The Chow groups and the motive of the Hilbert scheme of points on a surface

Mark Andrea A. de Cataldo^{∗} and Luca Migliorini^{†}

Abstract

We compute the Chow motive and the Chow groups with rational coefficients of the Hilbert scheme of points on a smooth algebraic surface.

1 Introduction

In this paper we compute the Chow motive and the Chow groups with rational coefficients of the Hilbert scheme X^{[n]} of n points on a nonsingular algebraic surface X. The result holds over an arbitrary field. The space X^{[n]} is a remarkably rich object which finds itself, for reasons that are perhaps still mysterious, at the croassroads of geometry, representation theory and mathematical physics. In the context of testing the S-duality conjecture, using Göttsche’s calculation of the Betti numbers of X^{[n]}, Vafa and Witten suggested, for reasons stemming from orbifold cohomology, that the singular cohomology groups of these Hilbert schemes should be naturally linked to infinite dimensional graded Lie algebras. This fact was firmly established by Nakajima and Grojnowski, independently. Some of the contributors to this circle of ideas are: Fogarty [7], Briancon [1], Iarrobino [14], Ellingsrud-Stromme [5], Göttsche [9], Göttsche-Sorgel [10], Cheah [3], Vafa-Witten [18], Nakajima [16], Grojnowski [12]. The reader is referred to the beautiful lectures [17] and to our paper [4] for background, further results and references to the literature. In our paper [4] we proved, directly, a precise form of the decomposition theorem for the socalled Hilbert-Chow map. We detected, via the action of the Lie algebra, certain subvarieties of X^{[n]} carrying the topological information necessary to understand the additive structure of singular cohomology. Subsequently, we looked directly at certain related correspondences. In this paper we show how these correspondences identify the Chow groups with rational coefficients of X^{[n]} with the Chow groups with rational coefficients of a certain collection of products of symmetric products of the surface X. Using this result we determine the Chow motive with rational coefficients of X^{[n]}. Voevodsky’s theories of motivic cohomology incorporate the formalisms of Tate twists and shifts of complexes. Once the Chow motive of X^{[n]} is computed, it is easy to compute the motive in these more general theories. Our result, read in these theories, harmonizes the shifts present in the decomposition theorem mentioned above and the Tate twists in the mixed Hodge structure of X^{[n]}. There is also a generating function for the motives in question. The main tool used in our calculation is the Gysin formalism of Fulton-MacPherson. None of the tools used in the various derivations of the structure of the singular cohomology of X^{[n]} in the literature seems applicable in the context of cycles. In fact, this cycle-theoretic approach can be easily supplemented to give another proof of Göttsche’s formula for the Betti numbers. The fact, due to Ellingsrud and Stromme, that punctual Hilbert schemes admit affine cellular decompositions is essential to our approach. The outline of the paper is as follows. §2 is devoted to fixing the notation and to introducing the varieties naturally associated with X^{[n]} and its natural stratification given by partitions. §3 reviews the basic results from intersection theory that we need. §4 reviews well-known facts concerning the Gysin formalism of correspondences in a “non-complete” situation. §4.1 discusses correspondences in this “refined” formalism. §4.2 discusses the composition of correspondences in the context of the refined formalism. §4.3 overviews the situation for quotient varieties. In §4.4 we define the natural map Γ̂_{∗} via the correspondences Γ̂. The main result of the paper is Theorem 5.4.1, which states that Γ̂_{∗} is an isomorphism. The injectivity statement is Corollary 5.1.5. The surjectivity statement, Corollary 5.3.2, is the heart of the present paper. Corollary 5.4.2 and Remark 5.4.3 are standard consequences for the Grothendieck groups also in the equivariant context. §6 is devoted to the identification, Theorem 6.2.1, of the motive of the Hilbert scheme X^{[n]} with a collection of motives of products of symmetric products of the surface X. Theorem 6.2.5 is the translation of Theorem 6.2.1 into Voevodsky’s categories. We give a “generating function” for this motivic structure at the end of §6.1.2.

∗Partially supported by N.S.F. Grant DMS 9701779.

†member of GNSAGA, supported by MURST funds, and by SFB237

Acknowledgments. Some ideas of N. Fakhruddin have influenced our approach to this question. We have asked him to co-author the present paper, but he has declined. The first author would like to thank Hong Kong University for its very generous hospitality while portions of this work have been carried out. We wish to thank M. Kumar, J.I. Manin and A. Vistoli for useful conversations. The commutative diagrams are produced by Paul Taylor’s

diagrams macro package, available in CTAN in TeX/macros/generic/diagrams/taylor.

2 Some special varieties and fibrations

In these section we review a few definitions and introduce the correspondences which will play a major role in the paper. Proofs or references for all the results stated here can be found in our previous paper [4]. Let X be an irreducible quasiprojective nonsingular surface defined over an algebraically closed field, X^{(n)} be its n-th symmetric product, X^{[n]} be the Hilbert scheme of 0-dimensional subschemes of X of length n, π : X^{[n]} → X^{(n)} be the Hilbert-Chow morphism. It is well known that X^{[n]} is nonsingular. 𝔓(n) denotes the set of partitions of n and p(n) its cardinality. If ν ∈ 𝔓(n), we denote by l(ν) its length, and define X_{ν}^{(n)} to be the locally closed subset of points in X^{(n)} of the type ν_{1}x_{1} + ⋯ + ν_{l(ν)}x_{l(ν)}, with x_{h} ∈ X and x_{i} ≠ x_{j} for every i ≠ j. Similarly, X_{ν}^{[n]} := (π^{−1}(X_{ν}^{(n)}))_{red}. Let \overline{X}_{ν}^{(n)} denote the closure of the stratum. It can be proved that \overline{X}_{ν}^{[n]} = (π^{−1}(\overline{X}_{ν}^{(n)}))_{red}. If ν = 1^{a_{1}} ⋯ n^{a_{n}}, then the finite group Σ_{ν} := Σ_{a_{1}} × ⋯ × Σ_{a_{n}} acts naturally on X^{l(ν)}. The quotient X^{(ν)} is isomorphic to X^{(a_{1})} ×⋯×X^{(a_{n})}. For the sake of uniformity of notation we shall denote X^{l(ν)} by X^{ν}. There is a natural Σ_{ν}-invariant map ν : X^{ν} → X^{(n)} whose image is \overline{X}_{ν}^{(n)}. This map descends to a map which we denote by the same symbol ν : X^{(ν)} → X^{(n)}. This map is the normalization map of \overline{X}_{ν}^{(n)}. We define correspondences Γ^{ν} and Γ̂^{ν} as follows:

Γ^{ν} = { (x_{1}, ⋯, x_{l(ν)}, J) ∈ X^{ν} × X^{[n]} : π(J) = ν_{1}x_{1} + ⋯ + ν_{l(ν)}x_{l(ν)} } ≃ (X^{ν} ×_{X^{(n)}} X^{[n]})_{red}.

Denote by X_{reg}^{ν} (resp. X_{reg}^{(ν)}) the set of points of X^{ν} (resp. X^{(ν)}) strictly of type ν.

Remark 2.0.1 The restriction Γ_{reg}^{ν} of Γ^{ν} to X_{reg}^{ν} is a Zariski locally trivial fibration with irreducible fibers and it is open and dense in Γ^{ν} (see [4] Lemma 3.6.1 and Remark 5.2.5). It follows that Γ^{ν} is irreducible of dimension n + l(ν).

The projection p_{1} will be denoted by p_{ν} in order to emphasize the dependence on ν. The correspondence Γ^{ν} is invariant under the action of Σ_{ν} on the first factor of the product. We can therefore define Γ̂^{ν} := Γ^{ν}/Σ_{ν}. We summarize these constructions with the diagram:

[figure: Γ^{ν}; q′; Γ̂^{ν}; P; X^{[n]}; X^{ν}; p_{ν}; q; X^{(ν)}; p̂_{ν}; ν; X^{(n)}; π]

where q and q′ are the quotient maps by the action of Σ_{ν}. The composition P ∘ q′ will still be denoted by P.

Remark 2.0.2 The restriction of P to Γ̂_{reg}^{ν} = Γ_{reg}^{ν}/Σ_{ν} identifies Γ̂_{reg}^{ν} with the locally closed stratum X_{ν}^{[n]}.

We introduce X = ∐_{ν∈𝔓(n)} X^{ν} and X̂ = ∐_{ν∈𝔓(n)} X^{(ν)}, Γ = ∐_{ν∈𝔓(n)} Γ^{ν}, Γ̂ = ∐_{ν∈𝔓(n)} Γ̂^{ν} and define p : Γ → X (resp. p̂: Γ̂ → X̂) to be the map induced by all the maps p_{ν} (resp. p̂_{ν}). The set of partitions has a natural partial order, which reflects the incidence relations of the strata:

Definition 2.0.3 Let ν, μ ∈ 𝔓(n). We say that ν ⪰ μ if there exists a decomposition {I_{1}, ⋯, I_{l(μ)}} of the set {1, ⋯ l(ν)} such that μ_{1} = ∑_{i∈I_{1}} ν_{i}, ⋯, μ_{l(μ)} = ∑_{i∈I_{l(μ)}} ν_{i}.

It is easily seen that μ ⪰ ν if and only if X_{ν}^{(n)} ⊆ \overline{X}_{μ}^{(n)}. Fix a total order ≥ on 𝔓(n) which is compatible with ⪰. For any μ ∈ 𝔓(n) we have the open subsets X_{≥μ}^{(n)} := ∐_{ν≥μ} X_{ν}^{(n)} ⊆ X^{(n)}. Similarly for X_{>μ}^{(n)}. Correspondingly, we have open sets X_{≥μ}, X_{>μ}, Γ_{≥μ}, Γ_{>μ}, X_{≥μ}^{[n]} and X_{>μ}^{[n]} obtained by base change followed by reduction. We have the corresponding quotients X̂_{≥μ}, X̂_{>μ}, Γ̂_{≥μ}, Γ̂_{>μ}. Similarly, with the symbol ≥ replaced by ⪰. The imbedding X_{μ}^{(n)} → X_{≥μ}^{(n)} is closed. We have corresponding closed imbeddings X_{μ} → X_{≥μ} and Γ_{μ} → Γ_{≥μ} obtained by base change followed by reduction. Note that Γ_{μ} ≠ Γ^{μ}. We thus have a diagram:

Γ_{μ}  \overline{P} X_{μ}^{[n]}
[figure]
Γ_{≥μ}  \overline{P} π  X_{≥μ}^{[n]}

[figure: X_{μ}; p_{≥μ}; X_{μ}^{(n)}]
[figure]
[figure: X_{≥μ}; p_{≥μ}; X_{≥μ}^{(n)}.; π]

3 Review of Intersection Theory

Our unique reference for this section is [8]. In what follows, the Chow groups A_{∗}(X) of an algebraic scheme X over a field are always taken with rational coefficients, even though many results hold with integer coefficients. We recall that given a regular imbedding i : X → Y of codimension d with normal bundle N_{X}Y and a morphism f : Y′ → Y from a pure ldimensional variety Y′, there are refined Gysin homomorphisms i^{!} : A_{∗}(Y′) → A_{∗−d}(X′) where X′ = X ×_{Y} Y′ . The construction of i^{!}([Y′]) goes as follows: the map X′ → Y′ is a closed imbedding and the normal cone C_{X′}Y′ is a pure l-dimensional subscheme of the pullback of N_{X}Y to X′ ; its cycle class is therefore equivalent to the flat pullback of a unique cycle class i^{!}([Y′]) ∈ A_{l−d}(X′). To define i^{!}(α) for α ∈ A_{∗}(Y′) one replaces Y′ with the support of α and maps the resulting class to A_{∗}(Y′).

Let f : X → Y be a morphism from a scheme X to a nonsingular variety Y. The graph morphism γ_{f} : X → X ×Y is a regular imbedding. Given maps X′ → X and Y′ → Y there is a refined Gysin morphism γ_{f}^{!} : A_{k}(Y′)⊗A_{l}(X′) → A_{k+l−dimY} (X′ ×_{Y} Y′). If X′ = X, Y′ = Y and both maps are the identity map, then we will denote γ_{f}^{!}(α ⊗ β), the image of α ⊗ β via the morphism A_{k}(Y) ⊗ A_{l}(X) → A_{k+l−dimY} (X), with the more suggestive piece of notation f^{∗}(α) ∩ β. If X′ = X, then we will use the notation f^{!}(α) ∩ β for the refined intersection.

Remark 3.0.4 Let α ∈ Z_{k}(Y′), β ∈ Z_{l}(X′) be irreducible cycles. If the irreducible components ξ_{i} of |α| ×_{Y} |β| have the expected dimension k + l − dimY, then γ_{f}^{!}(α ⊗ β) is a linear combination with strictly positive coefficients of the cycles ξ_{i}; see [8], §7.1. In particular, if Y′ is a closed subvariety of Y and f^{−1}(Y′) is irreducible of dimension dimY′ +dimX −dimY, then f^{!}([Y′]) ∩ [X] is a positive multiple of [f^{−1}(Y′)].

The following well-known lemma will be used in the sequel of the paper. We omit the simple proof which follows easily from the case of a trivial fibration (see [8], Ex. 1.10.2) and by noetherian induction using the basic properties of the refined Gysin formalism.

Lemma 3.0.5 Let X be a nonsingular irreducible variety and p : Γ → X a Zariski locally trivial fibration with fiber F admitting a cellular decomposition. Suppose {α_{i}}_{i∈I} is a set of classes in A_{∗}(Γ) whose restrictions generate A_{∗}(p^{−1}(x)) for every x ∈ X. Then {α_{i}}_{i∈I} is a set of generators of the A_{∗}(X)-module A_{∗}(Γ). In other words, the map Φ : A_{∗}(X)^{⊕I} → A_{∗}(Γ), defined by Φ({β_{i}}) = ∑_{i} p^{∗}(β_{i}) ∩ α_{i}, is surjective.

4 Correspondences

4.1 Correspondences via refined Gysin maps

The standard reference is [8], Remark 16.1, §6 and §8. Consider a diagram

[figure: Γ; p_{2}; X_{2}; X_{1}; p_{1}]

with X_{1} nonsingular and p_{2} proper. In section 3 we define a map

[figure: A_{k}(X_{1}) ⊗ A_{l}(Γ); p_{1}^{∗}(−) ∩ (−); A_{k+l−dimX_{1}}(Γ); p_{2∗}; A_{k+l−dimX_{1}}(X_{2}).]

The case we will mostly use is

p_{2∗}(p_{1}^{∗}(−) ∩ [Γ]) : A_{k}(X_{1}) ⟶ A_{k+dimΓ−dimX_{1}}(X_{2}),

where [Γ] is some fixed cycle. Most important for us will be the case of relative correspondences of the type

Γ  \overline{p_{2}} X_{2}
□
[figure: X_{1}; p_{1}; f; Y,]

where X_{1} is a nosingular variety, f is a proper morphism and Γ = X_{1} ×_{Y} X_{2}. If V → Y is a morphism, then we have a similar diagram by pulling back to V. Set, for brevity, X_{1,V} = X_{1} ×_{Y} V and similarly X_{2,V} = X_{2} ×_{Y} V and Γ_{V} = Γ ×_{Y} V. Note that, by the transitivity property of the fibre product, Γ_{V} = X_{1,V} ×_{V} X_{2,V}. Fix a cycle [Γ]. The refined intersection product gives a map p_{1}^{!}(−) ∩ [Γ] : A_{∗}(X_{1,V}) → A_{∗}(Γ_{V}).

Remark 4.1.1 It is important that we do not assume that X_{1} is complete, i.e. that we do not assume that p_{2} : X_{1} ×X_{2} → X_{2} is proper and that we do not factorize through A_{∗}(X_{1} × X_{2}). Otherwise, we would loose too much information: e.g. the diagonal in 𝔸^{1} × 𝔸^{1} is zero in A_{1}(𝔸^{1} × 𝔸^{1}), but it induces the identity map via the formalism of refined intersections.

Let i : Z → Y be a closed imbedding and j : U := Y \ Z → Y be the resulting open imbedding.

Lemma 4.1.2 Let [Γ] be any cycle in A_{∗}(Γ). The diagram

[figure: A_{∗}(X_{1,Z}); p_{1}^{!}(−) ∩ [Γ]; A_{∗}(Γ_{Z}); p_{2∗}; A_{∗}(X_{2,Z})]

[figure: A_{∗}(X_{1}); i_{∗}; p_{1}^{∗}(−) ∩ [Γ]; A_{∗}(Γ); p_{2∗}; A_{∗}(X_{2})]

[figure: A_{∗}(X_{1,U}); j^{∗}; p_{1}^{∗}(−) ∩ [Γ_{U}]; A_{∗}(Γ_{U}); p_{2∗}; A_{∗}(X_{2,U})]

[figure: 0; 0; 0]

is commutative and the columns are exact.

Proof. The exactness of the columns stems from [8], Proposition 1.8. The commutativity of the first column of diagrams stems from [8], Proposition 8.1.1.(c) and Theorem 6.2. The commutativity of the second column is obvious. □

Remark 4.1.3 Note that, even if Γ is reduced, Γ_{Z} may fail to be so. However, this causes no trouble in view of the obvious canonical isomorphism A_{∗}(−_{red}) → A_{∗}(−). In the sequel we shall often take fibre products. We shall always assume that we are taking the reduced structure. When we write that a diagram is “cartesian modulo nilpotents”, we mean that the fibre product is to be taken with the reduced structure. We shall always take care to define the cycle [Γ].

4.2 Composition of correspondences

This formalism of correspondences via refined Gysin maps extends easily to the case of composition of correspondences. The reference is [8], Remark 16.1, Proposition 16.1.2, §6 and §8.

Let X_{i}, i = 1, 2, 3 be nonsingular varieties. Let p_{ij} : X_{1} × X_{2} × X_{3} → X_{i} × X_{j} be the obvious projections. Let Φ ⊆ X_{1} × X_{2} and Γ ⊆ X_{2} × X_{3} be two irreducible cycles such that all the following maps are proper: |Φ| → X_{2}, Γ → X_{3}, p_{12}^{−1}(|Φ|) ∩ p_{23}^{−1}(|Γ|) → X_{1} × X_{3}, p_{13}(p_{12}^{−1}(|Φ|) ∩ p_{23}^{−1}(|Γ|)) → X_{3}. In addition, assume that p_{13}(p_{12}^{−1}(|Φ|) ∩ p_{23}^{−1}(|Γ|)) has the expected dimension.

We can therefore define a refined cycle Γ ∘ Φ in Z_{∗}(p_{13}(p_{12}^{−1}(|Φ|) ∩ p_{23}^{−1}(|Γ|)) and maps: Φ_{∗}, Γ_{∗} and (Γ ∘ Φ)_{∗}. We have Γ_{∗} ∘ Φ_{∗} = (Γ ∘ Φ)_{∗}.

4.3 Quotient varieties

The formalism recalled so far extends to the case of quotient varieties. See [8], Ex. 16.1.13. In this case rational coefficients are necessary. The basic set-up we need is as follows. Z is a nonsingular integral variety; Z′ is an integral variety; p_{Z} : Z′ → Z is a surjective morphism; G is a finite group acting on Z and Z′, compatibly with p_{Z}. There is a commutative diagram:

[figure: Z′; p′; Z′_{1}; Z; p_{Z}; p; Z_{1}; p_{Z_{1}}]

where Z_{1} = Z/G, p is a quotient Galois morphism of degree |G|, Z′_{1} = Z′/G and p′ is a quotient Galois morphism of degree |G|. Note that it follows that p and p′ are separable. Define [Z′] and [Z′_{1}] to be the corresponding fundamental classes.

We have that p′^{∗}[Z′_{1}] = [Z′]. We can define a Gysin-type map as follows:

Φ_{[Z′_{1}]} : A_{∗}(Z_{1}) ⟶ A_{∗}(Z′_{1}) , z_{1} ⟶ \frac{1}{|G|} p′_{∗}(p_{Z}^{∗}(p^{∗}z_{1}) ∩ [Z′]).

We need the following elementary fact which can be checked directly using the definition of p^{∗} for quotient maps; see [8], Ex. 1.7.6.

Lemma 4.3.1 Let R be a variety endowed with an action of a finite group G on it. Denote the quotient p : R → S. Let σ : A_{∗}(R) → A_{∗}(R), z → ∑_{γ∈G} γ_{∗}z, be the so-called symmetrization operator. Then σ = p^{∗}p_{∗}.

Lemma 4.3.2 We have the following commutative diagram:

[figure: A_{∗}(Z′); p′_{∗}; A_{∗}(Z′_{1}); A_{∗}(Z); Φ_{[Z′]}; p_{∗}; A_{∗}(Z_{1}); Φ_{[Z′_{1}]}]

Proof. For ease of notation we denote a class p_{Z}^{∗}(a) ∩ p^{′∗}z′_{1}, by a · p^{′∗}z′_{1}. Let z ∈ A_{∗}(Z) and z′ ∈ A_{∗}(Z′). Define σ : z → ∑_{γ∈G} γ_{∗}z and σ′ : z′ → ∑_{γ∈G} γ_{∗}z′. By virtue of Lemma 4.3.1, we have that σ = p^{∗}p_{∗} and that σ′ = p′^{∗}p′_{∗}. It follows that p′^{∗}p′_{∗}(z · p′^{∗}z′_{1}) = σ′(z · p′^{∗}z′_{1}) = ∑_{γ∈G} γ_{∗}(z · p′^{∗}z′_{1}) = (by the proper push-forward property) = ∑_{γ∈G} (γ_{∗}z · γ_{∗}p′^{∗}z′_{1}) = (since p′^{∗}z′_{1} is G-invariant) = ∑_{γ∈G} (γ_{∗}z · p′^{∗}z′_{1}) = σ(z) · p′^{∗}z_{1} = (p^{∗}p_{∗}z)·p′^{∗}z′_{1} = (since p^{∗}p_{∗}z is G-invariant) = \frac{1}{|G|}[∑_{γ∈G} γ_{∗}(p^{∗}p_{∗}z] · p′^{∗}z′_{1} = \frac{1}{|G|}σ′(p^{∗}p_{∗}z)·p^{′∗}z′_{1} = \frac{1}{|G|}p′^{∗}p′_{∗}(p^{∗}p_{∗}z·p′^{∗}z′_{1}). Since p′^{∗} is injective, we have that p′_{∗}(z·p′^{∗}z′_{1}) = \frac{1}{|G|}p′_{∗}(p^{∗}p_{∗}z·p′^{∗}z′_{1}) which is what we wanted to prove. □

4.4 The morphisms Γ_{∗} and Γ̂_{∗}

Let X be an irreducible nonsingular quasi projective surface defined over an algebraically closed field. Recalling the notation introduced in §2, we summarize the results of this section by stating that the following diagram is commutative:

[figure: A_{∗}(Γ) = ⊕_{ν}A_{∗}(Γ^{ν}); A_{∗}(Γ̂) = ⊕_{ν}A_{∗}(Γ̂^{ν}); A_{∗}(X^{[n]})]

[figure: A_{∗}(X) = ⊕_{ν}A_{∗}(X^{ν}); Γ_{∗} = ⊕_{ν}Γ_{∗}^{ν}; A_{∗}(X̂) = ⊕_{ν}A_{∗}(X^{(ν)}).; Γ̂_{∗} = ⊕_{ν}Γ̂_{∗}^{ν}]

By abuse of notation, the corresponding maps into A_{∗}(X^{[n]}) will be denoted with the same symbol, e.g. Γ_{∗} : A_{∗}(X) → A(X^{[n]}) and Γ̂_{∗} : A_{∗}(X̂) → A(X^{[n]}).

5 The Chow groups of X^{[n]}

Let X be an irreducible nonsingular quasi projective surface defined over an algebraically closed field. The reader should keep in mind §2.

5.1 The injectivity of Γ̂_{∗}

We shall freely use the formalism developed for correspondences over quotient varieties in §4.3.

Lemma 5.1.1 Let Z be an irreducible algebraic scheme of dimension n, quotient of a nonsingular algebraic scheme via a finite group. Let V and W be pure-dimensional cycles on Z of dimensions k and l. Let f : Z → Y be a proper morphism of algebraic schemes. If dim f(|V | ∩ |W |) < k + l − n, then f_{∗}(V · W) = 0.

Proof. There is a well-defined refinement of the product in A_{k+l−n}(|V | ∩ |W |) which must map to zero. See [8], Ex. 8.3.12. □

The following lemma is essentially a reformulation of [6]. Before stating it, we introduce some notation: Let ν be a partition of n. For \underline{x} ∈ X_{reg}^{(ν)} we set F_{ν} = (π^{−1}(\underline{x}))_{red}, which we identify via P with p_{ν}^{−1}(\underline{x}) (cf. 2.0.2). We set m_{ν} := (−1)^{n−l(ν)} ∏_{j=1}^{l(ν)} ν_{j}. We denote by [F_{ν}] · X_{ν}^{[n]} ∈ A_{0}(F_{ν}) the refined intersection defined by the closed imbeddings of F_{ν} and X_{ν}^{[n]} in X_{≥ν}^{[n]}.

Lemma 5.1.2 deg ([F_{ν}] · X_{ν}^{[n]}) = m_{ν}.

Proof. The case ν = n^{1} is precisely the main result in [6]. The general case follows from the Künneth formula. □

In the following two propositions we compute the compositions ^{t}Γ̂^{ν} ∘ Γ̂^{μ}. Here ^{t}− denotes, as usual, the transposed correspondence; see [8], §16.1.

Proposition 5.1.3 Let μ ≠ ν be two distinct partitions of n. Then ^{t}Γ̂^{ν} ∘ Γ̂^{μ} = 0 in A_{l(μ)+l(ν)}(X^{(μ)} × X^{(ν)}).

Proof. Consider X^{(μ)} × X^{[n]} × X^{(ν)} together with the natural projections p, π and q to X^{(μ)} × X^{[n]}, X^{(μ)} × X^{(ν)} and X^{[n]} × X^{(ν)}, respectively. By virtue of Lemma 5.1.1, it is enough to show that dim π(p^{−1}Γ̂^{μ} ∩ q^{−1} ^{t}Γ̂^{ν}) < l(μ) + l(ν). One checks directly that dim π(p^{−1}Γ̂^{μ} ∩ q^{−1} ^{t}Γ̂^{ν}) = dim (\overline{X}_{μ}^{(n)} ∩ \overline{X}_{μ}^{(n)}). Since μ ≠ ν, this last dimension is strictly less than 2 min (l(μ), l(ν)). □

Proposition 5.1.4 ^{t}Γ̂^{ν} ∘ Γ̂^{ν} = m_{ν}Δ_{X^{(ν)}} in A_{2l(ν)}(X^{(ν)} × X^{(ν)}).

Proof. The cycle ^{t}Γ̂^{ν} ∘ Γ̂^{ν} is supported on the diagonal of X^{(ν)} × X^{(ν)} which is irreducible of the expected dimension, therefore ^{t}Γ̂^{ν} ∘ Γ̂^{ν} = c Δ_{X^{(ν)}}, with c ∈ ℚ. Let \underline{x} ∈ X_{reg}^{(ν)}. Since the map Γ̂_{reg}^{ν} → X_{reg}^{(ν)} is flat, we have that Γ̂_{∗}^{ν}([\underline{x}]) = [F_{ν}]. By virtue of Lemma 5.1.2 and of the projection formula we have that

c = deg (^{t}Γ̂_{∗}^{ν} ∘ Γ̂_{∗}^{ν}([\underline{x}])) = deg (^{t}Γ̂_{∗}^{ν}([F_{ν}])) = deg ([F_{ν}] · X_{ν}^{[n]}) = m_{ν}.
□

Corollary 5.1.5 The natural map

Γ̂_{∗} = ⨁_{ν∈𝔓(n)} Γ̂_{∗}^{ν} : A_{∗}(X̂) = ⨁_{ν∈𝔓(n)} A_{∗}(X^{(ν)}) ⟶ A_{∗}(X^{[n]})

is injective.

Proof. Propositions 5.1.4 and 5.1.3 imply that ^{t}Γ̂_{∗} ∘ Γ̂_{∗} is a non zero multiple of the identity being the map induced by a non zero multiple of the diagonal. □.

Remark 5.1.6 An argument identical to the one given above shows that Corollary 5.1.5 holds if we replace all spaces by the ones obtained by base change with respect to any open immersion U → X^{(n)}.

5.2 A surjectivity statement

The following surjectivity statement is essential to proving Proposition 5.3.1 which, in turn, constitutes the surjectivity part of the main result of this paper, Theorem 5.4.1. The proof will be given at the end of this section, after a series of preparatory results.

Proposition 5.2.1 Let μ ∈ 𝔓(n): the map P_{∗}(p_{≥μ}^{!}(−) ∩ [Γ_{≥μ}]) : A_{∗}(X_{μ}) → A_{∗}(X_{μ}^{[n]}) is surjective.

In order to prove Proposition 5.2.1 we need to make explicit the combinatorics involved. Let l be an integer, and ℜ_{l} be the set of decompositions ρ = {I_{1}, ⋯, I_{r}} of the set {1, ⋯, l} into disjoint subsets, i.e. {1, ⋯, l} = I_{1} ∐ ⋯ ∐ I_{r}. For ρ ∈ ℜ_{l} define D_{ρ} = {(x_{1}, ⋯, x_{l}) ∈ X^{l} : x_{i} = x_{j} ⇔ i, j ∈ I_{k} for some k}. For ρ = {1}, ⋯ {l} we have D_{ρ} = X_{reg}^{l} = X^{l} \ Diagonals. For ρ = {1, ⋯, l}, we have that D_{ρ} is the small diagonal.

Let 𝔓_{⪯ν}(n) = { μ ∈ 𝔓(n) such that μ ⪯ ν}. Define Q_{ν} : ℜ_{l(ν)} → 𝔓_{⪯ν}(n) by setting Q_{ν}({I_{1}, ⋯, I_{r}}) = (∑_{i∈I_{1}} ν_{i}, ⋯, ∑_{i∈I_{l}} ν_{i}). Note that this partition is not ordered, i.e. ∑_{i∈I_{1}} ν_{i} ≱ ∑_{i∈I_{2}} ν_{i} ⋯ ≱ ∑_{i∈I_{l}} ν_{i}, but see Remark 5.2.4.

Let ν : X^{ν} → \overline{X}_{ν}^{(n)} be the already mentioned map defined by ν(x_{1}, ⋯, x_{l(ν)}) = ν_{1}x_{1} + ⋯ + ν_{l(ν)}x_{l(ν)}. Note that ν^{−1}(X_{ν}^{(n)}) = X_{reg}^{ν}. Note that l(Q_{ν}({I_{1}, …, I_{r}})) = r. The proof of the following lemma is immediate:

Lemma 5.2.2

X^{ν} ×_{X^{(n)}} X_{μ}^{(n)} ≠ ∅ ⇔ ν ⪰ μ and (X^{ν} ×_{X^{(n)}} X_{μ}^{(n)})_{red} = ∐_{ρ∈Q_{ν}^{−1}(μ)} D_{ρ}.

Let Γ^{ν} be the correspondences introduced in §2.

Remark 5.2.3 The group Σ_{ν} acts on ℜ_{l(ν)} and Q_{ν} is Σ_{ν}-invariant. In particular, Σ_{ν} acts on Q_{ν}^{−1}(μ).

Remark 5.2.4 For ν ∈ 𝔓(n) we can fix a total order on the set ℜ_{l(ν)} so that we always have ∑_{i∈I_{1}} ν_{i} ≥ ∑_{i∈I_{2}} ν_{i} ⋯ ≥ ∑_{i∈I_{r}} ν_{i}. Let ρ ∈ ℜ_{l(ν)} and μ := Q_{ν}(ρ). We can identify D_{ρ} with X_{reg}^{μ} via the map δ_{ρ} : X_{reg}^{μ} → X^{ν} sending the point (y_{1}, ⋯, y_{r}) ∈ X_{reg}^{μ} to (x_{1}, ⋯, x_{l(ν)}) defined by x_{j} = y_{i} precisely if j ∈ I_{i}. In this case ν ∘ δ_{ρ} = μ_{|X_{reg}^{μ}} : X_{reg}^{μ} → X^{(n)}, hence (Γ^{ν} ×_{X^{ν}} D_{ρ})_{red} = Γ_{|X_{reg}^{μ}}^{μ} =: Γ_{reg}^{μ}. These identifications will always be tacitly made in the sequel.

Remark 5.2.5 p_{μ} : Γ_{reg}^{μ} → X_{reg}^{μ} is a Zariski locally trivial fibration. The fiber is isomorphic to the product of punctual Hilbert schemes H_{μ_{1}} × ⋯ × H_{μ_{l}} and it admits a cellular decomposition. See [9], Lemma 2.1.4. and [5].

Let (ν, ρ) be such that Q_{ν}(ρ) = μ. The pair (ν, ρ) defines a set of partitions ν^{i} ∈ 𝔓(μ_{i}) as follows: if ρ = {I_{1}, ⋯, I_{l(μ)}}, then μ_{i} = ∑_{j∈I_{i}} ν_{j} and therefore ν^{i} = {ν_{j}}_{j∈I_{i}} is a partition of μ_{i}.

Definition 5.2.6 Define the open sets

U_{ρ} = {(x_{1}, ⋯, x_{l(ν)}) ∈ X^{ν}, such that the subsets {x_{j}}_{j∈I_{i}} are pairwise disjoint}.

In other words we allow only points belonging to the same I_{j} to collapse: in particular U_{ρ} ⊇ D_{ρ}.

Lemma 5.2.7 Let (ν, ρ) be such that Q_{ν}(ρ) = μ, and ν^{i} ∈ 𝔓(μ_{i}) be the corresponding set of partitions. There is a canonical isomorphism

Γ_{|U_{ρ}}^{ν} = ∏ Γ_{|U_{ρ}}^{ν^{i}}.

Proof. Let (x_{1}, ⋯, x_{l(μ)}, I) ∈ Γ_{|U_{ρ}}^{ν}. From the definition of U_{ρ} it follows that I is the product of ideals I_{1}, I_{2}, ⋯, I_{r} of lengths μ_{1}, μ_{2}, ⋯, μ_{r} supported on {x_{j}}_{j∈I_{1}}, {x_{j}}_{j∈I_{2}}, …, {x_{j}}_{j∈I_{r}}, whence the obviously bijective map from Γ_{|U_{ρ}}^{ν} to ∏ Γ_{|U_{ρ}}^{ν_{i}}. □

We have the following diagram

[figure: Γ_{reg}^{μ}; Γ_{|U_{ρ}}^{ν}; D_{ρ} = X_{reg}^{μ}; p_{μ}; δ_{ρ}; U_{ρ}.; p_{ν}]

which is cartesian modulo nilpotents (see Remark 4.1.3) and a map γ_{ν,ρ} : A_{∗}(X_{reg}^{μ}) → A_{∗}(Γ_{reg}^{μ}) defined by γ_{ν,ρ}(β) = p_{ν}^{!}(β) ∩ [Γ^{ν}] (cf. §3).

Lemma 5.2.8 Let α_{ν,ρ} := γ_{ν,ρ}([X_{reg}^{μ}]) ∈ A_{∗}(Γ_{reg}^{μ}) and β ∈ A_{∗}(X_{reg}^{μ}). Then

γ_{ν,ρ}(β) = p_{μ}^{∗}(β) ∩ α_{ν,ρ}.

Proof. It follows from the associativity property of refined products, [8], Proposition 8.1.1.a).

□

What has been done so far can be summarized by the following:

Lemma 5.2.9 Let μ be a partition, denote by q̃: Γ_{reg}^{μ} → X_{μ}^{[n]} the quotient map by the action of Σ_{μ}. Let W ⊆ A_{∗}(Γ_{reg}^{μ}) be the A_{∗}(X_{reg}^{μ})-submodule generated by the classes {α_{ν,ρ}}_{ρ∈Q_{ν}^{−1}(μ)}.

The surjectivity of the map P_{∗}(p_{≥μ}^{!}(−) ∩ [Γ_{≥μ}]) : A_{∗}(X_{μ}) → A_{∗}(X_{μ}^{[n]}) is equivalent to the surjectivity of the restriction of q̃_{∗} : A_{∗}(Γ_{reg}^{μ}) → A_{∗}(X_{μ}^{[n]}) to W.

Proof. By virtue of lemma 5.2.2

X_{μ} = ∐_{ρ∈Q_{ν}^{−1}(μ)} D_{ρ} = ∐_{ρ∈Q_{ν}^{−1}(μ)} X_{reg}^{μ},

this last identification is made using the map described in 5.2.4. The following diagram, where the equality ∑ γ_{ν,ρ} = ∑ p_{μ}^{∗}( ) ∩ α_{ν,ρ} is a consequence of 5.2.8, commutes:

[figure: A_{∗}(X_{μ}); P_{∗}(p_{≥μ}^{!}( ) ∩ [Γ_{≥μ}]); A_{∗}(X_{μ}^{[n]}); ⊕_{ρ∈Q_{ν}^{−1}(μ)}A_{∗}(X_{reg}^{μ}); ∑ γ_{ν,ρ} = ∑ p_{μ}^{∗}( ) ∩ α_{ν,ρ}; A_{∗}(Γ_{reg}^{μ}),; q̃_{∗}]

whence the statement. □

We now study a special case which is crucial to the proof of Proposition 5.2.1. We consider μ = n^{1}. For every partition ν there is only one δ_{ρ} = δ : X → X^{l(ν)} and p := p_{n^{1}} : Γ^{n^{1}} → X is a Zariski locally trivial fibration, whose fibre is isomorphic to the length n punctual Hilbert scheme H_{n}. For each ν ∈ 𝔓(n) we have α_{ν} = p_{ν}^{!}([X]) ∩ [Γ^{n^{1}}] and the map P : A_{∗}(X)^{⊕p(n)} → A_{∗}(Γ^{n^{1}}) sending the collection {β_{ν}} to the class ∑_{ν} p^{∗}(β_{ν}) ∩ α_{ν}.

Lemma 5.2.10 The map P is surjective

Proof. By virtue of Lemma 3.0.5, it is enough to prove that the restrictions α̂_{ν} of the classes α_{ν} to any fibre generate A_{∗}(H_{n}). We may assume, without loss of generality that X is projective, for the restrictions α̂_{ν} do not change. Let i_{x} : {point} → X be the imbedding of a point x. Note that α̂_{ν} = i_{x}^{!}([Γ^{ν}]). Let g : H_{n} → X^{[n]} be the closed embedding and define β̂_{ν} := g_{∗}α̂_{ν} ∈ A_{∗}(X^{[n]}). Consider the pairing A_{∗}(X^{[n]}) × A_{∗}(X^{[n]}) → ℚ given by taking (a, b) → deg a · b = ∫_{X^{[n]}} a · b, where the last product is the product in the Chow ring. Note that this pairing is almost never perfect. It descends to algebraic equivalence.

CLAIM. deg (β̂_{ν} · [\overline{X}_{μ}^{[n]}]) = 0 if and only if μ ≠ ν.

Proof of the CLAIM. Consider Γ^{ν} and X^{ν} ×\overline{X}_{ν}^{[n]} as a pair of cycles in X^{ν} ×X^{[n]}. They define two families of cycles on X^{[n]}: Γ_{τ}^{ν} and \{X^{ν} × \overline{X}_{ν}^{[n]}\}_{τ}, τ ∈ X^{ν}. See [8], §10. In particular, we have Γ_{y}^{ν} = i_{y}^{!}([Γ^{ν}]). If y ∈ X_{reg}^{ν}, then Γ_{y}^{ν} = [(p_{ν}^{−1}(y))_{red}]. Recalling that we have the canonical δ_{ρ} : X → X^{ν}, we see that β̂_{ν} = Γ_{x}^{ν}. In other words: Γ_{y}^{ν} and β̂_{ν} belong to the same family of cycles. By construction, β̂_{ν} can be represented, modulo algebraic equivalence, by a cycle supported at (p_{ν}^{−1}(y))_{red}. Note that β̂_{ν} · [\overline{X}_{μ}^{[n]}] ∈ A_{l(ν)−l(μ)}(X^{[n]}). If l(ν) < l(μ), or if l(ν) > l(μ), then the degree is zero for trivial reasons. If l(ν) = l(μ), but μ ≠ ν, then the degree is still zero. In fact, using refined intersection products in the context of algebraic equivalence, we can represent β̂_{ν} as a cycle supported on a fiber π^{−1}(y). This fiber does not meet \overline{X}_{μ}^{[n]}. Finally, if ν = μ, then, by virtue of [8], Corollary 10.1 and Proposition 10.2, we have that

deg (β̂_{ν} · [\overline{X}_{μ}^{[n]}]) = deg (Γ_{y} · [\overline{X}_{μ}^{[n]}]) = deg([p_{ν}^{−1}(y)] · [\overline{X}_{ν}^{[n]}]) = m_{ν}.

The CLAIM follows easily. By virtue of Corollary 5.1.5 (see also [4]), the classes [\overline{X}_{μ}^{[n]}] are independent. Jointly with the CLAIM just proved, this shows that the classes β̂_{ν} are independent. It follows that the classes α̂_{ν} are independent. Their number is p(n), which is equal to the dimension of A_{∗}(H_{n}) (cf. [4], for example). This proves the lemma. □

Let us go back to the situation dealt with in Lemma 5.2.7: the diagram after that lemma and 5.2.10 imply immediately:

Lemma 5.2.11 The restriction of the cycle α_{ν,ρ} := p_{ν}^{!}([D_{ρ}]) ∩ [Γ^{ν}] ∈ A_{∗}(Γ^{μ}) to a fibre of p_{μ} is the cycle α̂_{ν^{1}} × ⋯ × α̂_{ν^{r}} ∈ A_{∗}(H_{μ_{1}} × ⋯ × H_{μ_{l}}).

Let \underline{η} = (η^{1}, ⋯, η^{r}) be a multipartition of μ. By this we mean that η^{i} ∈ 𝔓(μ_{i}). Set l_{i} = l(η^{i}) and define the open sets U\underline{η} ⊆ ∏ X^{η^{i}} by

U\underline{η} = \{(x_{1}^{(1)}, ⋯, x_{l_{1}}^{(1)}, ⋯, x_{1}^{(r)}, ⋯, x_{l_{r}}^{(r)}) such that x_{k}^{(i)} ≠ x_{l}^{(j)} if i ≠ j\}.

In other words we allow collisions only among points of the same group. X_{reg}^{μ} can be identified with the closed subset of points in U\underline{η} such that x_{k}^{(i)} = x_{l}^{(j)} if and only if i = j. According to [8], 8.1.4, the diagram

[figure: ∏ Γ_{|X_{reg}^{μ}}^{μ_{i}}; ∏ Γ^{η^{i}}; X_{reg}^{μ}; ×p_{μ_{i}}; ∏ X^{η^{i}}; ×p_{η^{i}}]

produces classes α\underline{η} ∈ A_{∗}(∏ Γ_{|X_{reg}^{μ}}^{μ_{i}}), α\underline{η} = (×p_{η^{i}})^{!}([X_{reg}^{μ}]) ∩ [∏ Γ^{η^{i}}] whose restrictions to a fibre is the cycle α̂_{η^{1}} × ⋯ × α̂_{η^{r}} ∈ A_{∗}(H_{μ_{1}} × ⋯ × H_{μ_{l}}).

Lemma 5.2.12 The set \{α\underline{η}\} generates A_{∗}(Γ_{reg}^{μ}) as a A_{∗}(X_{reg}^{μ})-module.

Proof. Since H_{μ_{1}} ×⋯×H_{μ_{r}} has a cellular decomposition, A_{∗}(H_{μ_{1}} × ⋯ × H_{μ_{r}}) = A_{∗}(H_{μ_{1}})⊗ ⋯ ⊗ A_{∗}(H_{μ_{r}}) has dimension ∏ p(μ_{i}). The set in question is therefore a basis of this space, and the statement is a direct consequence of Lemma 3.0.5, Remark 5.2.5 and Lemma 5.2.11.

□

Remark 5.2.13 We have already observed that a couple (ν, ρ) with Q_{ν}(ρ) = μ determines a multipartition \underline{ν}_{ρ} of μ; Lemma 5.2.7 implies that α_{\underline{ν}_{ρ}} = α_{ν,ρ}.

Remark 5.2.14 The set \{α_{ν,ρ}\} is only a subset of \{α\underline{η}\} as shown by the following example

Example 5.2.15 Let n = 4, and μ = 2^{2}. There are 4 multipartitions: (2, 2), (2, 1^{2}), (1^{2}, 2),(1^{2}, 1^{2}) and 4 corresponding cycles which restrict to a basis for A_{∗}(H_{2} × H_{2}). The couples (ν, ρ) such that Q_{ν}(ρ) = 2^{2} are: if ν = 1^{4}, then there are 3 different ρ’s which are conjugate by Σ_{ν} and give the same (0-dimensional) cycle corresponding to the multipartition (1^{2}, 1^{2}); if ν = 2^{2}, then there is only one ρ which gives the 2-dimensional cycle corresponding to (2, 2); if ν = 2 · 1^{2}, then the only ρ is {1}, {2, 3} which induces the multipartition (2, 1^{2}). This is consistent with the fact that X^{2·1^{2}} ×_{X^{(4)}} X_{2^{2}}^{(4)} contains only one component D_{ρ} given by points of type x_{2} = x_{3}. Note also that the map D_{ρ} → X_{2^{2}}^{(4)} is 2 : 1, the quotient map by Σ_{2^{2}}.

This can be easily explained: note first that the group Σ_{μ} acts on the set of multipartitions of μ. Given a multipartition \underline{η} = (η^{1}, ⋯, η^{r}) of μ, let η^{i} = η_{1}^{i} ≥ ⋯ ≥ η_{l_{i}}^{i}. The sequence of the η_{j}^{i} is a partition ν ∈ 𝔓(n), let l = ∑ l_{i} be its length. A permutation σ ∈ Σ_{l}, reordering the sequence in a non-increasing one, is identified up to left multiplication with Σ_{ν}. Define the subsets I_{i} := {σ((∑_{k=0}^{i−1} l_{k}) + 1), ⋯, σ(∑_{k=0}^{i} l_{k})}. We get {I_{i}} ∈ ℜ_{l}, and ∑_{k∈I_{i}} ν_{k} = μ_{i}. This associates with \underline{η} a couple (ν, ρ) such that ρ ∈ Q_{ν}^{−1}(μ). A different choice of the permutation σ gives the same partition ν whereas ρ is changed by the action of Σ_{ν} (cf. Remark 5.2.3). Starting from the couple (ν, ρ), instead, gives, as we have already observed, a multipartition η̂ of μ. Note that this multipartition depends on the way the I_{i}’s were ordered. We thus have:

Lemma 5.2.16 η̂ and \underline{η} are in the same Σ_{μ}-orbit.

Finally, let q : D_{ρ} = X_{reg}^{μ} → X_{μ}^{(n)} and q̃: Γ_{reg}^{μ} → X_{μ}^{[n]} be the quotient maps by Σ_{μ}. It follows from [8], Ex. 1.7.6, that q^{∗} (resp. q̃^{∗}) identify A_{∗}(X_{μ}^{(n)}) (resp. A_{∗}(X_{μ}^{[n]})) with A_{∗}(D_{ρ})^{Σ_{μ}}, (resp. A_{∗}(Γ_{reg}^{μ})^{Σ_{μ}}). By virtue of lemma 4.3.1, if α ∈ A_{∗}(D_{ρ}), then q^{∗}q_{∗}α = ∑_{σ∈Σ_{μ}} σ · α . Similarly, if α ∈ A_{∗}(Γ_{reg}^{μ}), then q̃^{∗}q̃_{∗}α = ∑_{σ∈Σ_{μ}} σ · α.

Lemma 5.2.17 The images under the map q̃^{∗}q̃_{∗} of the A_{∗}(X_{reg}^{μ})-submodules generated by \{α\underline{η}\} and \{α_{ν,ρ}\} coincide.

Proof. We fix our attention on a single Σ_{μ}-orbit. Set α_{ν,ρ} = α_{0}. Let H denote the stabilizer of α_{0} in Σ_{μ}, and choose a set g_{1} = e, g_{2}, …, g_{r} of representatives for Σ_{μ}/H, so that the orbit of α_{0} is α_{0}, g_{2}α_{0}, …, g_{r}α_{0}. Given a cycle α = ∑_{i=1}^{r} p^{∗}(β_{i})g_{i}α_{0}, let β = (∑_{i=1}^{r} g_{i}^{−1}p^{∗}(β_{i}))α_{0}.

Then q̃^{∗}q̃_{∗}α = q̃^{∗}q̃_{∗}β. In fact, q̃^{∗}q̃_{∗}α = ∑_{g,j} gp^{∗}(β_{j})gg_{j}α_{0} = ∑_{i} g_{i}(∑r ; j=1^{h∈H} hg_{j}^{−1}p^{∗}(β_{j}))g_{i}α_{0},

while q̃^{∗}q̃_{∗}β = ∑_{g} g(∑_{j} g_{j}^{−1}p^{∗}(β_{j}))gα_{0} = ∑_{i}(∑r ; j=1^{h∈H} g_{i}hg_{j}^{−1}p^{∗}(β_{j}))g_{i}α_{0}. □

We are now in the position to prove Proposition 5.2.1. Lemma 5.2.12 and Lemma 5.2.17 imply that the restriction of the push-forward map q̃_{∗} to the A_{∗}(X_{reg}^{μ})-submodule of A_{∗}(Γ_{reg}^{μ}) generated by the classes {α_{ν,ρ}} is surjective. This implies Proposition 5.2.1 by virtue of Lemma 5.2.9.

5.3 The surjectivity of Γ̂_{∗}

Proposition 5.2.1 implies easily the following surjectivity result:

Proposition 5.3.1 Let [Γ] be the fundamental cycle of Γ. The map

P_{∗}(p^{∗}(−) ∩ [Γ]) = Γ_{∗} : A_{∗}(X) = ⨁_{ν∈𝔓(n)} A_{∗}(X^{ν}) → A_{∗}(X^{[n]})

is surjective.

Proof. By virtue of Lemma 4.1.2 we have, for every μ ∈ 𝔓(n), a commutative diagram with exact columns:

[figure: A_{∗}(X_{μ}); p_{≥μ}^{!}(−) ∩ [Γ_{≥μ}]; A_{∗}(Γ_{μ}); P_{∗}; A_{∗}(X_{μ}^{[n]})]

[figure: A_{∗}(X_{≥μ}); i_{∗}; p_{≥μ}^{∗}(−) ∩ [Γ_{≥μ}]; A_{∗}(Γ_{≥μ}); P_{∗}; A_{∗}(X_{≥μ}^{[n]})]

[figure: A_{∗}(X_{>μ}); j^{∗}; p_{>μ}^{∗}(−) ∩ [Γ_{>μ}]; A_{∗}(Γ_{>μ}); P_{∗}; A_{∗}(X_{>μ}^{[n]})]

[figure: 0; 0; 0.]

We proceed by decreasing induction on μ, proving that P_{∗}(p^{∗}(−) ∩ [Γ_{≥μ}]) : A_{∗}(X_{≥μ}) → A_{∗}(X_{≥μ}^{[n]}) is surjective. The statement in the case μ = 1^{n} is clearly true since X_{1^{n}}^{[n]} = X_{1^{n}}^{(n)} = X_{1^{n}}/Σ_{n}. Suppose now that the surjectivity of P_{∗}(p^{∗}(−) ∩ [Γ_{>μ}]) : A_{∗}(X_{>μ}) → A_{∗}(X_{>μ}^{[n]}) has been established. By virtue of Proposition 5.2.1, the map in the first line of the diagram is surjective, whence the surjectivity of the second line and the statement. □

Corollary 5.3.2 The natural map

Γ̂_{∗} : A_{∗}(X̂) = ⨁_{ν∈𝔓(n)} A_{∗}(X^{(ν)}) ⟶ A_{∗}(X^{[n]})

is surjective.

Proof. Immediate from Proposition 5.3.1 and §4.4.

Remark 5.3.3 Corollary 5.3.2 holds if we replace all spaces by the ones obtained by base change with respect to any open immersion U → X^{(n)}. In fact, it is sufficient to use the corollary together with the standard exact sequence [8], Proposition 1.8.

5.4 The isomorphism of Chow groups

The main result of this paper now follows easily. It does not hold with ℤ coefficients.

Theorem 5.4.1 Let X be an irreducible nonsingular algebraic surface defined over an algebraically closed field. The natural map

Γ̂_{∗} = ⨁_{ν∈𝔓(n)} Γ̂_{∗}^{ν} : A_{∗}(X̂) = ⨁_{ν∈𝔓(n)} A_{∗}(X^{(ν)}) ⟶ A_{∗}(X^{[n]})

is an isomorphism.

Proof. Injectivity and surjectivity are proved in Corollary 5.1.5 and 5.3.2, respectively. □

The following is an immediate consequence of Theorem 5.4.1. Let Y be a scheme. We denote by K_{o}(Y) the Grothendieck group of coherent sheaves on Y and we define K_{o}(Y)_{ℚ} := K_{o}(Y) ⊗_{ℤ} ℚ.

Corollary 5.4.2 There is a natural isomorphism of Grothendieck groups with ℚ-coefficients

Γ̂_{∗} = ⨁_{ν∈𝔓(n)} Γ̂_{∗}^{ν} : K_{o}(X̂)_{ℚ} = ⨁_{ν∈𝔓(n)} K_{o}(X^{(ν)})_{ℚ} ⟶ K_{o}(X^{[n]})_{ℚ}.

Proof. It follows immediately from Theorem 5.4.1 and [8], Corollary 18.3.2. □

Remark 5.4.3 Corollary 5.4.2 and the localization theorem for equivariant K-theory [19], imply that one has an isomorphism K_{o}^{Σ_{n}}(X^{n})_{ℚ} ≃ K_{o}(X^{[n]})_{ℚ}. In a paper in preparation, we construct this isomorphism directly, i.e. without the localization theorem. We expect similar statements to hold for the higher K-theory and for the equivariant derived category. If the so-called n!-conjecture, [13], were proved, then the paper [2] would imply, in characteristic zero, a weaker version of Theorem 5.4.1.

Remark 5.4.4 Theorem 5.4.1 and Corollary 5.4.2 hold if we replace X and X^{[n]} by the corresponding open sets obtained by base change with respect to any open immersion U → X^{(n)}. See Remark 5.1.6 and 5.3.3.

Remark 5.4.5 Note that, by virtue of [8], Ex. 1.7.6, A_{∗}(X^{(ν)}) = A_{∗}(X^{ν})^{Σ_{ν}}. However, unless X itself has a cellular decomposition, the natural map A_{∗}(X)^{⊗l(ν)} → A_{∗}(X^{(ν)}) is not necessarily surjective. This contrasts singular cohomology, where one has Künneth Formula.

Remark 5.4.6 With minor modifications, which we leave to the interested reader, Theorem 5.4.1 holds: 1) for a geometrically irreducible smooth surface X defined over any field, and 2) in the analytic context where X is a smooth complex analytic surface (see [8], Ex. 19.2.5).

6 The motive of X^{[n]}

Let X be an irreducible nonsingular quasi projective surface defined over an algebraically closed field.

6.1 The correspondences Δ_{ν}

We freely use refined products over quotient varieties and the formalism of correspondences between quotient varieties together with their standard properties. See §4. Let ν ∈ 𝔓(n) and consider (X_{ν}^{[n]} ×_{X^{(n)}} X_{ν}^{[n]})_{red}. Being the quotient of a Zariski locally trivial fibration with irreducible fiber by the action of a finite group, this space is a 2n-dimensional irreducible locally closed subset of X^{[n]} × X^{[n]}. Let D^{ν} ⊆ X^{[n]} × X^{[n]} be its closure:

D^{ν} := \overline{\{(a, b) ∈ X^{[n]} × X^{[n]} | π(a) = π(b) ∈ X_{ν}^{(n)}\}} ∈ Z_{2n}(X^{[n]} × X^{[n]}).

Note that the image of D^{ν} under either projection is the closed stratum \overline{X}_{ν}^{[n]}.

Remark 6.1.1 D^{ν} is an irreducible component of (\overline{X}_{ν}^{[n]} ×_{X^{(n)}} \overline{X}_{ν}^{[n]})_{red}. The latter contains all the D^{μ} such that μ ⪯ ν precisely as its irreducible components.

Let ν ∈ 𝔓(n). The components of p_{12}^{−1}(^{t}Γ̂^{ν}) ∩ p_{23}^{−1}(Γ̂^{ν}) in X^{[n]} × X^{(ν)} × X^{[n]} are all of the expected dimension. The refined formalism mentioned above, Remark 3.0.4 and Remark 6.1.1 show that the following is a well-defined 2n-dimensional cycle with rational coefficients

Δ_{ν} := \frac{1}{m_{ν}} Γ̂^{ν} ∘^{t}Γ̂^{ν} ∈ Z_{2n}(X^{[n]} × X^{[n]}) ⊗_{ℤ} ℚ

with the property that Δ_{ν} is supported precisely on (\overline{X}_{ν}^{[n]} ×_{X^{(n)}} \overline{X}_{ν}^{[n]})_{red} and that

Δ_{ν} = ∑_{ν⪰ν′} ϵ_{ν′}^{ν}D^{ν′},

where the numbers ϵ_{ν′}^{ν} are nonzero rational numbers with the same sign as m_{ν}.

Lemma 6.1.2 Let ν and μ be partitions of n. Then

Δ_{ν} ∘ Δ_{μ} = δ_{νμ}Δ_{ν} ∈ Z_{2n}(X^{[n]} × X^{[n]}) ⊗_{ℤ} ℚ.

where δ_{νμ} is the usual Krönecker function. In particular, the Δ_{ν} are mutually orthogonal projectors and:

Δ_{ν}_{∗} : A_{∗}(X^{[n]}) → A_{∗}(X^{[n]}), Δ_{ν}_{∗} ∘ Δ_{μ}_{∗} = δ_{μν}Δ_{ν}_{∗}.

Proof. By the associativity of the composition of correspondences we have

Δ_{ν} ∘ Δ_{μ} = \frac{Γ̂^{ν} ∘^{t}Γ̂^{ν}}{m_{ν}} ∘ \frac{Γ̂^{μ} ∘^{t}Γ̂^{μ}}{m_{μ}} = \frac{Γ̂^{ν}}{m_{ν}} ∘ (\frac{^{t}Γ̂^{ν} ∘ Γ̂^{μ}}{m_{μ}} ) ∘^{t}Γ̂^{μ}.

We conclude by Proposition 5.1.3 and Proposition 5.1.4. □

Recall that F_{μ} denotes the reduced fiber (π^{−1}(\underline{x}))_{red}, where \underline{x} ∈ X_{μ}^{(n)}.

Lemma 6.1.3 Let ν and μ be partitions of n.

(i) If ν ⪰ / μ, then Δ_{ν}_{∗}([F_{μ}]) = 0. If ν = μ, then Δ_{ν}_{∗}([F_{μ}]) = [F_{μ}]. (ii) If ν ⪰ / μ, then D^{ν}_{∗}([F_{μ}]) = 0. If ν = μ, then Δ_{ν}_{∗}([F_{μ}]) = c_{ν}[F_{ν}], where ℚ ∋ c_{ν} ≠ 0.

Proof. We compute Δ_{ν}_{∗} using refined intersections so that the classes [F_{μ}], which are non zero in A_{∗}(F_{μ}), may be zero in A_{∗}(X^{[n]}). Of course this does not happen if, for example, X is projective. If ν does not refine μ, then p^{−1}(F_{μ}) ∩ Δ_{ν} = ∅, and the first parts of (i) and (ii) follow at once. Note that since we have proved the first parts of (i) and (ii), (i) implies (ii) by consideration of supports. We now prove the second part of (i). Let μ = ν. Note that Δ_{ν}_{∗}([F_{ν}]) = \frac{1}{m_{ν}} Γ̂_{∗}^{ν}(^{t}Γ̂_{∗}^{ν}([F_{ν}])). The zero cycle ^{t}Γ̂_{∗}^{ν}([F_{ν}]) is supported at the point ν^{−1}(\underline{x}) ∈ X^{(ν)}. By

virtue of the projection formula, its degree equals deg ([F_{ν}] · [\overline{X}_{ν}^{[n]}]) = m_{ν}; see Lemma 5.1.2. In other words^{t}Γ̂_{∗}^{ν}([F_{ν}]) = m_{ν}[ν^{−1}(\underline{x})]. We apply Γ̂_{∗}^{ν} and we find the conclusion. □

Note that the diagonal Δ = D^{1^{n}}. The main reason for introducing the correspondences Δ_{ν} is the following

Proposition 6.1.4 The map (∑_{ν∈𝔓(n)} Δ_{ν} − Δ)_{∗} is the zero map. In particular, ∑_{ν} Δ_{ν}_{∗} = Δ_{∗} = Id_{A_{∗}(X^{[n]})}.

Proof. This follows immediately from Theorem 5.4.1 which implies that^{t}Γ̂_{∗} is the inverse of Γ̂_{∗} and from the fact that Γ̂_{∗} ∘^{t}Γ̂_{∗} = ∑_{ν∈𝔓(n)} Δ_{ν}_{∗}. □

Proposition 6.1.5 ∑_{ν} Δ_{ν} − Δ = 0 in Z_{2n}(X^{[n]} × X^{[n]}) ⊗_{ℤ} ℚ.

Proof. Without loss of generality, we may assume that X is projective. Note that, for every ν ≠ 1^{n}, D_{∗}^{ν} is identically zero on A_{0}(X^{[n]}). It follows, by virtue of Lemma 6.1.3, that we can write ∑_{ν∈𝔓(n)} Δ_{ν} − Δ = ∑_{ν≠1^{n}} p_{ν}D^{ν}, where p_{ν} are suitable rational numbers. Seeking a contradiction, assume that there is at least one partition μ for which p_{μ} ≠ 0. Choose a partition θ which is maximal, with respect to the partial ordering on partitions, among all partitions μ for which p_{μ} ≠ 0. We have, again by virtue of Lemma 6.1.3, that

0 = (∑_{ν} p_{ν}D^{ν})_{∗}[F_{θ}] = p_{θ}D_{∗}^{θ}([F_{θ}]) ≠ 0.

The contradiction stems from the projectivity assumption, for then no effective cycle can be trivial. □

6.2 The structure and the generating function of motives

Let X be an irreducible nonsingular projective surface defined over an algebraically closed field. The correspondence Γ̂^{ν} defines a morphism, which by abuse of notation we denote by the same symbol, of effective Chow motives with rational coefficients:

Γ̂^{ν} : (X^{(ν)}, Δ_{X^{(ν)}})(n − l(ν)) ⟶ (X^{[n]}, Δ_{X^{[n]}}).

See [15], page 459. The Tate-type shift is in accordance with the usual convention (ℙ^{1}, Δ_{ℙ^{1}})(1) = (ℙ^{1}, p), where p = ℙ^{1} × {0}. This morphism admits, in the category of Chow motives with rational coefficients, a right inverse. This is given, again by abuse of notation, by ^{t}Γ̂^{ν}. By virtue of [15], page 453, we can split-off a direct summand:

Γ̂^{ν} : (X^{(ν)}, Δ_{X^{(ν)}})(n − l(ν)) ≃ (X^{[n]}, Δ_{ν}).

By virtue of Lemma 6.1.2, the projectors Δ_{ν} are mutually orthogonal so that we can split-off a direct summand for each partition:

Γ̂ := ⨁_{ν∈𝔓(n)} Γ̂^{ν} : ⨁_{ν∈𝔓(n)}(X^{(ν)}, Δ_{X^{(ν)}})(n − l(ν)) ≃ ⨁_{ν∈𝔓(n)}(X^{[n]}, Δ_{ν}) = (X^{[n]}, ∑_{ν∈𝔓(n)} Δ_{ν}).

The following is the main theorem of this section and follows immediately from what above and Proposition 6.1.5.

Theorem 6.2.1 Let X be an irreducible nonsingular projective surface defined over an algebraically closed field. There is a natural isomorphism of effective Chow motives with rational coefficients

(X^{[n]}, Δ_{X^{[n]}}) ≃ ⨁_{ν∈𝔓(n)}(X^{(ν)}, Δ_{X^{(ν)}})(n − l(ν)).

Remark 6.2.2 Theorem 6.2.1 and Theorem 6.2.5 below hold over any field. We leave the necessary but easy modifications to the reader.

Remark 6.2.3 Over the complex numbers, Theorem 6.2.1 gives immediately the structure of the singular cohomology H^{∗}(X^{[n]}, ℚ), together with its Hodge structure. The resulting isomorphisms coincide with the ones obtained in [4].

Remark 6.2.4 After this paper was submitted, L. Göttsche has informed us that he has calculated, by an entirely different method [11], the class of X^{[n]} in the Grothendieck ring of varieties over an algebraically closed field of characteristic zero. As a consequence, he calculates the class of the motive of X^{[n]} in the Grothendieck ring of Chow motives. This result is implied, by Theorem 6.2.1.

Voevodsky [20] has defined several motivic categories over a field k: DM_{gm}^{eff}(k) (cf. [20], 2.1.1), DM_{−}^{eff}(k) (cf. [20], 3.1.12), DM_{−,et}^{eff}(k) (cf. [20], 3.3), DM_{h}(k)_{ℚ} (cf. [20] and [21]). These are, essentially, categories of bounded complexes of pre-sheaves on certain sites. There is an obvious formalism of shifts of complexes and a formalism of Tate-type shifts. In addition, there are natural transformations between the category of effective Chow motives with rational coefficients and these other categories (see [20], 2.1.4):

Chow^{eff}(k)_{ℚ} → DM_{gm}^{eff}(k) → DM_{−}^{eff}(k) → DM_{−,et}^{eff}(k) → DM_{h}(k)_{ℚ}.

The formalism holds for quotient varieties. Let us denote by M(Z) the object in any of the above categories corresponding to a variety Z. Theorem 6.2.1 and the natural transformations mentioned above imply the following

Theorem 6.2.5 Let X be an irreducible nonsingular projective surface defined over an algebraically closed field. Then

M(X^{[n]}) ≃ ⨁_{ν∈𝔓(n)} M(X^{(ν)})(n − l(ν))[2n − 2l(ν)].

If we use the dual Tate twist, then we can re-write the conclusion of the theorem as follows:

M(X^{[n]})(n)[2n] ≃ ⨁_{ν∈𝔓(n)} M(X^{(ν)})(l(ν))[2l(ν)].

This is formally reminiscent of 1) the decomposition theorem for the Douady-Barlet morphism [4], where shifts for complexes of sheaves appear and 2) the Hodge structure of X^{[n]} (see [4], for example), where Tate shifts appear. It is by staring at those two formulas that we became convinced that the statement of Theorem 6.2.5 should hold. Theorem 5.4.1 and Theorem 6.2.1 were the means to realize this isomorphism.

It is amusing to note that there is a generating function for this picture, the one that generates partitions. In fact, skipping the bookeeping details, the formula is as follows:

∑_{n=0}^{∞} X^{[n]}(n)[2n] q^{n} = ∏_{m=1}^{∞} \frac{1}{(1 − X_{m}(1)[2] q^{m})},

where it is understood that in the r.h.s. we identify monomials of type X_{1}^{a_{1}} ⋯ X_{t}^{a_{t}} with X^{(ν)}, where ν = 1^{a_{1}} ⋯ t^{a_{t}} ∈ 𝔓(t).

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