Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants

Contemporary Mathematics Hilbert scheme intersection numbers, Hurwitz numbers, and Gromov-Witten invariants Wei-Ping Li 1, Zhenbo Qin 2, and Weiqiang Wang 3 Abstract. Some connections of the ordinary intersection numbers of the Hilbert scheme of points on surfaces to the Hurwitz numbers for P 1 as well as to the relative Gromov-Witten invariants of P 1 are established. 1. Introduction In [LQW2], a direct algebraic relation between the equivariant cohomology ring of the Hilbert scheme C 2 of n points on the affine plane and the Gromov- Witten theory of the complex projective line P 1 has been found. This Gromov- Witten/Hilbert correspondence was formulated in the operator formalism of Fock space. In concrete terms, for partitions λ, µ of n, the so-called N-point function F λ,µz 1,..., z N of disconnected stationary Gromov-Witten invariants of P 1 relative to 0, P 1 studied extensively by Okounkov and Pandharipande [OP1] is identified precisely with a generating function of the equivariant intersection numbers of Hilbert schemes. Furthermore, the insertions associated to the point class in P 1 at the marked points in the Gromov-Witten theory corresponds precisely to the insertions of the equivariant Chern characters on the Hilbert scheme side. These results will be reviewed in Sect. 2. The results of [LQW2] has been generalized to the minimal resolution associated to a cyclic subgroup of SL 2 C [QW]. It is natural to ask if there is any natural connection between the ordinary intersection numbers on Hilbert schemes and Gromov-Witten invariants of P 1. We may also ask for a natural connection between the intersection numbers on Hilbert schemes and Hurwitz numbers. Note that the stationary Gromov-Witten invariants on P 1 have been known cf. [OP1] to be equivalent to the Hurwitz numbers for P 1. Thus, these two questions are closely related. The modest goal of this note is to establish some precise relations of these sorts without really computing these 2000 Mathematics Subject Classification. Primary 14C05, 14N35; Secondary 14H30, 17B69. 1 Partially supported by the grant HKUST6114/02P. 2 Partially supported by an NSF grant. 3 Partially supported by an NSA grant. 1 c 0000 copyright holder

2 LI, QIN, AND WANG numbers explicitly. It has been believed that there should be some deep geometric connections behind and beyond the Gromov-Witten/Hilbert correspondence of [LQW2] and of this note. The recent remarkable works [MNOP, OP2] seem to be providing a right geometric framework. Let X be a smooth projective complex surface, and X be the Hilbert scheme of n points on X. Let H X be the total cohomology of X with complex coefficients. It is by now well known [Na1, Na2, Gro] that H X = H X n=0 is a Fock space of a Heisenberg algebra generated by the geometrically constructed operators a m α EndH X with m Z and α H X. Given an ideal I in the cohomology ring H X, define I to be the subspace of H X spanned by Heisenberg monomial classes a n1 α 1 a nb α b 0 where α i I for some i and n 1,..., n b > 0 with l n l = n. By the Lemma 2.7 i in [LQW1], I is an ideal in the cohomology ring H X. Recall for a partition λ of n, we have the Heisenberg monomial class a λ 1 X H X see 2.5 for a definition. One main result in this paper is the following theorem. Theorem 1.1. Let 2 k n, µ n, and ςz = e z/2 e z/2. Then, 1.1 a 1 n k k1 X a µ 1 X c λ k,µ a λ 1 X mod I λ n lλ=lµ+1 k where c λ k,µ is the coefficient of zk 1 in k 1! z λ F ςz λ,µz, and I = 4 H i X. We remark that the structure constants c λ k,µ above can be reduced by the Theorem 2.12 in [LQW1] to the affine plane case and then the Theorem 4.10 in [Lehn] becomes applicable, while the 1-point disconnected series F λ,µ z is given by Proposition 4.1 in [LQW2] which generalizes the 1-point connected series Fλ,µ z calculated earlier in [OP1]. Theorem 1.1 reveals a very interesting connection between these seemingly unrelated numbers, and can be regarded as the numerical aspect of the Gromov-Witten/Hilbert correspondence discovered in [LQW2]. Theorem 1.1 is verified in Sect. 4. The main idea in the proof is to use results from [LQW1, LQW2], together with the geometric realization of Heisenberg monomial classes. Such a geometric realization up to a possible ambiguity of the exact scalar multiple seems to be folklore, and we provide a detailed proof of it in Sect. 3. Our Theorem 1.1 handles the cup product a 1 n k k1 X a µ 1 X mod I

HILBERT, HURWITZ, AND GROMOV-WITTEN 3 for the partition 1 n k k of n. By Lemma 2.11 in [LQW1] also cf. [Lehn], the quotient ring H X /I is generated by the n 1 cohomology classes a 1 n k k1 X mod I, 2 k n. It follows that the cup product a ν 1 X a µ 1 X mod I for an arbitrary partition ν n can be obtained by repeatedly applying Theorem 1.1. In other words, the structure constants of a ν 1 X a µ 1 X mod I are also related to Gromov-Witten invariants. We might ask further how the structure constants in the cup product a ν 1 X a µ 1 X for two arbitrary partitions ν and µ of n, not modulo the ideal I, are related to Gromov-Witten theory. A natural class of surfaces for such a potential study could be the minimal resolutions [QW], or the toric surfaces studied by Nakajima and [LQW3] which are the total spaces of line bundles over P 1. In Sect. 5, we establish a precise identity between some general structure constants under the cup product of the Hilbert schemes of points on the affine plane and certain Hurwitz numbers for P 1 Theorem 5.2. A theorem of Lehn-Sorger [LS] and independently of Vasserot [Vas] on the cohomology ring H C 2 is essential in its proof. We refer to [GJV] for more references on single Hurwitz numbers and a recent development which relates double Hurwitz numbers to Gromov-Witten theory. Acknowledgments. The authors thank Dan Edidin and Ravi Vakil for stimulating discussions, and the referee for many valuable comments. 2. Hilbert schemes and Gromov-Witten invariants for P 1 In this section, we recall from [Na1, Na2, Gro] the geometric constructions of Heisenberg operators on the cohomology of Hilbert schemes of points on surfaces. We will also review results from [Vas, LQW2] concerning Heisenberg operators on the equivariant cohomology of Hilbert schemes of points on the affine plane and their relations with the Gromov-Witten theory of the projective line. 2.1. Let X be a smooth projective complex surface, and let X be the Hilbert scheme of points in X. An element in X is represented by a length-n 0-dimensional closed subscheme of X. Fix n > 0. For m 0, let Q [m,m] = and define Q [m+n,m] to be the closed subset: 2.1 {ξ, x, η X [m+n] X X [m] ξ η and SuppI η /I ξ = {x}}. The linear operator a n α EndH X

4 LI, QIN, AND WANG with α H X is defined by 2.2 a n αa = p 1 [Q [m+n,m] ] ρ α p 2a for a H X [m], where p 1, ρ, p 2 are the projections of X [m+n] X X [m] to X [m+n], X, X [m] respectively. Define a n α EndH X to be 1 n times the operator obtained from the definition of a n α by switching the roles of p 1 and p 2. We also set a 0 α = 0. These linear operators a m α EndH X with m Z and α H X satisfy the Heisenberg algebra commutation relation: 2.3 [a m α, a n β] = m δ m, n αβ Id HX. The space H X is the Fock space i.e. an irreducible module over the Heisenberg algebra generated by the linear operators a m α with a highest weight vector X 0 = 1 H 0 X [0] = C. It follows that H X is linearly spanned by all the Heisenberg monomial classes a n1 α 1 a nk α k 0 where k 0, n 1,..., n k > 0, and α 1,..., α k H X. For a partition λ = 1 m1 2 m2... where m i stands for the multiplicity of part i in λ, we introduce the notations: λ = r 1 rm r, lλ = r 1 m r, 2.4 2.5 z λ = r 1 r m r m r!, a λ 1 X = 1 a r 1 X mr 0 z λ r 1 where 1 X H 0 X is the fundamental cohomology class of X. 2.2. Next, we work with the equivariant cohomology of the Hilbert scheme C 2. Let us fix an action on C 2 by the torus T = C by letting the action on the affine coordinate functions w and z of C 2 be sw, z = sw, s 1 z. It induces an action on C 2 with finitely many fixed points parametrized by partitions of n [ES]. Let HT C2 be the equivariant cohomology of C 2 with complex coefficients. Without loss of information it is convenient to work with HT 2nC2 instead of the whole equivariant cohomology group HT C2.

HILBERT, HURWITZ, AND GROMOV-WITTEN 5 For example, a ring product on the n-th component HT 2nC2 can be defined in terms of the equivariant cup product: 2.6 t n w 1 w 2 = w 1 w 2, w 1, w 2 H 2n T C 2 where t is the character associated to the 1-dimensional standard module of T. The algebra HT 2nC2 was shown in [Vas] to be isomorphic to the class algebra associated to the n-th symmetric group. Put H C2,T = n=0 H 2n T C 2. In the equivariant setup, the Heisenberg operators a T mα EndH C2,T with m Z and α H T C2 can also be defined geometrically [Vas]. Set p m = a T m[{0} C] p λ = a T λ[{0} C] where [{0} C] denotes the equivariant fundamental cycle or the equivariant fundamental cohomology class of the equivariant closed subscheme {0} C C 2, and a T λ [{0} C] is defined similarly as in 2.5. It is proved in [Vas] see also Proposition 2.2, [LQW2] that the operators p m, m Z, acting on the space H C 2,T satisfy the Heisenberg commutation relation: 2.7 [p m, p n ] = mδ m, n Id, and that the space H C 2,T becomes the Fock space over the Heisenberg algebra with highest weight vector 0 = 1 HT 0 X[0] = C. For λ = 1 m1 2 m2..., we have p λ = 1 p mr r 0. z λ r 1 There is a natural nondegenerate bilinear form satisfying the property: 2.8, n : H 2n T C 2 H 2n C 2 C p λ, p µ n = 1 z λ δ λ,µ, λ, µ n. This induces a natural bilinear form, on the Fock space H C2,T : u n, v n := u n, v n n n n n where u n, v n H 2n T C2. T

6 LI, QIN, AND WANG 2.3. Following [LQW2], we recall the N-point function G λ,µ z 1,..., z N of equivariant intersection numbers on the Hilbert schemes C 2. First of all, let 2.9 O C 2 = π O Zn be the rank-n tautological bundle over C 2, where Z n = {ξ, x C 2 C 2 x Suppξ} is the universal codimension-2 subscheme in C 2 C 2 and π is the projection from Z n to C 2. Then O C 2 is T -equivariant. Let ch k,t be the k-th component of the T -equivariant Chern character of O C 2. Put 2.10 ch k = t n k ch k,t H2n T C 2. Imitating the ordinary cohomolgy case as in [Lehn], we define the Chern character operator G z in EndH X by sending a H n to a k 0 z k ch k for each n, where z is a variable. The N-point function with variables z 1,..., z N of equivariant intersection numbers on Hilbert schemes is defined as G λ,µ z 1,..., z N := z k1 1... zk N k 1,...,k N N p λ, ch k 1 ch k 2... ch k N p µ. The N-point function can be reformulated as the expectation value of operators in the bosonic Fock space which can be conveniently identified with H C 2,T thanks to 2.7 as follows: 2.11 where G λ,µ z 1,..., z N = p λ, Hz 1... Hz N p µ Hz := 1 ςz k Z+ 1 2 e kz E k,k is an operator defined in terms of the standard basis elements E k,k in the completed infinite-rank general linear Lie algebra ĝl which acts on the bosonic Fock space H C 2,T see [MJD, LQW2, Wang], and ςz := e z/2 e z/2. The proof of 2.11 uses the Chern character operator G z in an essential way. 2.4. Following [OP1], we recall here the N-point function F λ,µz 1,..., z N of disconnected stationary Gromov-Witten invariants of P 1 relative to 0, P 1. Let λ, µ be partitions of n. Let M g,n P 1, λ, µ be the moduli space which parameterizes genus-g, N-pointed relative stable maps with possibly disconnected domain and with monodromy λ, µ at 0, P 1. The foundational aspects of relative Gromov-Witten theory were developed by A.M. Li-Y. Ruan [LR], J. Li

HILBERT, HURWITZ, AND GROMOV-WITTEN 7 [Li], E. Ionel-T. Parker [IP], etc. The relative Gromov-Witten invariant we need is defined by integration over the virtual fundamental class: P 1 N N λ, τ ki pt, µ := ψ ki i ev i pt. [M g,n P1,λ,µ] vir Here pt denotes the class in H 2 P 1 poincaré dual to a point, ev i is the evaluation morphism at the i-th marked point, ψ i is the first Chern class of the i-th cotangent line bundle L i over the moduli space. The N-point function of Gromov-Witten invariants is then defined as F λ,µz 1,..., z N := k 1,...,k N = 2 λ, P 1 N N τ ki pt, µ z ki+1 i. Here by convention τ 2 pt = 1. The Fock space formalism developed in [OP1] allows them to express the N- point function F as the expectation value of operators on the bosonic Fock space which is again identified with H C2,T as follows: 2.12 where F λ,µz 1,..., z N = p λ, E 0 z 1 E 0 z N p µ E 0 z := k Z+ 1 2 e kz E k,k + 1 ςz Id. 2.5. The operator Hz arises naturally from our study of Hilbert schemes while E 0 z plays a fundamental role in Okounkov-Pandharipande s approach to Gromov-Witten theory. These two operators are related to each other by 2.13 Hz = 1 ςz E 0 z 1 ςz Id In view of 2.11, 2.12 and 2.13, we have a Gromov-Witten/Hilbert correspondence, that is, a very simple way of converting between the N-point function F of relative Gromov-Witten invariants on P 1 and G of equivariant intersection numbers on C 2.. 3. Geometric representations of Heisenberg monomial classes In this section, we describe geometric representations of Heisenberg monomial classes. These geometric representations will be used in the next section. Let n = k n i be a partition of n. Fix closed submanifolds X 1,..., X k of X in general position so that any subset of the X i s meet in the expected dimension. Define W n 1, X 1 ;... ; n k, X k X to be the closed subset consisting of all ξ X which admit filtrations with ξ = ξ k... ξ 1 ξ 0 = lξ i = lξ i 1 + n i

8 LI, QIN, AND WANG and 3.1 for 1 i k. Let SuppI ξi 1 /I ξi = x i X i W n 1, X 1 ;... ; n k, X k 0 W n 1, X 1 ;... ; n k, X k consist of all ξ W n 1, X 1 ;... ; n k, X k such that the points x 1,..., x k in 3.1 are distinct. Lemma 3.1. Let n 1,..., n k, X 1,..., X k be as in the preceding paragraph. Denote W n 1, X 1 ;... ; n k, X k 0 and W n 1, X 1 ;... ; n k, X k by W 0 and W respectively. Let W c = W W 0. Then, dimw c < dimw 0 = 2n i 2 + dimx i. Proof. Note that the points ξ W 0 are of the form ξ = η 1 +... + η k with lη i = n i and Suppη i = x i X i for every i. So dimw 0 = 2n i 2 + dimx i. Next, without loss of generality, we may assume that a point ξ W c satisfies x 1 =... = x k1 def = y 1,..., x k1+...+k r 1+1 =... = x k1+...+k r def = y r where r 1, k 1 2, k 2,..., k r 1, k 1 +... + k r = k, and the points y 1,..., y r are distinct. Since the submanifolds X 1,..., X k are in general position, these ξ s either do not exist at all, or form a subset of dimension at most [ k1 ] k 1 2 n i 1 + dimx i 4k 1 1 +... = < + [ 2 kr ] k r n k1+...+k r 1+i 1 + dimx k1+...+k r 1+i 4k r 1 2n i 2 + dimx i 2n i 2 + dimx i. r 2k i 1 It follows that dimw c < k 2n i 2 + dimx i = dimw 0. We conclude from Lemma 3.1 that W n 1, X 1 ;... ; n k, X k is a union of irreducible closed subsets among which the closure of W n 1, X 1 ;... ; n k, X k 0 is the only irreducible component having the largest dimension. The following generalizes the well-known fact that a 1n 1 X corresponding to the partition λ = 1 n is the fundamental cohomology class 1 X of X.

HILBERT, HURWITZ, AND GROMOV-WITTEN 9 Proposition 3.2. Let l, k 0, s i 0 1 i l, n i > 0 1 i k. 4 Let α 1,..., α k H i X be represented by the submanifolds X 1,..., X k X respectively such that X 1,..., X k are in general position. Then, the Heisenberg monomial class l k a i 1 X si a ni α i 0 s i! is represented by the closure of 3.2 W 1, X;... ; 1, X;... ; l, X;... ; l, X; n }{{}}{{} 1, X 1 ;... ; n k, X k 0. s 1 times s l times Proof. We use induction on l is i + k n i. When l is i + n i = 0, the conclusion is trivially true. In the following, assuming that the conclusion is true whenever l is i + k n i < n, we shall prove that it still holds when l is i + n i = n. There are two cases: s i > 0 for some i, or n i > 0 for some i. Since the proof of these two cases are similar, we only prove the first case. So let s i > 0 for some i. Without loss of generality, we may assume that i = l, i.e., s l > 0. Denote 3.2 and the Heisenberg monomial in Proposition 3.2 by Ws 0 l and A sl respectively. By the induction hypothesis, A sl 1 is represented by the closure Ws 0 l 1. By 2.2 and 2.1, a l1 X A sl 1 is represented by p 1 [Q [n,n l] X ] p 2 W 0 sl 1 where p 1, p 2 are the projections of X X [n l], and Q [n,n l] X = {ξ, η X X [n l] ξ η and SuppI η /I ξ is a point}. Since a l 1 X A sl 1 = s l A sl, we see that A sl is represented by 1 p 1 [Q [n,n l] X ] p s 2 W 0 sl 1 3.3. l Note that the image p 1 Q [n,n l] X p 1 2 W 0 sl 1 is contained in the closed subset W 1, X;... ; 1, X;... ; l, X;... ; l, X; n }{{}}{{} 1, X 1 ;... ; n k, X k ; l, X s 1 times s l 1 times whose real dimension is precisely the cohomology degree of A sl, and W 0 s l = W 1, X;... ; 1, X;... ; l, X;... ; l, X; n }{{}}{{} 1, X 1 ;... ; n k, X k ; l, X 0. s 1 times s l 1 times

10 LI, QIN, AND WANG By Lemma 3.1, p 1 [Q [n,n l] X ] p 2 W 0 sl 1 is a multiple of W 0 s l. By 3.3, it remains to show that the multiplicity is equal to s l. Indeed, take a general element ξ W 0 s l. Then ξ = l s i ξ i,j + j=1 where lξ i,j = i, lξ i = n i, Suppξ i,j = x i,j, Suppξ i = x i, all the points x i,j, x i are distinct, x i X k if and only if i = k, and ξ i,j X k for all i, j, k. Now, p 1 1 ξ Q [n,n l] X p 1 2 W 0 sl 1 consists of exactly s l different pairs ξ, ξ ξ l,j, 1 j s l. Recall from [Na2, Lehn, LQW1] that as in the case of the projective surface X, Heisenberg operators a m 1 C 2 with m > 0 on the space def H C 2 = H C 2 n=0 can be defined similarly. The following is analogous to Proposition 3.2. Proposition 3.3. Let l 0 and s i 0 1 i l. Then, the Heisenberg monomial class l a i 1 C 2 si 0 s i! is represented by the closure of 3.4 W 1, C 2 ;... ; 1, C 2 ;... ; l, C 2 ;... ; l, C 2 0. }{{}}{{} s 1 times s l times Proof. Follows from the same argument as in the proof of Proposition 3.2. ξ i 4. Proof of Theorem 1.1 First of all, we outline our main idea used in the proof. Using the results in [LQW1], we shall convert 1.1 to a cup product in the cohomology ring H C 2 of the Hilbert scheme C 2. Then, we shall apply our results in [LQW2] where algebraic relations between the equivariant cohomology ring of C 2 and the Gromov-Witten theory of P 1 have been established. Let notations be the same as in Theorem 1.1. To convert 1.1 to a cup product in H C 2, we define the Heisenberg monomial a µ 1 C 2 H C 2 as in 2.5. By Theorem 2.12 in [LQW1], the quotient ring H X /I is isomorphic to the cohomology ring H C 2. Moreover, we see from Theorem 4.10 in [Lehn] and the proof of Theorem 2.12 in [LQW1] that via this isomorphism, the cohomology classes and a 1 n k k1 X mod I a µ 1 X mod I

HILBERT, HURWITZ, AND GROMOV-WITTEN 11 correspond to the classes 1 k 1 k 1! ch k 1 and a µ1 C 2 in H C 2 respectively. Here, ch k 1 stands for the k 1-th component of the Chern character of the rank-n tautological vector bundle O C 2 defined in 2.9. Hence to prove our formula 1.1 is equivalent to show that 1 k 1 k 1! ch k 1 a 4.1 µ1 C 2 = c λ k,µ a λ 1 C 2 λ n lλ=lµ+1 k where c λ k,µ coincides with the coefficient of zk 1 in k 1! z λ F ςz λ,µz. Next we rewrite the left-hand-side of 4.1 by using the equivariant cohomology ring H T C2. It is well-known that H T C2 is a C[t]-module, and there is a natural ring isomorphism Let H T C 2 /t = H C 2. Ψ : H T C 2 H T C 2 /t = H C 2 be the natural projection. Assume that the partition µ of n is given by µ = 1 s1 2 s2... l s l. Let W 0 µ C 2 be the closure of 3.4. Then, W 0 µ is a T -equivariant closed subscheme of C 2. Moreover, by Proposition 3.3, z µ r 1 s r! a µ1 C 2 is the cohomology class corresponding to W 0 µ. Similarly, z µ r 1 s r! at µ1 C 2 is the equivariant cohomology class corresponding to W 0 µ. So and 4.2 Note that in H T C2. It follows that 4.3 Hence we see from 4.2 that Ψa T µ1 C 2 = a µ 1 C 2 1 k 1 k 1! ch k 1 a µ1 C 2 = 1 k 1 k 1! Ψ ch k 1,T at µ1 C 2. [C 2 ] = t 1 [{0} C] a T µ1 C 2 = t 1 lµ a T µ[{0} C] = t lµ p µ. 1 k 1 k 1! ch k 1 a µ1 C 2

12 LI, QIN, AND WANG is equal to 4.4 1 k 1 lµ k 1! Ψ t lµ ch k 1,T p µ. 4.5 Recall from Sect. 2 that {p λ λ n} is a linear basis of HT 2nC2. Put ch k 1 p µ = λ n d λ k,µ p λ where all the coefficients d λ k,µ are independent of t. So by 2.6 and 2.10, 4.6 ch k 1,T p µ = t n+k 1 ch k 1,T p µ = t k 1 ch k 1,T p µ = t k 1 λ n d λ k,µ p λ. Combining this with 4.4 and 4.3, we conclude that 1 k 1 k 1! ch k 1 a µ1 C 2 = 1 k 1 lµ k 1! Ψ t k 1 lµ d λ k,µ p λ λ n 4.7 = k 1! Ψ t k 1 lµ+lλ d λ k,µ a T λ1 C 2. λ n Since Ψa T λ 1 C 2 = a λ1 C 2 0 for each λ n, we obtain from 4.7 that 4.8 = 1 k 1 k 1! ch k 1 a µ1 C 2 k 1! d λ k,µ a λ 1 C 2. λ n lλ=lµ+1 k By 4.8, to finish the proof of 4.1, it remains to show that the structure constant d λ k,µ in 4.5 is the coefficient of zk 1 in the expansion of the series By 2.11, 2.12 and 2.13, we have z i p λ, i z λ ςz F λ,µz. ch i p µ n = = 1 F ςz λ,µz 1 1 ςz F λ,µz Combining these observations with 4.5 and 2.8, we get d λ k,µ = z λ p λ, = Coeff z k 1 ch k 1 p µ n zλ ςz F λ,µz ςz F λ,µ δ λ,µ z λ ςz δ λ,µ. z λ ςz.

HILBERT, HURWITZ, AND GROMOV-WITTEN 13 Finally, since lλ = lµ + 1 k and k 2, λ µ. constant d λ k,µ in 4.8 is the coefficient of zk 1 in z λ ςz F λ,µz. This completes the proof of Theorem 1.1. Therefore, the structure 5. Relation to the Hurwitz numbers of P 1 Consider a possibly disconnected cover of P 1 of degree-n with k +1 branching points. The ramifications at the branching points are specified by k + 1 partitions λ 0, λ 1,..., λ k of n. Denote by H P1 n λ 0,..., λ k the Hurwitz number of possibly disconnected covers of P 1 of degree n with k + 1 ramification points of type λ 0, λ 1,..., λ k. We have the following well-known lemma. Lemma 5.1. The Hurwitz number H P1 n λ 0,..., λ k is equal to the number of k + 1-tuples of permutations s 0,..., s k in S n satisfying the two conditions below divided by n!: i s i is of cycle type λ i for each i; ii s 0 s 1... s k = 1. The Riemann-Hurwitz formula provides the genus constraint of a Hurwitz cover: 5.1 2g 2 + 2n = n lλ i. Next, we consider the cup product i=0 a λ 11 X a λ k1 X in the cohomology ring of C 2. Denote by c λ0 λ 1,...,λ the coefficient of a k λ 01 X in a λ 11 X a λ k1 X when expressed as a linear combination of the Heisenberg monomial classes. By cohomology degree reasons, we have 5.2 n lλ 0 = n lλ i. Setting g = 1 lλ 0, we see that n, g, λ 0, λ 1,..., λ k satisfy the identity 5.1. Recall that z λ was defined in 2.4. Theorem 5.2. Let λ 0, λ 1,..., λ k be partitions of n satisfying 5.2. Then, c λ0 λ 1,...,λ = z k λ 0 HP1 n λ 0, λ 1,..., λ k. Proof. By the assumption 5.2, Theorem 1.1 in [LS] and Theorem C in [Vas], we see that c λ0 λ 1,...,λ is equal to the number of conjugacy class C k λ 0 in the multiplication of the conjugacy classes C λ 1,..., C λ k of S n, denoted by k [C λ 0] C λ i.

14 LI, QIN, AND WANG Since the conjugacy class C λ 0 consists of n!/z λ 0 elements, [C 1 n ]C λ 0 k C λ i = n! z λ 0 [C λ 0] k = n! c λ0 λ z 1,...,λ k. λ 0 On the other hand, we conclude from Lemma 5.1 that H P1 n λ 0, λ 1,..., λ k = 1 n! [C 1 n ]C λ 0 C λ i k C λ i. Now the theorem follows from putting the above together. References [ES] G. Ellingsrud, S.A. Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 1987, 343-352. [GJV] I. Goulden, D. Jackson, R. Vakil, Towards the geometry of double Hurwitz numbers, math.ag/0309440. [Gro] I. Grojnowski, Instantons and affine algebras I: the Hilbert scheme and vertex operators, Math. Res. Lett. 3 1996, 275 291. [IP] E.-M. Ionel, T. Parker, Relative Gromov-Witten invariants, Ann. Math. 157 2003, 45-96. [Lehn] M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 1999, 157 207. [LS] M. Lehn, C. Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 2001, 345-357. [LR] A.M. Li, Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I, Invent. Math. 145 2001, 151-218. [Li] J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 2002, 199-293. [LQW1] W.-P. Li, Z. Qin, W. Wang, Ideals of the cohomology rings of Hilbert schemes and their applications, Trans. of AMS 356 2004, 245-265. [LQW2] W.-P. Li, Z. Qin, W. Wang, Hilbert schemes, integrable hierarchies, and Gromov-Witten theory, Intern. Math. Res. Notices 40 2004, 2085-2104. [LQW3] W.-P. Li, Z. Qin, W. Wang, The cohomology rings of Hilbert schemes via Jack polynomials, In: J. Hurtubise and E. Markman eds., Algebraic Structures and Moduli Spaces, CRM Proc. and Lect. Notes 38 2004, 249-258. [MJD] T. Miwa, M. Jimbo and E. Date, SOLITONS. Differential equations, symmetries and infinite dimensional algebras, originally published in Japanese 1993, Cambridge University Press, 2000. [MNOP] D. Maulik, N. Nekrasov, A. Okounkov, R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory, II, math.ag/0406092. [Na1] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math. 145 1997, 379 388. [Na2] H. Nakajima, Lectures on Hilbert schemes of points on surfaces. Univ. Lect. Ser. 18, Amer. Math. Soc. 1999. [OP1] A. Okounkov, R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and completed cycles, math.ag/0204305. [OP2] A. Okounkov, R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, math.ag/0411210. [QW] Z. Qin, W. Wang, Hilbert schemes of points on the minimal resolution and soliton equations, math.ag/0404540. [Vas] E. Vasserot, Sur l anneau de cohomologie du schéma de Hilbert de C 2, C. R. Acad. Sci. Paris, Sér. I Math. 332 2001, 7 12.

HILBERT, HURWITZ, AND GROMOV-WITTEN 15 [Wang] W. Wang, Vertex algebras and the class algebras of wreath products, Proc. London Math. Soc. 3 88 2004, 381-404. Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong Kong E-mail address: mawpli@ust.hk Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address: zq@math.missouri.edu Department of Mathematics, University of Virginia, Charlottesville, VA 22904 E-mail address: ww9c@virginia.edu