View of A partial resolution of the punctual Hilbert scheme of a nonsingular surface

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A PARTIAL RESOLUTION OF THE PUNCTUAL HILBERT SCHEME OF A NONSINGULAR

SURFACE

MARTIN G. GULBRANDSEN

1. Introduction

The punctual Hilbert scheme of a nonsingular surface is a variety whose closed points correspond to subschemes of finite lengthn, say, supported at a fixed point on the surface. It is singular in general. A less singular model has been suggested by A. S. Tikhomirov [8], namely a certain component of the variety parameterizing flags ξ1 ⊂ ξ2 ⊂ · · · ⊂ ξn of subschemes, where each ξi

has lengthi and is supported at the chosen point. It is not obvious, however, how to determine whether a given flag belongs to this particular component.

In this paper we show that a necessary, and at least for n ≤ 7 sufficient, condition is that the associated filtration of idealsI¹⊃I²⊃ · · · ⊃Inhas the multiplicative propertyIiIj ⊆Ii+j. The variety parameterizing such flags can be algorithmically computed. In particular we find that the suggested model for the punctual Hilbert scheme is singular forn=5. This corrects an assertion of S. A. Tikhomirov’s paper [9], where nonsingularity is erroneously claimed forn= 5. In [8], A. S. Tikhomirov showed that the model is nonsingular for n≤4, a result we also obtain here.

In sections 2–4 we construct a scheme parameterizing flags of subschemes in a more general setting. In sections 5–6 we specialize to the case of a nonsingular surface.

I would like to thank Geir Ellingsrud for many valuable discussions. Also I thank Roy Skjelnes and the anonymous referee for useful comments.

2. Punctual Hilbert schemes of flags

Letk be an algebraically closed field. By a scheme we shall mean a locally Noetherian scheme overk. Product of schemes means product overkthrough- out. IfY¹andY²are closed subschemes of a third schemeX, the expression

Received January 23, 2003; in revised form December 11, 2003.

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Y1∩Y2denotes their scheme theoretic intersection andY1⊆Y2means scheme theoretic inclusion. By a map of schemes we always mean a morphism in the category of schemes.

Let(A,ᒊ)be a local Artiniank-algebra of finite type. ThenX = SpecA is a projective scheme, hence the Hilbert scheme Hilbⁿ(X)parameterizing subschemesξ ⊂Xof lengthnexists [5].

Introduce the following notation: For a map of schemesf:Y →Y, let fX:Y ×X−→Y ×X

denote the product off with the identity map on X. Furthermore, for any schemeY, let

iY:Y −→Y×X denote the closed immersion obtained by identifying

Y ∼=Y ×Spec(A/ᒊ)⊂Y ×X.

To make formulas slightly more readable, we writei_∗^Y in place of (iY)∗ for push forward alongiY.

We want to construct a scheme Flagⁿ(X)parameterizing complete flags of subschemes

ξ1⊂ · · · ⊂ξn⊂X such that eachξi has lengthi.

Deﬁnition2.1. TheHilbert functor of complete flagsinXof lengthnis the contravariant functor

Flagⁿ(X):Sch_k −→Sets

from the category of locally Noetherian schemes overkto the category of sets that associates to a schemeT the set ofn-tuples of families

T ×Spec(A/ᒊ)=W¹⊂ · · · ⊂Wn⊂T ×X, withWi being defined by the ideal sheafJi ⊂OT×X, such that

(I) eachWi is flat and finite of degreeioverT

(II) i_T^∗(Ji/Ji+1)is an invertible sheaf onT fori =1,2, . . . , n−1.

Remark2.2. Fork-valued points, condition (II) is automatic, thus a scheme representing Flagⁿ(X)does parameterize complete flags of subschemes inX. In fact, ak-valued point consists of subschemesξi ⊂ Xof lengthi, defined by ideals

In⊂ · · · ⊂I1=ᒊ⊂A.

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The sheafi_T^∗(Ji/Ji+1)is now nothing but thek-vector spaceIi/(Ii+1+ᒊIi). Consider the obvious inclusions

Ii+1⊆ᒊIi +Ii+1⊆Ii.

By Nakayama’s lemma, the rightmost inclusion must be strict. By the assumption on lengths, the leftmost inclusion must then be an equality, that is, ᒊIi ⊆Ii+1. Thus

Ii/(Ii+1+ᒊIi)=Ii/Ii+1

which is one-dimensional.

Similarly one can show that condition (II) is automatic for any reduced locally Noetherian base schemeT, but we shall not need this fact.

In the next section we shall prove the following result.

Theorem2.3. There exists a schemeFlagⁿ(X)representingFlagⁿ(X). 3. Construction of Flagⁿ(X)

We construct Flagⁿ(X)by induction onn. Forn=1 we clearly have Flag¹(X)=

Speck, with universal family

Z1=Speck×Speck⊂Speck×X.

The main idea is the following: A closed point in Flagⁿ(X)corresponds to a filtration of idealsI¹⊃ · · · ⊃In. Consider a closed point inP(In/ᒊIn), that is a vector space quotient

In/ᒊIn−→k−→0.

Such a quotient is also a homomorphism ofA-modules, hence the kernel of the composite

In−→In/ᒊIn−→k

is an idealIn+1. The extended filtrationI1⊃ · · · ⊃In⊃In+1defines a closed point in Flagⁿ⁺¹(X), and conversely any point arises in this way. The rest of this section is a straightforward globalization of this “fibrewise” construction.

Suppose now, for some fixedn, there exists a schemeF = Flagⁿ(X)representing Flagⁿ(X), and let

Z1⊂ · · · ⊂Zn⊂F ×X

denote the universal flag, withZidefined by the ideal sheafIi ⊂OF×X. Define the coherentOF-module

En=i_F^∗In

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and let

π:P(En)−→F

denote the structure map. We want to show thatP(En)represents Flagⁿ⁺¹(X) by exhibiting a universal flag

Z¹⊂ · · · ⊂Zn+1⊂P(En)×X.

Fori=1, . . . , n, simply let

Zi =π_X⁻¹(Zi)⊂P(En)×X which, sinceZi is flat overF, is defined by the ideal sheaf

Ii =π_X^∗Ii. Furthermore, we define

Zn+1⊂P(En)×X by the ideal sheafIn+1, constructed as follows: Let (1) φ1:In −→i_∗^P^(Eⁿ⁾iP^∗(En)In=i_∗^P^(Eⁿ⁾π^∗En

be the canonical surjection and let

(2) φ2:i_∗^P^(Eⁿ⁾π^∗En −→i_∗^P^(Eⁿ⁾O(1)

be the map obtained by applyingi∗^P(Eⁿ⁾to the universal quotient (3) π^∗En−→O(1)−→0

onP(En). Then defineIn+1to be the kernel ofφ2◦φ1. The horizontal row in the following diagram is then exact:

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i∗^P^(Eⁿ⁾π^∗En φ1

↓^φ²

0−−→In+1−−→In−−→ i∗^P^(Eⁿ⁾O(1)−−→0

By the short exact sequence in (4) we see thatiP^∗(En)(In/In+1)is invertible, hence condition (II) in definition 2.1 is fulfilled. The same exact sequence may be rewritten

0−→i_∗^P^(Eⁿ⁾O(1)−→OZn+1 −→OZn −→0

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from which we see thatZn+1is flat and finite of degreen+1 overP(En), hence condition (I) is satisfied as well.

The following theorem ends the induction step and thus proves theorem 2.3:

Theorem3.1.The flagZ1 ⊂ · · · ⊂ Zn+1 constructed above has the fol- lowing universal property: For any schemeT and anyT-valued point

T ×Spec(A/ᒊ)=W1⊂ · · · ⊂Wn+1⊂T ×X ofFlagⁿ⁺¹(X), there exists a unique map

f:T −→P(En)

such thatWi =f⁻¹(Zi)for eachi. HenceP(En)representsFlagⁿ⁺¹(X). Proof. LetJi ⊂OT×Xbe the sheaf of ideals definingWi. By the induction hypothesis we have assumed thatF represents Flagⁿ(X), so the families W¹, . . . , Wndetermine a unique mapg:T →F such thatWi = g_X⁻¹(Zi)for i=1, . . . , n. SinceZi is flat overF, the inverse imageg⁻_X¹(Zi)is defined by g_X^∗Ii, henceJi =g^∗_XIi. We want to show thatgextends uniquely to a mapf in the diagram

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P(En)

f ↓^π

T−−→^g F

such thatf_X⁻¹(Zn+1)=Wn+1, or equivalentlyf_X^∗(In+1)= Jn+1. Extending gto a mapf in the diagram (5) is equivalent to giving a quotient

(6) g^∗En−→L −→0

whereL is an invertible sheaf onT. In fact,f is then the unique map such that (6) is obtained by applyingf^∗to the universal quotient (3).

Uniqueness: Assume there exists anfin diagram (5) such thatf_X^∗(In+1)= Jn+1. We want to show that this determines the quotient (6) uniquely. This can be seen by applyingf^∗iP^∗(En)to diagram (4). Firstly, applyingiP^∗(En)to the map φ1in (1) we obtain the identity map on

(7) iP^∗(En)In=iP^∗(En)π_X^∗In=π^∗i_F^∗In=π^∗En.

Furthermore, applyingi_P^∗_(E_n₎toφ2in (2) we recover the universal quotient (3).

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Thus, the result of applyingi_P^∗_(E_n₎to diagram (4) is the following diagram:

π^∗En

↓

i_P^∗_(E_n₎In+1−−→i_P^∗_(E_n₎In−−→O(1)−−→0 Now applyingf^∗and using the identityi_T^∗f_X^∗=f^∗i_P^∗_(E_n₎, we obtain

g^∗En

↓

i^∗_TJn+1−−→i_T^∗Jn−−→ L −−→0 whereL =f^∗O(1). Hencef corresponds to the quotient (8) i_T^∗Jn−→i_T^∗(Jn/Jn+1)−→0 and is thus uniquely determined by the familiesWi.

Existence: Simply defineL =i_T^∗(Jn/Jn+1)and letf be the unique map corresponding to the quotient (8). This makes sense, sinceL is invertible by assumption. It remains only to check that we havef_X^∗In+1=Jn+1. For this, applyf_X^∗to the short exact sequence in (4) to obtain

(9) f_X^∗In+1−→Jn−→i_∗^TL −→0.

Now observe that the canonical mapJn/Jn+1 → i_∗^TL is an isomorphism, under which the rightmost map in (9) may be identified with the canonical map Jn → Jn/Jn+1. Thus the kernel isf_X^∗In+1 = Jn+1, that is,f_X⁻¹(Zn+1) = Wn+1.

Proposition3.2. The schemeFlagⁿ(X)is connected.

Proof. Iff:X →Y is a closed continuous surjective map of topological spaces, it is elementary thatXis connected if bothY and the fibers off are.

We apply this to the structure map

P(En)−→Flagⁿ(X).

This map is proper and the fibers are projective spaces. Hence Flagⁿ⁺¹(X)= P(En)is connected if Flagⁿ(X)is. The conclusion follows by induction onn.

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4. Punctual Hilbert schemes of multiplicative flags

Deﬁnition4.1. Ak-valued point in Flagⁿ(X), corresponding to a filtration of ideals

In ⊂ · · · ⊂I1=ᒊ⊂A ismultiplicativeif we haveIiIj ⊆Ii+j for alli+j ≤n.

We next construct a subscheme of Flagⁿ(X), parameterizing only multiplicative flags inX.

Deﬁnition4.2. TheHilbert functor of multiplicative complete flagsinX of lengthnis the contravariant functor

Multⁿ(X):Sch_k −→Sets

from the category of locally Noetherian schemes overkto the category of sets that associates to a schemeT the set ofn-tuples of families

T ×Spec(A/ᒊ)=W¹⊂ · · · ⊂Wn⊂T ×X, withWi being defined by the ideal sheafJi ⊂OT×X, such that

(I) eachWi is flat and finite of degreeioverT (II) i_T^∗(Ji/Ji+1)is an invertible sheaf onT for alli (III) JiJj ⊆Ji+j for alli+j ≤n.

We want to show that the conditionJiJj ⊆ Ji+j is closed, in the strong sense that Multⁿ(X)is a closed subfunctor of Flagⁿ(X). This is a consequence of the following lemma:

Lemma4.3. Letπ:Y →Sbe a morphism of locally Noetherian schemes and letW, Z ⊆Y be closed subschemes such thatZis flat and finite overS. Then there exists a uniqueS-scheme

i:S −→S such that

(I) Z×SS ⊆W×SS

(II) ifT → S is anyS-scheme satisfyingZ×S T ⊆ W ×S T then there exists a unique morphismg:T →S overS.

Furthermore,iis a closed immersion.

Proof. Suppose the lemma holds wheneverSis affine. Then we may apply the lemma to eachSαin an affine open cover{Sα}ofS. Thus there exists closed

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immersionsiα:S_α →Sα, uniquely determined by properties (I) and (II) when replacingS,WandZwithSα,W∩SαandZ∩Sα. Again applying the lemma to an affine open cover of each intersectionSα∩Sβ, we see that the immersions {iα}agree on the overlaps. Hence they may be glued to form the required closed immersioni:S →S. Thus we may assumeSis affine.

SinceZ is finite overS, Z is affine as well. Then we may choose a free presentation

(10) O_Zⁿ−→^φ OZ−→OZ∩W −→0

whereZ∩Wdenotes the scheme theoretic intersection. Letf:T →Sbe any morphism, and letZ = Z×ST andW =W ×ST. We claim the condition Z⊆Wis equivalent to requiringf^∗π∗φ =0: Form the fibre square

Y ×ST −−−−→^f Y

↓

π ^π↓

T −−−−→^f S

Then applyingf^∗ to (10) gives a free presentation of the structure sheaf of Z∩W:

O_Zⁿ −−−→^f^∗^φ OZ−→OZ∩ W −→0

Thus the conditionZ⊆ W, or equivalentlyZ∩W = Z, is the same thing as requiringf^∗φ = 0. Now the restriction ofπ to Zis finite, hence affine, so f^∗φ = 0 if and only if π∗f^∗φ = 0. Furthermore, as Z is flat over S, π∗f^∗φ=f^∗π∗φ. HenceZ⊆Wif and only iff^∗π∗φ =0 as claimed.

SinceZis flat and finite overS,

(11) π∗O_Zⁿ −−−→^π^∗^φ π∗OZ

is a map of locally free sheaves of finite rank onS. Thusπ∗φ can be locally represented by a matrix of regular functions, hence its vanishing locus has a canonical structure of a closed subschemei:S → S. Theni^∗π∗φ = 0, soi has property (I). Furthermore, if a morphismf:T →Ssatisfiesf^∗π∗φ =0, then the image inOT of the ideal sheaf definingS ⊂ Sis zero, which says thatf factors throughi. Soihas property (II).

Theorem4.4. Multⁿ(X)is a closed subfunctor ofFlagⁿ(X).

Proof. Let S denote a scheme and hS its functor of points. Consider a

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cartesian diagram

h −−−→Multⁿ(X)

↓ ↓

hS −−−→Flagⁿ(X)

wherehis the fibre product functor. We claim there exists a closed subscheme S ⊆Sand an isomorphismh∼=hS such that the maph→hSis compatible with the inclusion maphS →hS.

The image of a morphismT →Sunder the given maphS →Flagⁿ(X)is a flag

(12) W1⊂ · · · ⊂Wn ⊂X×T .

LetJi ⊂OX×T denote the ideal sheaf corresponding toWi. By definition,his the subfunctor ofhS whoseT-valued points are the morphismsT →Ssuch that the corresponding flag (12) has the multiplicative property

(13) JiJj ⊆Ji+j for all i+j ≤n.

Thus our claim is that there is a closed subschemeS ⊆ Ssuch thatT → S factors throughS if and only if property (13) holds. This can be seen as follows:

The image of the identity map id_S under the given maphS →Flagⁿ(X)is a flag

(14) Z1⊂ · · · ⊂Zn ⊂X×S

overS, withZicorresponding to some ideal sheafIi ⊂OX×S. For any morph- ismT → S, the corresponding flag (12) is just the pullback of the flag (14) alongT →S. Thus the existence ofS ⊆ Sis a consequence of lemma 4.3, applied toY =X×S,W =V (IiIj)andZ =Zi+j, for eachiandj.

Corollary4.5.There exists a closed subschemeMultⁿ(X) ⊆ Flagⁿ(X) representingMultⁿ(X).

Remark4.6. The scheme Multⁿ(X)can be constructed more explicitly in the same fashion that we constructed Flagⁿ(X): Consider the universal flag

Z¹⊂ · · · ⊂Zn⊂Flagⁿ(X)×X, withZi defined by the ideal sheafIi. Denote by

W¹⊂ · · · ⊂Wn ⊂Multⁿ(X)×X

their restriction to Multⁿ(X), withWi defined by the ideal sheafJi. In section 3 we constructed Flagⁿ⁺¹(X)asP(En), whereEn =i_F^∗In. Thus Multⁿ⁺¹(X)is

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the maximal subscheme ofP(En)such that the restriction of the universal flag has the multiplicative property. This is precisely the universal property of

π:P(Fn)−→Multⁿ(X) where

Fn =Jn n−1

i=0Ji+1Jn−i,

considered as a coherent sheaf on Multⁿ(X) ∼= W¹ ⊂ Multⁿ(X)×X. Thus we have an isomorphism Multⁿ⁺¹(X)∼=P(Fn)over Multⁿ(X). The universal multiplicative flag

W¹⊂ · · · ⊂Wn+1⊂Multⁿ⁺¹(X)×X

is defined by idealsJ1⊃ · · · ⊃Jn+1whereJi =π_X^∗Ji fori≤n, whereas Jn+1is the kernel of the canonical map

Jn−→i_∗^P^(Fⁿ⁾O(1)

whereO(1)now denotes the tautological invertible sheaf onP(Fn). Proposition4.7. The schemeMultⁿ(X)is connected.

Proof. Using the construction of Multⁿ(X)in remark 4.6, the proof of 3.2 can be repeated.

5. Punctual Hilbert schemes of points on a nonsingular surface

For the rest of this text we consider the following situation: Assume k has characteristic zero. Fix an algebraic surfaceSoverk and a nonsingular point p∈S. LetOS,pdenote the local ring atpand letᒊ_p ⊂OS,pdenote its maximal ideal. Any subschemeξ ⊂Sof lengthnand supported atpis contained in the (n−1)’st infinitesimal neighbourhoodX =SpecOS,p/ᒊⁿ_p. Thus the scheme Hilbⁿ(X)parameterizes lengthnsubschemes ofSsupported atp. We let

H (n)=Hilbⁿ(X)red

denote the underlying reduced subscheme. We suppress S and p from the notation, as the definition ofH (n)only depends on the(n−1)’st infinitesimal neighbourhood ofp, whose isomorphism class is independent of the choices ofSandp.

It is well known thatH (n)is irreducible and has dimensionn−1 (proved by Briançon [1] over the complex numbers, see e.g. Ellingsrud and Lehn [2]

for a proof in a more general setting). However, it is singular in general. For instance,H (3)is isomorphic to the projective cone over the twisted cubic in

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P³. In the rest of this paper we present work towards finding a natural resolution of singularities ofH (n).

Following Le Barz [7], we make the following definition:

Deﬁnition5.1. A subschemeξ ⊂S, supported atp, iscurvilinearif there exists a curveCwhich containsξand is nonsingular atp.

It is well known ([1], [6]) that the subset ofH(n)consisting of curvilinear subschemes is open, dense and nonsingular. The following result is also well known:

Lemma 5.2. Let ξ ⊂ S be a subscheme supported at a point p. If ξ is curvilinear, there is a unique flag

ξ1⊂ · · · ⊂ξn−1⊂ξ

withξi of lengthi. In fact,ξi is the intersection ofξ with the(i−1)’st infin- itesimal neighbourhood ofpinS.

Proof. SupposeC is a nonsingular curve throughpcontainingξ, locally defined by the idealJ ⊂ OX,p. Letξi ⊂ ξbe a subscheme of lengthiand let I ⊂Ii ⊂OX,pbe the ideals definingξandξi. Then we have_ᒊ_pⁱ ⊆Ii, hence

J +ᒊ_pⁱ ⊆I +ᒊ_pⁱ ⊆Ii.

But the left hand side is the ideal defining the(i−1)’st infinitesimal neighbourhood ofpinC, which has colengthisinceCis nonsingular. Since the right hand side idealIi has colengthi also, the inclusions are actually equalities.

In particularIi =I +ᒊ_pⁱ, which shows thatξi is uniquely determined as the intersection ofξwith the(i−1)’st infinitesimal neighbourhood ofpinS.

Define

HF(n)=Flagⁿ(X)red

which is a reduced scheme whose closed points correspond to flags of subschemes inSsupported atp. The canonical map

Flagⁿ(X)−→Hilbⁿ(X) induces a map

ρn:HF(n)−→H(n).

Proposition5.3.There is a unique component HF(n)⊆HF(n)which is mapped birationally ontoH (n)byρn.

Proof. LetU ⊆H (n)be the open subset corresponding to curvilinear subschemes. By lemma 5.2, the fibreρ_n⁻¹(ξ)is a single point for every (closed)

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pointξ ∈U. Henceρnis bijective overU. Sinceρnis proper andUis nonsingular, Zariski’s main theorem [4, prop. 4.4.1] shows thatρnis an isomorphism overU. Thus the closureHF(n)ofρ_n⁻¹(U)inHF(n)is the unique component mapping birationally ontoH (n).

Denote by

ρ_n:HF(n)−→H(n)

the restricted map. We call this a partial resolution ofH(n). This construction has been studied by Tikhomirov in [8], where he proves thatρ_nis a resolution of singularities forn≤4. The problem addressed in the next section is how to determine whether a given flag belongs to the componentHF(n). This leads us to a different proof of Tikhomirov’s result (theorem 6.1) and also the new result thatHF(5)is singular (theorem 6.2).

Define

HMF(n)=Multⁿ(X)red

which is a reduced scheme whose closed points correspond to multiplicative flags of subschemes inSsupported atp. Since Multⁿ(X)is a closed subscheme of Flagⁿ(X), we find that HMF(n) is a closed subscheme of HF(n). The motivation for studyingHMF(n)is the following observation:

Proposition5.4. Any (closed) point in HF(n)is multiplicative, hence HF(n)is contained in HMF(n).

Proof. Denote byU ⊆H (n)the open set consisting of curvilinear points.

LetV ⊆ HF(n)denote the inverse image ofU by the mapρ_n:HF(n) → H (n). By definition,HF(n)is the closure ofV inHF(n).

First consider a (closed) point inV, that is, a flag ξ1⊂ · · · ⊂ξn

withξncurvilinear. Then, ifξi corresponds to the idealIi ⊂OX,pwe have Ii =ᒊ_pⁱ +In for all i

by lemma 5.2. Then it is obvious thatIiIj ⊆Ii+j.

ThusV ⊂HMF(n). SinceHMF(n)is closed inHF(n)andHF(n)is the closure ofV, we haveHF(n)⊂HMF(n).

Question5.5. Is the converse to proposition 5.4 true, i.e. do we have an equalityHF(n) = HMF(n)? As HF(n)is a component of HF(n), this is equivalent to asking whetherHMF(n)is irreducible.

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The calculations in section 6 show that the answer to the question is positive forn≤ 7. For highernwe do not know. We remark thatHMF(n)is at least connected, by proposition 4.7.

6. Examples

To describeHMF(n), we follow the construction of Multⁿ(X)in remark 4.6.

More explicitly, letU = SpecAbe an affine open subset of Multⁿ(X). We want to describe an affine open cover for the inverse image ofUin Multⁿ⁺¹(X), denoted Multⁿ⁺¹(X)

U. With notation as in remark 4.6, the familyWiis defined overU by the idealJi ="(U×X,Ji)in the affine coordinate ring ofU×X. Then

Multⁿ⁺¹(X)

U =P(M) where

M ="(U,Fn)=Jn n−1

v=0Jv+1Jn−v

considered as anA-module. To give concrete equations forP(M), choose a free presentation

A^{r (g}−−−→^ij⁾ A^{s (f}−−−→^j⁾ M −→0. ThenP(M)=ProjRwhere

(15) R =A[t1, . . . , ts]

jg1jtj, . . . ,

jgrjtj .

ThusP(M)is covered by the affine open subsetsVi = SpecRi whereRi is the degree 0 part of the localizationRti. The universal quotient is the homomorphism

M⊗Ri −→Ri −→0

sendingfj ⊗1 toTj = tj/ti (in particular fi ⊗1 → 1). Hence, on Vi the universal flag is defined by ideals

J¹⊃ · · · ⊃Jn+1

whereJv =JvRi forv≤n, and

Jn+1=(Tjfi −fj)j=i+_n−1

v=0Jv+1Jn−v Ri.

As long as the ringsRi are nilpotent-free, this gives an algorithm for com- puting an open cover ofHMF(n). Otherwise we should divide by the nilrad- ical to get the underlying reduced scheme. It turns out that in all our examples, i.e. whenevern≤7, Multⁿ(X)is already reduced, henceHMF(n)= Multⁿ(X). We do not know whether this is true for arbitraryn.

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Clearly, Mult²(X) = HMF(2) ∼= H (2) ∼= P¹. The next result describes HMF(3) andHMF(4). We are going to use the following (well known and easy to derive) classification of punctual subschemes of length 2 and 3 on a nonsingular surface: For a suitable choice of local parameters, any subscheme of length two may be defined by an ideal of the form

(x, y²)⊂OS,p.

Thus any such subscheme is curvilinear. For subschemes of length three, there are two types: Firstly there are the curvilinear ones, which for a suitable choice of local parameters may be defined by an ideal of the form

(x, y³)⊂OS,p.

Secondly there is just one non curvilinear subscheme of length three, namely the first infinitesimal neighbourhood ofp, defined by

ᒊ²_p=(x², xy, y²)⊂OS,p.

Theorem 6.1. For n = 2 and3 the sheafFn is locally free of rank 2, hence HMF(n+1)is aP¹-bundle over HMF(n). In particular, HMF(3)and HMF(4)are nonsingular.

Proof. Any point inHMF(2)is curvilinear, henceF2has rank two everywhere. Thus it is locally free.

A punctual subscheme of length 3 is either the first order infinitesimal neighbourhood ofpor it is curvilinear. Consider a point inHMF(3), that is a filtration of ideals

I3⊂I2⊂I1=ᒊ_p. IfI³is curvilinear, then

I3/(I1I3+I2²)=I3/I1I3

is two dimensional as before. If not, thenI3 = (x², xy, y²). For a suitable choice of local parameters we may assumeI²=(x, y²). Then

I1I3+I2²=(x², y³, xy²) and hence

I³/(I¹I³+I2²)= xy, y² is two dimensional. ThusF3has rank two everywhere.

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The surfaceHMF(3)can be determined completely. In fact it is isomorphic to the minimal ruled surfaceF3. For this, letR = k[a⁰, a¹], thenHMF(2) = H (2)=ProjRwith universal family defined by the ideal

(16) J =(a1y−a0x, x², xy, y²)⊂R⊗kOX,p.

Then the sheafF2corresponds to the gradedR-moduleN with generators f =a1y−a0x

h=xy

g=x² k=y²

wheref has degree 1 and the rest have degree 0. The relations are a1h=a0g a1k =a0h.

From this we conclude that N is isomorphic to R(−1)⊕R(2) in positive degrees, wheref generates the summand corresponding toR(−1), andg,h andkgenerate the summand corresponding toR(2). Thus

F2=OP¹(−1)⊕OP¹(2) and the associated projective bundle isF₃.

Finally, we remark thatHF(4)is reducible, so HMF(4) = HF(4)is not the only component. In fact, above the rational curve inHF(3) = HMF(3) consisting of filtrations of the form

ᒊ_p² =I3⊂I2⊂I1=ᒊ_p

whereI2varies freely in aP¹, every fibre inHF(4)is aP². Thus the inverse image of this curve has dimension 3, which therefore cannot be contained in the irreducible three dimensional varietyHMF(4). To give an explicit example, the ideals

(x², xy, y³)⊂(x², xy, y²)⊂(x², y)⊂(x, y) define a point inHF(4)which is not multiplicative.

Forn=5 we obtain the following, which corrects [9, Theorem 1].

Theorem6.2. HMF(5)is singular along a curve, but irreducible.

Proof. We compute the restriction ofHMF(5)to a particular open affine chartU4 ⊂ HMF(4). By the same method one can compute an open cover explicitly.

With notation as in equation (16), letU2 ⊂ HMF(2) be the open affine subset defined bya0=0. Then

U2=Speck[a]

(16)

wherea=a1/a0, and the universal flag is defined by the ideals (17) J1=(x, y) J2=(ay−x, y²).

Carrying through the recipe given above, we find HMF(3)

U2 =Projk[a][b⁰, b¹]

where the generatorsbi correspond toti in equation (15). We define the open affineU3 ⊂ HMF(3)byb0 = 0, then the universal flag onU3is defined by idealsJ1⊃J2⊃J3, whereJ1andJ2are the ideals in (17) and

J3=(b(ay−x)−y², (ay−x)x, (ay−x)y)

whereb=b1/b0. (We should really writeJ1k[a, b] andJ2k[a, b] in place of J¹andJ², but this shouldn’t cause any confusion.) Since

a ((ay−x)y)−(ay−x)x=(ay−x)²∈J2²

we find thatU3trivializesF3and HMF(4)

U3 =Projk[a, b][c0, c1].

where again the new coordinatesci correspond toti in equation (15). Define U4⊂HMF(4)byc0=0, then the universal flag is defined overU4by

J4=(c(b(ay−x)−y²)−(ay−x)y, b(ay−x)y−y³, (ay−x)²) wherec=c1/c0, together withJ1, J2, J3as above.

Now we are in position to describe the restriction ofHMF(5)toU⁴. The module

M =J4/(J1J4+J2J3) is generated by

f =c(b(ay−x)−y²)−(ay−x)y g=b(ay−x)y−y³

h=(ay−x)²

and the elementbh−cf is contained inJ2J3, thus HMF(5)

U4 =Projk[a, b, c][F, G, H]/(bH −cF ).

In fact, since this is irreducible, reduced and of dimension four, the found relationbh−cf is the only one.

(17)

Thus HMF(5)

U4 is irreducible and singular along a curve. Repeating the calculations while movingU⁴around proves the statement.

By the same procedure one may test the irreducibility ofHMF(n), and hence question 5.5, for highern. The explicit calculations get rather involved, but with the aid of the computer program Singular [3], using a primary decomposition algorithm, it has been verified that HMF(n) is irreducible for n ≤ 7, and also that Multⁿ(X)is already reduced. At 8 points we stopped due to lack of computer power.

REFERENCES

1. Briançon, Joël,Description deHilbⁿC{x, y}, Invent. Math. 41 (1977), 45–89.

2. Ellingsrud, Geir, and Lehn, Manfred,Irreducibility of the punctual quotient scheme of a surface, Ark. Mat. 37 (1999), 245–254.

3. Greuel, G.-M., Pfister, G., and Schönemann, H.,Singular 2.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de.

4. Grothendieck, Alexander,Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. (1961), 167.

5. Grothendieck, Alexander,Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249–276.

6. Iarrobino, Anthony A.,Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (1977), no. 188, viii+112.

7. Le Barz, Patrick,Platitude et nonplatitude de certains sous-schémas deHilb^kP^N, J. Reine Angew. Math. 348 (1984), 116–134.

8. Tikhomirov, Alexander S.,A smooth model of punctual Hilbert schemes of a surface, Trudy Mat. Inst. Steklov. 208 (1995), 318–334.

9. Tikhomirov, Sergei A.,Punctual Hilbert schemes of small length in dimensions 2 and 3, Mat.

Zametki 67 (2000), no. 3, 414–432.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO P.O. BOX 1053 N-0316 OSLO NORWAY

E-mail:martingu@math.uio.no