Moduli Spaces of Stable Pairs in Donaldson-Thomas Theory
aa r X i v : . [ m a t h . AG ] M a r Moduli Spaces of Semistable Pairs inDonaldson − Thomas Theory
Malte WandelFreie Universit¨at Berlin, Leibniz Universit¨at Hannovercurrent address:Research Institute of Mathematical SciencesKyoto UniversityKitashirakawa Oiwake cho, Sakyo-kuKyoto 606-8502, JAPANe-mail: [email protected] 25, 2018
Abstract
Let ( X, O X (1)) be a polarized smooth projective variety over the complex num-bers. Fix D ∈ coh( X ) and a nonnegative rational polynomial δ . Using GIT wecontruct a coarse moduli space for δ -semistable pairs ( E , ϕ ) consisting of a coher-ent sheaf E and a homomorphism ϕ : D → E . We prove a chamber structure resultand establish a connection to the moduli space of coherent systems constructedby Le Potier in [LeP] and [LeP2].Keywords: moduli spaces, curve counting, stable pairs, coherent systemsMCS: 14D20, 14D22, 14N35
Contents Construction 155 Variation of the Stability Parameter 206 Coherent Systems 22Bibliography 24
String theorists are highly interested in counting curves on Calabi − Yau threefolds. Thiscan be done by integrating over a virtual cycle on the moduli space of these curves (cf.[LT]). The arising moduli problems are not compact and there have been different ap-proaches to their compactification (cf. [PT]). Following Pandharipande and Thomas weconsider pairs ( E , s ) consisting of a coherent sheaf E with one dimensional support anda section s ∈ H ( E ). Such a pair is called stable if firstly the sheaf is pure and secondly s considered as a homomorphism O X → E is generically surjective. Thus a stable pair( E , s ) provides us with a Cohen − Macaulay curve C E = supp E and a finite number ofpoints on this curve, namely the cokernel of s .In [LeP2] the moduli space for such stable pairs is contructed. More generally Le Potierconsidered so-called coherent systems (Γ , E ) consisting of a sheaf E together with a sub-space Γ ⊆ H ( E ). Since the pairs introduced above can only cover the case of irreduciblecurves there is a need for generalizations of the notion of stable pairs. We want a section s for every irreducible component of C E . Thus we should consider pairs ( E , ϕ ) with ahomomorphism ϕ : O rX → E or even more generally with a homomorphism ϕ : D → E for an arbitrary but fixed coherent sheaf D . There is a generalized notion of stabilityfor such pairs.In this article we will construct a coarse moduli space for semistable pairs on an arbi-trary polarized smooth projective variety and relate these moduli spaces to the modulispaces of coherent systems in the case D = O rX .We will now give a more detailed overview of the content of this article. In the first sec-tion we will define the generalized notion of stability depending on a parameter δ ∈ Q [ x ]and discuss some basic properties of semistable pairs. Next in Section 2 we will provethe boundedness of the family of δ -semistable pairs which will enable us to define the pa-rameter space for our moduli problem in Section 3. The core of this article is containedin Section 4 where we perform the GIT construction of the moduli space. In Section 5we prove the usual chamber structure result summarizing how the moduli spaces changeif we vary the parameter δ . Last but not least we show in Section 6 that the modulispace of coherent systems by Le Potier can be obtained from the moduli space of stablepairs as a quotient by a group action.Notations: The main guidelines for the notations used in this article are [Har] and[HL]. By a scheme one should always think of a scheme of finite type over some fixedalgebraically closed field k of characteristic zero. If V is a finite dimensional k -vector2pace we let P ( V ) := ( V \{ } ) / ∼ be the set of lines passing through the origin. For anyvector v ∈ V we denote its equivalence class in P ( V ) by [ v ]. Finally by PGL n we denotethe quotient GL n +1 /k ⋆ Id. Thus PGL n is the automorphism group of P n In order to define pairs we first fix some notation. Let ( X, O X (1)) be a polarized smoothprojective variety, D a coherent O X -module and δ a rational polynomial ≥
0, i.e., δ ( m ) ≥ ∀ m ≫ . Definition 1.1
A pair ( E , ϕ ) consists of a coherent O X -module E and a homomorphism ϕ : D → E . By P = P E we denote the Hilbert polynomial of E . This is a polynomialof degree d = dim E and its leading coefficient is just the rank of the sheaf E whichwe denote by r or r E . We call a pair pure if E is pure. A homomorphism of pairs α : ( E , ϕ ) → ( E ′ , ψ ) is a homomorphism of O X -modules α : E → E ′ such that there is ascalar λ ∈ k making the following diagramm commute D λ · id / / ϕ (cid:15) (cid:15) D ψ (cid:15) (cid:15) E α / / E ′ . In the obvious way we define the notion of an isomorphism of pairs . Lemma 1.2
Let ( E , ϕ ) be a pair. Then for any λ ∈ k ⋆ ( E , λϕ ) is isomorphic to ( E , ϕ ) .Proof: Obvious. (cid:3)
Definition 1.3
We define the Hilbert polynomial of a pair ( E , ϕ ) to be P ( E ,ϕ ) := P E + ǫ ( ϕ ) δ, where ǫ ( ϕ ) = 1 if ϕ = 0 and otherwise. For any subsheaf F ⊆ E we define the inducedhomomorphism ϕ ′ to be equal to ϕ if im ϕ ⊆ F and otherwise. For the correspoding uotient G = E / F the induced homomorphism ϕ ′′ is defined to be the composition of ϕ with the quotient map. It is if and only if im ϕ ⊆ F . Thus we easily see that theHilbert polynomial of pairs is additive in exact sequences. The reduced Hilbert polynomial of a pair ( E , ϕ ) is defined as p ( E ,ϕ ) = P ( E ,ϕ ) r E . Definition 1.4
A pair ( E , ϕ ) is called (semi)stable with respect to δ (or δ -(semi)stable) if for every saturated submodule F ⊆ E of rank r ′ we have P ( F ,ϕ ′ ) ( ≤ ) r ′ p ( E ,ϕ ) . Remark : For δ = 0 this condition does not depend on ϕ at all and we get the usualstability condition for sheaves. Thus the corresponding moduli-space is constructed anddiscussed in [HL]. Therefore we will consider only strictly positive stability parameters δ from now on.Similarly, if the homomorphism ϕ is trivial the stability condition does not depend on δ and is again equivalent to the usual stability condition for sheaves. Thus during theconstruction of the moduli space of stable pairs we will restrict to pairs with nontrivialhomomorphism ϕ .There is another good reason for not considering pairs with trivial homomorphism.Assume we have constructed a moduli space M of δ -semistable pairs allowing trivialhomomorphisms. Fix an arbitrary semistable pair ( E , ϕ ) with nontrivial ϕ and considerthe family ( E , λϕ ) λ ∈ k parametrized by the affine line. Assume that ( E ,
0) is as wellsemistable. This family would lead to a classifying map f : A → M . For every λ ∈ k ⋆ we get the same point f ( λ ) ∈ M (cf. 1.2). Thus for λ = 0 we have to end up with thesame point. But this time we have a point with trivial homomorphism. Thus we seethat in such a case our moduli space would degenerate and parametrize sheaves only. Proposition 1.5
A pair is δ -(semi)stable if and only if for every pure quotient G of E of rank r ′′ with induced homomorphism ϕ ′′ we have: P ( G ,ϕ ′′ ) ( ≥ ) r ′′ p ( E ,ϕ ) . Proof:
Let G be a pure quotient of E . Recall that by definition we have an exact sequence0 → F → E → G → F a saturated submodule of E . Also recall, that the rank and the Hilbert polynomialof pairs are additive. Now the claim follows just by substituting P G and r G by theappropriate expressions of F . (cid:3) Lemma 1.6
Let α : ( E , ϕ ) → ( E ′ , ψ ) be a nontrivial homomorphism between two stablepairs of the same reduced Hilbert polynomial. Then α is an isomorphism. roof: On F = im α there are two induced homomorphisms: the first coming from E called ϕ ′′ , the second induced by ψ on E ′ called ψ ′ . It is easy to compute that ψ ′ istrivial if and only if ϕ ′′ is trivial. Thus the Hilbert polynomial of the pairs ( F , ϕ ′′ ) and( F , ψ ′ ) coincide. Since ( E , ϕ ) and ( E ′ , ψ ) have the same reduced Hilbert polynomial thestability assumption yields F = E ′ and ker α = 0. (cid:3) Proposition 1.7
Let ( E , ϕ ) be a δ -semistable pair with reduced Hilbert polynomial p .There is a filtration E ( E ( · · · ( E r = E , such that every factor gr i ( E ) = E i / E i − with its induced homomorphism is stable withreduced Hilbert polynomial p . Such a filtration is called a Jordan − H¨older filtration andthe graded object gr( E ) := ⊕ i gr i ( E ) is independent of the choice of the filtration. Itnaturally inherits an induced homomorphism gr( ϕ ) : D → gr( E ) .Proof: If ( E , ϕ ) is stable there is nothing to prove. Otherwise there is a subsheaf F ⊆ E with induced homomorphism ϕ ′ such that p ( F ,ϕ ′ ) = p . It is therefore semistable. Choosesuch a subsheaf F which is maximal with this property. Then the quotient E / F is stablewith reduced Hilbert polynomial p by construction. We can now proceed in the sameway with F constructing another subsheaf and so on. We end up with a filtration withstable factors which has to be finite since the rank is decreasing in every step. For aproof of the remaining assertions we refer to [HL] Proposition 1.5.2 or [HL2] Proposition1.13. (cid:3) Remark:
One can easily see that for every semistable pair ( E , ϕ ) with nontrivial ϕ the homomorphism gr( ϕ ) of the graded object is nontrivial and its image is containedin exactly one summand of gr( E ). Definition 1.8
Two δ -semistable pairs ( E , ϕ ) and ( E ′ , ψ ) are called S-equivalent if theirgraded objects (gr( E ) , gr( ϕ )) and (gr( E ′ ) , gr( ψ )) are isomorphic. From now on until the end of this article we will only consider semistable pairs with non-trivial homomorphism ϕ . Thus whenever we say a pair is semistable, we mean semistablewith nontrivial homomorphism. For later use we want to formulate a characterisationof the stability condition which is equivalent to the one given before if we restrict to thecase of nontrivial homomorphisms. We first need another definition. Definition 1.9
For every pair ( E , ϕ ) and every exact sequence → F → E → G → we define ǫ ( F ) := ( if im ϕ ⊆ F otherwise , and ǫ ( G ) := 1 − ǫ ( F ) . emma 1.10 A pair ( E , ϕ ) with ϕ = 0 is δ -(semi)stable if and only if for every saturatedsubsheaf F ⊆ E of rank r ′ the following inequality holds: P F + ǫ ( F ) δ ( ≤ ) r ′ r ( P + δ ) . Proof:
Obvious. (cid:3)
Proposition 1.11
Let ( E , ϕ ) be a δ -semistable pair. Then E is pure.Proof: Let ( E , ϕ ) be a semistable pair and let T := T d − ( E ) denote the maximal subsheafof strictly smaller dimension. This is saturated and of rank zero. Thus the semistabilitycondition yields P T + ǫ ( T ) δ ≤ r ′ r ( P + δ ) = 0 , or equivalently P T ≤ − ǫ ( T ) δ ≤
0, hence T = 0. (cid:3) Proposition 1.12 If deg δ ≥ dim X for every pair ( E , ϕ ) the following two assumptionsare equivalent: (i) ( E , ϕ ) is δ -semistable, (ii) ϕ is generically surjective and E is pure.Proof: (i) ⇒ (ii): If ϕ was not generically surjective, there would exist a saturated module F satisfying im ϕ ⊆ F * E (for instance the saturation of im ϕ ). Now semistability yields P F r F + δr F ≤ P E r E + δr E . For deg δ ≥ deg P E it follows that r E ≤ r F , thus we get r E = r F because F ⊆ E andwe conclude F = E , because F was assumed to be saturated. This contradicts theassumption im ϕ ⊆ F * E .(ii) ⇒ (i): If ϕ is generically surjective then for every saturated submodule F ⊆ E wehave im ϕ
6⊆ F , hence ǫ ( F ) = 0. Now if ( E , ϕ ) was not semistable there would exist asaturated submodule F satisfying the destablizing condition: P F > r ′ r ( P + δ ) . Since E is pure we have r ′ = 0. Thus for deg δ > d we get δ < δ = d if we compare the leading coefficients of the polynomials. (cid:3) Remark:
By this proposition we see, that for large δ the moduli space we are inter-ested in can be realized as some Quot-scheme parametrizing certain quotients of ourfixed module D . Thus from now on we will assume deg δ < dim X .6 roposition 1.13 If a pair ( E , ϕ ) can be deformed into a pure pair, then there is a puresheaf H and a morphism ψ : E → H satisfying ker ψ = T ( E ) . In particular if we set ϕ H = ψ ◦ ϕ we get a pure pair ( H , ϕ H ) .Proof: The condition on ( E , ϕ ) says there is a smooth connected curve C and a flatfamily ( E C , ϕ C ) on C × X such that ( E , ϕ ) ∼ = ( E , ϕ ) for some closed point 0 ∈ C and( E t , ϕ t ) pure for all t = 0. In particular E deforms into a pure sheaf. Now our claimfollows from [HL], Proposition 4.4.2. (cid:3) Remark : Note that a priori ϕ H could be trivial. This might occur if im ϕ is con-tained in T d − ( E ) ⊆ E . Therefore in order to show that such a ( H , ϕ H ) is semistable weshould first show that ϕ H is infact nontrivial. In this section we prove that the set of δ -semistable pairs with fixed Hilbert polynomialis bounded. Furthermore we deduce an important stability criterion which we will needfor the construction of the moduli space in Section 4. Proposition 2.1
Let P and δ ≥ be polynomials. Then there is a constant C dependingonly on P and D such that for every O X -module E occuring in a δ -semistable pair wehave µ max ( E ) ≤ C . In particular, the family of pairs which are semistable with respectto any stability parameter δ having the fixed Hilbert polynomial P is bounded.Remark: Note that the uniform bound is independent of δ . Proof:
Let µ P denote the slope of P and δ denote the coefficient of δ in degree d −
1. Itis δ ≥
0. If F is a submodule of E of rank r ′ satisfying im ϕ ⊆ F then the semistabilitycondition yields µ F + δ r ′ ≤ µ P + δ r . And since r ′ ≤ r we have µ F ≤ µ P . Now let F ⊆ E be an arbitrary submodule. Wehave an exact sequence 0 → F → F + im ϕ → G → , where G = im ϕ/ ( F ∩ im ϕ ) is a quotient of im ϕ , so a fortiori a quotient of D . Set H := F + im ϕ and note that H contains im ϕ , so we can apply the first part of the proof.By the additivity of the degree we have: µ F = µ H r H − µ G r G r F ≤ µ P r H − µ min ( D ) r G r F ≤ µ P r − µ min ( D ) r G r F . µ min ( D ) we find a uniform bound C for µ F by setting C := max { µ P , µ P r − µ min ( D ) r, µ P r − µ min ( D ) r } . (cid:3) Next we need a result to get an estimate for the global sections of certain sheaves.It follows from a more general result due to Simpson (cf. [Sim], Lemma 1.5). Thereforewe define α i to be the i-th coefficient of P O X , the Hilbert polynomial of the structuresheaf. For any O X -module E we define b µ ( E ) := µ E + α d − α d . Note that b µ ( E ) is just the coefficient of P E /r in degree d − Proposition 2.2
Let E be a coherent O X -module of rank r and dimension d and let C := r ( r + d ) / . Then h ( E ( m )) r ≤ r − r · d ! [ b µ max ( E ) + C − m ] d + + 1 r · d ! [ b µ ( E ) + C − m ] d + . Proof: [HL], Corollary 3.3.8. (cid:3)
Corollary 2.3
If in addition E is semistable we have h ( E ( m )) r ≤ d ! [ b µ ( E ) + C − m ] d + . Proof:
Since E is semistable we have µ max ( E ) ≤ µ ( E ). (cid:3) Now we’re able to proof a very important stability criterion which we will need forthe GIT construction of the moduli space. It corresponds logically to [HL], Theorem4.4.1 and we will follow their proof closely. First we fix a stability parameter δ . Let m be an integer such that δ ( m ) ≥ ∀ m ≥ m . Proposition 2.4
There is an integer m such that for every m ≥ m and every pure pair ( E , ϕ ) with Hilbert polynomial P and rank r the following three assertions are equivalent: (i) ( E , ϕ ) is δ -(semi)stable, (ii) P ( m ) ≤ h ( E ( m )) and for all subsheaves F ⊆ E of rank < r ′ < r we have h ( F ( m )) + ǫ ( F ) δ ( m ) ( ≤ ) r ′ r (cid:0) P ( m ) + δ ( m ) (cid:1) , (iii) for all quotients E → G of rank < r ′′ < r we have r ′′ r (cid:0) P ( m ) + δ ( m ) (cid:1) ( ≤ ) h ( G ( m )) + ǫ ( G ) δ ( m ) . emark: Note that we may assume m ≥ m . Proof: (i) ⇒ (ii): Since the family of semistable pairs ( E , ϕ ) is bounded, there is aninteger m such that for any E occuring in such a pair we have P ( m ) = h ( E ( m )). Nowlet F ⊆ E be an arbitrary subsheaf of rank r ′ . To show (2) we may assume F to besaturated. First we assume that max { µ P , µ P r − µ min ( D ) } = µ P = µ E and distinguishtwo cases:(1) b µ ( F ) ≥ b µ ( E ) − C · r − δ ,(2) b µ ( F ) < b µ ( E ) − C · r − δ ,where C := r ( r + d ) / b µ ( F ) is boundedfrom below. Since b µ ( F ) = ( µ F + α d − ) /α d we easily find that µ F is aswell boundedfrom below. Since we are talking about saturated subsheaves only by Grothendieck’sproposition (cf. Proposition 2.4) the family of subsheaves of type (1) is bounded. Thusthe set of Hilbert polynomials of this family is finite and by taking m sufficiently largewe have h ( F ( m )) = P ( F ( m )) and P ( F ( m )) + ǫ ( F ) δ ( m )( ≤ ) r ′ r [ P ( m ) + δ ( m )] ⇔ P ( F ) + ǫ ( F ) δ ( ≤ ) r ′ r [ P + δ ] . Now we consider subsheaves of type (2). Note that b µ max ( F ) ≤ b µ max ( E ) ≤ µ ( E ) by theassumption made at the beginning. Set C ′ := r ′ ( r ′ + d ) / >
0. Bythe estimate of Proposition 2.2 we have h ( F ( m )) r ′ ≤ r ′ − r ′ · d ! [ b µ max ( F ) + C ′ − m ] d + + 1 r ′ · d ! [ b µ ( F ) + C ′ − m ] d + ≤ r ′ − r ′ · d ! [ b µ ( E ) + C ′ − m ] d + + 1 r ′ · d ! [ b µ ( E ) + C ′ − C · r − δ − m ] d + = m d d ! + m d − ( d − (cid:16) b µ ( E ) + C ′ − r ′ ( C · r − δ ) | {z } =: A (cid:17) + O ( d − O ( d −
2) stands for polynomials in m of degree ≤ d −
2. Since r ′ ≤ r we have A = b µ ( E ) − C ′ − rr ′ C − δ r ′ < b µ ( E ) − − δ r ′ . Now as we have said before b µ ( E ) is the coefficient of P/r in degree d −
1. Thus forsufficiently large m we get1 r ′ (cid:0) h ( F ( m )) + ǫ ( F ) δ ( m ) (cid:1) < m d d ! + m d − ( d − b µ ( E ) − < P ( m ) r < P ( m ) + δ ( m ) r Now if max { µ P , µ P r − µ min ( D ) } = µ P r − µ min ( D ) we can run through all the argumentsbefore substituting b µ ( E ) by b µ ( E ) r − µ min ( D ) + ( r − α d − α d
9n the inequalities of the distinction of the cases.(ii) ⇒ (iii): For any quotient E → G we let
F ⊆ E denote the corresponding kernel. By(ii) we get: h ( G ( m )) + ǫ ( G ) δ ( m ) ≥ h ( E ( m )) − h ( F ( m )) + δ ( m ) − ǫ ( F ) δ ( m )( ≥ ) 1 r (cid:16) rP ( m ) − r ′ P ( m ) + rδ ( m ) − r ′ δ ( m ) (cid:17) = r ′′ r (cid:16) P ( m ) + δ ( m ) (cid:17) . (iii) ⇒ (i): Let E be an arbitrary sheaf satisfying (iii). We denote by G min the minimaldestabilizing quotient sheaf of E . Since it is semistable it has the nice property that µ min ( G min ) = µ ( G min ) = µ ( G max ). By (iii) and Proposition 2.2 we have P ( m ) + δ ( m ) r − ǫ ( G min ) δ ( m ) r ′′ ≤ h ( G min ( m )) r ′′ ≤ d ! [ b µ ( G min ) + C − m ] d + . Thus b µ min ( E ) = b µ ( G min ) is bounded from below which is equivalent to b µ max ( E ) beingbounded from above. Hence we find the family of sheaves E satisfying (iii) to be bounded.Now we want to apply Proposition 1.5. Thus let G denote an arbitrary quotient. Wehave either b µ ( G ) > b µ ( E ) + δ /r but then we get a strict inequality in the semistabilitycondition or b µ ( G ) ≤ b µ ( E ) + δ /r , i.e., b µ ( G ) is bounded from above and we can oncemore apply Grothendieck’s result to get the boundedness of the family of the quotientsin question. Again for large m we have h ( G ( m )) = P ( G ( m )) and P ( G ( m )) + ǫ ( G ) δ ( m )( ≥ ) r ′′ r [ P ( m ) + δ ( m )] ⇔ P ( G ) + ǫ ( G ) δ ( ≥ ) r ′′ r [ P + δ ] . This finally shows (iii) ⇒ (i) and finishes the proof. (cid:3) Remark:
The proof shows that in (ii) and (iii) equality holds if and only if the subsheafor the quotient, resp., is destabilizing.
We will now define the moduli functor for stable pairs and present the parameter spacewhich is somewhat more complicated than just a quot-scheme. Furthermore we discussthe natural linearizations on the parameter space and state the central theorems of thisarticle, the existence of the moduli space of stable pairs.Throughout this section we will fix a stability parameter δ , a rational polynomial P anda polarized smooth projective scheme ( X, O X (1)). Definition 3.1
We define a functor M X,δ ( D , P ) : (Sch /k ) ° → (Sets) as follows: Let S be a k -scheme of finite type. Define M X,δ ( D , P )( S ) to be the set ofisomorphism classes of pairs ( E , ϕ ) consisting of a coherent S -flat sheaf E on X × S coming with the projections π X onto X and π S onto S ) and a homomorphism π ⋆X D →E , such that for every closed point s ∈ S the pair ( E s , ϕ | π ⋆X D s ) is δ -semistable withHilbert polynomial P . For every morphism of k -schemes f : S ′ → S we obtain a map M X,δ ( D , P )( f ) : M X,δ ( D , P )( S ) → M X,δ ( D , P )( S ′ ) by pulling back E and ϕ by Id X × f .We define a subfunctor M sX,δ ( D , P ) concerning only stable pairs. Definition 3.2
A scheme M X,δ ( D , P ) is called a coarse moduli space of δ -semistablepairs , if it correpresents the functor M X,δ ( D , P ) . As we have seen before the family of isomorphism classes of sheaves with Hilbert poly-nomial P occuring in a δ -semistable pair is bounded. Therefore there is an m such thatfor all integers m ≥ m the sheaves E are m -regular. Now we fix one m ≥ m , m , m (for notations see Proposition 2.4) until the end of Section 4 and note that we mayassume that D is m -regular as well.Set V = k P ( m ) . Let Q := Quot X ( V ⊗ O X ( − m ) , P ) be Grothendieck’s Quot-schemeparametrizing quotients q : V ⊗ O X ( − m ) → E with Hilbert polynomial P . Recall thatfor large l we have the very ample line bundles L l on Q which come from pullbacks ofthe universal bundle on Q × X . Furthermore let N := P (cid:0) Hom( H ( D ( m )) , V ) (cid:1) be thespace of morphisms a : H ( D ( m )) → V which is polarized by O N (1). Lemma 3.3
Let ( E , ϕ ) be a δ -semistable pair with Hilbert polynomial P . To this pairwe can then associate in a natural way a pair ( a, q ) ∈ N × Q such that the induced map H ( q ( m )) is an isomorphism and q ◦ a = ϕ ◦ ev : H ( D ( m )) ⊗ O X ( − m ) a (cid:15) (cid:15) ev / / D ϕ (cid:15) (cid:15) V ⊗ O X ( − m ) q / / E . Here ‘ ev ’ denotes the natural evaluation map.Proof: Since E is m -regular E ( m ) is globally generated. Since the Hilbert polynomial of E is P, we have h ( E ( m )) = P ( m ) = dim V . If we choose any isomorphism V → H ( E ( m ))we find a surjection ρ : V ⊗ O X → E ( m ) . Since taking tensor products is a right exact functor we obtain by tensoring with O X ( − m ) another surjection ρ ( m ) : V ⊗ O X ( − m ) → E and hence a point q in Q . On the other hand we consider ϕ : D → E . Tensoringwith O X ( m ), applying the global section functor and again choosing some isomorphism V → H ( E ( m )) we obtain a map a := H ( ϕ ( m )) : H ( D ( m )) → V. E , ϕ ) is assumed to be semistable the homomorphism ϕ is not zero.Thus a is not zero as well. Furthermore Lemma 1.2 says that ( E , ϕ ) and ( E , λϕ ) areisomorphic pairs, thus a is a well-defined point in N . It is clear from the constructionthat q ◦ a = ϕ ◦ ev and H ( q ( m )) is an isomorphism. (cid:3) Proposition 3.4
There is a closed subscheme Z ′ ⊆ N × Q such that for every pair ( a, q ) ∈ N × Q the composition q ◦ a factors through the evaluation map ev (and thusinduces a homomorphism ϕ ) if and only if ( a, q ) ∈ Z ′ : H ( D ( m )) ⊗ O X ( − m ) a (cid:15) (cid:15) ev / / D ϕ y y V ⊗ O X ( − m ) q (cid:15) (cid:15) E . Proof:
We’ll have a look at the relative version of the upper diagramm on N × Q × X :0 / / K / / f ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ H ( D ( m )) ⊗ π ⋆X O X ( − m ) ev / / ˜ q ◦ ˜ a (cid:15) (cid:15) π ⋆X D ˜ ϕ u u / / π ⋆Q E . Here ˜ q denotes the pullback of the universal quotient on Q × X and ˜ a denotes the pull-back of the universal homomorphism on N .Now ˜ q ◦ ˜ a factors through ev if and only if f vanishes. Indeed, the if direction followsfrom the fundamental theorem on homomorphisms, the only if direction is an immediateconsequence of the exactness of the upper row of the diagramm. Now the claim followsfrom [Schm], Proposition 2.3.5.1 and [GS], Lemma 3.1. (cid:3) The set of points ( a, q ) where E = q ( V ⊗ O X ( − m )) is pure forms an open subset U ⊆ Z ′ .We let Z be its closure in Z ′ . Now SL( V ) acts diagonally on Z : g · ( a, q ) = ( g ◦ a, q ◦ g − ( − m )) . Now L l and O N (1) inherit natural linearizations of this SL( V )-action. Thus for integers n , n the very ample line bundles O Z ( n , n ) := π ⋆Q L ⊗ n l ⊗ π ⋆N O N ( n ) as well inheritnatural SL( V )-linearizations. For some fixed l we will choose n , n to satisfy n n = P ( l ) · δ ( m ) − δ ( l ) · P ( m ) P ( m ) + δ ( m ) . Definition/Lemma 3.5
We define
R ⊆ Z to be the subset consisting of points ( a, q ) corresponding to pairs which are δ -semistable and the induced map H ( q ( m )) is an iso-morphism. This is an open and SL( V ) -invariant subset of Z . Furthermore there is anopen subset R s ⊆ R corresponding to δ -stable pairs. roof: Consider the universal homomorphism ˜ q ◦ ˜ a : H ( D ( m )) ⊗ π ⋆X O X ( − m ) → π ⋆Q E on Z × X and the projection π : Z × X → Z which is a projective morphism since X isprojective. Let us consider the set A of polynomials P ′ occuring as a Hilbert polynomialof a quotient which destabilizes a pair corresponding to a point ( a, q ) ∈ Z . For everysuch polynomial P ′ its slope µ ′ satisfies µ ′ ≤ µ P + δr P . By [Gro], 2.5 the set A is finite. For every P ′ we consider the relative quot-scheme Q ( P ′ ) := Quot Z× X/ Z ( π ⋆Q E , P ′ ). It comes with a morphism f P ′ : Q ( P ′ ) → Z . We denotethe image of f by Z ( P ′ ). It is a closed subset of Z . Now it is clear that a point ( a, q ) ∈ Z corresponds to a semistabe pair if and only if it is not in the finite and therefore closedunion S P ′ ∈ A Z ( P ′ ). (cid:3) Proposition 3.6
If a scheme M is a categorical quotient for the SL( V ) -action on R ,then it correpresents the functor M X,δ ( D , P ) .Proof: Let M be such a categorical quotient of R . We have to show that for any k -scheme of finite type S there is a map M X,δ ( D , P )( S ) → Hom(
S, M ). Therefore let S be such a k -scheme and ( E , ϕ ) be an element of M X,δ ( D , P )( S ). Thus V E := p ⋆ ( E ⊗ q ⋆ O X ( m ))is a locally free sheaf on S of rank P ( m ) = dim V . We obtain a quotient on X × S : ̺ E : p ⋆ V E ⊗ q ⋆ O X ( − m ) → E . Let F E := I som( V ⊗ O S , V E ) be the frame bundle of V E together with the natural pro-jection π : F E → S and the universal trivialisation of V E f : V ⊗ O F E → π ⋆ V E . Now from the quotient q E := ( id X × π ) ⋆ ̺ E ◦ π ⋆F E f : O X ( − m ) ⊗ V ⊗ O F E → π X E on X × F E we obtain its classifying morphismΦ E : F E → Q. On the other hand Lemma 3.3 states that the homomorphism ϕ yields a homomorphismof vector spaces a : H ( D ( m )) → V . Lifting this map to GL( V ) and composing with theuniversal automorphism τ : O GL( V ) ⊗ V → O GL( V ) ⊗ V we obtain a homomorphism onGL( V ) O GL( V ) ⊗ H ( D ( m )) → O GL( V ) ⊗ V. a : GL( V ) → N. Clearly there is a natural action of GL( V ) on F E and GL( V ) itself. Moreover it isnot hard to see that S is a categorical quotient for the GL( V )-action on the productGL( V ) × F E . Alltogether we get a diagramm where vertical arrows are the maps fromthe categorical quotients: GL( V ) × F E Ξ a × Φ E / / (cid:15) (cid:15) N × Q (cid:15) (cid:15) S / / M It follows from the construction that the map Ξ a × Φ E is GL( V )-equivariant and there-fore yields a map between the categorical quotients S → M . (cid:3) The following result forms the center of this article. The proof will take all of Sec-tion 4.
Theorem 3.7
For sufficiently large l the subset of points in the closure ¯ R of R whichare (semi)stable with respect to the SL( V ) -linearization coincides with the subset of pointscorresponding to δ -(semi)stable pairs. Theorem 3.8
Let ( X, O X (1)) be a polarized smooth projective variety, D a coherent O X -module and δ a nonegative rational polynomial. Then there exists a coarse mod-uli space M X,δ ( D , P ) of δ -semistable pairs. Two pairs correspond to the same pointin M X,δ ( D , P ) if and only if they are S-equivalent. Moreover there is an open subset M sX,δ ( D , P ) ⊆ M X,δ ( D , P ) corresponding to stable pairs. It is a fine moduli space of δ -stable pairs, i.e., it represents the functor M sX,δ ( D , P ) .Proof: The existence of the moduli space follows easily from Theorem 3.7 and [GIT],Theorem 1.10 or [Schm], Theorem 1.4.3.8. The first remaining thing is to show thatthe coarse moduli space in fact parametrizes S-equivalence classes. Similarly as in theproofs of Proposition 3.3 in [HL2] and Lemma 4.1.2 in [HL] one can show that the orbitof any semistable pair ( E , ϕ ) corresponding to a point in R ss also contains the gradedobject (gr( E ) , gr( ϕ )) and the orbits of these graded objects are closed. Thus S-equivalentpairs are mapped to the same point and since a geometric quotient map separates closedorbits we are done.The second and last open statement is the fact that M s := M sX,δ ( D , P ) is indeed a finemoduli space. This is equivalent to the existence of a universal family on X × M s .Following Section 4.6 in [HL] (in particular Proposition 4.6.2) the only thing we have todo is to show that there is a line bundle on R s on which the center Z := k ⋆ · Id ⊆ GL( V )acts with weight 1. Such a line bundle can be named explicitly. Just remember that R was a subset of the product N × Q = Quot X ( V ⊗O X ( − m ) , P ) × P (cid:0) Hom( H ( D ( m )) , V ) (cid:1) .Now O N (1) has Z -weight 1. (cid:3) Construction
Finally we present the calculations showing that under appropriate choices the notionof δ -stability coincides with GIT-stability. Proposition 4.1
Let ( a, q ) be a point in ¯ R . For sufficiently large l ( a, q ) is (semi)stablein the GIT sense with respect to O Z ( n , n ) if and only if the following holds: Set W := H ( O X ( l − m )) and q ′ := H ( q ( l )) : V ⊗ W → H ( E ( l )) . Then for every nontrivialproper subspace U of V we have: dim U [ n P ( l ) − n ] ( ≤ ) P ( m )[dim( q ′ ( U ⊗ W )) n − ǫ ( U ) n ] , (1) where ǫ ( U ) = 1 if U ⊆ im a and otherwise.Proof: In order to apply the Hilbert − Mumford criterion we have to look at 1-parametersubgroups λ : G m → SL( V ). Such a λ is completely determined by giving a basis v , . . . , v p of V and a weight vector ( γ , . . . , γ p ) ∈ Z p satisfying γ ≤ · · · ≤ γ p and P γ i = 0. The action of λ is then given by λ ( t ) · v i = t γ i v i .Now we look at a point ( a, q ) ∈ ¯ R represented by homomorphisms q : V ⊗ O X ( − m ) → E and a : H ( D ( m )) → V and we denote by ϕ : D → E the corresponding framing. Forthe moment we fix an l ≥ m and let W := H ( O X ( l − m )) and ̺ := h ( E ( l )) = P ( l ).Now q induces the homomorphisms q ′ = H ( q ( l )) and q ′′ : Λ ̺ ( V ⊗ W ) → det H ( E ( l )).Let w , . . . , w t be a basis of W . We then get a basis of Λ ̺ ( V ⊗ W ) by elements of theform u IJ = ( v i ⊗ w j ) ∧ · · · ∧ ( v i ̺ ⊗ w j ̺ )with multiindices I and J satisfying i α ≤ i α +1 and j α < j α +1 if i α = i α +1 . Now theaction of λ on Λ ̺ ( V ⊗ W ) is given by λ ( t ) · u IJ = t γ I u IJ with γ I := X α γ i α . Now µ ( q ′′ , λ ) is given by − min { γ I |∃ I, J with q ′′ ( u IJ ) = 0 } . But a slightly better formu-lation is possible. We set ψ ( i ) = dim( q ′ ( h v , . . . , v i i ⊗ W )). Then we have µ ( q ′′ , λ ) = − p X i =1 γ i (cid:0) ψ ( i ) − ψ ( i − (cid:1) . Observe that ψ ( i ) − ψ ( i −
1) is always equal to one or zero. Thus the right hand sidesums up all the γ i such that dim q ′ ( h v , . . . , v i i ⊗ W ) is increasing. By the surjectivityof q (or q ′ ) we know that there are ̺ such γ i . Because the γ i are in increasing order this γ I must be the smallest such that q ′ ( h v , . . . , v i i ⊗ W ) = H ( E ( l )).Next we want to determine µ ( a, λ ). Look at the following identification:Hom( U, V ) ∼ = U ∨ ⊗ Va : w j X i α ij v i ↔ X i,j α ij w ∨ j ⊗ v i .
15e now deduce easily that µ ( a, λ ) = max { γ i |∃ j with α ij = 0 } = min { γ i | im a ⊆ h v , . . . , v i i} = γ τ where τ = min { i | im a ⊆ h v , . . . , v i i} .Now by the Hilbert − Mumford criterion ( q, a ) is (semi)stable if and only if n · µ ( q ′′ , λ ) + n · µ ( a, λ )( ≥ )0 , i.e., (2) n · p X i =1 γ i (cid:0) ψ ( i ) − ψ ( i − (cid:1) − n · γ τ ( ≤ )0 . Fixing a basis v , . . . , v p for the moment we can consider the left hand side as a linearform on the set of weight vectors. Thus it is enough to check the inequality for thespecial weight vectors γ ( i ) = ( i − p, . . . , i − p | {z } i , i, . . . , i | {z } p − i ) , i = 1 , . . . , p − . Indeed, every weight vector can be expressed as a finite nonnegative linear combinationof the γ ( i ) . Now for such a γ ( i ) we have γ ( i ) τ = ( i − p if im a ⊆ h v , . . . , v i i i otherwise . In other words we have γ ( i ) τ = i − ǫ ( i ) p where ǫ ( i ) := 1 if im a ⊆ h v , . . . , v i i and 0otherwise. On the other hand we have p X i =1 γ i (cid:0) ψ ( i ) − ψ ( i − (cid:1) = ̺i − p i X k =1 (cid:0) ψ ( k ) − ψ ( k − (cid:1) = ̺i − pψ ( i ) . Alltogether our inequality now reads: i · ( n ̺ − n ) ( ≤ ) p · ( n ψ ( i ) − ǫ ( i ) n ) . In particular this inequality does not contain no weights anymore. It does not evendepend on the fixed basis we chose but on the subspaces spanned by this basis. Thus apoint ( q, a ) is (semi)stable if and only if for every nontrivial subspace U ⊆ V we have:dim U · ( n ̺ − n ) ( ≤ ) dim V · (cid:0) n dim( q ′ ( U ⊗ W )) − ǫ ( U ) n (cid:1) where ǫ ( U ) := 1 if im a ⊆ U and 0 otherwise. But we have the identifications ̺ = P ( l )and dim V = P ( m ) . (cid:3) For every nontrivial subspace U ⊆ V we denote by F U the subsheaf of E generatedby U . Lemma 4.2
For every GIT-semistable point ( a, q ) the induced morphism H ( q ( m )) : V → H ( E ( m )) is injective. In particular dim( V ∩ H ( E ( m ))) ≤ h ( E ( m )) where V ∩ H ( E ( m )) denotes the preimage of H ( E ( m )) in V . Furthermore q ′ is injective and for every sub-space U ⊆ V we have dim( q ′ ( U ⊗ W )) ≤ h ( F U ( l )) , where F U := q ( U ⊗ O X ( − m )) . roof: Let U ⊆ V denote the kernel of H ( q ( m )), then the generated subsheaf F U iszero (and so is ǫ ( F U )), thus by the upper inequality dim U ≤
0. Note that by the choiceof n , n we have n ̺ − n ≥ (cid:3) Proposition 4.3
For sufficiently large l a point ( a, q ) is GIT-(semi)stable if and onlyif for every nontrivial subspace U ⊆ V we have the following inequality of polynomialsin l : dim U · ( n P ( l ) − n ) ( ≤ ) P ( m ) · ( n P F U ( l ) − ǫ ( F U ) n ) . (3) Proof:
By Prop 4.1 it is enough to show that (1) is equivalent to (3) for every U .First of all we note that the family of these subsheaves F U generated by some subspaceof U ⊆ V is bounded. Thus by taking l large enough all the F U are globally generatedand we have P F U ( l ) = h ( F U ( l )) = dim( q ′ ( U ⊗ W )) ∀F U . We’ll now have a look at thefollowing diagram: H ( D ( m )) ⊗ O X ( − m ) a / / ev (cid:15) (cid:15) V ⊗ O X ( − m ) q (cid:15) (cid:15) D ϕ / / E . It is easy to see, that im a ⊆ U ⇒ im ϕ ⊆ F U . Thus if ǫ ( U ) = 1, we have ǫ ( F U ) = 1 andin this case we easily see that (1) is equivalent to (3).So the only interesting case is ǫ ( U ) = 0 and ǫ ( F U ) = 1. Now let ( a, q ) be GIT-(semi)stable with ǫ ( U ) = 0 and ǫ ( F U ) = 1. We claim that for every nontrivial subspacewe also have inequality (3). Let U be a nontrivial subspace of V and let U ′ := U ⊕ im a .Note that ǫ ( U ′ ) = 1 and U ′ generates the same subsheaf F U . Thus we get the desiredinequality with dim U replaced by dim U ′ . But since dim U ≤ dim U ′ our claim follows.Conversely let ( a, q ) be a point satisfying (3) for every nontrivial subspace U ⊆ V . Now(1) is equivalent to P ( m ) · n >
0. But this is clear by the definition. (cid:3)
Proposition 4.4
For sufficiently large l a point ( a, q ) is GIT-(semi)stable if and onlyif for every nontrivial subspace U ⊆ V we have the following inequality of polynomials: P · (cid:0) dim U + ǫ ( F U ) δ ( m ) (cid:1) + δ · (cid:0) dim U − ǫ ( F U ) P ( m ) (cid:1) ( ≤ ) P F U · (cid:0) P ( m ) + δ ( m ) (cid:1) . (4) Proof:
Again because the family of such subsheaves is bounded we can find an l suchthat inequality (3) holds if and only if it holds as an inequality of polynomials in l . Nowsubstitute n n = P ( l ) · δ ( m ) − δ ( l ) · P ( m ) P ( m ) + δ ( m ) . We then easily derive the required inequality. (cid:3)
Theorem 4.5
For sufficiently large l if a point ( a, q ) is GIT-semistable then the corre-sponding pair ( E , ϕ ) is semistable (with respect to δ ) and H ( q ( m )) is an isomorphism.In particular every GIT-semistable point corresponds to a pair with pure sheaf. roof: First we will drop the restriction to subsheaves generated by subspaces U ⊆ V .Let F be an arbitrary subsheaf of E . Set U := V ∩ H ( F ( m )). Then the subsheafgenerated by U is contained in F . Now if ǫ ( F U ) = 1 we also have ǫ ( F ) = 1 and weeasily get the required inequality. Conversely if ǫ ( F ) = 1 by definition of U we have ǫ ( U ) = 1, hence ǫ ( F U ) = 1. Thus from now on we can consider arbitrary subsheavestogether with the subspace U := V ∩ H ( F ( m )).Passing to the leading coefficient of the polynomials in the inequality (4) we have:dim U + ǫ ( F ) δ ( m ) ≤ r ′ r ( P ( m ) + δ ( m )) . We now assign to any submodule F its corresponding quotient G . Note that sincedim U ≤ h ( F ( m )) we find dim( V /U ) ≤ h ( G ( m )). We have h ( G ( m )) + ǫ ( G ) δ ( m ) ≥ dim( V /U ) + ǫ ( G ) δ ( m )= dim V + δ ( m ) − (dim U + ǫ ( F ) δ ( m )) ≥ P ( m ) + δ ( m ) − r ′ r ( P ( m ) + δ ( m )) ≥ r ′′ r ( P ( m ) + δ ( m )) . Since ( a, q ) ∈ ¯ R , it deforms into a pair with torsion free sheaf. By Proposition 1.13there is a torsion free sheaf H together with a homomorphism ψ : E → H satisfying P ( H ) = P ( E ) and ker( ψ ) = T ( E ). Next we want to show that ( H , ϕ H ) is semistable.As noted after Proposition 1.13 the corresponding homomorphism ϕ H might be trivial.But here we will use the following lemma. Lemma 4.6
Let l be chosen as in Proposition 4.4 and for every GIT-semistable point ( a, q ) let T denote the maximal subsheaf of strictly smaller dimension of E (cf. [HL],Definition 1.1.4). Then we have im ϕ
6⊆ T , i.e., ǫ ( T ) = 0 .Proof : Suppose ǫ ( T ) to be 1 and look at the leading coefficients of (4). Since T is torsionthe leading coefficient of P T is zero. Hence we have r · (dim U + δ ( m )) ≤ δ ( m ) is clearly positive we have a contradiction. (cid:3) Continuation of the proof of
Theorem 4.5: Thus we have seen that ϕ H is nontrivialand we can now continue the proof by showing that H is semistable. If π : H → G H isany quotient of H , let G denote the image of E by π ◦ ψ . This is a quotient of E whichis contained in G H . Hence we have h ( G H ( m )) ≥ h ( G ( m )) and ǫ ( G H ) = 0 ⇒ ǫ ( G ) = 0 . And even if ǫ ( G H ) = 1 and ǫ ( G ) = 0 we get the following inequalities: h ( G H ( m )) + ǫ ( G H ) δ ( m ) ≥ h ( G ( m )) + ǫ ( G ) δ ( m ) (5) ≥ r ′′ r ( P ( m ) + δ ( m )) = r G H r ( P ( m ) + δ ( m )) .
18y Proposition 2.4 ( H , ϕ H ) is semistable and therefore m -regular. By taking G H = H we find in fact equality in (5) everywhere. This shows, that h ( ψ ( E )( m )) = h ( H ( m )) = P ( m ) and since H is globally generated we find ψ to be surjective. Since E and H havethe same Hilbert polynomial ψ is in fact an isomorphism. Hence E is (semi)stable.Because of Lemma 4.2 H ( q ( m )) is injective and surjectivity now follows easily fromdimension reasons. (cid:3) Theorem 4.7
Let ( E , ϕ ) be a δ -(semi)stable pair such that q induces an isomorphism V → H ( E ( m )) . Then the corresponding point ( a, q ) is (semi)stable in the GIT-sense.Proof: By Proposition 2.4 we have h ( F ( m )) + ǫ ( F ) δ ( m ) ( ≤ ) r ′ r ( P ( m ) + δ ( m ))for every subsheaf F of E of rank r ′ satisfying 0 < r ′ < r = rk ( E ). If ( E , ϕ ) is stablethen the inequality is strict. Now if U is an arbitrary subspace of V and F U denotes thesubsheaf generated by U we have U ⊆ V ∩ H ( F U ( m )) and dim U ≤ h ( F U ( m )). Thuswe get the following strict inequality:dim U + ǫ ( F U ) δ ( m ) < r ′ r ( P ( m ) + δ ( m )) . This is a strict inequality of the leading coefficients of the desired inequality.If ( E , ϕ ) is semistable but not stable, again strict inequality holds except the case of adestabilizing semistable subsheaf F . But such an F is semistable with the same reducedHilbert polynomial. Hence F has the same slope as E and by the choice of m we have P F ( m ) = h ( F ( m )). The destablizing condition on F says: P F + ǫ ( F ) δ = r ′ r ( P + δ ) . Now let U := V ∩ H ( F ( m )) and note that dim U = h ( F ( m )) = P F ( m ) because q induces an isomorphism V → H ( E ( m )). The terms (let’s call them (1) and (2)) wewant to show to be equal are:(1) P F ( P ( m ) + δ ( m )) = (cid:0) r ′ r ( P + δ (cid:1) − ǫ ( F ) δ (cid:1)(cid:0) P ( m ) + δ ( m ) (cid:1) = P · r ′ r (cid:0) P ( m ) + δ ( m ) (cid:1) + δ · (cid:0) r ′ r ( P ( m ) + δ ( m )) − ǫ ( F )( P ( m ) + δ ( m )) (cid:1) , (2) P · (cid:0) dim U + ǫ ( F ) δ ( m ) (cid:1) + δ · (cid:0) dim U − ǫ ( F ) P ( m ) (cid:1) = P · (cid:0) P F ( m ) + ǫ ( F ) δ ( m ) (cid:1) + δ · (cid:0) P F ( m ) − ǫ ( F ) P ( m ) (cid:1) . By applying the destabilizing condition evaluated at m to the coefficients standing infrontof P and δ shows that they are indeed the same. Thus the point ( a, q ) corresponding to( E , ϕ ) is semistable but not stable. (cid:3) Variation of the Stability Parameter
In this section we want to study how the moduli space M X,δ ( D , P ) changes if we varythe stability parameter δ . Therefore we fix P and D . Lemma 5.1
Define W to be the set of all subsheaves im ϕ occuring as the image of ahomomorphism ϕ in a δ -semistable pair for any δ . Then W is bounded.Proof: For every semistable pair ( E , ϕ ) there is a corresponding point ( a, q ) ∈ N × Q (cf.Lemma 3.3) such that q ◦ a = ϕ ◦ ev. On N × Q × X there is the universal homomorphism(cf. Proposition 3.4) ˜ q ◦ ˜ a : H ( D ( m )) ⊗ π ⋆X O X ( − m ) → π ⋆Q ˜ E . It has the universal property that its fibre at the point ( a, q ) ∈ N × Q (which is a ho-momorphism of sheaves on X ) is just q ◦ a . Thus we find that every sheaf in W occursas a fibre of the sheaf im(˜ q ◦ ˜ a ). Hence W is bounded. (cid:3) The next result is a refinement of Proposition 1.12.
Proposition 5.2
There is a rational polynomial δ max of degree less than dim X suchthat for every δ > δ max and every pair ( E , ϕ ) the following two assumptions are equiva-lent: (i) ( E , ϕ ) is δ -semistable, (ii) ϕ is generically surjective.Proof: We follow closely the proof of Proposition 1.12:(i) ⇒ (ii): If ϕ was not generically surjective then the saturation F of im ϕ would be aproper subsheaf of E . Semistability yields p F ≤ p E + δ · ( 1 r E − r F ) | {z } =: c . (6)Now one can easily see that − < c <
0, because 1 ≤ r F ≤ r E . Since the leadingcoefficients of p F and p E agree we may choose a δ of degree less than dim X such thatthe inequality (6) is violated. Since the set of image sheaves of semistable pairs isbounded there are only finitely many polynomials we have to consider. Hence we maychoose a δ max working for all of these.(ii) ⇒ (i): Conversely, let ( E , ϕ ) be a pair, where ϕ is generically surjective. Then forany subsheaf F ⊆ E we have ǫ ( F ) = 0. Thus if ( E , ϕ ) is not δ max -semistable then F satisfies µ F > µ E + δ max , r E > µ E , (7)20here δ max , shall denote the coefficient of δ max in degree d −
1. Note that the set ofsheaves E occuring in a pair ( E , ϕ ) where ϕ is generically surjective is certainly bounded.Therefore the set of subsheaves satisfying (7) is bounded and we may choose δ max , bigenough to contradict the first inequality in (7) for all F . (cid:3) Lemma 5.3
For any δ ≥ we consider the set S δ of all subsheaves occuring as a δ -destabilizing subsheaf of any pair ( E , ϕ ) which is δ ′ -semistable for some δ ′ . Let S := S δ ≥ S δ . Then S is bounded.Proof: If δ ≤ δ max then the destabilizing condition reads µ F > µ E + δ ( 1 r E − ǫ ( F ) r F ) . But the right hand side is bounded from below by µ E − δ max , . Thus for every sheaf in S there is a uniform bound µ F > µ E − δ max , . If δ > δ max then by the proposition abovethere are no destabilizing subsheaves. Now the claim follows from [Gro], Lemme 2.5. (cid:3) Corollary 5.4
There are only finitely many polynomials occuring in a destabilizing con-dition for a semistable pair in M X,δ ( D , P ) for any δ . From these results one may deduce easily the following theorem which summarizes howstability depends on the parameter δ . Theorem 5.5
There are finitely many critical values δ , . . . , δ s ∈ Q [ z ] , / / | | · · · | δ δ s ∞ such that setting δ := 0 and δ s +1 := ∞ the following properties hold true: (i) For i = 0 , . . . , s and δ, δ ′ ∈ ( δ i , δ i +1 ) , one has R ( s ) sδ = R ( s ) sδ ′ . (ii) For i = 0 , . . . , s and δ ∈ ( δ i , δ i +1 ) , there are the inclusions R ssδ ⊆ R ssδ i ∩ R ssδ i +1 , R sδ ⊇ R sδ i ∪ R sδ i +1 . (iii) For i = 0 , . . . , s and δ ∈ ( δ i , δ i +1 ) , one has M X,δ ( D , P ) = M sX,δ ( D , P ) . Remark:
Of course we have δ s = δ max . Alltogether these inclusions yield a nice diagramoften called the chamber structure of the stability parameter: M X ( D , P ) z z ttttttttt $ $ ❏❏❏❏❏❏❏❏❏ M sX ( D , P ) z z ttttttttt % % ▲▲▲▲▲▲▲▲▲▲ M X,δ ( D , P ) M X,δ ( D , P ) M X,δ s ( D , P ) M X,δ s +1 ( D , P ) , where we set M iX ( D , P ) := M X,δ ( D , P ), for some δ ∈ ( δ i , δ i +1 ) , i = 0 , . . . , s . The diagonalmaps are easily obtained by the universal property of the categorical quotient.21 Coherent Systems
Definition 6.1 A coherent system is a pair (Γ , E ) consisting of a coherent sheaf E on X and a vector subspace Γ ⊆ H ( E ) of dimension r . Two coherent systems (Γ , E ) and (Γ ′ , E ′ ) are called isomorphic if there is an isomorphism of sheaves E → E ′ such that thecorresponding map of global sections maps Γ isomorphically onto Γ ′ .Remark: As a simple consequence we deduce that for isomorphic systems we havedim Γ = dim Γ ′ .Next we want to introduce a notion of stability for coherent systems. Just like thestability of pairs it depends on a parameter, a nonnegative rational polynomial α . Definition 6.2
Let (Γ , E ) be a coherent system. For a subsheaf F ⊆ E we define Γ ′ :=Γ ∩ H ( F ) . A coherent system (Γ , E ) is called α -(semi)stable if for every nontrivialsaturated subsheaf F ( E we have the following inequality of rational polynomials: dim Γ ′ · α + P F ( ≤ ) r F r E (cid:8) dim Γ · α + P E (cid:9) . Remark:
Consider the case dim Γ = 0. Then our stability condition reduces to the usualone for sheaves. Thus from now on we may assume dim Γ > ev (Γ , E ) : Γ ⊗ O X → E provides a pair ( E , ev (Γ , E ) ) which we want to call the corresponding pair to the system (Γ , E ). Note that the evaluation map, of course, is injective on global sections.Conversely any pair ( E , ϕ ) consisting of a coherent sheaf E and a map ϕ : O rX → E yieldsa coherent system (im( H ( ϕ )) , E ). If in addition H ( ϕ ) is injective the dimension ofim( H ( ϕ )) is, of course, equal to r .Just as we have done in Sections 2-4 in the case of δ -(semi)stable pairs one can definea moduli functor for α -(semi)stable coherent systems and construct the moduli spaceSyst X,α ( r, P ) of α -semistable coherent systems (Γ , E ), where P is the Hilbert polynomialof E and dim Γ = r . This construction is done by Le Potier in [LeP] and [LeP2]. In thefollowing passage we want to show that Syst X,α ( r, P ) can be obtained as a quotient bysome GL r -action of a moduli space of semistable pairs which we constructed in Section4. We need the following results: Lemma 6.3
Let (Γ , E ) be a α -(semi)stable coherent system. Set δ := dim Γ · α . Thenthe corresponding pair ( E , ev (Γ , E ) ) is δ -(semi)stable.Proof: Let F ( E be a saturated subsheaf. Assume im(ev (Γ , E ) ) ⊆ F . Thus Γ ⊆ H ( F )and we have Γ ′ = Γ ∩ H ( F ) = Γ. Now (Γ , E ) is α -(semi)stable and we can apply thestability condition to F :dim Γ · α + P F ( ≤ ) r F r E (cid:8) dim Γ · α + P E (cid:9) . δ = dim Γ · α we get the required inequality for F .Now assume im(ev (Γ , E ) ) * F . Since dim Γ ′ and α are nonnegative the required inequalitythis time follows immediately: P F ≤ dim Γ ′ · α + P F ( ≤ ) r F r E (cid:8) dim Γ · α + P E (cid:9) . (cid:3) Set D = O rX and consider the set R ⊆ Q × N of Definition/Lemma 3.5. Now Lemma6.3 above says that all semistable coherent systems of type ( r, P ) are parametrized by R . Consider the SL r ( k ) × SL( V )-action on R . Again there is a natural linearization ofthis action in O R ( n , n ) where we choose n , n as before. Analogously to Proposition4.1 we calculate the weights for this linearization. Proposition 6.4
For every GIT-semistable point ( a, q ) ∈ R the corresponding coherentsystem (Γ , E ) is semistable.Proof: Let
F ⊆ E , Γ ′ := Γ ∩ H ( F ) and j := dim Γ ′ . Fix a basis x , . . . , x j of Γ ′ andextend it to a basis x , . . . , x r of Γ. Furthermore set U := V ∩ H ( F ( m )) and i := dim U .Clearly we have a (Γ ′ ⊗ H ( O X ( m ))) ⊆ U . We divide into two cases.Case 1 ○ : im a * U . Choose a basis v , . . . , v i of U and extend it to a basis v , . . . , v p of V . Consider the weight vectors pr · ( − j, . . . , − j | {z } r − j , r − j, . . . , r − j | {z } j ) and ( i − p, . . . , i − p | {z } i , i, . . . , i | {z } p − i ) . Let λ be the formal one-parameter subgroup associated to these vectors. One easilyderives µ ( a, λ ) = i − j · pr . (8)From the semistability condition on the point ( a, q ) we derive as in (2) − i · P ( l ) + p · P F ( l ) + µ ( a ) δ ( m ) · P ( l ) − p · δ ( l ) p + δ ( m ) ≥ δ ( m ) p ( µ ( a ) − i ) − i + r F r E ( p + δ ( m )) . We plug in equation (8) and end up with the desired semistability condition on F .Case 2 ○ : im a ⊆ U . Now we have Γ ′ = Γ, thus r = j . Therefore we choose the weightvectors (0 , . . . ,
0) and ( i − p, . . . , i − p | {z } i , i, . . . , i | {z } p − i )23nd find µ ( a, λ ) = i − p. Proceeding just as in case 1 ○ we again derive the necessary semistability condition. (cid:3) Proposition 6.5
Consider a point ( a, q ) ∈ R such that the corresponding coherentsystem is semistable. Then ( a, q ) is GIT-semistable.Proof: We now have to consider all the one parameter subgroups of SL r ( k ) × SL( V )and show that the corresponding weight is nonnegative. Since it is nearly impossible tostudy all kinds of such subgroups it is more convenient to restrict to special ones. Thuswe apply a result by A. Schmitt (cf. [Schm2], Theorem 3.4) allowing us to restrict toone parameter subgroups corresponding to weight vectors of the special form (cid:0) r ( j − r, . . . , j − r, j, . . . , j ); 1 p ( i − p, . . . , i − p, i, . . . , i ) (cid:1) , such that the corresponding weight is exactly i − j · p/r . Now we are almost done. Justas in Proposition 6.4 we see that the weight is nonnegative if and only if (9) is fulfilled.Therefore we first look at the leading coefficient of the left hand side of (9) wich isnonnegative if and only if j · α + P F ≤ r F r E ( P E + r · α ) , (10)where F is the subsheaf of E generated by the subspace spanned by the first i elementsof V corresponding to the weight vector. Now since j ≤ dim Γ ′ this is a strict inequalityunless (Γ ′ , F ) is destabilizing. In this case we proceed just as we did at the ending ofSection 4 observing that in this case j = dim Γ ′ . (cid:3) As the last ingredient we now state a powerful tool for the construction of quotientsif we have a group action of a product of groups.
Theorem 6.6
Let X be a projective k -scheme with group actions σ : G × X → X and τ : H × X → X of reductive groups G and H coming with linearizations ¯ σ and ¯ τ .Suppose these group actions do commute such that there is an action of the product σ × τ : G × H × X → X . Set Q σ := X (cid:12) ¯ σ G and let π : X ss ¯ σ → Q σ be the quotient map.Then there is a natural H -action γ on Q σ together with a linearization ¯ γ such that theset of ¯ σ × ¯ τ -semistable points is π − (( Q σ ) ssγ ) and X (cid:12) ¯ σ × ¯ τ ( G × H ) ∼ = Q σ (cid:12) ¯ γ H Proof: [Schm], Theorem 1.5.3.1. (cid:3)
Alltogether we have proven the following result:
Theorem 6.7
There is a rational map M X,r · α ( O rX , P ) Syst
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