q n ) r j,i for all 1 j k, 1 i n. −→ j ≤ ≤ ≤ ≤ ≤ Proposition 3.4 (Fulton, [F1, Prop. 8.1]). Assume that r is permissible and let w be the corresponding permutation. Then Dr has codimension `(w) in F (n1,...,nk) and its class in the Chow ring is given by a universal polynomial in the Chern classes

of the bundles Qni . ˜ e With this preparation we can study now the locus Ωw(t) defined in Lemma 3.1.

Construction 3.5. Let k = card i ei i =1,...,n 1 .Noticethat1 e { − | − } − 1≤ n 1 en 1, because of the condition (2.1.2) on the multiindex e.Set ··· ≤ − − − e0=en= 0 and define a partition of [0,n] as follows:

i0 =0,ij=min i i ei ij 1 eij 1 +1 , for 1 j k, ik+1 = n. { | − ≥ − − − } ≤ ≤

Let nj = ij eij ,forj=0,1,...,k, and let F (n1,...,nk) be the corresponding partial flag variety.− 1 Over e(t)=π− t d e all the maps U {}×HQ − V Kn∗ 1,K
n∗ 1 Kn∗ 2,...,K2∗ K1∗ ⊗O→ − − → − → are surjective. By the definition of i1,...,ik, on each of the intervals

[1,i1 1], [i1,i2 1],...,[ik 1,ik 1], [ik,n 1] − − − − − i ei is constant, equal respectively to 0,n1,...,nk, and the corresponding bundles − ˜ e Ki∗ are all isomorphic. Therefore the rank conditions defining Ωw(t) inside e(t) are U

rank (Vp K∗) rw(q, p), 1 p n, q i ,...,ik . ⊗O→ q ≤ ≤ ≤ ∈{1 } Define recursively r =(rj,p)1 p n, 1 j k as follows: ≤ ≤ ≤ ≤ r ,p =min rw(i ,p),n ,1 p n, 1 { 1 1} ≤ ≤

rj,p =min rw(ij,p),rj 1,p + nj nj 1 , 1 p n, 2 j k. { − − − } ≤ ≤ ≤ ≤ Claim. The conditions

rank (Vp K∗) rj,p, 1 p n, 1 j k ⊗O→ ij ≤ ≤ ≤ ≤ ≤ e define the same degeneracy locus Ω˜ (t) in e(t). w U

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Proof. For each m =1,...,k, define Am to be the degeneracy locus determined by

rank (Vp K∗) rw(ij,p), 1 p n, 1 j m, ⊗O→ ij ≤ ≤ ≤ ≤ ≤ and Bm to be the locus determined by the same conditions, but with rw(ij,p) replaced by rj,p. We prove by induction on m that Am = Bm as schemes. For m = 1, there is nothing to prove if rw(i ,p) n , by the definition of r ,p. 1 ≤ 1 1 On the other hand, if rw(i1,p) >n1,thenAm =Bm = e(t), so we are done in this case too. U Assume now that Am 1 = Bm 1. To finish the proof we show that − − rank (Vp K∗ ) rw(im,p) ⊗O→ im ≤ and

rank (Vp K∗ ) rm,p ⊗O→ im ≤ give in fact the same degeneracy condition at points in Am 1. Thisisobviousif − rw(im,p) rm 1,p + nm nm 1. Hence we may assume ≤ − − − rm,p = rm 1,p + nm nm 1

rank (Vp Ki∗ ) rm 1,p + nm nm 1 +1 ⊗O→ m ≥ − − − at some point in A .SinceV K factors through K and K m 1 p i∗m 1 i∗m i∗m K is surjective, we− have ⊗O→ − → i∗m 1 − rank(V K ) rank(V K ) rank( ker(K K )) p i∗m 1 p i∗m i∗m i∗m 1 ⊗O→ − ≥ ⊗O→ − → − = rank(Vp Ki∗ ) (nm nm 1) ⊗O→ m − − − rm 1,p +1, ≥ − contradicting the assumption Am 1 = Bm 1. We conclude that if rm,p

Restricting the bundles to e(t) defines an “evaluation at t” morphism U Φe(t): e(t) F ( n ,...,nk), U −→ 1 e and Ω˜ (t) is the preimage under Φe(t) of the degeneracy locus D F (n ,...,nk). w r ⊂ 1 Lemma 3.6. (rj,p) form a permissible collection of rank numbers.

e Proof. We construct a permutationw ˜ Sn as follows: ∈ For each j =0,1,...,k+ 1, define sets Wj (w)by

W(w)= ,Wj(w)= w(1),...,w(ij) . 0 ∅ { } Also we define sets Zj (w), and ordered sets Z˜j(w), such that (a) card Zj(w)=ij nj 1, ˜ − − (b) card Zj(w)=nj nj 1, ˜ ˜ − − (c) Zj (w) Zj0 (w)= if j = j0, k+1 ˜∩ ∅ 6 (d) j=1 Zj(w)=[1,n], by the following recursive procedure: S

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˜ ˜ Let Z0(w)=Z0(w)= .IfZi(w) has been already defined for i =0,1,...,j 1, let ∅ − j 1 − Zj(w)=Wj(w) Z˜i(w) . \ i=1 ! [ Arrange Zj(w) in increasing order:

Zj(w)= zj,1 <

z˜nj 1+1 := zj,1, z˜nj 1+2 := zj,2, ..., z˜nj := zj,nj nj 1 , − − − − and ˜ Zj(w):= z˜nj 1+1 < z˜nj 1+2 <

Note that it may happen thatz ˜nj 1 > z˜nj 1+1 ! e − − e Now definew ˜ (q)=˜zq, for all 1 q n. By definition, if (q, p) ss(˜w ), then e e ≤ ≤ ∈E w˜ (q) > w˜ (q + 1), and so we have necessarily q n1,...,nk . Furthermore, one ∈{ } e checks easily that rj,p = rw˜e (nj ,p), for all 1 p n, 1 j k. Hencew ˜ satisfies the conditions in Definition 3.3 and the Lemma≤ ≤ is proved.≤ ≤

e Example 3.7. Let n = 10, e =(1,2,3,4,5,6,2,3,2). We want to computew ˜0 and e w˜1 for the permutations 1 2345678910 w = 0 1098765432 1   and 12345678 9 10 w = . 1 45371986102   We have k =2,i0=0,i1=7,i2=9,i3=10,n0=0,n1=5,n2=7,n3= 10, and Z (w )= 10, 9, 8, 7, 6, 5, 4 ; Z˜ (w )= 4, 5, 6, 7, 8 , 1 0 { } 1 0 { } Z (w )= 9, 10, 3, 2 ; Z˜ (w )= 2, 3 , 2 0 { } 2 0 { } Z (w )= 9, 10, 1 ; Z˜ (w )= 1, 9, 10 , 3 0 { } 3 0 { } and therefore 12345678910 w˜e = . 0 45678231910   Similarly, Z (w )= 4, 5, 3, 7, 1, 9, 8 ; Z˜ (w )= 1, 3, 4, 5, 7 , 1 1 { } 1 1 { } Z (w )= 8, 9, 6, 10 ; Z˜ (w )= 6, 8 , 2 1 { } 2 1 { } Z (w )= 9, 10, 2 ; Z˜ (w )= 2, 9, 10 , 3 1 { } 3 1 { } and 12345678910 w˜e = . 1 13457682910  

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Note that since the construction in Lemma 3.6 involves arranging each of the e sets Z1,...,Zk+1 in increasing order, the resulting permutationw ˜ will satisfy the inequality `(˜we) `(w). We now want≤ to estimate `(w) `(˜we), i.e. the number of inversions in w that are killed by the procedure described− above.

e n 1 Lemma 3.8. `(w) `(˜w ) − ei. − ≤ i=1 Proof. Denote as usual by w0 Pthe permutation of longest length 12... n 1 n w = . 0 nn 1... −21  −  It is clear from the construction that `(w) `(˜we) `(w ) `(˜we), − ≤ 0 − 0 for every w Sn, therefore it suffices to prove the Lemma for w0. We will count∈ the number of inversions killed in each step of the procedure described in Lemma 3.6. First Z (w )= n, n 1, ..., n i +1 , 1 0 { − − 1 } i hence, when ordering it increasingly, we destroy 1 inversions. Next we have 2   Z (w )= n i +1+n , ..., n 1,n,n i,n i 1, ..., n i +1 , 2 0 { − 1 1 − −1 −1− − 2 } i2 i1 and when ordering it we destroy − + ei (i2 i1) more inversions. 2 1 −   In general a similar counting shows that when ordering Zj(w0)wedestroy ij ij 1 − − +eij 1(ij ij 1) inversions for each j =1,...,k+ 1. Hence 2 − − −   k+1 e ij ij 1 (3.8.1) `(w0) `(˜w0)= − − + eij 1 (ij ij 1) . − 2 − − − j=1 X    On the other hand, since i ei is constant on [ij 1,ij 1], for each j =1,...,k+1, we have − − − i 1 j −

ei = eij 1 +(eij 1 +1)+ +(eij 1 +(ij ij 1 1)) − − ··· − − − − i=ij 1 (3.8.2) X− ij ij 1 = − − + eij 1 (ij ij 1). 2 − − −   If we add up the identities (3.8.2) for j =1,...,k+ 1 and compare with (3.8.1), we get

n 1 e − `(w ) `(˜w )= ei. 0 − 0 i=1 X

We will need one more auxiliary result.

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Lemma 3.9. Let e =(e1,...,en 1)be a multiindex satisfying the conditions (2.1.1) − n 1 and (2.1.2) of Lemma 2.1. Assume in addition that i=1− ei 1. Then we have the following: ≥ n 1 P (i) i=1− ei(ei ei 1) 1. (ii) We have equality− − in≥(i) iff there exist integers 1 h l n 1 such that P ≤ ≤ ≤ − ei =0, for i [1,h 1] [l +1,n 1] and ei =1, for i [h, l], i.e., e consists of a (possibly empty)∈ string− of∪ zeroes, followed− by a string of∈ 1’s, followed by a (possibly empty) string of zeroes. We denote such a multiindex by ehl. (iii) Let e = ehl be a multiindex as in (ii), let w Sn be any permutation and let w˜e be the permutation constructed in Lemma 3.6.∈ Then n 1 e − `(w) `(˜w )= ei = l h +1 − − i=1 X iff w(h) > max w(h+1),...,w(l),w(l+1) , and in this case w˜e is the permutation { } 1 ... h 1 h ... l l+1 l+2 ... n . w(1) ... w(h− 1) w(h +1) ... w(l+1) w(h) w(l+2) ... w(n)  −  Proof. (i)and(ii) follow from the identity

n 1 − 1 2 2 2 2 ei(ei ei 1)= [e1+(e2 e1) + +(en 1 en 2) +en 1], − − 2 − ··· − − − − i=1 X while (iii) is an easy consequence of the construction ofw ˜e in Lemma 3.6.

4. Gromov-Witten invariants and the quantum multiplication map To define the quantum multiplication for cohomology classes on F we will need the Gromov-Witten invariants associated to Schubert varieties. Following [Be2], we will define these invariants as certain intersection numbers on the hyperquot schemes. In order to check that this gives the correct invariants, a “general position” result is needed.

Theorem 4.1. (i) Let Y be a fixed subvariety of Hd. For any w S,acorre- 1 ∈ sponding general translate of Ωw F ,andt P, the intersection Y Ωw(t) is either empty or has pure codimension⊂ `(w) in∈Y . In particular, for any∩ permuta- 1 tions w1,...,wN Sn, points t1,...,tN P , and general translates of Ωwi F , ∈N ∈ ⊂ the intersection i=1 Ωwi (ti) is either empty or has pure codimension `(wi) in H . d T P (ii) Moreover, if t1,...,tN are distinct, then for general translates of the Ωwi N N the intersection i=1 Ωwi (ti) is either empty or has pure codimension i=1 `(wi) N in and is the Zariski closure of Ωw (ti). HQd T i=1 i P Proof. (i) This follows from a Bertini-typeT theorem of Kleiman ([Kl]), since F is a homogeneous space. A very short proof can be found also in [Be2, Lemma 2.2]. (ii) The proof is by induction on (d1,...,dn 1) and is similar to that in [Be2, Lemma 2.2 A]. −

If d1 = = dn 1 =0,then d = Hd = F and (ii) reduces to the statement that general··· translates− of SchubertHQ varieties on F intersect transversely. Assume n 1 that the statement is true for all f N − such that fi di,1 i n 1, and ∈ ≤ ≤ ≤ − fj

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N Let c = i=1 `(wi) and denote for each (nonzero) multiindex e =(e1,...,en 1), satisfying the conditions (2.1.1) and (2.1.2) of Lemma 2.1, − P N Te := he( e) Ωw (ti) . U ∩ i i=1 ! \ By (i) and Theorem 2.3 (ii), we only need to prove the inequality

codim Te >c, HQd for every e.Sinceheis birational onto its image, this will follow if we show that N 1 codim e he− Ωwi (ti) >c (dim d dim e) U − HQ − U i=1 ! (4.1.1) \
n 1 − = c +1 ei(1 + ei ei 1). − − − i=1 X By Lemma 3.1 N N 1 1 1 ˜ e he− Ωwi (ti) = π− (P Ωwi (ti)) Ωw (ti) × ∪ i i=1 i=1 \
\   Since each Ω˜ e (t ) is supported on π 1( t )andthepointst ,...,t are wi i − i d e 1 N distinct, the only intersections that may{ contribute}×HQ − are of the type N 1 1 (*) π− (P Ωwi (ti)) , × i=1 \
and (after possibly renumbering the points) N 1 − 1 1 ˜ e (**) π− (P Ωwi (ti)) Ωw (tN ). × ∩ N i=1 \
For intersections of type (*) the codimension estimate (4.1.1) follows easily from the induction hypothesis, as π is a smooth map. It remains to deal with the intersections of type (**). Notice first that any such 1 intersection is supported on π− ( tN d e). If we let { }×HQ − N 1 − 1 W := π− ( tN Ωw (ti)) , { }× i i=1 \
then W has codimension c `(wN )in e(tN), and by the Construction 3.5, Lemma 3.6, Proposition 3.4, and Kleiman’s− TheoremU again (partial flag varieties are homo- geneous!), we get that W Ω˜ e (t ) has codimension `(˜we )inW. Therefore (**) wN N N ∩ e has codimension c `(wN )+`(˜w )+1in e. By Lemma 3.8 − N U n 1 e − (4.1.2) c `(wN )+`(˜w )+1 c+1 ei, − N ≥ − i=1 X and by Lemma 3.9 (i) n 1 n 1 n 1 n 1 − − − − (4.1.3) ei(1 + ei ei 1)= ei(ei ei 1)+ ei 1+ ei. − − − − ≥ i=1 i=1 i=1 i=1 X X X X The required inequality follows by combining (4.1.2) and (4.1.3).

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With exactly the same arguments as in [Be2, Cor. 2.3 and Cor. 2.4] we get

`(w) 1 Corollary 4.2. The class of Ωw(t) in CH ( d) is independent of t P and the flag V V . HQ ∈ • ⊂ n(n 1) n 1 Let = − +2 − di = dim( ). D 2 i=1 HQd N Corollary 4.3. If i=1P`(wi)= and t1,...,tN are distinct, then the number N D of points in Ωw (ti) can be computed as the degree of the product Ωw (t ) i=1 Pi 1 1 · Ωw (tN) in the Chow ring CH∗( ), hence it is independent of ti and the ···· N T HQd (general) translates of Ωwi .

The corollaries imply that, for general translates of Ωw1 ,...,ΩwN ,wehavea well-defined intersection number N N number of points in i=1 Ωwi (ti), if i=1 `(wi)= , Ωw ,...,Ωw := D h 1 Nid 0, otherwise. ( T P

This is the Gromov-Witten invariant associated to the classes Ωw1 ,...,ΩwN. Definition 4.4. The quantum multiplication map is the linear map

mq :Sym(H∗(F;Z)[q1,...,qn 1]) H∗(F ; Z)[q1,...,qn 1] − → − given by N mi mi d mq( Ωwi q )=q q Ωw,Ωw1,...,ΩwN dΩw0 w . P h i · i=1 d n 1 w Sn ! Y ∈XN − X∈ m n 1 mi Here q denotes as usual the monomial i=1− qi . The following is an immediate consequenceQ of Theorem 4.1 (see e.g. [Be2]). 1 Lemma 4.5. The map mq induces the identity on Sym (H∗(F ; Z)[q1,...,qn 1]). − Equivalently, if d =(0,...,0), every Gromov-Witten invariant Ωw1 , Ωw2 d involv- ing only two Schubert6 classes vanishes. h i

The map mq defined above induces a pairing 2 Sym (H∗(F ; Z)[q1,...,qn 1]) H ∗ ( F ; Z )[q1,...,qn 1] − −→ − and there are by now several successful approaches to prove that this gives a new ring structure on H∗(F ; Z)[q1,...,qn 1] ([BM], [KM], [LT1], [LT2], [RT]). We explain below why our seemingly ad-hoc− definition of quantum multiplication agrees with the general one. In general Gromov-Witten invariants are defined algebro-geometrically as inter- section numbers on Kontsevich’s space of stable pointed maps,whichexistsforevery smooth variety (see [KM]). Since F is homogeneous, the three-point invariant (in the notation of [KM]) F I (Ωw Ωw Ωw ) h 0,3,di 1 ⊗ 2 ⊗ 3 gives the number of maps from P1 to F of multidegree d, mapping three given

points (e.g. 0, 1and )toΩw1,Ωw2 and Ωw3 respectively (see [FP, Lemma 14] for ∞ F a proof of this). Hence I (Ωw Ωw Ωw ) coincides with the three-point h 0,3,di 1 ⊗ 2 ⊗ 3 invariant Ωw1 , Ωw2 , Ωw3 d of F as defined above. Since the quantum multiplica- tion can beh given only ini terms of three-point invariants, it follows that we have

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defined in Definition 4.4 the correct map mq. Note however that the invariants F Ωw ,...,Ωw and I (Ωw Ωw ) are not equal when N = 3 (they h 1 Nid h 0,N,di 1 ⊗···⊗ N 6 are solutions to different enumerative problems).

Definition 4.6. H∗(F ; Z)[q1,...,qn 1], together with the multiplication mq,is called the (small) quantum cohomology− ring of F . Recall from Section 0 the presentation

H∗(F ; Z) ∼= Z[x1,x2,...,xn]/(R1(n),...,Rn(n)). There is an induced map

mq : Z[x1,x2,...,xn,q1,q2,...,qn 1] H∗(F;Z)[q1,...,qn 1], − → − which is surjective (see [ST, Lemma 2.1] for an easy argument by induction on degree). Let Iq be the kernel of this map. Then Iq is an ideal with respect to the usual polynomial ring structure on Z[x1,...,xn,q1,...,qn 1]. Thus mq defines a − Z[q1,...,qn 1]-algebra structure on H∗(F ; Z)[q1,...,qn 1] which is isomorphic to − − Z[x1,x2,...,xn,q1,q2,...,qn 1]/Iq. − 5. The quantum cohomology ring of F

Since the generators xi of the usual cohomology ring can be written linearly

in terms of the “basic” Schubert classes Ωsi , to determine the ring structure on

H∗(F ; Z) one should know how to calculate a product of the type Ωsi Ωw. · Let Mi,w denote the class given by the product Ωsi Ωw in the cohomology of F . Monk’s formula (see [Mo]) expresses this class in terms· of the additive basis given by Schubert varieties:

Mi,w = Ωw t , · jk t Xjk summed over all transpositions tjk of integers j i

We will prove a similar formula for the quantum multiplication of Ωsi and Ωw. Since the quantum multiplication is defined by specifying the Gromov-Witten

invariants as structure constants, we need to figure out Ωsi , Ωw, Ωw0 d for all the h ni 1 permutations w0 Sn such that `(w)+`(w0)+1=n(n 1)/2+2 i=1− di. For the rest of∈ this section we will always work with− general translates of the Schubert classes on F . By Theorem 4.1 and its two corollaries, P

Ωs , Ωw, Ωw = card Ωs (u) Ωw(v) Ωw (t) =deg Ωs (u) Ωw(v) Ωw (t) h i 0 id i ∩ ∩ 0 i · · 0 where u, v, t P1 are distinct points and the product in the second line is computed ∈

in CH∗( ). Moreover, we know that the (transverse) intersection in the first HQd line is contained in Hd. Assume now that when we allow some of the points to coincide, the intersection remains transverse (see Lemma 5.1). Then, by Corollary 4.3, its cardinality will still give

Ωs , Ωw, Ωw , h i 0 id however, some of the intersection points may lie now in the boundary. Let us analyze in more detail what happens if we let u = v only and keep t distinct. Recall the evaluation map 1 ev : P H F, ev(u, f)=f(u). × d →

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It is easy to see that this map is smooth. In particular, its restriction evu to u H { }× d is also smooth. Let Y denote the intersection Ωsi Ωw inside F . By choosing suitable general translates, we may assume that Y is∩ irreducible, of codimension `(w)+1inF, and the intersection 1 Ωs (u) Ωw(u) Ωw (t)=ev− (Y ) Ωw (t) i ∩ ∩ 0 u ∩ 0 is either empty, or it consists of finitely many reduced points in Hd. We claim that it is in fact empty. Indeed, we can interpret a point in the above intersection as 1 amapf:P F such that f(u) Y and f(t) Ωw0 . However, if such a map exists, then there→ is an entire 1-dimensional∈ family∈ of them, obtained by composing f with the automorphisms of P1 which fix u and t. (This is exactly the argument giving Lemma 4.5.) We conclude therefore that the intersection of the closures

(5.1.1) Ωs (u) Ωw(u) Ωw (t) i ∩ ∩ 0 is supported on the boundary of d. What is more remarkable is thatHQ the explicit description in Sections 2-4 allows us to calculate the contribution in the boundary as well. Assume that d =(0,...,0), and let Z denote the intersection (5.1.1). 6 Lemma 5.1. (i) Z is either empty or has pure codimension = n(n 1)/2+ n 1 D − 2 i=1− di in d. (ii) Z is containedHQ in h ( (u)), the union over all the indices e as in ehl ehl ehl hl P U 1 Lemma 3.9 (ii),withh i l. (recall that e(u)=π− ( u d e)). ≤S ≤ U { }×HQ − (iii) If Z he ( e (u)) is nonempty, then w satisfies the condition ∩ hl U hl w(h) > max w(h +1),...,w(l),w(l+1) { } of Lemma 3.9 (iii). Moreover, in this case there is an equality (of sets)

ehl 1 Z he ( e (u)) = he Ω˜ (u) π− u Ωw (t) . ∩ hl U hl hl w ∩ { }× 0 Proof. (i)Itsufficestoshowthat 


(5.1.2) he( e) Ωs (u) Ωw(u) Ωw (t) U ∩ i ∩ ∩ 0 has codimension at least in d, for every e =(0,...,0) satisfying the conditions (2.1.1)-(2.1.2). By repeatingD theHQ reasoning in the6 proof of Theorem 4.1 (ii), we are reduced to proving that 1 1 1 h− (Ωs (u)) h− (Ωw(u)) h− (Ωw (t)) e i ∩ e ∩ e 0 has codimension at least n 1 − (5.1.3) +1 ei(1 + ei ei 1) D − − − i=1 X in e. Using Lemma 3.1, we can again treat this as a union of several types of U intersections. This time however, due to the fact that not all the points in P1 are distinct, there are three types that may contribute. The first two of these are the types (*) and (**) discussed in the proof of 4.1 (ii), and we have already seen that n 1 their codimension in e is greater than +1 i=1− ei(1 + ei ei 1), so they are actually empty. TheU only (possibly) nonemptyD − intersection is then− − P ˜ e ˜ e 1 1 (***) W := Ωs (u) Ωw(u) π− (P Ωw0 (t)), i ∩ ∩ ×

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which is contained in e(u). The same argument as in 4.1 (ii)showsthat U n 1 e e − (5.1.4) codim e W +1 (`(si) `(˜si )) (`(w) `(˜w )) ei, U ≥D − − − − ≥D− i=1 X e where the second inequality in (5.1.4) follows from the two estimates `(si) `(˜si ) 1 e n 1 − ≤ (which is obvious) and `(w) `(˜w ) i=1− ei (Lemma 3.8). Using now Lemma 3.9 (i), we obtain from (5.1.4) that− the≤ bound (5.1.3) holds as well for the codimension P of W in e. The statementsU (ii)and(iii) are immediate from Lemma 3.9 (ii), (iii). Indeed, if the intersection Z he( e(u)) is nonempty, all the inequalities we have used in the proof of (i) to estimate∩ U its codimension must in fact be equalities.

Corollary 5.2. Ωs , Ωw, Ωw =length(Z). h i 0 id As a warm-up (and because this is the main step for the proof of Theorem 5.6), we will compute now the Gromov-Witten invariants and prove the quantum Monk formula for a special class of permutations. For m n, k m 1, let αk,m Sn be the cycle ≤ ≤ − ∈ 12 ... m km k+1 ... m 1 mm+1...n 12... m− km−k+2 ... m− m k+1 m+1...n  − − −  Proposition 5.3. mq(Ωs Ωα )=mq(Mi,α )+δi,m 1qimq(Ωα s ),where i k,m k,m − k,m· i δi,m 1 is the Kronecker symbol. − Proof. Let d be a multiindex and let w be a permutation such that `(w)= D− `(si) `(αk,m). We want to evaluate Ωsi , Ωαk,m , Ωw d by the procedure described above.− h i By Lemma 5.1 (ii), (iii), there may be a nontrivial contribution in the boundary only if i = m 1 and it comes from − ˜ em 1 1 hem 1 Ωα − (u) π− ( u Ωw(t)) . − k,m ∩ { }×

em 1   Moreover,α ˜k,m− = αk,m sm 1, by Lemma 3.9 (iii). In particular, note that · − (m 1,p)isnotin ss(αk,m sm 1), for every 1 p n. On the other hand, π : − 1 E is an· open− immersion, whose≤ image≤ contains 1 H em 1 P d em 1 P d em 1 U − → ×HQ − − × − − (see Remark 2.5). From these observations and the Construction 3.5, it follows that

em 1 Ω˜ − (u)=Ω (u) u . αk,m αk,m sm 1 em 1 d em 1 · − ∩U − ⊂{ }×HQ − − Therefore, by Theorem 4.1,

˜ em 1 1 Ωα − (u) π− ( u Ωw(t)) = Ωαk,m sm 1 (u) Ωw(t). k,m ∩ { }× · − ∩

By Lemma 4.5, this intersection is empty, unless d = em 1 and Ωw is the comple- − mentary Schubert variety for Ωαk,m sm 1 , in which case it is a single point. More- 1 · − over, in this case em 1 = P F and hem 1 is an embedding. We conclude that U − ∼ × − Ωsm 1 , Ωαk,m , Ωw d = 1. The formula in the proposition is now immediate. h − i The following example, which was pointed out to us by W. Fulton, shows that in general the boundary contributions will come as well from the deeper strata he ( e ), with h l +1 2. hl U hl − ≥

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Example 5.4. Let n = 3, and let w0 = (321) be the permutation of longest length.

We want to compute mq(Ωs1 Ωw0 ). For dimension reasons, the only Gromov-Witten invariants that may be nonzero are

Ωs , Ωw , Ωs , Ωs , Ωw , Ωs , Ωs , Ωw , Ωs , h 1 0 1 i(1,0) h 1 0 1 i(0,1) h 1 0 2 i(1,0)

Ωs , Ωw , Ωs , Ωs , Ωw , Ωw . h 1 0 2 i(0,1) h 1 0 0 i(1,1) By the same reasoning as in Proposition 5.3, one computes easily

Ωs , Ωw , Ωs = Ωs , Ωw , Ωs = Ωs , Ωw , Ωs =0, h 1 0 1 i(0,1) h 1 0 2 i(1,0) h 1 0 2 i(0,1) and

Ωs , Ωw Ωs =1. h 1 0 1 i(1,0)

To analyze Ωs1 , Ωw0 , Ωw0 (1,1), recall that we have described completely in Exam- h i 1 ple 2.6 the boundary of (1,1).Ifu, t P are distinct points, the intersection HQ ∈ Ωs (u) Ωw (u) Ωw (t) 1 ∩ 0 ∩ 0 is supported on D D E. Sinces ˜(0,1) = s , there will be no contribution from 1 ∪ 2 ∪ 1 1 D . On the other hand,s ˜(1,0) = (123) andw ˜(1,0) = (231), hence on D 1 H 2 1 0 1 ∼= P (0,1) the contribution is given by ×

1 u Ωw(1,0) (u) P Ωw0 (t) { }× ˜0 ∩ × By Lemma 4.5, this last intersection is empty. We must still deal with E.Wehave

(1,1) (1,1) w˜0 = s1, s˜1 = (123), ˜ 1 hence Ωw0 (u) is the locus inside the P -bundle 1 π− ( u F) u F { }× →{ }× 1 where V K∗ vanishes, and Ω˜ s (u)=π− ( u F). The intersection 1 → 2 1 { }× 1 Ω˜ w (u) π− ( u Ωw (t)) 0 ∩ { }× 0 is then obviously a single point (the first term is a section, while the second is a fibre). Hence Ωs , Ωw , Ωw =1,and h 1 0 0 i(1,1)

mq(Ωs1 Ωw0 )=q1mq(Ω(231))+q1q2mq(Ω(123)). A similar reasoning gives the general quantum Monk formula. As we mentioned in the introduction, this formula was obtained first (in a different way) in [FGP].

Theorem 5.5 (quantum Monk formula). For every w Sn and every 1 i n 1, ∈ ≤ ≤ − mq(Ωs Ωw)= mq(Ωw t )+ qh ...ql 1mq(Ωw t ), i · jk − · hl t t Xjk Xhl where the first term is the one in the usual Monk formula, while the second term is summed over all transpositions thl of integers h i

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Proof. We start by recalling some of the notation introduced earlier in the paper. Let h, l be integers such that 1 h i l n 1, and let ehl be the multiindex (0,...,0,1...,1,0...,0) defined≤ in Lemma≤ ≤ ≤ 3.9. −

Let ehl Xehl be the locally closed subset defined in Construction 2.2, and, as in 3.5,U let ⊂ (u) be its restriction to π 1( u ). ehl − d ehl Let U { }×HQ −

Y = X 1 1 X , j 1 P d e P d e j × ×HQ − hl ···× ×HQ − hl 1 and let pj : Yj+1 Yj and %j : Yj+1 P d e be the projections. → → ×HQ− hl It is easily seen from the construction of Xehl that pj is the identity map for j [0,n 2] [h 1,l 1], while for h 1 j l 1, the projection pj realizes ∈ − \ − − − ≤ ≤ − j ˇ d ehl Yj+1 as the P -bundle P(%j∗ 1 j+1− )overYj. Recall that we have denoted in 2.4 − S by Uj the locally closed subscheme of Yj+1 determined by

rank(Kj V ∗)=j, j =1,...,i+1, → and

rank(Kj Qj )=0,j=1,...,i. → +1 1 Let Uj(u):=Uj %j− (u d e ). (In particular, ehl (u)=Un 2(u), and ∩ { }×HQ − hl U − %n 2 = π.) One sees easily that the restriction − pj : Uj(u) U j 1 ( u ) −→ − is a P1-bundle projection, for h 1 j l 1. More precisely, − ≤ ≤ − ˇ d ehl Uj(u) = P(%j∗ 1 j+1− /Kj). ∼ − S Therefore the restriction

π : ehl (u) u d e U −→ { }×HQ − hl factors as a succession of P1-bundle projections

π = ph 1 pl 1, − ◦···◦ − and its image, U0(u), satisfies

u Hd e U0(u) u d e . { }× − hl ⊂ ⊂{ }×HQ − hl n 1 Let w0 Sn be a permutation such that `(w)+`(w0)+1 = n(n 1)/2+ − di, ∈ − i=1 let u, t P1 be distinct points, and let ∈ P Z = Ωs (u) Ωw(u) Ωw (t). i ∩ ∩ 0 By Lemma 5.1 (i) and Corollary 5.2, Z is either empty or has pure dimension zero, and Ωs , Ωw, Ωw =length(Z). To evaluate this, we only have to look at h i 0 id Z he ( e (u)) , ∩ hl U hl for all 1 h i l n 1, by Lemma 5.1 (ii). Finally, by (iii) in Lemma 5.1, we may assume≤ ≤ that≤ w≤ satisfies− (5.5.1) w(h) > max w(h +1),...,w(l),w(l+1) , { } and

ehl 1 Z he ( e (u)) = he Ω˜ (u) π− u Ωw (t) . ∩ hl U hl hl w ∩ { }× 0 


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Hence we need to decide when

ehl 1 (5.5.2) Ω˜ (u) π− u Ωw (t) w ∩ { }× 0 ˜ ehl is nonempty. Recall that by Construction 3.5 and Lemma
3.6, Ωw (u) is defined inside e (u) by the conditions U hl (5.5.3)

d ehl rank Vp π∗( − )∗ r ehl (j, p),j [1,n] [h, l],p [1,n], ⊗O→ Sj ≤ w˜ ∈ \ ∈ (rank (Vp K∗) r ehl (j 1,p),j[h, l],p [1,n], ⊗O→ j ≤ w˜ − ∈ ∈ wherew ˜ehl is the permutation 1 ... h 1 h ... l l+1 l+2 ... n w(1) ... w(h− 1) w(h +1) ... w(l+1) w(h) w(l+2) ... w(n)  −  from Lemma 3.9 (iii). We consider first the projection

pl 1 : Ul 1(u) U l 2 ( u ) . − − −→ − If w(l)

˜ ehl 1 (5.5.4) pl 1 Ωw (u) (ph 1 pl 2)− u Ωw (t) − ∩ − ◦···◦ − { }× 0 is empty for dimension reasons. Therefore the same holds for (5.5.2).
However, if w(l) >w(l+ 1), then it follows from the exact sequence

d ehl ∗ 0 ( 1) pl∗ 1 l − /Kl 1 (Kl/Kl 1)∗ 0 →O − → − S − → − → 1   on the P -bundle

pl 1 : Ul 1(u) U l 2 ( u ) − − −→ − ˜ ehl that pl 1(Ωw (u)) is given inside Ul 2(u) by the rank conditions (5.5.3), except that − −

rank (Vp K∗) r ehl (l 1,p), 1 p n, ⊗O→ l ≤ w˜ − ≤ ≤ are changed into e r e (l 1,p), if 1 p

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˜ ehl the restriction of π to Ωw (u) is birational onto its image, and

˜ ehl (5.5.6) π Ωw (u) = Ωw thl+1 (u) Uh 1(u), · ∩ −   where thl+1 is the transposition interchanging h and l +1(wehaveusedherethat e w˜ hl sl 1 sh =w thl+1). By Lemma 4.5, · − ··· · Ωw t (u) u Ωw (t) · hl+1 ∩{ }× 0

is empty (hence (5.5.2) is empty as well) unless d = ehl,andΩw0 is the complemen- tary Schubert cycle to Ωw t .Inthiscase · hl+1 1 e (u)=π− ( u F), Uhl { }×

and, by Theorem 2.3 (ii)(b), hehl is an isomorphism onto its image. Therefore e 1 length(Z) = length(Ω˜ hl (u)) π− u Ω (t) , w d e w0 ∩ − hl { }× and the latter can be computed as

ehl deg Ω˜ (u) π∗ u Ωw (t) w · { }× 0 1   in the Chow ring of π− ( u F). By the projection formula, { }× ˜ ehl ˜ehl π Ωw (u) π∗ u Ωw (t) =π (Ωw (u)) u Ωw (t) , ∗ · { }× 0 ∗ · { }× 0 and from (5.5.6) 

˜ ehl π (Ωw (u)) u Ωw (t)=Ωwthl+1 (u) u Ωw (t)=Ωwthl+1 Ωw =1, ∗ ·{ }× 0 · ·{ }× 0 · · 0 the products in the previous line being all computed in the Chow ring of F . Sum- marizing, we have proven that

Ωs , Ωw, Ωw =0, h i 0 id

unless d = ehl, w satisfies (5.5.1) and (5.5.5), and Ωw0 is the complementary Schu- bert cycle to Ωw t ,inwhichcase · hl+1 Ωs ,Ωw,Ωw e =1. h i 0ihl Replacing l +1byl, this gives exactly the formula we were seeking. Finally, we can give now a presentation by generators and relations for the quantum cohomology ring. th For all 1 k m n we denote by Rk(m)thek elementary symmetric ≤ ≤ ≤ function in the variables x1,...,xm. Also, denote by Ak(m) the class corresponding to Rk(m) in the cohomology of F . Notice that Ak(n) = 0 for all 1 k n,since ≤ ≤ H∗(F;Z)∼=Z[x1,x2,...,xn]/(R1(n),R2(n),...,Rn(n)). In the quantum cohomology ring however, mq(Rk(n)) does not vanish anymore. In fact, consider for k m n the following polynomials in Z[x1,...,xn,q1,...,qn 1]: ≤ ≤ − m 1 − ∂2 ∂4 Rk0 (m):= qi Rk(m) + qiqj Rk(m) ∂xi∂xi ∂xi∂xi ∂xj∂xj i=1 +1 i

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Theorem 5.6. (i) For all 1 k m n, ≤ ≤ ≤ (*) mq (Rk (m)) = mq(Ak(m)) mq(R0 (m)). − k In particular mq(Rk(n)) + mq(Rk0 (n)) = 0 for k =1,...,n, i.e. the polynomials q Rk(n):=Rk(n)+Rk0 (n)

are in the ideal Iq. q q (ii) Iq is generated by R1(n),...,Rn(n). Proof. (i) We will proceed by induction on m. First notice that for k =1wehaveR0(m) = 0 for all 1 m n and 1 ≤ ≤ therefore the formula (*) follows from the fact that mq is the identity map on 1 Sym (H∗(F ; Z)[q1,...,qn 1]) (Lemma 4.5). This also takes care of the case m =1. − We assume now that (*) is true for all k and m0 with 1 k m0

mq(Rk(m)) = mq(Rk(m 1)) + mq(xmRk 1(m 1)). − − − By the induction hypothesis,

mq(Rk(m)) = mq(Ak(m 1)) mq(Rk0 (m 1)) (5.6.1) − − − + mq(xmAk 1(m 1)) mq(xmRk0 1(m 1)). − − − − − st Recall the cycle αk,m, m n, k m 1. Its Schubert polynomial is the k 1 ≤ ≤ − − elementary symmetric function in variables x1,...,xm 1, − Sα (x)=Rk 1(m 1), k,m − −

as can be easily seen. Hence Ak 1(m 1) = Ωαk,m in H∗(F ; Z), by Theorem 0.3. − − Recall also that xm =Ωsm Ωsm 1, for all 1 m n, with the convention − − ≤ ≤ Ωsm =0, for m =0orn. The special case of the quantum Monk formula (Proposition 5.3) gives now

mq(xmAk 1(m 1)) = mq(Ωsm Ωαk,m ) mq(Ωsm 1 Ωαk,m ) (5.6.2) − − − − = mq(Bk,m) qm 1mq(Ωαk,m sm 1 ), − − · − where by Bk,m we have denoted the class of xm Ak 1(m 1) in H∗(F ; Z). · − − Obviously, Bk,m = Ak(m) Ak(m 1). Moreover, since αk,m sm 1 = αk 1,m 1, − − · − − − we can use again Theorem 0.3 to conclude that Ωαk,m sm 1 = Ak 2(m 2). With these observations (5.6.2) becomes · − − − (5.6.3) mq(xmAk 1(m 1)) = mq(Ak(m) Ak(m 1)) qm 1mq(Ak 2(m 2)). − − − − − − − − By the inductive assumption,

(5.6.4) mq(Ak 2(m 2)) = mq(Rk 2(m 2) + Rk0 2(m 2)). − − − − − − Combining (5.6.1), (5.6.3), (5.6.4), and the easy polynomial identity (5.6.5)

Rk0 (m)=Rk0(m 1) + xmRk0 1(m 1) + qm 1Rk 2(m 2) + qm 1Rk0 2(m 2), − − − − − − − − − we get the formula (*). (ii) is a consequence of (i) and [ST, Theorem 2.2].

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In [GK], Givental and Kim conjectured that the generators for the ideal Iq are n k the coefficients Ck(n)ofλ − ,k=1,...,n in the expansion of the determinant:

x1 + λq1 00... 00 1x+λq 0... 00 −2 2 01x3+λq3 ... 00 − ......

0000... xn 1 +λqn1 − − 0000... 1 xn +λ −

The following lemma shows that this is the same as the presentation of Theorem 5.6. q Lemma 5.7. Ck(n)=Rk(n), for all k =1,...,n. q Proof. The polynomials Rk(n) satisfy the recursion q q q q Rk(n)=Rk(n 1) + xnRk 1(n 1) + qn 1Rk 2(n 2) − − − − − − (combine (5.6.5) above with the recursion Rk(n)=Rk(n 1) + xnRk 1(n 1) satisfied by the elementary symmetric polynomials). By expanding− the− Givental-− Kim determinant along the last column, one sees that the polynomials Ck(n)satisfy the same recursion, and the lemma follows.

Appendix: Existence and local properties of hyperquot schemes The proofs we give here are taken from [C-F2]. Although rather lengthy, they are straightforward following the same line as for the corresponding results in the case of the Quot functor. The ideas are due to Grothendieck ([Gr]). We follow the presentation by Koll´ar ([Kol]), adapted to our case. As usual, the following two general results will be used in an essential way. The proofs in [Mu, Lectures 14 and 8] require only minor modifications. Theorem A. Let X be a projective variety, be a vector bundle on X and be asubsheafof with fixed Hilbert polynomial EP . Then there exists an integerGN, depending onlyE on P , such that for every m N the following hold: (A1) Hi(X, (m)) = 0 for i 1. ≥ (A2) (m) isG generated by global≥ sections. G Theorem B. Let X be a projective scheme and let S be any scheme. Let be a coherent sheaf on X S. For every numerical polynomial P there exists aG locally × closed subscheme iP : SP , S with the following property: → Given any morphism g : T S , the pull-back (id, g)∗ on X T is flat over T with Hilbert polynomial P if−→ and only if g can be factoredG as ×

i P g : T S P S. −→ −→

Proof of Theorem 1.1. Given , n and P =(P1,...,Pn 1), it follows from Theorem A that there is an N such thatE for every sequence of flat− quotients

π∗  Qn 1  Q1 E − ···

with χ(Xs,Qjs(m)) = Pj (m), if we let

j =ker π∗ Qn j , E { E − }

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i then H (Xs, j (a)) = 0 for all i 1,a N,s Sand also j (a) is generated by E s ≥ ≥ ∈ E s global sections. (Note that each j is flat.) We may choose N large enoughE so that (N) is generated by global sections and i E i we have H (X, (N)) = 0 for i 1 as well, hence H (Xs, s(N)) = 0 for all s S. E ≥ E ∈ Since all Qj are flat over S, the sequences

0 js(N) s(N) Q(n j) (N) 0 →E →E → − s → i are all exact for every s S,soweseethatH(Xs,Qjs(N)) = 0 for i 1. Let f : X S ∈S be the projection. Let Y be the partial≥ flag variety parametrizing× successive−→ quotients

0 H (X, (N))  Gn 1 G1, E − ··· with rank(Gj)=Pj(N). The discussion above implies that the sheaf f Qn j(N) ∗ − is locally free of rank Pn j (N), for each j =1,...,n 1. Moreover, there are surjective maps − −

0 H (X, (N)) S =f πX∗ (N)f Qn 1(N) f Q1(N). E ⊗O ∗ E ∗ − ··· ∗ Thus each element of (X, )(S) determines a morphism Φ : S Y and so we Fn,P E −→ get a morphism of functors n,P (X, ) h Y ,wherehY is the functor of points associated to Y .Let F E −→

0 1, , n 1 , H (X, (N)) Y U →··· →U − → E ⊗O be the universal sequence of subbundles on Y . Consider the two projections

Y π1 Y X π 2 X ←− × −→ and form the compositions

0 gj : π1∗ j H ( X, (N)) X Y π 2∗ ( N ) . U −→ E ⊗O × −→ E Let j =coker(gj), j =1,...,n 1. By Theorem B applied to j( N), there is a K − ij K − largest locally closed subscheme ZPn j , Y with the property that ij∗ j ( N), as − → K − sheaf on ZPn j X,isflatoverZPn j with Hilbert polynomial Pn j (m). Let ZP − × − − be the scheme-theoretic intersection of ZP1 ,...,ZPn 1. We claim that n,P (X, ) is represented by Z . Indeed, given a sequence of flagged− quotients on XF S E P × π∗  Qn 1  Q1, E − ···

flat over S with Hilbert polynomials Pn 1(m),...,P1(m) respectively, we have seen that there is a morphism Φ : S Y such− that −→ 0 0 Φ∗[ j H (X, (N)) Y]=[f j(N) H (X, (N)) S]. U → E ⊗O ∼ ∗E → E ⊗O From the diagram

f S S X π X X ←−−−− × −−−−→ Φ (Φ,id)

 π1  π 2 Y Y  X X y ←−−−− ×y −−−−→

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we get

(Φ,id)∗π1∗ j f∗f j(N) U ∗E

(Φ,id)∗gj ψ   (Φ,id)∗π∗ (N) π∗ (N). y 2E XEy But Imψ = j(N), since j (N) are generated by sections for every s S.This E E s ∈ implies that (Φ,id)∗ j =Qn j(N). Hence (Φ,id)∗ j( N) are all flat over S with K ∼ − K − Hilbert polynomials Pn j . Therefore Φ factors through ZP and we see that there is − an injective morphism of functors p : (X, ) h Z . By construction, every Fn,P E −→ P morphism S Z P gives an element of n,P (X, )(S). This provides an inverse for p and proves−→ the claim. F E

As for the projectivity, we only need to check that ZP is proper. But this follows at once from the valuative criterion (using the case of the Quot scheme).

For the proof of Theorem 1.2 we shall use the standard techniques of deformation theory. For the convenience of the reader we reproduce part of the exposition in [Kol, Sec. I.2]. We treat first the local case. Let (A, mA)and(B,mB) be local Artin rings with residue field k such that B is an extension of A by the ideal J: 0 J B A 0, → → → → B with mBJ =0.LetT be a k-algebra and let T be a Noetherian ring flat over B A B k B such that T B k = T .LetT =T BA.DenotebyM=M the free ⊗ ∼ j ⊗ rank rT-module and let F k = F , M be a submodule. Let Q = M/F and let p : M Q be the projection. Let → → g R r G F 0 −→ −→ −→ be a presentation of F by free T -modules. In what follows the extensions of the above T -modules and morphisms to T A and T B will receive corresponding upper indices. A A j A A A A A Let F , M be such that Q = M /F is flat over A and F A k = F . Flatness implies→ that the presentation of F lifts to a presentation of F A⊗:

A g A RA r G A F A 0 −−−−→ −−−−→ −−−−→

 r  g  R G F 0 , y −−−−→ y −−−−→ y −−−−→ where the vertical arrows are specializations. Pick arbitrary liftings gB : GB M B and rB : RB GB (recall that R, G, M are free modules). This gives→ a commutative diagram→

B g B RB r G B F B 0 −−−−→ −−−−→ −−−−→ (*)

 A  g A  RA r GA FA 0 . y −−−−→ y −−−−→ y −−−−→

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Here F B M B is the image of GB. Note that the top row need not be a presen- tation and⊂ that the quotient QB = M B/F B need not be flat. The following result (Lemma I.2.2 in [Kol]) gives the conditions under which the lifting QB is flat.

B Lemma C. (i) To the diagram (∗) one can associate a morphism e(g ):R Q J , which is independent of the choice of rB . −→ ⊗ e(gB)+r (ii) The module E(gB):=coker[R Q J + G ] canbewrittenasan extension −→ ⊗ ( ) 0 Q J E(gB ) F 0, † → ⊗ → → → B 1 hence we have an element [E(g )] ExtT (F, Q J). (iii) Any different lifting gB : GB∈ M B canbewrittenas⊗ gB =gB+φB,where φB Hom(GB,M J). This gives a→ natural extension isomorphism ∈ ⊗ E(φB ):E(gB) E ( g B ) . −→ B 1 B In particular, [E(g )] ExtT (F, Q J) is independent of the choice of g and de- pends only on the data∈ over A. It is denoted⊗ by [EB(QA)] and called the obstruction. (iv) The following are equivalent: (a) QB is flat over B. (b) e(gB)=0. B B B (c)g r =0for a suitable choice of r (i.e. the top row in (∗) is a complex). (v) For◦ a suitable choice of gB, the quotient QB is flat over B if and only if [EB(QA)] = 0. In this case, there are affine linear isomorphisms set of quotients QB = splittings of ( ) = HomT (F, Q J). which are flat over B ∼ { † } ∼ ⊗   rA g A A 1 A 1 A A (vi) If R1 G 1 F 0 is a different presentation of F , then there is a natural extension−→ isomorphism−→ −→ E(gB) E(gB).Thus[EB(QA)] depends only ∼= 1 on QA. Also, if [EB(QA)] = 0, then the principal homogeneous space structure on the set of quotients QB which are flat over B depends on QA only. Remark D. The following special case will be used later. Take A = k and B = k + J.Then B B B T =T Bk+T BJ=T+T kJ. ⊗ ⊗ ⊗ k B k k In this case there is always the trivial flat lifting of F to Ftriv = F + F k J. This implies that we have a natural isomorphism ⊗ set of quotients QB = HomT (F, Q J), which are flat over B ∼ ⊗   with the trivial extension as the zero vector. We will be interested in lifting suc- cessive quotients

pn 1 pn 2 p1 A − A − A A M  Qn 1  Qn 2  ...Q1 . − − A A A A ji A Let Fi := ker(M  Qn i) and denote by Fi , Fi+1 the inclusions. Let − → n 1 n 2 − f − K := ker[ HomT (Fi,Qn i J) HomT (Fi,Qn i 1 J)], − ⊗ −→ − − ⊗ i=1 i =1 M M

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n 1 n 2 where f is defined by f( φi i=1− )= pn i 1 φi φi+1 ji i=1− . Then Lemma C (v) implies that { } { −− ◦ − ◦ } set of successive quotients = K. QB which are flat over B ∼ { i }  We will pass now to the global situation. Let X be a smooth projective variety, let be a rank r vector bundle on X, E and let P =(P1(m),...,Pn 1(m)) be numerical polynomials. Assume that we are given a sequence of quotient− sheaves

pn 1 pn 2 p1 − −  n 1  n 2  ... 1 E Q − Q − Q with χ(X, j(m)) = Pj (m) and let x Qn,P (X, ) be the corresponding point. Let Q ∈H E

i =ker(  n i) E E Q − ji and denote by i , i+1 the corresponding sheaf inclusions. Consider theE map→E of sheaves n 1 n 2 − f − om ( i, n i) om ( i, n i 1), H OX E Q − −→ H OX E Q − − i=1 i =1 M M given ( for any open set U X )by ⊂ n 1 n 2 f( φi i=1− )= pn i 1 φi φi+1 ji i=1− { } { −− ◦ − ◦ } and let := ker f. The followingK proposition is the main step for the proof of Theorem 1.2 (compare with [Kol, Prop. I.2.5 and Thm I.2.8] ). Proposition E ([Lau1, Prop. 2.5]). (i) The Zariski tangent space to (X, ) HQn,P E at x is isomorphic to H 0(X, ). K 1 (ii) Assume in addition that Ext ( i, n i)=0, for i =1,...,n 1.If H1(X, )=0,then (X, ) is smoothE Q at− x. − K HQn,P E

Proof. Let Uα = Spec(Tα) be an affine covering of X such that Uα is trivial. k k k k E| Denote by F ,M,Q ,K the Tα-modules obtained by restricting i, , i iα α iα α E E Q and respectively to Uα. ToK prove (i)weletA=k,B=k[]/2.Let Bbe the pull-back of to X Spec(B). The tangent space to at x is givenE by the set of all liftingsE × HQ pB pB B n 1 n 2 p1 B − B − B B  n 1  n 2  ... 1 E Q − Q − Q which are flat over Spec(B) and specialize to

pn 1 pn 2 p1 − −  n 1  n 2  ... 1 E Q − Q − Q over X Spec(k). × B As we remarked earlier, we can always take over Uα the trivial liftings Qiα = k B Qiα k B and they glue together to give trivial liftings i ,flatoverBand such ⊗ B B Q that the maps i+1 i are all surjective. By Remark D, any other flat lifting Q →Q 0 of the successive quotients is obtained from elements φα H (Uα,Kα) which glue together. This gives (i). ∈

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(ii)TakeBAto be a quotient of local Artin rings with residue fields equal to k. It is well known that the local ring x, of at x is regular if and only O HQ HQ if, for every such quotient, the induced map Homk( x, ,B) Homk( x, ,A) is surjective. This says that we need to check the followingO HQ assertion:→ O HQ For every sequence of flagged quotients

pA pA A n 1 n 2 p1 A − A − A A  n 1  n 2  ... 1 E Q − Q − Q on X Spec(A), which are flat over Spec(A) and specialize to × pn 1 pn 2 p1 − −  n 1  n 2  ... 1 E Q − Q − Q over X Spec(k), there exists a lifting × pB pB B n 1 n 2 p1 B − B − B B  n 1  n 2  ... 1 E Q − Q − Q B such that each i is flat over Spec(B). Pick presentationsQ RA GA F A 0 iα → iα → iα → A A B of the restrictions of the sheaves i to Uα := Uα Spec(A) and liftings giα : GB M B . By Lemma C weE get extensions E(×gB ) that glue together over iα −→ iα iα Uαβ = Uα Uβ to give global extensions ∩ B 0 i E ( g iα ) n i 0 . −→ E −→ { } −→ Q − −→ 1 Since Ext ( i, n i)=0fori=1,...,n 1, all the above global extensions split, so E Q − − B B the same is true for the affine ones and we can choose liftings giα such that Qiα are B B flat over Spec(B)andQiα  Qi 1α for all i =2,...,n 1. Over each Uαβ we have − − two different liftings, hence by Remark D we get an element φαβ Kαβ J. φαβ is a cocycle, so it gives a class in H1(X, ), which vanishes by assumption.∈ ⊗ Therefore{ } K B n 1 B B φαβ = φα φβ, with φα Kα J. Lift φα to φ − Hom(G ,Q J). The − ∈ ⊗ α ∈ i=1 iα iα ⊗ new maps gB = gB φB give local liftings that patch together. iα iα − α L Proof of Theorem 1.2. By the previous proposition it suffices to show that for every sequence of successive quotient sheaves

pn 1 pn 2 p1 − − V ∗1  n 1  n 2  ... 1 0 P Q − Q − Q 1 1 such that χ(P , i(m)) = (m +1)i+dn i,wehaveExt( i, n i)=0,fori= Q 1 1 − E Q− 1,...,n 1, and H (P , )=0,with as above. Since the sheaves i are torsion- free on a− smooth curve,K it follows thatK they are locally-free and weE can identify 1 1 1 Ext ( i, n i) with H (P , n i i∗). Since there is a surjection E Q − Q − ⊗E 1 1 1 1 H (P ,V∗1 i∗) H ( P , n i i∗) 0 , P ⊗E −→ Q − ⊗E −→ 1 1 it is enough to prove that H (P , i∗) = 0. Note that the map j∗ : VP1 i∗ , obtained by dualizing the sheaf inclusion,E is generically surjective. −→ E Next we consider the map

n 1 − 1 V ∗ V = om 1 (V ∗1 ,V∗1) om 1 ( i, n i), ⊗ ⊗OP H OP P P −→ H OP E Q − i =1 M

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given by n 1 ψ pn i ψ ji i=1− . →{ − ◦ ◦ } One can easily check that the above map factors through , and moreover, it is generically surjective onto . Therefore, we can conclude theK two desired vanishing results from the easy K

Lemma F. Let be a coherent sheaf on P1 such that there exists a generically F surjective map from a trivial bundle on P1 to .ThenH1(P1, )=0. F F A standard computation using exact sequences and Riemann-Roch shows that n 1 0 1 n(n 1) − h (P , )= − +2 di. K 2 i=1 X By Proposition E, all irreducible components of d are smooth, of dimension n(n 1) n 1 HQ − := 2− +2 i=1 di. In particular Hd is an open subset in d. D As for the irreducibility, since we have already shown smoothness,HQ it is enough P to prove that d is connected. First notice that, by Theorem 2.3, successive quotients whichHQ do not correspond to morphisms to F vary in families of (base)

dimension strictly less than . (Without knowing the irreducibility of d e, D HQ − the proof of 2.3 shows only that e has pure dimension; of course, this does not U affect the above conclusion.) Therefore, any connected component will intersect Hd non-trivially and it suffices to show that Hd is connected. 1 Fix t P and let Φ : Hd F be the “evaluation at t” morphism Φ([f]) = f(t). ∈ → Pick x F and g SL(n, C). Translation by g induces an isomorphism between ∈ 1 ∈ 1 the fibres Φ− (x)andΦ− (gx), given by [f] [gf]. Since SL(n, C) acts transitively on F , it follows that all the fibres of Φ are isomorphic.→ Each fibre can be viewed as the moduli space of based holomorphic maps P1 F ,withfixedmultidegree.By 1 → [MM, Corollary 5.19] Φ− (x) is connected, hence Hd is connected as well. References

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Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 Current address: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730 E-mail address: [email protected]

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