Hilbert schemes and stable pairs: GIT and derived category wall crossings

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scientifique

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 139 Fascicule 3 2011

pages 297-339

HILBERT SCHEMES AND STABLE PAIRS: GIT AND DERIVED

CATEGORY WALL CROSSINGS

Jacopo Stoppa & Richard P. Thomas

HILBERT SCHEMES AND STABLE PAIRS:

GIT AND DERIVED CATEGORY WALL CROSSINGS by Jacopo Stoppa &Richard P. Thomas

Abstract. — We show that the Hilbert scheme of curves and Le Potier’s moduli space of stable pairs with one dimensional support have a common GIT construction.

The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations.

We explain why this is not enough to prove the “DT/PT wall crossing conjecture”

relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce’s theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristics of these moduli spaces.

When the 3-fold is Calabi-Yau the identity is the Euler-characteristic analogue of the DT/PT wall crossing conjecture, but for general 3-folds it is something different, as we discuss.

Résumé(Schémas de Hilbert et paires stables : GIT et croisements de murs de caté- gories dérivées)

Nous montrons que le schéma de Hilbert de courbes et l’espace de modules de Le Potier de paires stables à support à une dimension, ont une construction GIT commune. Les deux espaces correspondents aux chambres de par et d’autre d’un mur dans l’espace de linéarisations GIT.

Nous expliquons pourquoi cela ne suffit pas pour prouver la « conjecture de croi- sement de murs DT/PT » qui relie les invariants dérivés de ces espaces de modules quand la variété sous-jacente est un 3-fold. Nous donnons, ensuite, une introduction simple à une petite partie de la théorie de Joyce sur les croisements de murs de ce type,

Texte reçu le 25 mars 2009, révisé le 12 octobre 2009, accepté le 6 novembre 2009.

Jacopo Stoppa, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy • E-mail :[email protected]

Richard P. Thomas, Department of Mathematics, Imperial College London • E-mail :[email protected]

BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2011/297/$ 5.00

©Société Mathématique de France

et nous nous en servons pour donner une brève démonstration d’une identité reliant les caractéristiques d’Euler de ces espaces de modules.

Quand le 3-fold est de type Calabi-Yau, l’identité est le pendant, pour la carac- téristique d’Euler, de la conjecture de croisement de murs DT/PT, mais dans le cas général elle s’avère être différente de celle-ci, comme nous l’expliquons.

1. Introduction

This paper is motivated by the conjectural equivalence between two curve counting theories on smooth complex projective threefoldsX: the one studied in [17] and the stable pairs of [21].

MNOP and stable pairs invariants are sheaf-theoretic analogues of Gromov- Witten invariants, sometimes called DT and PT invariants respectively. The space of stable maps to X is replaced by suitable moduli spaces of sheaves supported on curves in X.

Fix β ∈ H2(X,Z) and n ∈ Z. In MNOP theory we integrate suitable classes against the virtual fundamental class of the Hilbert schemeIn(X, β)of subschemesZ ofX in the class[Z] =β with holomorphic Euler characteristic χ(

O

Z) =n. The virtual fundamental class comes from thinking ofIn(X, β)as a moduli space of sheaves of trivial determinant – namely the ideal sheavesIZ

with Chern character (1.1) �

1,0,−β,−n+β.c1(X) 2

� ∈ H⁰(X)⊕H²(X)⊕H⁴(X)⊕H⁶(X).

For stable pair theory we work instead with stable pairs (F, s): F is apure sheaf onX with Chern character(0,0, β,−n+β.c1(X)/2), ands:

O

X →F is a section with 0-dimensional cokernel. A special case of the work of Le Potier [14] constructs the fine moduli space Pn(X, β) as a projective scheme. The virtual fundamental class comes from thinking [21] of Pn(X, β) as a moduli space of objects of the derived category of coherent sheaves onX (with trivial determinant) – namely the complexes I^•:={

O

X →F} with Chern character (1.1).

Roughly speaking, we think ofIn(X, β)as parameterising pure curves plus points (free and embedded) on X. Any Z ∈ In(X, β) contains a maximal Cohen-Macaulay curve C ⊆ Z (the pure curve: recall that Cohen-Macaulay means no embedded points) such that the kernel of

O

Z →

O

Cis 0-dimensional (the points). Equally loosely we think of stable pairs as parameterising Cohen- Macaulay curves (the support of the sheafF) and free pointson the curve(the cokernel of the sections).

Over the Zariski-open subset of Cohen-Macaulay curves C with no free or embedded points, the moduli spacesIn(X, β)andPn(X, β)are isomorphic: the

o

stable pair

O

X →

O

C determines and is determined by the kernel ideal sheaf IC. Indeed I^•is quasi-isomorphic toIC.

WhenωX ∼=

O

X, i.e. X is a Calabi-Yau threefold, MNOP and stable pair invariants take a particularly simple form. Then the virtual dimension is zero and we get invariants by taking the length of the 0-dimensional virtual cycle:

I_m,^vir_β =

�

[Im(X,β)]^vir

1, and

P_m,^vir_β =

�

[Pm(X,β)]^vir

1.

In this case the deformation-obstruction theories [24, 21, 9] used to define the virtual cycles areself dual in the sense of [2]. This implies thatI_m,β^vir , P_m,β^vir are in fact weighted Euler characteristics:

I_m,β^vir =e(Im(X, β), χ^B), P_m,β^vir =e(Pm(X, β), χ^B).

Here the weighting function is Behrend’s integer-valued constructible function χ^B [2], which assigns to each point of the moduli space the multiplicity with which it contributes to the invariants. At smooth points of the moduli space, χ^B≡(−1)^dim.

We can also form their generating series Z_β^I,vir(X)(t) :=�

m∈Z

I_m,β^vir t^m and Z_β^P,vir(X)(t) := �

m∈Z

P_m,β^vir t^m. The conjectural equivalence between the MNOP and stable pair invariants in the Calabi-Yau case is then the following.

Conjecture 1.2. — [21]ForX a Calabi-Yau threefold, Z_β^P,vir(X) = Z_β^I,vir(X)

Z₀^I,vir(X).

Equivalently, for each m ∈ Z we have the following identity (where the right hand side is a finite sum):

(1.3) I_m,β^vir = P_m,β^vir + I_1,0^vir·P_m^vir_−1,β + I_2,0^vir·P_m^vir_−2,β + · · · .

HereZ₀^I,vir(X)is the generating series of virtual counts of zero dimensional subschemes ofX. By [17,3,15, 16] it is in fact

Z₀^I,vir(X)(t) =M(−t)^e(X),

where M(t) is the MacMahon function, the generating function for 3- dimensional partitions.

Using Kontsevich-Soibelman’s identities for χ^B [13], now proved in some cases [12], it should now be possible to extend what follows to the weighted

Euler characteristics I_m,β^vir and P_m,β^vir . But in this paper we content ourselves with working with the unweighted Euler characteristics

Im,β :=e(Im(X, β)) and Pm,β:=e(Pm(X, β)), which are not deformation invariant. Form their generating series

Z_β^I(X)(t) := �

m∈Z

Im,βt^m and Z_β^P(X)(t) := �

m∈Z

Pm,βt^m. In Sections4and5we will give two different proofs of the following topological analogue of Conjecture 1.2 (first proved by Toda [26] in the Calabi-Yau case, as discussed below).

Theorem 1.4. — Let X be a smooth projective threefold. Then Z_β^P(X) = Z_β^I(X)

Z₀^I(X).

Equivalently for each m ∈ Z we have the following identity (where the right hand side is a finite sum):

Im,β = Pm,β + I1,0·Pm−1,β + I2,0·Pm−2,β + · · · .

Here the Ik,0 = e(Hilb^kX) are the Euler characteristics of the Hilbert schemes of points on X, and Z₀^I(X) is their generating series. By [5] this is

Z₀^I(X) =M(t)^e(X).

In fact we prove a little more. Fixing a Cohen-Macaulay C in class β, define In,C to be the Euler characteristic of the subset of In(X, β)consisting of subschemes whose underlying Cohen-Macaulay curve isC (this is naturally a projective scheme, see below). Similarly let Pn,C be the Euler characteristic of the subset (in fact projective scheme) ofPn(X, β)of pairs supported onC.

Theorem 1.5. — Let C⊂X be a Cohen-Macaulay curve in a smooth projec- tive threefold. Then

In,C=Pn,C+e(X)Pn−1,C+e(Hilb²X)Pn−2,C+· · ·+e(HilbⁿX)P0,C. We explain in Section 4.2 how to deduce Theorem 1.4 from this by “inte- grating” over the space of Cohen-Macaulay curves C.

Naively one should think of the above identities as reflecting the decompo- sition of In(X, β)into a union of the subset of pure Cohen-Macaulay curves with no free or embedded points, the subset with one free or embedded point, the subset with two points, etc. Birationally such a decomposition is given by (1.6) below.

o

Remark. — We emphasise that the identity of Euler characteristics in The- orem 1.4 holds forany threefold. (Only the χ^B-weighted Euler characteristic motivation requires the Calabi-Yau condition.) From a virtual class perspective it is very surprising that the result should hold in the non-Calabi-Yau case, as we discuss at the end of Section 4. We also show there how to use Theorem 1.5 to add incidence conditions, getting a topological analogue of the DT/PT conjecture with insertions for general threefolds.

An intermediate step in our proof is given in Section 3, which may also be interesting in its own right. The main result there is Theorem3.3, which shows that the moduli spaces In(X, β) and Pn(X, β) are related by a special wall crossing in the sense of Geometric Invariant Theory. This result holds in any dimension. In fact a single wall is crossed, so we get natural morphismsϕI, ϕP

to the moduli scheme of semistable objects modulo S-equivalence (pointwise this coincides with the set of polystable objects) on the wall, which we call SSn(X, β),

(1.6) In(X, β)

ϕI

��

Pn(X, β)

ϕP

��

SSn(X, β) = �n

k=0I_n−k^pur(X, β)×S^kX.

(The latter equality is a stratification by locally closed subschemes). The mor- phisms ϕI, ϕP have a simple geometric meaning: they separate the pure and torsion part of a subscheme (respectively the support and cokernel of a stable pair). This produces a Cohen-Macaulay curve inI_n^pur_−k(X, β)(the subset of the Hilbert scheme consisting of subschemes of pure dimension 1) and a finite num- ber of points with multiplicity. Compatibly with the general theory of GIT wall crossings [6, 23], on passing from In(X, β) to Pn(X, β) each irreducible com- ponent either disappears, appears, or undergoes a birational transformation.

(Since [6,23] only work with normal quotients they only see this behaviour for connected components.)

Remark. — As we note at the end of Section 4, our methods also give a relation between the Euler characteristics of single fibres of ϕI, ϕP. This is a punctual analogue of Theorem 1.5. Let C be a Cohen-Macaulay curve in A³ andpa closed point. Then one has

(1.7) �

n≥0

e(ϕ⁻_I¹(C, np))tⁿ=M(t)�

n≥0

e(ϕ⁻_P¹(C, np))tⁿ.

This holds whetherplies onC or not; in the latter case it is Cheah’s formula for the punctual Hilbert schemes Hilbⁿ(p),

�

n≥0

e(Hilbⁿ(p))tⁿ=M(t).

The more fundamental ingredient in our proof is Joyce’s motivic Ringel-Hall algebra [10]. Theorem 1.4 reflects identities between elements of this algebra naturally associated to the wall crossing morphismsϕI, ϕP.

In fact it was immediately clear from [21] that for Calabi-Yau 3-foldsX, some kind of wall crossing formula for the Euler characteristics Im,β, Pm,β should follow from the full weight of Joyce’s theory (in particular [11]). Working in the abelian subcategory ofD^b(X)given by tilting the usual one by the subcategory of dimension zero sheaves one could then try to unravel the formulae in [11] in this special case.

This is hard, however, and would use results from most of Joyce’s intimi- dating theory. This is because the wall crossing of [21], described in Section 4.4 below, relates ideal sheaves of 1-dimensional subschemes of X to stable pairs and 0-dimensional sheaves (see for instance (4.8, 4.9) below). These 0- dimensional sheaves are strictly semistable, and have automorphisms. Joyce has a theory counting (in the sense of Euler characteristics, but yielding ra- tional numbers) such things, and a wall crossing involving certain sums over graphs. Then one should relate this count to the count of ideal sheaves of 0-dimensional subschemes. Toda is not afraid of Joyce’s work and has carried this out [26], as well as a related (and even harder) set of wall crossings in [27].

A quicker and apparently simpler proof was shown to us by Tom Bridgeland [4], but using the full force of Kontsevich and Soibelman’s theory [13] (which is conjectural in parts). This meant that his proof was also something of a mystery to us. Nagao also found a proof for small crepant resolutions of affine toric Calabi-Yau 3-folds using all of Joyce’s theory, and one using Kontsevich- Soibelman [18].

Our original intention was to understand Bridgeland’s work from first prin- ciples, reducing it to a down-to-earth analysis of the GIT wall crossing between moduli spaces derived in Section 3. We soon discovered why that was impos- sible, but managed to find a way to make a similar proof work (at the level of Euler characteristics) without using Kontsevich-Soibelman’s algebra homomor- phism, making do instead with Joyce’s (proved, and more easily understood) virtual Poincaré polynomials on motivic Ringel-Hall algebras. In particular, we do not need to use Joyce’s counting invariants for strictly semistable objects, nor his remarkable theory of virtual indecomposables.

So we decided to use this paper as an opportunity to give an elementary introduction to a small part of Joyce’s work, and explain why it is necessary

o

(and GIT is not sufficient) to handle wall crossing formulae. We illustrate this with a couple of proofs of the wall crossing formula, one along the lines of Bridgeland’s, the second in Section 5 remarkably short and simple and closer to the philosophy of stability conditions.

Bridgeland has now found a way to carry out his proof using Joyce’s work and the weightingχ^B, thus proving Conjecture1.2without assuming any other conjectures. We decided that this paper still had some value, however, in the GIT construction, the introduction to Joyce’s work, and the simpler proof that we give in Section 5 avoiding Joyce’s virtual indecomposables. Finally the identity of Euler characteristics clearly has independent interest of its own, regardless of any virtual class motivation.

Acknowledgements. We mainly learnt about the work of Joyce and Kontsevich- Soibelman from talking to Tom Bridgeland, from his lectures and his preprint [4]. We also benefitted from stealing his work and that of Yukinobu Toda. We would like to thank Jim Bryan, Daniel Huybrechts, Dominic Joyce, Davesh Maulik, Kentaro Nagao, Rahul Pandharipande, Balázs Szendrői and Yukinobu Toda for useful conversations. Special thanks are due to David Steinberg and an anonymous referee for carefully reading the manuscript and making many useful suggestions. The first author thanks Simon Donaldson and the Royal Society for support during a visit to Imperial College London, and Gang Tian for a visit to BICMR, Beijing.

2. Coherent systems and pairs

Let(X,

O

X(1))be a smooth complex projective scheme of dimensionDwith a very ample line bundle. By a sheaf F we always mean a coherent sheaf of modules over

O

X. ItsHilbert polynomial with respect to

O

X(1) isPF(m) :=

χ(F(m)). This is a polynomial inmwith degreed= dim(F) = dim(Supp(F)), where Supp(F) ⊂ X is the closed subscheme defined by the kernel of the canonical section

O

X→End(F).

Letr(F)denote themultiplicity ofF, that isd!times the leading coefficient of the Hilbert polynomial PF,

PF(m) =r(F)m^d

d! +O(m^d−1).

Then thereduced Hilbert polynomial ofF is simply pF =PF

r .

The essential reference for the rest of this section is Le Potier [14].

Definition 2.1. — A coherent system on X is a pair (Γ, F), where F ∈ Coh(X) andΓ⊂H⁰(F). Amorphism f: (Γ^�, F^�)→(Γ, F) is a morphism of sheaves F^� →F such that f_∗(Γ^�)⊂Γ. The dimension dof a coherent system (Γ, F)is the dimension ofF; i.e. d= dim Supp(F),notdim(Γ).

From the next section we will restrict to the special casedim(Γ) = 1,d= 1.

Definition 2.2. — A coherent system (Γ, F)withh⁰(F)≥1,dim Γ = 1,d= 1 is called simply a pair. As a slight abuse of notation we also denote this by (F, s)or[s:

O

X →F], wheres∈H⁰(F)is a nonvanishing section (defined up to a nonzero scalar multiple).

A coherent system(Γ, F)does not have an intrinsically defined Hilbert poly- nomial. Roughly speaking we must prescribe a weight forΓ. LetQbe a positive rational polynomial.

Definition 2.3. — The reduced Hilbert polynomialof (Γ, F)with respect to Qis given by

p^Q_(Γ,F)= dim(Γ)

r(F) Q+pF.

Le Potier introduced Gieseker stability conditions for coherent systems.

Definition 2.4. — A coherent system (Γ, F) isQ-semistableif

• the sheafF is pure: it has no nontrivial subsheaves of dimension≤d−1,

• for any subsheafF^�⊂F with0< r(F^�)< r(F), settingΓ^�= Γ∩H⁰(F^�), one has the inequality of reduced Hilbert polynomials

p^Q_(Γ�,F�)≤p^Q_(Γ,F).

It isQ-stableif in addition the above inequality is strict.

LetPX=χ(

O

X(m))denote the Hilbert polynomial ofX.

Definition 2.5. — We say a coherent system is (semi)stable if it is PX- (semi)stable. Similarly a (semi)stable pairis simply aPX-(semi)stable pair.

One can prove that replacing PX by any rational, positive polynomial Q withdeg(Q)≥D in the above definition gives the same (semi)stable objects.

An important feature of a semistable (i.e. PX-semistable) coherent system (Γ, F)is that Γ must generate F generically. This is proved by applying the stability inequality to the subsheaf F^� ⊂ F given by the image of Γ in F to show that r(F^�) =r(F); see [14, Proposition 4.4].

In the case of interest to us one can show that semistability coincides with stability which in turn corresponds to a simple geometric condition.

o

Proposition 2.6 ([21] Lemma 1.3). — A pair s:

O

X → F is semistable if and only if it is stable, and this holds precisely when

• the 1-dimensional sheafF is pure,

• the section s has0-dimensional cokernel.

In this case Supp(F)is a Cohen-Macaulay curve C.

Let us now discuss the moduli problem. For any scheme S letF be a flat family of sheaves on X with Hilbert polynomial P parameterised by S, i.e.

a sheaf on X ×S which is flat over S. Let p, q denote projections to X, S respectively. One is tempted to define a family of coherent systems parame- terised bySsimply as a locally free subsheaf ofq_∗F. However the natural map q_∗F(s)→H⁰(Fs)for a closed points∈S need not be an isomorphism. This is why the Serre dual notion is used instead. Recall thatD:= dimX.

Definition 2.7. — Aflat family of coherent systems onX parameterised by S is a sheaf F on X ×S, flat over S, together with a locally free quotient of the sheaf Ext^D_q (F, p^∗ωX).

It is shown in [14, Lemma 4.9] that, for any closed point s∈S, Ext^D_q (F, p^∗ωX)(s)∼= Ext^D(Fs, ωX).

We now introduce the usual functor of families, S �→Syst_X,Q(P)(S),

mapping a scheme S to the set of isomorphism classes of flat families of Q- semistable coherent systems onX whose underlying sheaf has Hilbert polyno- mialP, parametrised byS. The discussion in [14, Section 4.3] shows that this is a well defined contravariant functor.

Theorem 2.8. — [14, Theorem 4.12]There is a projective schemeSyst_X,Q(P) co-representing the functor Syst_X,Q(P). Its closed points are in one-to-one correspondence with S-equivalence classes ofQ-semistable coherent systems.

Definition 2.9. — LetPβbe the Hilbert polynomialPβ(m) =m�

βc1(

O

X(1))+

n. Le Potier’s space of stable pairsPn(X, β)is the moduli spaceSyst_X,PX(Pβ).

Remark. — The moduli space Pn(X, β)is fine. This is because stable pairs have no nontrivial automorphisms. See [21, Section 2.3] for more details.

In order to prove the GIT wall crossing in the next section we need to take a different point of view. Namely we assume the conclusion of Proposition2.6 as thedefinition of stable pairs, and provide an ad hoc construction for their moduli space which slightly differs from Le Potier’s. This is because we need to make full use of the special features of thed= 1case.

3. GIT wall crossing

Throughout this sectionPβ denotes the Hilbert polynomial Pβ(m) =rm+n,

where the multiplicityr is given by�

βc1(

O

X(1)).

Let us clarify what we mean by a GIT wall crossing.

Definition 3.1. — A GIT wall crossing between two projective schemes

M

0,

M

1 is given by a projective scheme

N

with an action of a reductive algebraic group G, ample Q-linearisations

L

0,

L

1 for the action of G and t^∗∈(0,1)∩Qsuch that, ift∈[0,1]∩Qand

L

t= (1−t)

L

0+t

L

1, then

• for0≤t < t^∗,

N

^s(

L

t) =

N

^ss(

L

t) =

N

^ss(

L

0) and

N

//_L0G∼=

M

0;

• fort^∗< t≤1,

N

^s(

L

t) =

N

^ss(

L

t) =

N

^ss(

L

1) and

N

//_L1G∼=

M

1.

Clearly then

N

//_LtG∼=

M

0for0≤t < t^∗,

N

//_LtG∼=

M

1 fort^∗< t≤1.

Remark. — Note that, unlike for example [23] Lemma 3.2, we require semista- bility to coincide with stability away from t^∗.

A GIT wall crossing gives the following well known diagram.

Lemma 3.2. — In the situation of Definition 3.1there are inclusions

N

^s(

L

0)

�� ��

N

^s(

L

1)

��

��

N

^ss(

L

t^∗)

inducing morphisms

N

//_L0G

ϕ0

��

N

//_L1G

ϕ1

��

N

//_Lt∗G.

The fibre ofϕi overϕi(G·x)fori= 0,1 is given by

ϕ⁻_i¹(ϕi(G·x)) ={G·y:G·x∩G·y�=∅ in

N

^ss(

L

t^∗)}.

o

Proof. — The first part is an application of the Hilbert-Mumford criterion.

For the second part we only use the standard description of the quotient map

N

_→

N

//_Lt∗G in terms of closures of semistable orbits, and the fact that

N

^ss(

L

i) =

N

^s(

L

i)fori= 0,1.

While there exist many trivial GIT wall crossings we are only interested in those that give interesting morphismsϕ0, ϕ1.

The MNOP/stable pairs GIT wall crossing can be stated as follows. For any subscheme Z ∈ In(X, β) let Z^pur denote its maximal Cohen-Macaulay closed subscheme (soZ^pur is pure of dimension1, and

O

Z →

O

Z^pur has finite kernel). Also letI_n^pur(X, β)⊂In(X, β)be the locus of Cohen-Macaulay closed subschemes, and writeS^kX for thekth symmetric product.

Theorem 3.3. — There is a GIT wall crossing between the moduli spaces In(X, β) andPn(X, β)as in Definition3.1, for which the following holds.

Write

(3.4) In(X, β)

ϕI

��

Pn(X, β)

ϕP

��SSn(X, β)

for the wall crossing morphisms to SSn(X, β) :=

N

//_Lt∗G. Then there is a stratification by locally closed subschemes

(3.5) SSn(X, β) =

�n

k=0

I_n−k^pur(X, β)×S^kX, and on closed points ϕI is given by

ϕI([Z]) =�

Z^pur,�

p

len(IZ^pur/IZ)pp� . Similarly, for (F, s)∈Pn(X, β),

ϕP(F, s) =�

Supp(F),�

p

len(F/im(s))pp� .

Remark. — When X is the (projective over an affine) threefold given by a small resolution of the threefold ordinary double point {xy = zw} ⊂ C⁴ the same result has been proved by Nagao and Nakajima [19] using moduli of perverse sheaves.

Example. — Rigid curve. Let C ⊂ X be a smooth rigid curve in the class β. Then the Hilbert schemeI1+χ(OC)(X, β)contains an irreducible component I1(X, C)∼= BlCX. Similarly the space of stable pairsP_1+χ(OC)(X, β)contains a componentP1(X, C)∼=C, andSS1(X, β)contains a componentSS1(X, C)∼= X. Restricting to these components the triangle (3.4) becomes

BlCX

π

��

C��

�� i

X

where π,i are respectively the blow up and the inclusion. This may look odd to the reader familiar with [6], [23] asiis not birational. This happens because in this case the space

N

of Definition 3.1 is not connected. At least locally near C it has two connected components which can be identified respectively with pairs with onto section and pairs with pure support, as we explain below.

Each of the two GIT quotients selects only one component.

We will see that the relevant

N

for us contains two distinguished closed subschemes

N

I,

N

P, given by the closure of subschemes parametrising pairs with surjective section and with pure support, respectively. For simplicity assume thatCis the only curve in the classβ. Then in our case

N

I consists of structure sheaves of subschemes given by the union ofCplus a point (inX\C, or an embedded point along C) plus the obvious section, while

N

P consists of degree 1 line bundles plus section defined by a point on C. Therefore their intersection is empty. (This is not the general case; when the curve C moves and singularities develop its genus can change, bubbling off points.)

Example. — Surfaces. WhenX is a surface the triangle (3.4) takes a special form. Divisors on a smooth algebraic surface are Gorenstein curves, and stable pairs supported on a Gorenstein curveCare in one-to-one correspondence with closed subschemes ofCby sending a pair

O

C→F to the ideal sheafF^∗→

O

C. For a proof see [22, Proposition B.5].

For a given degreeβthere exists a uniqueg(given by the adjunction formula 2g−2 = β·(KX +β)) such that I_1−g(X, β)is the moduli space of divisors in the class β, with a universal divisor D. Let Hilbⁿ(D/I1−g(X, β)) denote the relative Hilbert scheme of points on the fibres ofD→I1−g(X, β). By the result mentioned above, for anyn≥0 there is a one-to-one correspondence

P1−g+n(X, β)→Hilbⁿ(D/I1−g(X, β))

o

which is in fact an isomorphism of schemes ([22, Proposition B.8]). The triangle (3.4) then becomes

I1−g+n(X, β) ∼= I1−g(X, β)×Hilbⁿ(X)

µ ��

Hilbⁿ(D/I1−g(X, β))

��I_1−g(X, β)×SⁿX.

Here we are decomposing the Hilbert scheme of curvesI1−g+n(X, β)on a sur- face into a divisorial part I_1−g(X, β) and a punctual part Hilbⁿ(X) – see for example [7]. Thenµ:I1−g+n(X, β)→I1−g(X, β)×SⁿXis the Hilbert to Chow morphismHilbⁿ(X)→SⁿX times by the identity on I1−g(X, β).

We start by constructing the objects of Theorem 3.3. We build on [14, Chapter 4], but taking advantage of the very special cased= deg(Pβ) = 1. In brief, the master space

N

is going to be a closed subscheme of aQuotscheme times a projective space. Then a suitable choice of linearisation will pick out either stable pairs or structure sheaves of 1-dimensional subschemes.

3.1.

N

and G. — There is a general boundedness result for the family of isomorphism classes of Q-semistable coherent systems (Γ, F) with prescribed Hilbert polynomialPF, [14, Theorem 4.11]. In fact we will only use the follow- ing easier result.

Lemma 3.6. — The set of isomorphism classes of sheaves underlying isomor- phism classes of stable pairs [s:

O

X→F] withPF =Pβ is bounded.

Proof. — The set of (the structure sheaves of) Cohen-Macaulay curvesCsup- porting stable pairs of Hilbert polynomialPβ is bounded, since they all lie in the union of a finite number of Hilbert schemes of curves (since arithmetic genus is bounded on curves in the classβ). By the cohomological characterisation of boundedness (see for instance [8, Lemma 1.7.6]) this gives a uniform m such that

Hⁱ(X,

O

C(m−i)) = 0

for i > 0 (recall that the opposite also holds, i.e. a uniform bound on the regularity implies boundedness when the family of Hilbert polynomials is finite).

Let the extension

0→

O

C→F →Q→0

define a sheafF underlying a stable pair. SinceHⁱ(Q(k)) = 0fori >0and all k, we get a surjection

Hⁱ(

O

C(k))→Hⁱ(F(k))→0

fori >0and allk. Choosingk=m−igivesHⁱ(F(m−i)) = 0and therefore a uniform bound on the regularity ofF. As we recalled this implies boundedness for the family of allFs.

Fix a positive integer m and let V be a fixed vector space of dimension Pβ(m). We form the rankPβ(m)locally free sheaf onX

H

=V ⊗

O

X(−m).

Consider the Quot scheme of quotients of

H

with Hilbert polynomialPβ, Q= Quot(

H

, Pβ).

By Lemma3.6we can choosem�0such that, for any stable pair[s:

O

X →F]

with PF = Pβ, the sheaf F(m) is globally generated and Hⁱ(F(m)) = 0 for i >0. Therefore the isomorphism class ofF corresponds to a closed point of Qform�0, and this is unique modulo the natural action of SL(V)onQ. In other words there is a surjective map

H⁰(F(m))⊗

O

X(−m)→F →0

and we can pick an isomorphism H⁰(F(m))∼= V. Indeed the group G that occurs in Theorem3.3isSL(V)form�0.

Let us now construct

N

. LetRmdenote them-th graded piece of the graded ring ofX,

Rm=H⁰(

O

X(m)).

Lemma 3.7. — For any sheafF and any positive integermthere is a canonical injection

F ⊂F(m)⊗R^∗_m.

Proof. — The morphismF→F(m)⊗R^∗_min question is the element of Hom(F, F(m)⊗R^∗_m)∼= Hom(Rm,Hom(F, F(m)))

given by tensorings∈Rm∼= Hom(

O

X,

O

X(m))byF. This induces an injection F ⊂F(m)⊗R^∗_mbecause

O

X(m)is globally generated.

Thus there is an induced inclusion

H⁰(F)⊂H⁰(F(m))⊗R_m^∗.

LetΓ denote the subspace of H⁰(F)generated bys (in this section we some- times write(Γ, F)for a pair(F, s)). By the above discussionΓcan be regarded as a point of the projective spaceP(V ⊗R^∗_m)of lines inV ⊗R^∗_m (uniquely up to the action ofSL(V)).

The upshot of this is that isomorphism classes of pairs [s:

O

X →F] with PF =Pβ are realised as particular geometric points of

B=P(V ⊗R^∗_m)×Q,

o

modulo the action ofSL(V)(and this includes structure sheaves of subschemes with their canonical section).

We spell out the conditions that a closed point (Φ, F)∈Bmust satisfy in order to arise from a pair(Γ, F)under the correspondence above:

• the natural mapV →H⁰(F(m))induced byV⊗

O

X(−m)→F must be an isomorphism;

• the subspace ΦofV ⊗R^∗_mmust correspond to a subspace ΓofH⁰(F);

• the resulting pair(Γ, F)must be stable.

The schemeBis too large for the second condition to arise from GIT stability.

Instead our scheme

N

will be a suitable closed subscheme of B, as we now explain.

The product X×Q (with projections p, q to X,Q respectively) carries a universal quotient sheafF. Lemma3.6implies that form�0 all stable pairs withPF =Pβcorrespond to geometric points (unique up to the natural action ofSL(V)) of the projective bundle

P:= Proj⊕^kS^k�

Ext^D_q (F, p^∗ωX)� of rank one locally free quotients ofExt^D_q (F, p^∗ωX)overQ.

The evaluation mapV ⊗

O

X×Q →F(m)inducesV⊗

O

Q→q_∗(F(m)). By relative Serre duality for the morphismq, its dual is a map

Ext^D_q (F(m), p^∗ωX)→V^∗⊗

O

Q.

Let T ⊂ Q denote the open subscheme where this morphism is surjective;

by duality this is precisely the open subscheme where the natural map V → H⁰(F(m))is an isomorphism. The surjective morphisms of sheaves overT

Ext^D_q (F(m), p^∗ωX)|^T ⊗Rm

�� ��

V^∗⊗Rm⊗

O

T Ext^D_q (F, p^∗ωX)|^T induce, for m�0,SL(V)-equivariant closed immersions overT (3.8) P(V ⊗R^∗_m)×T

��

P|^T

��P�

q_∗(F(m))⊗R_m^∗�

|^T.

By definition ofT the left arrow is in fact an isomorphism, so we regardP|^T as a closed subscheme ofP(V⊗R^∗_m)×T, and we can rewrite the above conditions on a closed point (Φ, F)∈P(V ⊗R^∗_m)×Qin a different form:

• F ∈T;

• (Φ, F)∈P(V ⊗R^∗_m)×T lies inP|^T;

• (Φ, F)is stable.

We will see that these conditions do arise from GIT stability in the following space.

Definition 3.9. — The space

N

is the scheme-theoretic closure of the locally closed subscheme

P|^T ⊂P(V ⊗R^∗_m)×T⊂B.

Lemma 3.10. — Form�0closed subschemesZ⊂X with Hilbert polynomial PZ = Pβ also correspond to closed points of

N

, unique up to the action of SL(V).

Proof. — We only need to replace the set of isomorphism classes of stable pairs with PF =Pβ with its union with the set of structure sheaves

O

Z with their canonical section 1 :

O

X →

O

Z such thatPZ =Pβ.

The reason why we compactify P|^T by embedding into B and taking the closure rather than working directly withPwill be explained below.

3.2.

L

0 and

L

1. — For l � m, the Quot scheme Q admits the familiar Grothendieck embedding

ιl:Q�→Gr(V ⊗H⁰(

O

X(l−m)), Pβ(l))

into the Grassmannian ofPβ(l)dimensional quotients. Pulling back the Plücker line bundle onGr(V ⊗H⁰(

O

X(l−m)), Pβ(l))(the very ample generator of the Picard group) gives a very ample line bundle onQwhich we denote byι^∗_l

O

Gr(1).

We also write

O

_P(1)for the hyperplane bundle of P(V ⊗R^∗_m).

Definition 3.11. — Let c0, c1 be rational numbers with 0< c0<1

r < c1.

Then, for i= 0,1 andl �m, define ampleQ-linearisations for the action of SL(V)on

N

by

L

i:=

O

_P(l)�ι^∗_l

O

Gr(1)^⊗ci|N.

Remark. — The relative tautological bundle U onP is relatively ample, so the linearisationU(l)⊗

O

Gr(1)^⊗c is ample onPforc=c(l)�0. However for GIT we need to work with a fixed c, independent ofl. This is why we needed to find a different compactification of P|^T where ampleness of

L

i, i = 0,1 is evident. This approach is due to Le Potier, [14, Section 4.8].

o

3.3. Application of the Hilbert-Mumford criterion. — Since

N

is a projective scheme and the linearisations

L

i,i= 0,1are ample we can apply the Hilbert- Mumford criterion to find the (semi)stable points for the action. Any 1- parameter subgroup λ: C^∗ �→ SL(V) is uniquely determined by the decom- position into weight spaces V =�

k∈ZVk. For any quotient ρ:V ⊗

O

X(−m)→F →0

withPF =Pβ, define increasing filtrations of V,F respectively by V_≤k =�

j≤k

Vj,

F_≤k =ρ(V_≤k⊗

O

X(−m)).

ForΦ∈P(V ⊗R^∗_m)we set

Φ_≤k = Φ∩(V_≤k⊗R^∗_m) and

Φk = Φ_≤k/Φ_≤k−1.

(Of coursedim(Φ_≤k)can only jump once!) The quotient objects for the filtra- tion ofF are

Fk=F_≤k/F_≤k−1 so we find onto maps

ρk:Vk⊗

O

X(−m)→Fk→0 and inclusions

Φk⊂Vk⊗R^∗_m.

Lemma 3.12. — Let(Φ, F)be a closed point of

N

. Then

tlim→0λ(t)·(Φ, F) = (⊕^kΦk,⊕^kFk)

and the Hilbert-Mumford weight µ^Li((Φ, F), λ)is given by the quantity 1

dim(V)

�

k

�dim(V)�

ciPF_≤k(l)−dim(Φ_≤k)l�

−dim(V_≤k)�

ciPF(l)−l��

. Proof. — The sheaf part of the statement is standard, see for instance Lemmas 4.4.3 and 4.4.4 of [8]. But theΦpart of the statement can be seen as a special case of this, regardingΦk as a subsheaf of the sheafVk⊗R^∗_mover a point. We should only be careful about signs: when we rearrange as in [8] Lemma 4.4.4 to compute the Hilbert-Mumford weight, the sign of terms coming fromΦ_≤k is theopposite of the sign of terms coming fromF_≤k, since we take “subsheaves”

(i.e. subspaces!) of the “sheaf” V ⊗R_m^∗, rather than quotients.

Remark. — By this Lemma the weight µ^Li((Φ, F), λ) equals the constant dim(V)⁻¹ multiplied by the sum of a finite number of quantities of the form (3.13) µⁱ(V^�) = dim(V)(ciPF^�(l)−dim(Φ^�)l)−dim(V^�)(ciPF(l)−l), where V^�⊂V,F^�⊂F is the subsheaf generated byV^�, and

Φ^�= Φ∩(V^�⊗R^∗_m).

Conversely, for any subspaceV^�⊂V, we may define a one parameter subgroup λ ∈ SL(V) by λ(t) = t^{dim(V�)−dim(V)}idV^� ⊕ t^dim(V�)idV /V^�, whose Hilbert- Mumford weight µ^Li((Φ, F), λ) for i = 0,1 is precisely the quantity µⁱ(V^�) defined by (3.13).

Proposition 3.14. — A stable pair (F, s) with PF = Pβ and a marking H⁰(F(m))∼=V corresponds to a closed point of

N

which is stable under the action ofSL(V)linearised on

L

1.

Conversely, the semistable locus

N

^ss(

L

1) coincides with the stable locus

N

^s(

L

1), and a closed point of

N

^s(

L

1)corresponds to a stable pair(F, s)with PF =Pβ and a marking of H⁰(F(m)).

Therefore

N

^s(

L

1)has the good quotient Pn(X, β)for the action of SL(V).

Proof. — Let(Γ, F)be a stable pair. By Lemma 3.12(and the remark after its proof) a closed point (Φ, F) ∈

N

belongs to

N

^ss(

L

1) if and only if, for every subspace0�V^��V, we have

(3.15) dim(V)(c1PF^�(l)−dim(Φ^�)l)≥dim(V^�)(c1PF(l)−l).

Here F^� ⊂F denotes the subsheaf generated byV^�. Replacing ≥by >gives the corresponding statement for

N

^s(

L

1).

When(Φ, F) = (Γ, F)represents a stable pair, then settingΓ^� = Γ∩H⁰(F^�) in the above we claim that in fact we always have GIT stability:

dim(V)(c1PF^�(l)−dim(Γ^�)l)>dim(V^�)(c1PF(l)−l).

Since (Γ, F)is a stable pair, we know a priori thatF is pure, soV^� generates a 1-dimensional subsheaf with multiplicity r^� > 0. By the choice of c1 it is enough to prove the inequality of leading order terms

dim(V)(c1r^�−dim(Γ^�))>dim(V^�)(c1r−1).

Ifdim(Γ^�) = 0then we need to prove the inequality dim(V^�)

dim(V) <r^� r

Ç 1 1−c¹1r

å . But

rdim(V^�)

r^�dim(V) =1 + ^χ(F_r�m^�)

1 + ^χ(F)_rm ,

o

and since there are only finitely many possible r^� for a fixed r, the term is converging to1asm→ ∞uniformlyover allF^�⊂F, which implies the desired inequality for a uniform m�0. If on the other handdim(Γ^�) = dim(Γ), that is im(s) ⊂F^�, the assumption that (Γ, F) is a stable pair says that F/F^� is 0-dimensional and so r =r^�. Using this information, the inequality to prove becomes

dim(V)(c1r−1)>dim(V^�)(c1r−1), which follows fromdim(V^�)<dim(V)sincec1r >1.

For the converse, let us first show that for(Φ, F)∈

N

^ss(

L

1), the canonical mapV →H⁰(F(m))is an isomorphism, so indeed the quotientV⊗

O

X(−m)→ F →0 corresponds to a sheafF plus a marking of H⁰(F(m)). We only need to prove it is injective. Let V^� denote its kernel, so that the subsheaf F^� it generates is trivial. Substituting in (3.15) we get

−dim(V) dim(Φ^�)l≥dim(V^�)((c1r−1)l+c1χ(F)).

By definition c1 > ¹_r, so this is impossible unless V^� = {0}, as required. In particular

N

^ss(

L

1)⊂P(V ⊗R^∗_m)×T.

By the definition of

N

,(Φ, F)is a point in the closure ofP|^T inside B, which moreover by the above discussion lies inP(V ⊗R^∗_m)×T. ButP|^T is a closed subscheme of P(V ⊗R^∗_m)×T by the diagram of closed immersions (3.8), so (Φ, F) really lies in P|^T, i.e. Φ is some subspaceΓ ⊂H⁰(F) generated by a nontrivial section s. From now on we will therefore write (Γ, F) in place of (Φ, F).

It remains to show that(Γ, F)is a stable pair (so in particular, by the first part of the proof, (Γ, F)∈

N

^s(

L

1)). To see thatF is pure, supposeF^� ⊂F is a lower dimensional subsheaf. But then F^� must be 0-dimensional and so generated by its global sections, i.e. F^� is generated by some proper subspace V^� ⊂V. In the inequality (3.15) we would get

dim(V)(c1χ(F^�)−dim(Γ^�)l)≥dim(V^�)((c1r−1)l+c1χ(F)),

whereΓ^�= Γ∩H⁰(F^�). Sincec1>¹_r this can never be satisfied. To see thatΓ generatesFgenerically, suppose that the sectionsfactors throughF^�⊂F with multiplicity r^� (r^� > 0 since we have shown that F^� must be 1-dimensional).

Thendim(Γ^�) = 1and in (3.15) we get

dim(V)((c1r^�−1)l+c1χ(F^�))≥dim(V^�)((c1r−1)l+c1χ(F)).

This implies

dim(V)(c1r^�−1)≥dim(V^�)(c1r−1),

which may be rewritten as 1≤r^�

r −(c1r−1)

Ådim(V^�) dim(V) −r^�

r ã

. Now Ådim(V^�)

dim(V) −r^� r

ã

= r^� r

Ç 1

1 + ^χ(F)_rm −1 å

+χ(F^�) rm

1 1 +^χ(F)_rm .

Since 0 < r^� ≤ r and χ(F^�) ≤ χ(F)this term is O(m⁻¹)uniformly over all subsheaves F^� ⊂F. Therefore

1 +O(m⁻¹)≤r^� r ≤1

holds uniformly over all subsheavesF^� ⊂F. Since, for a fixedr, there are only a finite number of possibler^�, this impliesr^�=rform�0. It follows that the quotientF/F^� is0-dimensional as required.

This proof may be seen as a toy model of the Simpson-Le Potier estimates [14, Section 4.5], modified to take advantage of the assumptiond= 1.

Proposition 3.16. — A structure sheaf

O

Z withPZ =Pβ, its canonical sec- tion 1 :

O

X →

O

Z →0, and a marking of H⁰(

O

Z(m))∼=V corresponds to a closed point of

N

which is stable under the action ofSL(V)linearised on

L

0. Conversely the semistable locus

N

^ss(

L

0) coincides with the stable locus

N

^s(

L

0), and a closed point of

N

^s(

L

0) corresponds to a structure sheaf

O

Z

with PZ =Pβ and the canonical section1 :

O

X→

O

Z →0.

Therefore

N

^s(

L

0)has the good quotient In(X, β)for the action of SL(V).

Proof. — To a structure sheaf

O

Z we associate the point(Γ,

O

Z)∈

N

which corresponds to the subspace Γ generated by 1 :

O

X →

O

Z plus a marking for H⁰(

O

Z(m)). We claim that forV^� ⊂V generatingF^�⊂

O

Z the inequality

dim(V)(c0PF^�(l)−dim(Γ^�)l)>dim(V^�)(c0PF(l)−l)

always holds, whereΓ^� = Γ∩H⁰(

O

Z). As before this proves the first part of the Proposition. To prove the inequality, by the choice of c0 it is enough to prove the inequality of leading order terms,

dim(V)(dim(Γ^�)−c0r^�)<dim(V^�)(1−c0r).

Note that for a proper subsheaf F^� ⊂ F we cannot have dim(Γ^�) = 1as the canonical section is onto. But if dim(Γ^�) = 0 the above inequality is always satisfied since by definition of c0 one has1−c0r >0.

For the converse, start with(Φ, F)∈

N

^ss(

L

0). The inequality (3.17) dim(V)(c0PF^�(l)−dim(Φ^�)l)≥dim(V^�)(c0PF(l)−l)

o

holds for anyV^�⊂V generatingF^� ⊂FandΦ^�= Φ∩(V^�⊗R^∗_m)by the Hilbert- Mumford criterion. We must show that it implies that (Φ, F) corresponds to a structure sheaf with its canonical section and the choice of a marking (so in particular, by the first part of the proof, (Φ, F) ∈

N

^s(

L

0)). Taking leading coefficients in (3.17) we find

dim(V)(dim(Φ^�)−c0r^�)≤dim(V^�)(1−c0r).

Asdim(V)>dim(V^�)this can only hold if

dim(Φ^�)−c0r^�<1−c0r.

If dim(Φ^�) = 1this would imply r^� > r, which is absurd. On the other hand fordim(Φ^�) = 0the inequality is always satisfied, since by the choice of c0 we have 1−c0r >0. Now let

π:

N

^ss(

L

0)→

N

//_L0SL(V)

denote the quotient map. The argument above shows that there is an isomor- phism

π(

N

^ss(

L

0)∩P|^T)∼=In(X, β).

Therefore we obtain a locally closed immersion In(X, β)�→

N

//_L0SL(V)

with dense image. Since both schemes are proper and separated this is an isomorphism.

3.4. The GIT wall crossing (Theorem3.3).— Let(Φ, F) be any closed point of

N

, and fix the ampleQ-linearisation

L

t= (1−t)

L

0+t

L

1. To check whether (Φ, F)belongs to

N

^ss(

L

t)we will use the Hilbert-Mumford criterion. For any one parameter subgroupλ:C^∗�→SL(V)we have

µ^Lt((Φ, F), λ) = (1−t)µ^L0((Φ, F), λ) +tµ^L1((Φ, F), λ).

Since we already know that

N

^ss(

L

i) =

N

^s(

L

i) fori = 0,1 we may assume that(Φ, F)is stable with respect to

L

0and unstable with respect to

L

1, or vice virsa. In the first case say (the other is similar) fix a one parameter subgroup λfor which

(3.18) µ^L0((Φ, F), λ)·µ^L1((Φ, F), λ)<0.

Clearly, when (3.18) holds, convexity implies that for any choice of(Φ, F)and λthere exists a unique critical valuet,0< t <1, withµ^Lt((Φ, F), λ) = 0.

We claim that in fact there exists a unique critical valuet^∗,0< t^∗<1, such that when (3.18) holds one has

µ^Lt∗((Φ, F), λ) = 0

independently of (Φ, F)andλ. This will complete the proof.

By the remark following the proof of Lemma 3.12we may assume without loss of generality thatµ^Li((Φ, F), λ)has the special formµⁱ(V^�), i.e.

µ^Li((Φ, F), λ) = dim(V)(ciPF^�(l)−dim(Φ^�)l)−dim(V^�)(ciPF(l)−l) for i= 0,1. Reading through the proofs of Propositions3.14and 3.16we see that the inequality (3.18) can only hold when

• F^� is0-dimensional anddim(Φ^�) = 0, or

• F^� is1-dimensional,r=r^� anddim(Φ^�) = 1.

The only possible value fort^∗ is t^∗=

Å

1−µ¹(V^�) µ⁰(V^�)

ã−1

and we must show that this is independent of V^�, and moreover that the two values coincide.

Whendim(F^�) = dim(Φ^�) = 0we get µ⁰(V^�)

µ¹(V^�) =dim(V)c0χ(F^�)−dim(V^�)((c0r−1)l+c0χ(F)) dim(V)c1χ(F^�)−dim(V^�)((c1r−1)l+c1χ(F))

=(dim(V)−χ(F))c0−(c0r−1)l (dim(V)−χ(F))c1−(c1r−1)l

=rmc0−(c0r−1)l rmc1−(c1r−1)l,

where we have used dim(V^�) =χ(F^�). The last expression is clearly indepen- dent ofV^�. When dim(F^�) = dim(Φ) = 1 andr=r^� we get

µ⁰(V^�)

µ¹(V^�) =dim(V)((c0r−1)l+c0χ(F^�))−dim(V^�)((c0r−1)l+c0χ(F)) dim(V)((c1r−1)l+c1χ(F^�))−dim(V^�)((c1r−1)l+c1χ(F))

=(dim(V)−dim(V^�))(c0r−1)l+c0(dim(V)χ(F^�)−dim(V^�)χ(F)) (dim(V)−dim(V^�))(c1r−1)l+c1(dim(V)χ(F^�)−dim(V^�)χ(F))

=c0(dim(V^�)−χ(F^�))−(c0r−1)l c1(dim(V^�)−χ(F^�))−(c1r−1)l

=rmc0−(c0r−1)l rmc1−(c1r−1)l , as required.

Finally we prove the stratification (3.5) for SSn(X, β). By the argument above the closed points of

N

^ss(

L

t^∗)\

N

^s(

L

t^∗)are in one-to-one correspondence with pairs(F, s)withPF =Pβ such that either

• F is the structure sheaf of a subschemeZ⊂X which is not pure, or

• F is pure, but the sectionshas a nontrivial0-dimensional cokernelQ.

o