VANISHING THEOREMS AND CHARACTER FORMULAS FOR THE HILBERT SCHEME OF POINTS IN THE PLANE 2 0 0 MARK HAIMAN 2 n a Abstract. Inanearlierpaper[14],weshowedthattheHilbertschemeofpoints J intheplaneHn=Hilbn(C2)canbeidentifiedwiththeHilbertschemeofregular 6 orbits C2n//Sn. Using this result, together with a recent theorem of Bridgeland, 1 KingandReid[4]onthegeneralized McKaycorrespondence, weprovevanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We ] apply the vanishing theorems to establish (among other things) the character G formula for diagonal harmonics conjectured by Garsia and the author in [9]. In A particularweprovethatthedimensionofthespaceofdiagonalharmonicsisequal . to (n+1)n−1. h t a m [ 1. Introduction 1 v In this article we continue the investigation begun in [14] of the geometry of the 8 Hilbert scheme of points in the plane and its algebraic and combinatorial impli- 4 cations. In the earlier article, we showed that the isospectral Hilbert scheme has 1 Gorenstein singularities, thereby proving the “n! conjecture” of Garsia and the au- 1 0 thorandthepositivity conjectureforMacdonald polynomials. Hereweextendthese 2 resultsbyprovingvanishingtheoremsfortensorproductsoftautological vector bun- 0 dles over the Hilbert scheme H = Hilbn(C2) and its zero fiber Z (the preimage of / n n h the origin under the Chow morphism σ: H → SnC2). n t a Thealgebraic-combinatorial consequenceofthenewresultsisaseriesofcharacter m formulas for the spaces of global sections of the vector bundles in question. As a : particular case, we obtain the character formula for the ring of coinvariants of the v Xi diagonal action of the symmetric group Sn on C2n, or equivalently, for the space of diagonal harmonics. This character formula had been conjectured by Garsia r a and the author in [9], where we showed that it implies several earlier conjectures in [11] relating the character of the diagonal harmonics to q-Lagrange inversion, q- Catalannumbers,andq-enumerationofrootedforestsandparkingfunctions. Oneof these earlier conjectures, now proven, is that the dimension of the space of diagonal harmonics is (1) dimDH = (n+1)n−1. n Date: Jan 14, 2002. Key words and phrases. Macdonald polynomials, diagonal harmonics, coinvariants, Hilbert scheme, sheaf cohomology, vanishing theorem, McKay correspondence. Research supported in part by N.S.F. grants DMS-9701218 and DMS-0070772 and the Isaac Newton Institute. 1 2 MARK HAIMAN Another is that the Hilbert series of the doubly-graded space (DH )ǫ of S - n n alternating diagonal harmonics is given by the q,t-Catalan polynomial C (q,t) n from [9, 12]. Hence C (q,t) has positive integer coefficients. Recently, Garsia and n Haglund [8] gave a different proof of this fact, based on a combinatorial interpreta- tion of thecoefficients. Yet another conjecture in [11] was that the space of diagonal harmonics is generated by certain S -invariant polarization operators applied to the n space of classical harmonics. We prove this “operator conjecture” here, using our identification of the coinvariant ring with the space of global sections of a vector bundle on Z . n To describe our results further, we recall from [14] that H is isomorphic to n the Hilbert scheme of orbits C2n//S for the diagonal action of S on C2n. Full n n definitions will be given in Section 2; for now we merely fix terminology in order to announce our main theorems. On the Hilbert scheme H we have a natural n tautological vector bundle B of rank n, while on C2n//S we have a tautological n bundle P of rank n!, with an S action in which each fiber affords the regular n representation. We can view both B and P as bundles on H via the isomorphism n H ∼= C2n//S . The usual tautological bundle B is the pushdown to H of the n n n sheaf O of regular functions on the universal family F over H . The “unusual” Fn n n tautological bundle P may similarly be identified with the pushdown of the sheaf O of regular functions on the isospectral Hilbert scheme X , which is actually Xn n the universal family over C2n//S . n Ourfirstmainresult,Theorem2.1, isavanishingtheoremforthehighercohomol- ogygroupsHi(H ,P ⊗B⊗l),i >0ofthetensorproductofP withanytensorpower n of B. We also identify the space of global sections H0(H ,P ⊗B⊗l). The latter n turnsouttobethecoordinateringR(n,l)ofthepolygraph,asubspacearrangement defined in [14], which plays an important technical role there and again here. This identification of R(n,l) with H0(H ,P ⊗B⊗l) explains why the polygraph carries n geometric information abouttheHilbertscheme, an explanation whichwewereonly able to hintat in [14]. Ourtheorem extends vanishingtheorems of Danila [5] for the tautological bundle B and of Kumar and Thomsen [17] for the natural ample line bundles O (k), k > 0. Indeed, it implies the vanishing of the higher cohomology Hn groups Hi(H ,O(k)⊗B⊗l) for all k,l ≥ 0. This is an immediate corollary, since n the trivial bundle O is a direct summand of P, and the line bundle O (1) is the Hn Hn highest exterior power of B. Our second main result, Theorem 2.2, is a vanishing theorem for the same vector bundles on the zero fiber Z . The vanishing part of this second theorem follows n immediately from the first theorem, applied to an explicit locally free resolution of O described in [12] and reviewed in detail in Section 2, below. By examin- Zn ing the resolution more closely, we can also identify the space of global sections H0(Z ,P ⊗B⊗l). When l = 0 it turns out that H0(Z ,P) coincides with the coin- n n variant ring for the diagonal S action on C2n, yielding the applications to diagonal n harmonics. Character formulas for the spaces of global sections follow from our vanishing theorems by an application of the Atiyah–Bott Lefschetz formula [1]. For the diag- onal harmonics, the calculation completes a program outlined by Procesi, who first VANISHING THEOREMS AND CHARACTER FORMULAS 3 proposed this method of determining the character. To carry out the calculation, we need to know the characters of the fibers of P at distinguished torus-fixed points I on H . By our results in [14], these characters are given by the Macdonald poly- µ n nomials. The character formulas we obtain here are therefore expressed in terms of Macdonald polynomials. Specifically, they are symmetric functions with coefficients depending on two parameters q,t. By virtue of being characters, these symmetric functions are necessarily q,t-Schur positive, that is, they are linear combinations of Schur functions by polynomials or power series in q and t with positive integer coef- ficients. It develops that certain operator expressions considered in [2] are instances of our character formulas, whose positivity partially establishes [2, Conjecture V]. The full conjecture in [2] is slightly stronger than what we obtain here. Its proof us- ing the methods of this paper would require an improved vanishing theorem, which we offer as a conjecture at the end of Section 3. Among our character formulas is one for the polygraph coordinate ring R(n,l) as a doubly graded algebra. Specializing this, we get a formula for its Hilbert seriesH (q,t)intermsofsymmetricfunctionoperatorswhoseeigenfunctionsare R(n,l) Macdonald polynomials. Acombinatorial interpretation ofH (q,t)isimplicitin R(n,l) the basis construction for R(n,l) in [14]. It can be made explicit (although we will not do so here), yielding an identity between a combinatorial generating function and the expression involving Macdonald operators in Corollary 3.9, below. This is one of only two combinatorial interpretations known at present for q,t-(Schur) positive expressions arising from our character formulas. The other is the Garsia– Haglund interpretation of C (q,t) mentioned above. An important problem that n remains open is to combinatorialize all the character formulas. InSection 2, we give definitions andstate our two main theorems infull. We then apply Theorem 2.1 to deduce Theorem 2.2. We deduce the character formulas and the operator conjecture from the vanishing theorems in Sections 3 and 4. For the proof of Theorem 2.1, we combine results from [14] with a recent general theorem of Bridgeland, King and Reid [4]. This is done in Section 5. To complete this introduction, we preview the proof of Theorem 2.1. The Bridgeland–King–Reid theorem concerns the Hilbert scheme of orbits V//G, for a finite subgroup G ⊆ SL(V). The theorem has two parts. The first part (which we will not use) is a criterion for V//G to be a crepant resolution of singularities of V/G, meaning that V//G is non-singular and its canonical sheaf is trivial. The second (and for us, crucial) part says that when the criterion holds there is an equivalence of categories Φ: D(V//G) → DG(V). Here D(V//G) is the derived category of complexes of sheaves on V//G with bounded, coherent cohomology, and DG(V) is the similar derived category of G-equivariant sheaves on V. Our identification of C2n//S with H shows that the Bridgeland–King–Reid cri- n n terion holds for V = C2n, G = S . It is well-known that H is a crepant resolution n n of C2n/S = SnC2, which is why we don’t need the first part of their theorem. n By the second part, however, we have an equivalence Φ between the derived cate- gory D(H ) of sheaves on the Hilbert scheme and the derived category DSn(C2n) n of finitely generated S -equivariant modules over the polynomial ring C[x,y] in 2n n 4 MARK HAIMAN variables. In this notation, Theorem 2.1 reduces to an identity ΦB⊗l = R(n,l). De- noting the inverse equivalence by Ψ, we may rewrite this as ΨR(n,l) = B⊗l, which is the form in which we prove it. The advantage of this form is that there is no sheaf cohomology involved in the calculation of Ψ, only commutative algebra. Con- veniently, the commutative algebraic fact we need is precisely the freeness theorem for the polygraph ring R(n,l), which was the key technical theorem in [14]. Thus we use here both the geometric results from [14] and the main algebraic ingredient in their proof. In closing, let us remark that a number of important problems relating to this circle of ideas remain open. We have already mentioned the problem of combi- natorializing the rest of the character formulas. Another set of problems involves phenomenain threeor more sets of variables. We expect, for example, that theana- log of the operator conjecture should continue to hold in additional sets of variables x,y,... ,z. For exactly threesetsof variables, weremindthereaderof theempirical conjecture in [11] that the dimension of the space of “triagonal” harmonics should be (2) 2n(n+1)n−2, and that of its S -alternating subspace should be n (3) (3n+3)(3n+4)···(4n+1)/3·4···(n+1). Our present methods do not readily apply to these problems, as we make heavy use of special properties of the Hilbert scheme Hilbn(C2) that do not hold for Hilbn(Cd) with d ≥ 3. Another open problem is to generalize from S to other Weyl groups or n complexreflectiongroups. Suchageneralization willnotbeentirelystraightforward, as shown by some obstacles discussed in [11] and [14]. Finally, despite the strength of the vanishing theorems proven here, they surely are not the strongest possible. The conjecture at the end of Section 3 suggests one possible improvement. 2. Definitions and main theorems We denote by H the Hilbert scheme of points Hilbn(C2) parametrizing 0- n dimensionalsubschemesoflengthnintheaffineplaneover C. ByFogarty’s theorem [7], H is irreducible and non-singular, of dimension 2n. As a matter of notation, n if V(I) ⊆ C2 is the subscheme corresponding to a (closed) point of H , we refer to n this point by its defining ideal I ⊆ C[x,y]. Thus H is identified with the set of n ideals I such that C[x,y]/I has dimension n as a complex vector space. The multiplicity of a point P ∈ V(I) is the length of the Artin local ring (C[x,y]/I) . The multiplicities of all points in V(I) sum to n, giving rise to a P 0-dimensional algebraic cycle m P of weight m = n. We may view this i i i i i cycle as an unordered n-tuple [[P ,... ,P ]]∈ SnC2, in which each point is repeated 1 n P P according to its multiplicity. The Chow morphism (4) σ: H → SnC2 =C2n/S n n istheprojectiveandbirationalmorphismmappingeachI ∈ H tothecorresponding n algebraic cycle σ(I) = [[P ,... ,P ]]. 1 n VANISHING THEOREMS AND CHARACTER FORMULAS 5 We denote by F the universal family over the Hilbert scheme, n F ⊆ H ×C2 n n (5) π  Hn, y whose fiber over a point I ∈ H is the subscheme V(I) ⊆ C2. The universal family n is flat and finite of degree n over H , and hence is given by F = SpecB, where n n B = π O is a locally free sheaf of O -algebras of rank n. Here and elsewhere ∗ Fn Hn we identify any locally free sheaf of rank r with the rank r algebraic vector bundle whose sheaf of sections it is. Then B is the tautological vector bundle, the quotient of the trivial bundle C[x,y]⊗ O with fiber C[x,y]/I at each point I ∈ H . C Hn n If G is a finite subgroup of GL(V), where V = Cd is a finite-dimensional complex vector space, we denote by V//G the Hilbert scheme of regular G-orbits in V, as defined by Ito and Nakamura [15, 16]. Specifically, if v ∈ V has trivial stabilizer (as is true for all v in a Zariski open set), then its orbit Gv is a point of Hilb|G|(V), and V//Gis theclosure inHilb|G|(V)of thelocus of all such points. By definition, V//G is irreducible. The universal family over Hilb|G|(V) restricts to a universal family X ⊆ (V//G)×V (6) ρ  V//G. y The group G acts on X and on the tautological bundle P = ρ O . This action ∗ X makes P a vector bundle of rank |G| whose fibers afford the regular representation ofG. ThereisacanonicalChowmorphismV//G → V/G,whichcanbeconveniently defined as follows. Since P is a sheaf of O -algebras, it comes equipped with a V//G homomorphism O → P. This homomorphism is an isomorphism of O V//G V//G onto the sheaf of invariants PG. Geometrically, this means that the map X/G → V//G induced by ρ is an isomorphism. The canonical projection X → V induces a morphismX/G → V/G whosecompositewiththeisomorphismV//G ∼= X/Gyields theChowmorphism. TheChowmorphismisprojectiveandbirational,restrictingto anisomorphismontheopenlocusconsistingoforbitsGv forv withtrivialstabilizer. The case of interest to us is V = C2n, G = S , where S acts on C2n = (C2)n by n n permuting the cartesian factors. This is the same as the diagonal action of S on n the direct sum of two copies of its natural representation Cn. Coordinates on C2n will be denoted (7) x,y = x ,y ,... ,x ,y ; 1 1 n n then S acts by permuting the x variables and the y variables simultaneously. In n [13] we constructed a canonical morphism C2n//S → H such that the composite n n (8) C2n//S → H →σ SnC2 n n istheChowmorphismforC2n//S . ByTheorem5.1of[14],thecanonicalmorphism n is an isomorphism C2n//S ∼= H . n n 6 MARK HAIMAN Theuniversal family over C2n//S will bedenoted X . We identify C2n//S with n n n H by means of the canonical isomorphism, so that the projection ρ of the universal n family onto C2n//S becomes a morphism from X to H . We have a commutative n n n square X −−f−→ C2n n (9) ρ Hn −−σ−→ SnC2, y y in which X ⊆ H ×C2n is the set-theoretic fiber product, with its induced reduced n n scheme structure. In other words, X is the isospectral Hilbert scheme, as defined n in [14]. We again write P = ρ O , as we did above for a general V//G. Now we ∗ Xn regard P as a bundle on H rather than on C2n//S . Thus H has two different n n n “tautological” bundles, the usual one B and the unusual one P. The unusual tauto- logical bundle P has rank n!, with an S action affording the regular representation n on every fiber. Our notation for the various schemes, bundles and morphisms just described is identical to that in [14]. The two-dimensional torus group (10) T2 = (C∗)2 acts linearly on C2 as the group of 2×2 diagonal matrices. We write t−1 0 (11) τ = t,q 0 q−1 (cid:20) (cid:21) for its elements. Note that when a group G acts on a scheme V, elements g ∈ G act on regular functions f ∈ O(V) as gf = f ◦g−1. The inverse signs in (11) serve to make T2 act on the coordinate ring C[x,y] of C2 by the convenient rule (12) τ x = tx; τ y = qy. t,q t,q The action of T2 on C2 induces an action on the Hilbert scheme H and all other n schemes under consideration. In particular, T2 acts on the universal family F n and the isospectral Hilbert scheme X , so that the projections π: F → H and n n n σ: X → H are equivariant. Hence T2 acts equivariantly on the vector bundles n n B and P. There are induced T2 actions on various algebraic spaces, such as the coordinate ring C[x,y] of C2n, the space of global sections of any T2-equivariant vector bundle, or the fiber of such a bundle at a torus-fixed point in H . In these n spaces, the T2 action is equivalently described by a Z2-grading. Namely, an element f is homogeneous of degree (r,s) if and only if it is a simultaneous eigenvector of the T2 action with weight τ f = trqsf. Where there is an obvious natural double t,q grading, as in C[x,y], it coincides with the weight grading for the torus action. We have defined the bundles whose tensor products will be the subject of our vanishing theorems. The theorems also describe their spaces of global sections. To identify these spaces, we first need to recall the definition of the polygraph Z(n,l) from[14]. There,Z(n,l)wasdefinedasacertainunionoflinearsubspacesinC2n+2l, VANISHING THEOREMS AND CHARACTER FORMULAS 7 but it is better here to describe it first from a Hilbert scheme point of view. Let (13) W = X ×Fl /H n n n be the fiber product over H of X with l copies of the universal family F . The n n n scheme W is thus a closed subscheme of H ×C2n+2l, since we have X ⊆ H ×C2n n n n andF ⊆ H ×C2. WenowdefineZ(n,l) ⊆ C2n+2l tobetheimageoftheprojection n n of W on C2n+2l. To see that this agrees with the original definition in [14], let us identify the set Z(n,l) more directly. From (9), we see that a point of X is an ordered tuple n (14) (I,P ,... ,P )∈ H ×C2n 1 n n such that σ(I) = [[P ,... ,P ]]. In particular, this implies V(I) = {P ,... ,P } as a 1 n 1 n set. A point of F is a pair (I,Q) ∈ H ×C2 such that Q ∈ V(I). Hence a point of n W is a tuple (15) (I,P ,... ,P ,Q ,... ,Q ) ∈H ×C2n+2l 1 n 1 l n such that σ(I) = [[P ,... ,P ]] and Q ∈ {P ,... ,P } for all 1 ≤ i≤ l. Projecting 1 n i 1 n on C2n+2l, we see that (16) Z(n,l)= {(P ,... ,P ,Q ,... ,Q )∈ C2n+2l : Q ∈ {P ,... ,P } ∀i}. 1 n 1 l i 1 n Thisisequivalenttothedefinitionin[14]. TheschemeW isflatoverH andreduced n over the generic locus (the open set in H where the P are all distinct). Hence W n i is reduced. The set-theoretic description we have just given of the projection of W on Z(n,l) therefore also describes a morphism of schemes W → Z(n,l), in which we regard Z(n,l) as a reduced closed subscheme of C2n+2l. As in [14], the coordinate ring of the polygraph Z(n,l) will be denoted R(n,l). Writing (17) x,y,a,b = x ,y ,... ,x ,y ,a ,b ,... ,a ,b 1 1 n n 1 1 l l for the coordinates on C2n+2l, we see that R(n,l) is the quotient of the polynomial ring C[x,y,a,b] by a suitable ideal I(n,l). Given a global regular function on Z(n,l), we may compose it with the projection W → Z(n,l) to get a global regular functiononW,whichisthesamethingasaglobalsection ofP ⊗B⊗l onH . Hence n we have a canonical injective ring homomorphism (18) ψ: R(n,l) ֒→ H0(H ,P ⊗B⊗l). n We can now state our first vanishing theorem, which will be proven in Section 5. Theorem 2.1. For all l we have (19) Hi(H ,P ⊗B⊗l)= 0 for i> 0, and n (20) H0(H ,P ⊗B⊗l) = R(n,l), n where R(n,l) is the coordinate ring of the polygraph Z(n,l) ⊆ C2n+2l. 8 MARK HAIMAN The equal sign in (20) is to be understood as signifying that the homomorphism ψ in (18) is an isomorphism. Our second vanishing theorem is the analog of Theorem 2.1 for the restriction of the tautological bundles to the zero fiber Z = σ−1({0}) ⊆ H . In [12] we n n showed that the scheme-theoretic zero fiber is reduced, so there is no ambiguity as to the scheme structure of Z . The ideal of the origin {0}⊆ SnC2 = C2n/S is the n n homogeneousmaximalidealm = C[x,y]Sn intheringofinvariantsC[x,y]Sn. Pulled + back to H via σ, the elements of m represent global functions on H that vanish n n on Z . The bundle P ⊗B⊗l is a sheaf of O -algebras, so we have a canonical n Hn inclusion (21) H0(H ,O )⊆ H0(H ,P ⊗B⊗l). n Hn n Our choice of coordinates x,y,a,b on Z(n,l) identifies C[x,y] and C[x,y]Sn with subrings of R(n,l), in such a way that the diagram H0(H ,O ) ֒→ H0(H ,P ⊗B⊗l) n Hn n (22) σ∗ ψ C[x,xy]Sn ֒→ R(xn,l)   commutes. ItfollowsimmediatelythatψmapseveryelementoftheidealmR(n,l)to a section of P ⊗B⊗l that vanishes on Z . Composing ψ with restriction of sections n to the zero fiber, we get a well-defined homomorphism (23) ψ : R(n,l)/mR(n,l) → H0(Z ,P ⊗B⊗l). 1 n A priori, ψ need neither be injective nor surjective, but according to our next 1 theorem, it is both. Theorem 2.2. For all l we have (24) Hi(Z ,P ⊗B⊗l) =0 for i > 0, and n (25) H0(Z ,P ⊗B⊗l) = R(n,l)/mR(n,l), n where R(n,l) is the polygraph coordinate ring and m is the homogeneous maximal ideal in the subring C[x,y]Sn ⊆ R(n,l). Again, the equal sign in (25) signifies that the homomorphism ψ in (23) is an 1 isomorphism. In a sense, Theorem 2.2 is a corollary to Theorem 2.1. Its proof uses an O - Hn locally free resolution of O , which we now describe. Afterwards, we will prove Zn that Theorem 2.1 implies Theorem 2.2. The resolution we construct will be T2- equivariant. Towriteitdownweneedabitmorenotation. LetC andC denotethe t q 1-dimensional representations of T2 on which τ ∈ T2 acts by t and q, respectively. t,q We write (26) O = C ⊗ O , O = C ⊗ O t t C Hn q q C Hn for O with its natural T2 action twisted by these 1-dimensional characters. The Hn T2-equivariantsheavesO andO maybethoughtofascopiesofO withrespective t q Hn degree shifts of (1,0) and (0,1). VANISHING THEOREMS AND CHARACTER FORMULAS 9 There is a trace homomorphism of O -modules Hn (27) tr: B → O Hn defined as follows. Let α ∈ B(U) be a section of B on some open set U. Since B is a sheaf of O -algebras and also a vector bundle, there is a regular function Hn tr(α) ∈ O (U) whose value at I is the trace of multiplication by α on the fiber Hn B(I). The sheaf B is a quotient of C[x,y]⊗O , so it is generated by its global Hn sections xrys (i.e., they span every fiber). The trace map is given on these sections by n (28) tr(xrys) = p (x,y) = xrys. r,s i i def i=1 X Here we regard the symmetric function p ∈ C[x,y]Sn, called a polarized power- r,s sum, as a global regular function on H pulled back from SnC2 via the Chow n morphism. To verify (28) we need only check it on points I in the generic locus, where the fiber B(I) = C[x,y]/I is the coordinate ring of a set of n distinct points {(x ,y ),... ,(x ,y )} ⊆ C2. There it is clear that the eigenvalues of multiplication 1 1 n n by xrys in B(I) are just xrys,... ,xrys. In particular, 1 tr(1) =1, so 1 1 n n n 1 (29) tr: B → O n Hn is left inverse to the canonical inclusion O ֒→ B. Thus we have a direct-sum Hn decomposition of O -module sheaves, or of vector bundles, Hn (30) B = O ⊕B′, where B′ = ker(tr). Hn The projection of B on its summand B′ is given by id−1 tr, so from (28), we see n that B′ is generated by its global sections 1 (31) xrys− p (x,y). r,s n Here we can omit the section corresponding to r = s = 0, which is identically zero. Let J bethe sheaf of ideals in B generated by the global sections x and y and the subsheafB′. AnalternativewaytodescribeJ isasfollows. ThereareT2-equivariant sheaf homomorphisms O → B and O → B sending the generating section 1 in O t q t and O to x and y, respectively. Combining these with the inclusion B′ ֒→ B, we q get a homomorphism of sheaves of O -modules Hn (32) ν: B′⊕O ⊕O → B. t q Now composing 1⊗ν: B ⊗(B′⊕O ⊕O ) → B ⊗B with the multiplication map t q µ: B ⊗B → B, we get a homomorphism of sheaves of B-modules (33) ξ: B⊗(B′⊕O ⊕O ) → B, t q whose image is exactly J. Note that since x and y generate B as a sheaf of O - Hn algebras, the canonical homomorphism O → B/J is surjective. Thus B/J is Hn identified with a quotient of O , which turns out to be O . Hn Zn 10 MARK HAIMAN Proposition 2.3. Let J be the sheaf of ideals in B generated by x, y and B′. Then B/J is isomorphic as a sheaf of O -algebras to O . Hn Zn Let us recall the proof from [12, 13], skippingsome details. Denote by Z˜ the set- n theoreticpreimageπ−1(Z ),regardedasareducedclosedsubschemeoftheuniversal n family F . Clearly the regular functions x, y and p (x,y) for r+s > 0 vanish on n r,s Z˜ . By an old theorem of Weyl [26], the p generate C[x,y]Sn, so their vanishing n r,s defines Z as a subscheme of H . Hence Z˜ is defined set-theoretically by the n n n vanishing of x, y and the p , or equivalently of x, y, and every xrys − 1p . But r,s n r,s these sections generate J, so the subscheme of F defined by the ideal sheaf J ⊆B n coincides set-theoretically with Z˜ . We already know that B/J ∼= O for some n Z subscheme Z ⊆ H , and this shows that Z coincides set-theoretically with Z . Now n n F is flat and finite over the non-singular scheme H , hence Cohen-Macaulay. Since n n Z˜ projects bijectively on Z , it has codimension n+1 in F . But B′⊕O ⊕O is n n n t q locally free of rank n+1, so J is everywhere locally generated by n+1 elements. It follows that SpecB/J is a local complete intersection in F . Finally, one shows n that SpecB/J is generically reduced, hence reduced, which implies B/J ∼= O . Zn Our motive in reviewing this is to note that J is locally a complete intersection ideal in B generated by the image under ξ of any local basis of B′⊕O ⊕O . t q Hence the Koszul complex on the map ξ in (33) is a resolution of B/J ∼= O . Since Zn everything in the construction is T2-equivariant we deduce the following result. Proposition 2.4. We have a T2-equivariant locally O -free resolution Hn (34) ··· → B⊗∧k(B′⊕O ⊕O ) → ··· → B⊗(B′⊕O ⊕O ) → B → O → 0, t q t q Zn ξ where ξ is the sheaf homomorphism in (33). As in [12], it follows as a corollary that the scheme-theoretic zero fiber is equal to the reduced zero fiber, and that it is Cohen-Macaulay. Proof that Theorem 2.1 implies Theorem 2.2. Let V. denote the complex in (34) with the final term O deleted. The fact that (34) is a resolution means that Zn V. and O are isomorphic as objects in the derived category D(H ). Here and Zn n below we work inthe derived category of complexes of sheaves of O -modules with Hn bounded, coherent cohomology. Note that V. is a complex of locally free sheaves, each of which is a sum of direct summands of tensor powers of B. It follows from Theorem2.1 thatP ⊗B⊗l⊗V. isacomplex of acyclic objects fortheglobal section functor Γ on H , so we have n (35) RΓ(P ⊗B⊗l⊗V.) = Γ(P ⊗B⊗l⊗V.). Now P⊗B⊗l⊗V. is isomorphic to P⊗B⊗l⊗O in D(H ), so Hi(Z ,P ⊗B⊗l)= Zn n n RiΓ(P ⊗B⊗l⊗V.) is the i-th cohomology of the complex in (35). This complex is zero in positive degrees, so we deduce that Hi(Z ,P ⊗B⊗l) = 0 for i > 0, n which is the first part of Theorem 2.2. This is just the standard argument for the higher cohomology vanishing of a sheaf with an acyclic left resolution. Since