The Chow of S (inverted right perpendicular n inverted
left perpendicular) and the universal subscheme
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Negut, Andrei. "The Chow of S (inverted right perpendicular n
inverted left perpendicular) and the universal subscheme." Bulletin
Mathematique de la Societe des Sciences Mathematiques de
Roumanie 64, 3 (2020): 385-394.
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THE CHOW OF S
[n]AND THE UNIVERSAL SUBSCHEME
ANDREI NEGUT,
Abstract. We prove that any element in the Chow ring of the Hilbert scheme Hilbn of n points on a smooth surface S is a universal class, i.e. the
push-forward of a polynomial in the Chern classes of the universal subschemes on Hilbn× Skfor some k ∈ N, with coefficients pulled back from the Chow of Sk.
1. Introduction
We study the Hilbert scheme of n points Hilbn = S[n] on a smooth algebraic
surface over C. As Hilbn is smooth, we may consider the Chow groups A∗(Hilbn),
always with coefficients in Q throughout the present paper. One of the big sources of elements of A∗(Hilbn) are universal classes, see Definition 2.2. During
a conversation on Hilbert schemes, Alina Marian suggested that all elements of A∗(Hilbn) should be universal, and the purpose of the present note is to prove it.
Theorem 1.1. Any element of A∗(Hilbn) is a universal class.
When S is projective, this result follows from an explicit formula for the diagonal of Hilbn as a Chern class of the so-called Ext virtual bundle, which in turn can
be written in terms of universal classes (see [4] for S = P2, [1] for S with effective anti-canonical line bundle, [7] for S with trivial canonical line bundle, and [5] for the general case). Our proof is quite different from those above, and holds for quasi-projective S. We start from [2], which states that:
(1.1) A∗(Hilbn) ∼= k1≥...≥kt k1+...+kt=n M Γ∈A∗(St)sym Ck1,...,kt(Γ)
where Ck1,...,kt(Γ) are certain correspondences (see (2.11) for an explicit descrip-tion, as well as an explanation of the superscript “sym”) expressed in terms of the Heisenberg operators qk of [6] and [9]. The explicit description of these operators
in terms of l.c.i. morphisms from [11] allows us to show that they preserve the subrings of A∗(Hilbn) consisting of universal classes, thus implying Theorem
1.1. Moreover, this gives an algorithm for computing the universal classes corre-sponding to the various summands in (1.1), as we will explain on the last two pages. I would like to thank Alina Marian, Eyal Markman, Davesh Maulik, Georg Oberdieck, Junliang Shen, Richard Thomas and Qizheng Yin for many interesting
2010 Mathematics Subject Classification: 14C05
Key words: Hilbert schemes of points on surfaces, algebraic cycles
2 ANDREI NEGUT,
discussions on Hilbert schemes. I gratefully acknowledge the NSF grants DMS– 1760264 and DMS–1845034, as well as support from the Alfred P. Sloan Foundation.
2. Hilbert schemes
2.1. Let S be a smooth algebraic surface over C. Let Hilbn = S[n] denote the
Hilbert scheme which parametrizes length n subschemes of S, i.e. exact sequences: 0 → I → OS → Z → 0
(I will be an ideal sheaf) where length(Z) = n. There exists a universal subscheme: Zn ⊂ Hilbn× S
whose restriction to any {Z} × S is precisely Spec Z as a subscheme of S. Then: (2.1) 0 → In→ OHilbn×S→ OZn → 0
is a short exact sequence of coherent sheaves on Hilbn× S, flat over Hilbn. Let:
(2.2) Hilb =
∞
G
n=0
Hilbn
The Hilbert scheme Hilbn is well-known to be a smooth 2n dimensional variety, so
we may consider its Chow rings A∗(Hilbn), always with rational coefficients. Set:
A∗(Hilb) =
∞
M
n=0
A∗(Hilbn)
For any k ∈ N, we let π : Hilbn× Sk→ Hilbn denote the standard projection, and
let Zn(i)⊂ Hilbn×Skdenote the pull-back of Zn⊂ Hilbn×S via the i–th projection.
Definition 2.2. A universal class is any element of A∗(Hilbn) of the form:
(2.3) π∗ h P (..., chj(OZ(i) n ), ...) 1≤i≤k j∈N i
∀k ∈ N and ∀ polynomials P with coefficients pulled back from A∗(Sk), such that:
(2.4) supp P ⊂ Zn(1)∩ ... ∩ Zn(k)
(which implies that the push-forward in (2.3) is well-defined).1
Proposition 2.3. The set:
A∗univ(Hilbn) ⊆ A∗(Hilbn)
of all universal classes is a subring.
1The notion above is more general than either the small or big tautological classes considered
THE CHOW OF S AND THE UNIVERSAL SUBSCHEME 3
Proof. It is clear that any Q–linear combination of universal classes is universal, even if they are defined with respect to different k’s. This is because any class of the form (2.3) for a given k is also of the form (2.3) for k + 1. Indeed, we have: π∗ h P (..., chj(OZ(i) n ), ...) 1≤i≤k j∈N i = (π ◦ σ)∗ h σ∗P (..., chj(OZ(i) n ), ...) 1≤i≤k j∈N · ∆k,k+1 i
where ∆k,k+1is the pull-back of the codimension 2 diagonal in Sk+1involving the
last two factors, and σ : Hilbn× Sk+1→ Hilbn× Sk forgets the last copy of S. So
it remains to prove that the product of universal classes is also a universal class. This is a consequence of the identity:
π∗ h P (..., chj(OZ(i) n ), ...) 1≤i≤k j∈N i · ρ∗ h Q(..., chj(OZ(i) n ), ...) 1≤i≤l j∈N i = (2.5) = f∗ h P (..., chj(OZ(i) n ), ...) 1≤i≤k j∈N · Q(..., chj(OZ(i) n ), ...) k+1≤i≤k+l j∈N i
with all maps as in the following Cartesian square: Hilbn× Sk+l ρ0 π0 // f '' Hilbn× Sl ρ Hilbn× Sk π // Hilb n
(all the maps are identities on Hilbn, and we think of π0and ρ0as forgetting the first,
respectively last, factors of S). The identity (2.5) is a straightforward consequence of f = ρ ◦ π0 = π ◦ ρ0 and the base change formula π0∗ρ0∗= ρ∗π∗. Throughout the
present proof, we were able to use the push-forward maps π∗, σ∗ and ρ∗even if π, σ
and ρ were non-proper (in the case of quasi-projective S), because we only applied them to classes whose support is proper under the respective maps.
2.4. We will prove Theorem 1.1 by deducing it from another well-known description of A∗(Hilbn): the de Cataldo-Migliorini decomposition ([2]). To review this
con-struction, we must recall the Heisenberg algebra action introduced independently by Grojnowski ([6]) and Nakajima ([9]) on the Chow groups of Hilbert schemes. For any n, k ∈ N, consider the closed subscheme:
Hilbn,n+k=
n
(I ⊃ I0) s.t. I/I0 is supported at a single x ∈ So⊂ Hilbn× Hilbn+k
endowed with projection maps:
(2.6) Hilbn,n+k p− yy pS p+ && Hilbn S Hilbn+k
that remember I, x, I0, respectively. One may use Hilbn,n+k as a correspondence:
(2.7) A∗(Hilb)−→ Aqk ∗(Hilb × S) (recall the notation (2.2)) given by:
4 ANDREI NEGUT,
2The main result of [9] is that the operators q
k obey the commutation relations in
the Heisenberg algebra. More generally, we may consider: (2.9) qk1...qkt : A
∗(Hilb) → A∗(Hilb × St)
where the convention is that the operator qkiacts in the i-th factor of S
t= S×...×S.
Then associated to any Γ ∈ A∗(St), one obtains an endomorphism of A∗(Hilb): (2.10) qk1...qkt(Γ) = π1∗(π
∗
2(Γ) · qk1...qkt)
where π1, π2 : Hilb × St → Hilb, St denote the standard projections (the
non-properness of π1is not a problem for defining (2.10), because the support of qk1...qkt is proper over Hilb). One of the main results of [2] is the following decomposition: (2.11) A∗(Hilb) = k1≥...≥kt∈N M Γ∈A∗(St)sym qk1...qkt(Γ) · A ∗(Hilb 0)
where the superscript “sym” refers the part of A∗(St) which is symmetric with
respect to those transpositions (ij) ∈ St for which ki = kj. Since Hilb0= pt, we
have A∗(Hilb0) ∼= Q, and so Theorem 1.1 follows from (2.11) and the following:
Proposition 2.5. The endomorphisms (2.10) preserve the subrings A∗univ(Hilbn).
2.6. The remainder of our paper will be devoted to proving Proposition 2.5. The problem with doing so directly from the definition (2.8) is that the correspondences (2.6) are rather singular. The exception to this is the case k = 1, namely:
(2.12) Hilbn−1,n p− xx pS p+ %% Hilbn−1 S Hilbn (I ⊃xI0) p− {{ pS p+ ## I x I0
Above and hereafter, we will write I ⊃xI0if I ⊃ I0and I/I0 ∼= Cx. It is well-known
that Hilbn−1,n is smooth of dimension 2n. Consider the line bundle:
(2.13) L
Hilbn−1,n
L|(I⊃xI0)= Γ(S, I/I
0)
If E = [W → V] is a complex of locally free sheaves on a scheme X, then we define: (2.14) PX(E ) ,→ PX(V) := ProjX(Sym(V))
to be the closed subscheme determined by the image of the map: (2.15) ρ∗(W) → ρ∗(V) → O(1)
where ρ : PX(V) → X is the standard projection. In all cases considered in the
present paper, the closed subscheme (2.14) is a local complete intersection, cut out by the cosection (2.15). The following result is closely related to Lemma 1.1 of [3]:
2The transposed correspondences give rise to operators q
THE CHOW OF S AND THE UNIVERSAL SUBSCHEME 5
Proposition 2.7. Let In be the universal ideal sheaf on Hilbn× S, i.e. the kernel
of the map OHilbn×S OZn from (2.1). Then we have an isomorphism: (2.16) Hilbn−1,n ∼ = // p+×pS (( PHilbn×S(−I ∨ n ⊗ ωS) ρ Hilbn× S
The line bundle L on Hilbn−1,n is isomorphic to O(−1) on PHilbn×S(−I
∨ n ⊗ ωS).
We refer the reader to Section 4 of [10] for details on why (2.16) is a special case of (2.14). In a few words, there is a short exact sequence with W, V locally free: (2.17) 0 → W → V → In → 0
Then the notation −In∨ in (2.16) stands for the complex [V∨→ W∨]. Finally, ω S
denotes both the canonical line bundle on S and its pull-back to Hilbn× S.
2.8. Let us consider the following more complicated cousin of the scheme Hilbn,n+1:
(2.18) Hilbn−1,n,n+1=
n
(I, I0, I00) such that I ⊃xI0⊃xI00for some x ∈ S
o
where I ∈ Hilbn−1, I0 ∈ Hilbn and I00 ∈ Hilbn+1. We have shown in [10] that
Hilbn−1,n,n+1is smooth of dimension 2n + 1. Consider the line bundles:
(2.19) L, L0 Hilbn−1,n,n+1 L|(I⊃xI0⊃xI00)= Γ(S, I 0/I00), L0| (I⊃xI0⊃xI00)= Γ(S, I/I 0)
Consider also the proper maps which forget either I00or I:
(2.20) Hilbn−1,n,n+1 π− ww π+ Hilbn−1,n Hilbn,n+1 (I ⊃xI0⊃xI00) π− ww π+ (I ⊃xI0) (I0 ⊃xI00)
Let Γ : Hilbn,n+1 ,→ Hilbn,n+1×S be the graph of the map pS, and let L be the line
bundle (2.13) on Hilbn,n+1. We showed in [11] that there is a short exact sequence:
0 → L−1 → Γ∗(V∨) → Γ∗(W∨) with W, V as in (2.17). We will use the notation −Γ∗(I∨
n) + L−1 for the complex:
Γ∗(V∨)
L−1 −→ Γ ∗(W∨)
6 ANDREI NEGUT,
Proposition 2.9. ([11]) Let In be the universal ideal sheaf on Hilbn× S. Then:
(2.21) Hilbn−1,n,n+1 ∼ = // π+ ** PHilbn,n+1(−Γ ∗(I∨ n) ⊗ ωS+ L−1⊗ ωS) ρ Hilbn,n+1
The line bundle L0 on Hilbn−1,n,n+1is isomorphic to O(−1) on the projectivization.
In both (2.16) and (2.21), we considered projectivization P(∗) where ∗ is written as a K–theory class instead of as a complex of sheaves. The reason for this is that we are only interested in ∗ inasmuch as it helps us compute push-forwards. In fact, the definition of Chern/Segre classes implies that we have, for all k ≥ 0:
(2.22) (p+× pS)∗(c1(L)k) = (−1)kck+2 In⊗ ω−1S (2.23) π+∗(c1(L0)k) = (−1)kck+1 Γ∗(In) ⊗ ωS−1− L ⊗ ω −1 S
2.10. Our reason for defining the smooth schemes Hilbn−1,n,n+1 is that it allows
us to produce a resolution of the singular scheme Hilbn,n0, for any n < n0, in the following sense. Consider the following diagram of spaces and maps ([11]):
Hilbn,n+1,n+2 π− ww π+ %% Hilbn0−2,n0−1,n0 π− xx π+ (( Hilbn,n+1 p− . . . Hilbn0−1,n0 p+×pS Hilbn Hilbn0× S
for all n < n0. Then we have the following formula ([8]): (2.24) qk = (p+× pS)∗◦ (π+∗◦ π∗−)k−1◦ p∗−
Indeed, the right-hand side of (2.24) is a 2n + k + 1 dimensional cycle C supported on the 2n + k + 1 dimensional locus Hilbn,n+k. It is well-known that the latter
locus has a single irreducible component of top dimension, namely the closure of the locus U of pairs (I ⊃ I0) where I/I0 is isomorphic to a length k subscheme of a curve supported at a single point. But in this case, there exists a unique full flag of ideals I = I0⊃ I1⊃ ... ⊃ Ik= I0, which implies that C|U ∼= U , hence (2.24) follows.
2.11. For any t ≥ 0, define the universal subring:
(2.25) A∗univ(Hilbn× St) ⊂ A∗(Hilbn× St)
as the subring generated by the classes (2.3) for all k ≥ t, where one replaces π by the map Hilbn× Sk→ Hilbn× Stwhich forgets the last k − t factors. Then we let:
A∗univ(Hilbn,n+1× St) ⊂ A∗(Hilbn,n+1× St)
(2.26)
A∗univ(Hilbn−1,n,n+1× St) ⊂ A∗(Hilbn−1,n,n+1× St)
THE CHOW OF S AND THE UNIVERSAL SUBSCHEME 7
be the subrings generated by c1(L) (respectively c1(L0)) and the pull-backs of all
universal classes via the following maps, respectively: p−× IdSt : Hilbn,n+1× St→ Hilbn× St
π−× IdSt : Hilbn−1,n,n+1× St→ Hilbn−1,n× St
With this in mind, Proposition 2.5 is a consequence of (2.24) and the following:
Proposition 2.12. For any t ≥ 0, the maps (p− × IdSt)∗, (π− × IdSt)∗, (π+× IdSt)∗, (p+× pS× IdSt)∗ preserve the universal subrings, as defined above.
Indeed, formula (2.24) and Proposition 2.12 imply that if x ∈ A∗univ(Hilb), then y = qk1...qkt(x) ∈ A
∗
univ(Hilb × St), in the sense of (2.25). If we
multiply y by the pull-back of any Γ ∈ A∗(St) and then push it forward to Hilb,
it will remain in the subring of universal classes, and this establishes Proposition 2.5.
Proof. of Proposition 2.12: The statements about the pull-back maps (p−× IdSt)∗ and (π−× IdSt)∗ preserving the universal rings are obvious given definitions (2.26) and (2.27). Concerning the push-forward (π+× IdSt)∗, we must show that: (2.28) x ∈ A∗univ(Hilbn−1,n,n+1× St) ⇒ (π+× IdSt)∗(x) ∈ A∗univ(Hilbn,n+1× St) We have the following short exact sequence on Hilbn−1,n,n+1× S:
(2.29) 0 → L0⊗ (pS× IdS)∗(O∆) → OZn → OZn−1 → 0
where ∆ ⊂ S × S is the diagonal, and pS : Hilbn−1,n,n+1 → S is the map which
remembers the point x in (2.18). Then for any polynomial P in the Chern classes of Zn−1(i) , whose coefficients are pulled back from Hilbn,n+1× Sk to Hilbn−1,n,n+1× Sk:
P (..., chj(OZ(i) n−1 ), ...)1≤i≤kj∈N = P (..., chj(OZ(i) n − L 0⊗ (p S× proji)∗(O∆)), ...) 1≤i≤k j∈N = (2.30) = ∞ X a=0 c1(L0)a· (π+× IdSk)∗(Ra) ∈ A∗(Hilbn−1,n,n+1× Sk)
for various Ra∈ A∗(Hilbn,n+1×Sk) which are also polynomials in the Chern classes
of the universal subschemes. If we apply (π+× IdSk)∗to (2.30), we obtain: (2.31) (π+× IdSk)∗ h P (..., chj(OZ(i) n−1 ), ...)1≤i≤kj∈N i= ∞ X a=0 π+∗(c1(L0)a) · Ra LettingR
Sk−t denote the push-forward along the last k − t factors of S
k, we have: (π+× IdSt)∗ Z Sk−t P (..., chj(OZ(i) n−1 ), ...)1≤i≤kj∈N | {z }
x from the LHS of (2.28) is of this form
= ∞ X a=0 π+∗(c1(L0)a) Z Sk−t Ra
The implication (2.28) follows because the RHS above is a universal class: this is true for π+∗(c1(L0)a) because of formula (2.23), and for
R
Sk−tRa since Ra
is a polynomial in the Chern classes of the universal subscheme. Finally, the support condition (2.4) is satisfied becase the LHS of (2.30) is by assumption supported on the locus (In−1 ⊃x In ⊃x In+1) × (x1, ..., xk) ∈ Hilbn−1,n,n+1× Sk,
8 ANDREI NEGUT,
where x1, ..., xk ∈ supp In−1, hence the Ra’s in the RHS of (2.31) are supported
on the locus (In ⊃xIn+1)×(x1, ..., xk) ∈ Hilbn,n+1×Sk, where x1, ..., xk∈ supp In.
Concerning the push-forward (p+× pS× IdSt)∗, we must show that:
(2.32) y ∈ A∗univ(Hilbn−1,n× St) ⇒ (p+× pS× IdSt)∗(y) ∈ A∗univ(Hilbn× S × St) By a short exact sequence analogous to (2.29), for any polynomial P in the Chern classes of Zn−1(i) , whose coefficients are pulled back from Hilbn×Skto Hilbn−1,n×Sk:
(2.33) P (..., chj(OZ(i) n−1 ), ...)1≤i≤kj∈N = = ∞ X a=0 c1(L)a· (p+× pS× IdSk)∗(Ra) ∈ A∗(Hilbn−1,n× Sk)
for various Ra∈ A∗(Hilbn× S × Sk) which are polynomials in the Chern classes of
the universal subschemes. If we apply (p+× pS× IdSk)∗ to (2.33), we obtain: (2.34) (p+× pS× IdSk)∗P (..., chj(OZ(i) n−1 ), ...)1≤i≤kj∈N = ∞ X a=0 (p+× pS)∗(c1(L)a) · Ra hence: (p+×pS×IdSt)∗ Z Sk−t P (..., chj(OZ(i) n−1 ), ...)1≤i≤kj∈N | {z }
y from the LHS of (2.32) is of this form
= ∞ X a=0 (p+×pS)∗(c1(L)a) Z Sk−t Ra
The implication (2.32) follows because the RHS above is a universal class: this is true for (p+× pS)∗(c1(L)a) because of formula (2.22), and forRSk−tRasince Rais a
polynomial in the Chern classes of the universal subscheme. However, checking the support condition is non-trivial, so let us explain this. By assumption, the LHS of (2.33) is supported on the locus of points (In−1⊃xIn) × (x1, ..., xk) ∈ Hilbn−1,n×
Sk, where x
1, ..., xk∈ supp In−1. Therefore, the Ra’s which appear in the RHS of
(2.34) are supported on the locus of points (In, x) × (x1, ..., xk) ∈ Hilbn× S × Sk,
where x1, ..., xk ∈ supp In. However, a universal class also needs to be supported
on the locus of points where x ∈ supp In, but we are rescued because:
(p+× pS)∗(c1(L)a) = (−1)aca+2 In⊗ ωS−1
vanishes on the locus x /∈ supp In (this holds because the universal ideal sheaf I is
trivial on the locus x /∈ supp In, and ca+2(line bundle) = 0 for a ≥ 0).
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THE CHOW OF S AND THE UNIVERSAL SUBSCHEME 9
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MIT, Department of Mathematics, 77 Massachusetts Ave, Cambridge, MA 02139, USA Simion Stoilow Institute of Mathematics, Bucharest, Romania
