Generalized parabolic structures over smooth curves with many components and principal bundles over reducible nodal curves
aa r X i v : . [ m a t h . AG ] J u l Generalized parabolic structures oversmooth curves with many components andprincipal bundles over reducible nodal curves ´ANGEL LUIS MU ˜NOZ CASTA ˜NEDA ∗ Abstract
Let Y , . . . , Y l be smooth irreducible projective curves and let Y be its disjointunion. Given a semisimple linear algebraic group G and a faithful representation ρ : G ֒ → SL( V ) we construct a projective moduli space of ( κ, δ )-(semi)stable singu-lar principal G -bundles with generalized parabolic structure of type e . In case Y isthe normalization of a connected and reducible projective nodal curve X , there isa closed subscheme coarsely representing the subfunctor corresponding to descend-ing bundles. We prove that the descent operation induces a birational, surjectiveand proper morphism onto the schematic closure of the space of δ -stable singularprincipal G -bundles whose associated torsion free sheaf is of local type e . Keywords: principal bundles; generalized parabolic structures; reducible nodal curves.
Contents1 Introduction 22 Preliminaires 4 G -bundles with generalized parabolic structures . . . . 52.4 Some calculations in geometric invariant theory . . . . . . . . . . . . . . . 62.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 κ, δ )-semistability and Hilbert-Mumford semistability . . . . . . . 133.4 The moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ∗ Department of Mathematics, University of Le´on, Spain, email: a [email protected] Moduli space for generalized parabolic singular principal bundles 20 G -bundles . . . . . . . . . . . . . . . . . . . 235.3 Relation to the moduli space of principal G -bundles over a reducible nodalcurve. Specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 § Let X be a smooth projective curve over the field of complex numbers C , E a locallyfree sheaf on X and p ∈ X a closed point. A parabolic structure on E at p is just a flagof vector spaces (0) ⊂ E ⊂ · · · ⊂ E s ⊂ E p / m p E p together with weights 0 ≤ κ < κ < · · · < κ s < X (see [12, Theorem 4.1]).The concept of parabolic locally free sheaf can be generalized by considering weightedflags supported on divisors of the smooth projective curve X . These objects are calledgeneralized parabolic locally free sheaves and they where introduced by U. Bohsle in[2]. The importance of generalized parabolic locally free sheaves is not only the possiblelink to the space of representations of the topological fundamental groups but also thelink to the geometry of the moduli spaces of torsion free sheaves on nodal curves. Tobe more precise, U. Bohsle proved that if π : Y → X is the normalization map of areducible projective nodal curve then there exists a coarse projective moduli space forgeneralized parabolic locally free sheaves (the parabolic structure being supported on q + q = π − ( p )) on Y together with a morphism to the moduli space of torsion freesheaves on X of rank r and degree d making the former moduli space a desingularizationof the later provided ( r, d ) = 1 (see [3]).Likewise, generalized parabolic structures have been applied for studying the geome-try of the moduli space of Hitchin pairs over a reducible curve. In [5], U. Bhosle constructsa morphism between the moduli space of Hitchin pairs with generalized parabolic struc-ture over the normalization Y and the moduli space of Hitchin pairs over the reducedcurve X , showing that under certain condition this is a birrational morphism whoseimage contains all stable Higgs bundles.These ideas have also been applied to the more general problem of studying the com-pactification of the moduli space of principal G -bundles over an irreducible nodal curve.In [15], A. Schmitt realized that, once a faithful representation ρ : G ֒ → SL( V ) is fixed,every principal G -bundle can be seen as a pair ( E , τ ) formed by a locally free sheaf E and a non-trivial morphism of algebras τ : S • ( V ⊗ E ) G → O X . These objects are calledsingular principal G -bundles and they carry a semistability condition, which depends (apriori) on a positive rational parameter δ ∈ Q > . Then, the main result is that there ex-ists a coarse projective moduli space for δ -(semi)stable singular principal G -bundles andit coincides with the classical moduli space provided δ is large enough. This motivatedthe works [4, 17, 18], where U. Bohsle generalized the definition of singular principal G -bundles, as well as the δ -(semi)stability condition, over an irreducible nodal curve ina natural way and proved the existence of a projective moduli space for them, while A.Schmitt studied the asymptotic behavior of the δ -(semi)stability condition obtaining asimilar result as that of the smooth case. The study of the asymptotic behavior of the2 -(semi)stability condition becomes harder when the curve has singularities, and it wascarried out in [17, 18] by considering singular principal G -bundles on X as singular prin-cipal G -bundles with generalized parabolic structures on the normalization Y . Therefore,the moduli spaces of singular principal G -bundles with generalized parabolic structuresover a smooth projective curve play an important role in this problem.On the other hand, singular principal G -bundles with generalized parabolic structureshave been applied to the construction of a compactification of the moduli space of prin-cipal Higgs G -bundles over an irreducible nodal curve (see [7] for instance). In this case,A. Lo Giudice and A. Pustetto enlarge the category of principal Higgs G -bundles on thenodal curve to the category of singular principal G -bundles together with a Higgs field,which can be seen as singular principal G -bundles with generalized parabolic structuretogether with a Higgs field on the normalization of the nodal curve. Again, the modulispace of the last objects plays an important rol in the study of the moduli space of thefirst objects. Goal of the paper
Let X be a projective nodal curve with nodes x , . . . , x ν and l irreducible components,and π : Y = ` li =1 Y i → X its normalization. We fix an ample invertible sheaf O X (1) on X and we denote by O Y (1) the ample invertible sheaf obtained by pulling O X (1) backto Y . We denote by h the degree of O Y (1), by y i , y i the points in the preimage of the i th nodal point x i , by D i = y i + y i the corresponding divisor on Y and by D = P D i the total divisor. Let G be a semisimple linear algebraic group, ρ : G ֒ → SL( V ) afaithful representation of dimension r ∈ N , δ ∈ Q > and d ∈ Z . Let SPB( ρ ) δ − ( s ) sr,d bethe moduli space of δ -(semi)stable singular principal G -bundles of rank r and degree d over X (see [13]). Consider the set J ( r ) = { ( e , . . . , e ν ) ∈ N ν | ≤ e i ≤ r } . Then, thereis a stratification, SPB( ρ ) δ − ( s ) sr,d := S e ∈ J ( r ) SPB( ρ ) δ -(s)s r,d,e , where SPB( ρ ) δ -(s)s r,d,e parametrizessingular principal bundles, ( F , τ ), with F x i ≃ O e i X,x i ⊕ m r − e i x i . The goal of this paperis to construct a coarse projective moduli space, D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e , for ( κ, δ )-(semi)stabledescending singular principal G -bundles with generalized parabolic structures over Y = ` li =1 Y i of given type e supported on the divisors D i (see Theorem 5.6) together with amorphism (see Equation 26)Θ : D( ρ ) ( κ,δ )-(s)s r,d := a e ∈ J ( r ) D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e −→ SPB( ρ ) δ -(s)s r,d . We show that he restriction to each component Θ e : D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e −→ S e ′ ≤ e SPB( ρ ) δ -(s)s r,d,e ′ induces an isomorphism between a (functorialy well defined) dense open subscheme ofthe stable locus W e ⊂ D( ρ ) ( κ,δ )-s r,d ( e,r ) ,e and SPB( ρ ) δ -s r,d,e (see Theorem 5.11). Therefore, Θ e induces a birational surjective and proper morphism D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e ։ SPB( ρ ) δ -s r,d,e whenthe stable locus is dense inside D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e . Outline of the paper
In Section 2 we introduce the basic definitions of generalized parabolic swamps and gen-eralized parabolic singular principal G -bundles of given type, as well as the semistabilityconditions. In Section 3 we prove the existence of a coarse projective moduli space forgeneralized parabolic ( κ, δ )-(semi)stable swamps of given type. The main difficulty hereis to find the linearized projective embedding that makes the semistability condition tocoincide with the Hilbert-Mumford semistability. In Section 4 we prove the existenceof a coarse projective moduli space for ( κ, δ )-(semi)stable singular principal G -bundles.By [13, Theorem 5.5], this is a direct consequence of the results proved in Section 3. InSection 5, we construct the coarse moduli space for descending singular principal bundlesover the normalization, as well as the morphism Θ that relates it with the closure of thestable locus of the moduli space of singular principal bundles over the nodal curve.3 Let Y = ` li =1 Y i be a disjoint union of smooth projective and irreducible curves, j i : Y i ֒ → Y the natural embedding of the i th component, O Y (1) an ample invertible sheafand O Y (1) = j ∗ i O Y (1) the restriction of O Y (1) to the component Y i . Set h := deg( O Y )and h i := deg( O Y i ). Given a coherent sheaf on Y , we know that E = L li =1 j i ∗ ( E | i ), where E i := E | Y i . The multirank of E is defined as the tuple ( r , . . . , r l ) (where r i = rk( E i ))while the multidegree is defined as ( d , . . . , d l ) (where d i = deg( E i )). If r ∈ N andrk( E i ) = r for all i (we will say the rank is equal to r ), then P E ( n ) = αn + rχ ( Y ) + d ,where α = hr and d = P li =1 d i . Generalized parabolic structuresDefinition 2.1.
Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . A generalizedparabolic locally free sheaf of rank r , degree d and type e over Y is a tuple ( E , q , . . . , q ν )where E is a locally free sheaf of rank r and degree d , and q i is a quotient of dimension e i , E ( y i ) ⊕ E ( y i ) ։ R i , E ( y ij ) being the fibre of E over y ij .In order to abreviate the notation we will use the symbol q to refer to the tuple( q , . . . , q ν ). Denote by R := ⊕ R i the total vector space. Since the supports of thedivisors D i are disjoint we have Γ( D, E | D ) = L Γ( D i , E | D i ) = L ( E ( y i ) ⊕ E ( y i )). Fromthis, we can form the quotient q := ⊕ q i : Γ( D, E | D ) → R → Definition 2.2.
Let ( E , q ) and ( E ′ , q ′ ) be generalized parabolic locally free sheaves on Y . A homomorphism between them is a tuple ( f, u , . . . , u ν ) where f : E → E ′ is ahomomorphism of O Y -modules and u i : R i → R ′ i is a homomorphism of vector spacessuch that q ′ i ◦ ( f ( y i ) ⊕ f ( y i )) = u i ◦ q i , where f ( y ) denotes de induced linear map betweenthe fibers at y ∈ Y . Notation.
Given a tuple of natural numbers ( e , . . . , e ν ) ∈ N ν , we will denote by I ( e )the set { i ∈ { , . . . , ν } such that e i = 0 } of multitindices of non zero components. Definition 2.3.
Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . For each i ∈ I ( e ), fix κ i ∈ (0 , e i r ) ∩ Q . Let ( E , q ) be a generalized parabolic locally free sheaf ofrank r , degree d and type e . We define the κ -parabolic degree for any subsheaf F ⊆ E as κ -pardeg( F ) := deg( F ) − X i ∈ I ( e ) κ i re i dim q i ( F ( y i ) ⊕ F ( y i )) Remark 2.4.
Formally, we can take as κ i any rational number. Taking κ i = e i r werecover the definition given in [17]. On the other hand, tanking e i = r we recover thedefinition given in [18]. Thus, both are particular cases of the one considered in thiswork. Swamps with generalized parabolic structures
Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . Fix non negative integers a, b, c and an invertible sheaf L on Y . Definition 2.5.
A swamp with generalized parabolic structure of type ( a, b, c, L , e ) rank r and degree d is a triple ( E , q, φ ) where ( E , q ) is a generalized parabolic locally free sheafof rank r , degree d and type e , and φ : ( E ⊗ a ) ⊕ b → det( E ) ⊗ c ⊗ L is a non-zero morphism. Notation.
In order to be shorter, we will denote the tuple ( a, b, c, L , e ) that defines thetype of a generalized parabolic swamp by the symbol tp .4et φ : ( E ⊗ a ) ⊕ b → det( E ) ⊗ c ⊗ L be a swamp on Y and let ( E • , m ) be a weightedfiltration. For each E i denote by α i its multiplicity and by α the multiplicity of E . Definethe vector Γ := P t m i Γ ( α i ) , where Γ ( l ) = ( l − α, × l . . ., l − α, l, × α − l . . . , l ). Let us denote by J the set { multi-indices I = ( i , . . . , i a ) | I j ∈ { , . . . , t + 1 }} . Define µ ( E • , m, φ ) := − min I ∈ J { Γ α i + . . . + Γ α ia | φ | ( E i ⊗ ... ⊗ E ia ) ⊕ b = 0 } ,P κ ( E • , m ) := s X i =1 m i ( κ -pardeg( E ) α i − κ -pardeg( E i ) α ) . Definition 2.6.
Let δ ∈ Q > . For each i ∈ I ( e ), fix κ i ∈ (0 , e i r ) ∩ Q . A general-ized parabolic swamp ( E , q, φ ) of rank r degree d and type tp = ( a, b, c, L , e ) is ( κ, δ )-(semi)stable if for every weighted filtration ( E • , m ) of E , the inequality P κ ( E • , m ) + δµ ( E • , m, φ )( ≥ )0 holds true. Remark 2.7.
Observe that there is a positive integer A , depending only on the numericalinput data, r, a, b, c and L , such that it is enough to check the δ -semistability conditionfor weighted filtrations with m i < A . This follows from [8, Lemma 1.4] changing ranksby multiplicities.Let S be a scheme. Set S D i := S × D i ⊂ S × Y and let π S i : S × D i → S be theprojection onto the first factor. A family of generalized parabolic locally free sheavesparametrized by S is a tuple ( E S , q S ) where E S is a family of locally free sheaves on Y parametrized by S of rank r and degree d , and q S = ( q S , . . . , q Sν ), q Si : π Si ∗ ( E S | S Di ) → R i → e i on S . A family of generalizedparabolic swamps is a quadruple ( E S , q S , N S , φ S ) where ( E S , q S ) is a family of generalizedparabolic locally free sheaves of rank r and degree d , N S is an invertible sheaf on S , and φ S : ( E ⊗ aS ) ⊕ b → det( E S ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ S N S is a morphism of locally free sheaves on S × Y such that φ S | { s }× Y is non-zero for all s ∈ S . Finally, ( κ, δ )-(semi)stable familiesare families which are ( κ, δ )-(semi)stable fiberwise. Then, one can introduce the moduliproblem defined by the functor SGPS ( κ,δ ) − ( s ) sr,d, tp ( S ) = isomorphism classes of families of( κ, δ )-(semi)stable generalized parabolicswamps ( E S , q S , N S , φ S ) parametrizedby S with rank r, degree d and type tp . Singular principal G -bundles with generalized parabolic structures Let G be a semisimple linear algebraic group and let ρ : G ֒ → SL( V ) be a faithfulrepresentation. Definition 2.8.
A singular principal G -bundle over Y is a pair ( E , τ ) where E is alocally free sheaf and τ : S • ( V ⊗ E ) G → O Y is a non-trivial morphism of O Y -algebras. Definition 2.9.
Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . A singularprincipal G -bundle with a generalized parabolic structure over Y of rank r , degree d andtype e is a triple ( E , τ, q ) where ( E , q ) is a generalized parabolic locally free sheaf of rank r , degree d and type e , and ( E , τ ) is a singular principal G -bundle. Definition 2.10.
Let ( E , τ, q ) and ( G , λ, p ) be singular principal G -bundles with general-ized parabolic structure on Y . A morphism between them is a morphism of O Y -modules f : F → G compatible with both structures. The isomorphisms are the obvious ones.Following [13, Theorem 5.5], we can assign to any singular principal G -bundle aswamp of type ( a, b, , O Y ) for certain naural numbers a, b that depends only on the5umerical input data, isomorphism classesof singular principal G -bundles → isomorphism classesof swampsof type ( a, b, , O Y ) , ( E , τ, q ) ( V ⊗ E , ϕ τ , q ) (1)this map being injective. Thus, we can define, for any weighted filtration ( E • , m ), thesemistability function µ ( E • , m, τ ) as µ ( E • , m, ϕ τ ) (see [13, Definition 6.1]). Definition 2.11.
Let r ∈ N , d ∈ Z , e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r , and δ ∈ Q > . Foreach i ∈ I ( e ), fix κ i ∈ (0 , e i r ) ∩ Q . A generalized parabolic singular principal G -bundle ofrank r degree d and type e , ( E , q, τ ), is ( κ, δ )-(semi)stable if for every weigted filtration( E • , m ) of E , the inequality P κ ( E • , m ) + δµ ( E • , m, τ )( ≥ )0 holds true.Then, one can define a family as in the case of swamps and introduce the moduliproblem defined by the functor SPBGPS ( ρ ) ( κ,δ )-(s)s r,d,e ( S ) = isomorphism classes of familiesof ( κ, δ )-(semi)stable singularprincipal G -bundles withgeneralized parabolic structureon Y parametrized by S withrank r degree d and type e . (2) Some calculations in geometric invariant theory
Recall that a basis u := ( u , . . . , u p ) of the vector space U , together with a vector γ = ( γ , . . . , γ p ) ∈ N p such that γ ≤ . . . ≤ γ p and P pi =1 γ i = 0, defines a one parametersubgroup λ ( u, γ ) : C → SL( U ). Conversely, every one parameter subgroup of SL( U )arises in this way (see [19, Example 1.5.1.12]). Furthermore, every one parameter sub-group of SL( U ) determines a weighted flag ( U • , m ) of U and every weighted flag arises inthis way as well. It turns out that the Hilbert-Mumford function, µ ( − , λ ) depends onlyon the associated weighted flag of λ and not on λ itself (see [19, Proposition 1.5.1.35,Example 1.5.1.36]).We derive the explicit expression of the Hilbert-Mumford criterion (see [14, Theorem2.1, Proposition 2.3]) in some situations that will be important for our proposes. Similarcalculations can be found along [19], so we will skip some details. Let p, r be integers such that 1 ≤ e ≤ p −
1. Let G r := Grass e ( U ⊕ )be the Grassmannian of e -dimensional quotients of U ⊕ , U being a p -dimensional vectorspace, and let N be positive integer. The Grassmannian can be embedded into theprojective space through the Pl¨ucker embedding ι : G r ֒ → P ( ∧ e U ⊕ ). The group SL( U )acts on both spaces through the diagonal δ : SL( U ) ֒ → SL( U ⊕ ) in the obvious way,and ι is SL( U )-equivariant. If O (1) is the tautological invertible sheaf on P ( ∧ e U ⊕ ),then L := ι ∗ O (1) is a SL( U )-linearized very ample invertible sheaf. Let us compute thesemistability function of points in G r with respect to L .Let { u , . . . , u p } be a basis of the vector space U . Then, a basis of ∧ e U ⊕ is given bythe vectors u I,J := ( u i , ∧ . . . ∧ ( u i l , ∧ (0 , u j ) ∧ . . . ∧ (0 , u j e − l ) . Let λ : G m → SL( U )be a one parameter subgroup. Fix a basis u = { u , . . . , u p } and integers γ ≤ . . . ≤ γ p such that λ = λ ( u, γ ). Then, we have µ L ( τ, λ ( u, γ )) = s X i =1 i dim(Ker( τ )) − p dim(Ker( τ ) ∩ ( U i ⊕ U i )) = s X i =1 p dim τ ( U i ⊕ U i ) − ie, where ( U • , m ) is the weighted filtration associated to λ .6 .4.2.—Example 2 Let Y , . . . , Y l be smooth projective connected curves, and considertheir disjoint union, Y := F Y i . Let N , . . . , N l be invertible sheaves on Y , . . . , Y l respectively and denote by N := L N i the corresponding invertible sheaf on Y . Let r, n ∈ N and let U be a vector space of dimension p > r . Consider now, for each i , the projective space given by G i , N := P (Hom( V r U, H ( Y i , N i ( rn ))) ∨ ) , and define G , N = G , N × . . . × G l , N . Let b , . . . , b l ∈ N and consider the very ample invertiblesheaf on G given by L := π ∗ O G , N ( b ) ⊗ . . . ⊗ π ∗ l O G l , N ( b l ) with the obvious SL( U )-linearization. For the sake of clarity, we will use the symbol L i to denote the invertiblesheaf O G i , N (1). Clearly µ L ([ g ] , λ ) = P li =1 b i µ π ∗ i L i ([ g ] , λ ) = P li =1 b i µ L i ([ g i ] , λ ) , [ g i ]being the i -th component of [ g ]. Therefore the calculation of the semistability functionof points of G , N with respect to L is reduced to the calculation of the semistabilityfunction of points of G i , N with respect to L i . Let E be a locally free quotient sheaf ofrank r q : U ⊗ O Y ( − n ) → E → N . Restrictingto the i -th component, twisting by n , taking the r -th exterior power and taking globalsections we find the morphism H ( ∧ r ( q i ( n ))) : ∧ r U → H ( Y, N i ( rn )) , whose equivalenceclass defines a point [ H ( ∧ r ( q i ( n )))] ∈ G i , N . Now, a short calculation shows that µ L j ([ H ( ∧ r ( q j ( n )))] , λ ) = s X i =1 m i (rk( E i | Y j ) p − r dim( U i )) , ( U i , m i ) being the i th term of the weighted filtration associate to λ and E i | Y j the restric-tion to Y j of the saturated subsheaf generated by U i . Consider the same situation as in Example 2. Let L be an invertiblesheaf on Y , U a p -dimensional vector space and a, b, c, n ∈ N . Given an invertible sheaf N on Y we define the projective space G , N = P (Hom( U a,b , H ( Y, N ⊗ c ⊗ L ( na ))) ∨ ) , where U a,b := ( U ⊗ a ) ⊕ b . Consider the pair ( q, φ ) given by a locally free quotient sheaf ofrank r , q : U ⊗ O Y ( − n ) → E , whose determinant is isomorphic to N and a morphism φ : ( E ⊗ a ) ⊕ b → N ⊗ c ⊗ L . Let ∆ : U a,b ֒ → U ⊕ la,b be the diagonal linear map, and con-sider the morphism H (( q ( n ) ⊗ a ) ⊕ b ) ◦ ∆ : U a,b → H ( Y, ( E ⊗ a ) ⊕ b ⊗ O Y ( na )) . Twisting φ by O Y ( na ), we get H ( φ ( na )) : H ( Y, ( E ⊗ a ) ⊕ b ⊗ O Y ( na )) → H ( Y, N ⊗ c ⊗ L ( na )) . Composing both morphisms we get a point in G , N ,[ H ( φ ( na )) ◦ H (( q ( n ) ⊗ a ) ⊕ b ) ◦ ∆] : U a,b → H ( Y, N ⊗ c ⊗ L ( na ))] ∈ G , N . (3)Set p = dim( U ) and let u = ( u , . . . , u p ) be a basis of U . For any multiindex I =( i , . . . , i a ) with i j ∈ { , . . . , p } define u I := u i ⊗ . . . ⊗ u i a and u kI := (0 , . . . , , k ) u I , , . . . , . Then the elements u kI form a basis of U a,b and the group SL( U ) acts on G , N in theobvious way. We want to compute the semistability function for points T ∈ G , N ofthe form (3) with respect to the natural SL( U )-linearization of O G , N (1). Let λ : G m → SL( U ) be a one parameter subgroup. Then there exists a basis u , . . . , u p of U andintegers γ ≤ . . . ≤ γ p with P γ i = 0 such that λ ( z ) u i = z γ i u i , ∀ z ∈ G m . For anymultiindex I = ( i , . . . , i a ) consider u I and define γ I = γ i + · · · + γ i a . Then λ : G m → SL( U ) acts by λ ( z ) • u kI = z γ I • u kI , ∀ z ∈ G m , and we have µ ([ T ] , λ ) = − min { γ I | T ( u kI ) =0 } . Given a multiindex I = ( i , . . . , i a ) we want to compute γ I = γ i + · · · + γ i a for γ = ( i − p, . . . , i − p, i, . . . , i ). Set ν ( I, i ) := { j | i j ≤ i } . Then i , . . . , i ν ( I,i ) ≤ i and i ν ( I,i )+1 , . . . , i a > i , so γ I = ( i − p ) ν ( I, i )+ i ( a − ν ( I, i )) = ia − ν ( I, i ) p . A short calculationshows µ ([ T ] , λ ) = s X i =1 m i ( ν ( I, dim U i ) p − dim U i a ) , ( U • , m ) being the weighted flag associated to λ and I = ( i , . . . , i a ) is the multiindexgiving the minimum of the semistability function.7 Let r ∈ N , d ∈ Z , e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r and δ ∈ Q > . For each i ∈ I ( e ) fix κ i ∈ (0 , e i r ) ∩ Q . Fix non negative integers a, b, c and an invertible sheaf L on Y . Recallthat h := deg( O Y (1)) and h i := deg( O Y (1) | Yi ). It will be assumed that these data arefixed once and for all along this section.The main result of this section is Theorem 3.9 which shows the existence of a coarseprojective moduli space for ( κ, δ )-(semi)stable swamps with generalized parabolic struc-ture of given type tp = ( a, b, c, L , e ) and with rank and degree equal to r and d respec-tively. In order to do so, we have to consider the rigidified functor rig SGPS nr,d, tp ( S ) = isomorphism classes of tuples ( E S , q S , φ S , g S )where ( E S , φ S ) is a family of swampsparametrized by S with rank r and degree d ( E S , q S ) is a family of generalized paraboliclocally free sheaves and g S : U ⊗ O S → π S ∗ E S ( n )is a morphism such that the induced morphism U ⊗ O Y × S ( − n ) → E S is surjective , (4)where n ∈ N , U := C P ( n ) , P ( n ) = αn + rχ ( O Y ) + d and α = hr . Boundedness for generalized parabolic swamps
Let us denote by E d,r the family of locally free sheaves on Y of rank r and degree d . Recallthat a family of sheaves E ⊂ E d,r on Y is bounded if and only if there is a natural number n ∈ N such that for all n ≥ n and all locally free sheaves E ∈ E , h ( Y, E ( n )) = 0 and E ( n ) is globally generated. Boundedness for locally free sheaves appearing in ( κ, δ )-(semi)stable swamps with generalized parabolic structures (Proposition 3.2) will followfrom the next observation.Let E be a locally free sheaf over Y and let ( E • , m ) be a weighted filtration, with E • ≡ (0) ⊂ E ⊂ . . . ⊂ E s ⊂ E . Consider a partition of the multitindex I := (1 , , . . . , s ), I = I ⊔ I , let us say I = ( i , . . . , i t ) and I = ( k , . . . , k s − t ). Then, a simple calculation(see [8, Lemma 1.6] for the connected case) shows that( s X i =1 m i ) a ( α − ≥ µ ( E • , m, φ ) ≥ − ( s X i =1 m i ) a ( α − ,µ ( E • , m, φ ) ≥ µ ( E • , m , φ ) − ( s − t X i =1 m ,i ) a ( α − , (5)where E j = E i j . The following results are important direct consequences of Equation(5). Proposition 3.1.
A generalized parabolic swamp ( E , q, φ ) is ( κ, δ ) -(semi)stable if andonly if for any weighted filtration ( E • , m ) , such that par µ ( E i ) ≥ par µ ( E ) − C , where C = aδ + rν , the inequality P κ ( E • , m ) + δµ ( E • , m, φ )( ≥ )0 holds true.Proof. Let ( E • , m ) be a weighted filtration such that par µ ( E i ) < par µ ( E ) − C for all i . Since κ -pardeg( E i ) α − κ -pardeg( E ) α i < − C αα i , Equation (5) implies P κ ( E • , m ) + δµ ( E • m, φ ) ≥ Proposition 3.2.
The family of locally free sheaves of degree d and rank r appearing in ( κ, δ ) -(semi)stable swamps with generalized parabolic structure is bounded.Proof. By Equation (5) and a simple calculation, it follows that µ ( E ′ ) ≤ µ ( E ) + aδ + rν .Then, we conclude by [10, Lemma 2.5]. 8 emark 3.3. Let C ′ = αC . Note that if deg( E ) ≤ E ′ ) ≤ µ ( E ) + C ′ , and ifdeg( E ) > E ′ ) ≤ deg( E )+ C ′ . In both cases the degree of any subsheaf E ′ ⊂ E is bounded by a constant depending only on a, δ, r, h, ν, d . This in particular means thatfor any locally free sheaf E of rank r and degree d appearing in a ( κ, δ )-semistableswamp with generalized parabolic structure of type ( a, − , − , − , − ) (this means that thefirst component is fixed and equal to a but the others are left to be free) we have thatdeg( E | Y i ) is bounded from below and above by constants depending only on a, δ, α, ν, d which we will denote by A − ( a, δ, r, h, ν, d ) and A + ( a, δ, r, h, ν, d ), or just by A − and A + if there is no confusion. The Gieseker space and map
Our goal now is to construct the Gieseker space together with the Gieseker map, and toconstruct a representative for the moduli functor given in Equation (4). We will assumethat e i = 0 for each i = 1 , . . . ν . If e i = 0 for some index i , we will only have to drop thecorresponding Grassmannian in Equation (6) and Equation (8) below. Let H be an effective divisor of degree h in Y such that O Y ( H ) ≃ O Y (1). By Proposition 3.2 we know that there exists a natural number n ∈ N such that for every n ≥ n and every ( κ, δ )-(semi)stable generalized parabolicswamp of type tp = ( a, b, c, L , e ) of rank r and degree d we have H ( Y, E ( n )) = H ( Y, det( E ( rn ))) = H ( Y, det( E ) ⊗ c ⊗ L ⊗ O Y ( an )) = 0 and the locally free sheaves E ( n ) , det( E ( rn )) , det( E ) ⊗ c ⊗ L ⊗ O Y ( an ) are globally generated. Fix n ≥ n as above,and d = ( d , . . . , d l ) ∈ N l with d = P li =1 d i , and set p = rχ ( O Y ) + d + αn (recall α = hr ). Let U be the vector space C ⊕ p . We will use the notation U a,b for ( U ⊗ a ) ⊕ b .Denote by Q the quasi-projective scheme parametrizing equivalence classes of quotients q : U ⊗ π ∗ Y O Y ( − n ) → E where E is a locally free sheaf of uniform multirank r and mul-tidegree d = ( d , . . . , d l ) on Y and such that the induced map U → H ( Y, E ( n )) is anisomorphism. On Q × Y , we have the universal quotient q Q : U ⊗ π ∗ Y O Y ( − n ) → E Q . Since n > n , the sheaf H := H om O Q ( U a,b ⊗ O Q , π Q ∗ (det( E Q ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ Y O Y ( na )))is locally free. Consider the corresponding projective bundle π ′ : h = P ( H ∨ ) → Q andlet q h : U ⊗ π ∗ Y O Y ( − n ) → E h be the pullback of the universal locally free sheaf to h × Y .Now, the tautological invertible quotient on h , π ′ ∗ H ∨ → O h (1) →
0, induces a morphismon h × Y , s h : U a,b ⊗ O h → det( E h ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ Y O Y ( na ) ⊗ π ∗ h O h (1). From the uni-versal quotient we get a surjective morphism ( q ⊗ a h ) ⊕ b : U a,b ⊗ π ∗ Y O Y ( − na ) → ( E ⊗ a h ) ⊕ b . Denoting by K its kernel, we get a diagram0 / / K / / ( ( ◗◗◗◗◗◗◗◗ U a,b ⊗ π ∗ Y O Y ( − na ) / / s h ⊗ π ∗ Y id O Y ( − na ) (cid:15) (cid:15) ( E ⊗ a h ) ⊕ b / / E h ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ h O h (1)From [8, Lemma 3.1], it follows that there is a closed subscheme G ⊂ h over which s h ⊗ π ∗ Y id O Y ( − na ) factorizes through a morphism φ G : ( E ⊗ a G ) ⊕ b → det( E G ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ G N G ,π ∗ G N G being the pullback of the restriction of O h (1) to G . Then, on the scheme G × Y we have a family of swamps ( E G , N G , φ G ) parametrized by G . In order to include theparabolic structure, we need to consider the Grassmannian G r i := Grass e i ( U ⊕ ) of e i dimensional quotients of U ⊕ . Recall that ν is the number of nodes of the curve, so thatwe have ν divisors, D i = y i + y i , in the normalization Y . Define, Z := G × G r × · · · × G r ν (6)and denote by c i : Z → G r i the i th projection. Consider the pullback of the universalquotient of the Grassmannian G r i by the projection c i , q iZ : U ⊕ ⊗ O Z → R Z , and takethe direct sum q Z : U ⊕ ν ⊗ O Z → L ν R Z . Consider now the two natural projections9 × Y → G , Z × Y → Z. Denote by N Z the pullback of N G to Z , and by q Z , E Z and φ Z the pullbacks of the corresponding objects over G × Y to Z × Y . Considerthe morphisms π i : Z × { y i , y i } → Z × { x i } ≃ Z . For each i , there are quotients f i : U ⊕ × O Z → π i ∗ ( E Z | y i ,y i ) and we can form f : = L f i : U ⊕ ν × O Z → L π i ∗ ( E Z | y i ,y i ).Consider the following diagram,0 / / Ker( f ) / / q ′ % % U ⊕ ν × O Z f / / q Z (cid:15) (cid:15) L π i ∗ ( E Z | y i ,y i ) / / L ν R Z Denote by I d ⊂ Z the closed subscheme given by the zero locus of the morphism q ′ (see[8, lemma 3.1] again). Then the restriction of q Z to I d factorizes through q I d : ν M π i ∗ ( E Z | y i ,y i ) | I d = ν M π i I d ∗ ( E I d | y i ,y i ) → ν M R Z | I d = ν M R I d . Since f and q Z are diagonal morphisms we deduce that q I d is also diagonal. Therefore q I d is determined by ν morphisms q i I d : π i I d ∗ ( E I d | y i ,y i ) → R I d . Denote by ( E I d , N I d , φ I d )the restriction of ( E Z , N Z , φ Z ) to I d . Then we have a universal family of generalizedparabolic swamps, ( E I d , q I d , N I d , φ I d ), with rank r , multidegree ( d , . . . , d l ) and type tp = ( a, b, c, L , e ). Let us denote I ( r, d, κ, δ, tp ) = ( d , . . . , d l ) ∈ N l satisfying the condition d + . . . + d l = d andsuch that there exists a ( κ, δ )-semistable swampwith generalized parabolic structure of rank r multidegree ( d , . . . , d l ) and type tp (7)From Remark 3.3 it follows that for every multiindex ( d , . . . , d l ) ∈ I ( r, d, κ, δ, tp ) wehave A − ≤ d i ≤ A + , i = 1 , . . . , l . Thus, I ( r, d, κ, δ, tp ) is a finite set. Then we define I r,d, tp := a d ∈ I ( r,d,δ ) I d . We will show that there is a natural closed embed-ding of the parameter space I d into certain projective scheme which is SL( U )-equivariant.Fix a Poincare invertible sheaf P i on Y i × Pic d i ( Y i ) and let n ∈ Z . Define the sheaf G i = H om O Pic di ( Yi ) ( V r U ⊗ O Pic di ( Y i ) , π Pic di ( Y i ) ∗ ( P i ⊗ π ∗ Y i O Y i ( rn ))) . The natural numberwe have fixed satisfies n > n , therefore the above sheaf is locally free, and we canconsider the corresponding projective bundle on Pic d i ( Y i ), G i = P ( G i ∨ ). Note that thedeterminant map E I d V E I d | Y i = V ( E I d | Y i ) defines a morphism d i : I d → Pic d i ( Y i ).Consider now on I d × Y the universal quotient q I d : U ⊗ π ∗ Y O Y ( − n ) → E I d . Restrictingto the i th component, twisting by n and taking determinants we get V q i I d ( n ) : V r U ⊗ O I d × Y i → V r E I d | Y i ⊗ π ∗ Y i O Y i ( nr ). Let N i be an invertible locally free sheaf on I d suchthat V r E I d | Y i = ( d i × id Y i ) ∗ P i ⊗ π ∗ I d N i . Then, we have a point π I d ∗ ( V q i I d ( n )) ∈ G i • ( I d )for each i .Define now G = H om O Pic d ( Y ) ( U a,b ⊗ O Pic d ( Y ) , π Pic d ( Y ) ∗ ( P ⊗ c ⊗ π ∗ Y L ⊗ π ∗ Y O Y ( na ))) . For n > n , G is also locally free and we can consider the corresponding projective bundleon Pic d ( Y ), G = P ( G ∨ ). Consider now the universal quotient q I d : U ⊗ O I d × Y ( − n ) → E I d and the universal swamp φ I d : ( E ⊗ a I d ) ⊕ b → det( E I d ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ I d N I d . Let N be an invertible sheaf on I d such that det( E I d ) = ( d × id) ∗ P ⊗ π ∗ I d N and note U a,b ⊗ O I d × Y ≃ π ∗ I d ( U a,b ⊗ O I d ). Composing ( q I d ( n ) ⊗ a ) ⊕ b with the swamp φ I d , taking π I d ∗ ψ : U a,b ⊗ O I d → π I ∗ d π ∗ I d ( U a,b ⊗ O I d ) weget a point ψ ◦ ( π I d ∗ ( φ I d ◦ ( q I d ( n ) ⊗ a ) ⊕ b )) ∈ G • ( I d ).Altogether, with the obvious morphism to the Grasmannians, I d → G r × . . . × G r ν ,give us the so called Gieseker morphismGies : I d / / ( G × . . . × G l ) × Pic d ( Y ) G × ( G r × . . . × G r ν ) =: G . (8) Proposition 3.4.
The Gieseker morphism
Gies : I d → G is injective and SL( U ) -equivariant.Proof. Follows as in the connected case (see for instance [6, Lemma 4.3]).
Semistability
We will see that making n > n even larger, I d contains all ( κ, δ )-(semi)stable generalizedparabolic swamps of fixed type and fixed Hilbert polynomial. In order to show that thequotient I ( κ,δ )-(s)s d // SL( U ) exists and is projective we first find a linearized invertiblesheaf on G for which Gies − ( G (s)s ) = I ( κ,δ )-(s)s d and then we show that Gies | I ( κ,δ )-(s)s d is aproper morphism. The main auxiliary result is given in Subsection 3.3.2 (see Theorem3.5) regarding the sectional semistability condition. Let i , . . . , i ν ′ be the indices in I ( e ). Let b , . . . , b l , c, k i , . . . , k i ν ′ be positive integers and consider the ample invertible sheaf on G , O G ( b , . . . , b l , c, k i , . . . , k i ν ′ ). Consider the obvious linearization on it and let G ( s ) s bethe set of points which are (semi)stable with respect to the given linearization. Considera weighted flag ( U • , m ), where U • : (0) ⊂ U ⊂ . . . ⊂ U s ⊂ U , and m = ( m , . . . m s ).Let λ : G m → SL( U ) be a one parameter subgroup whose weighted flag is ( U • , m ). Let t be a rational point of I d and Gies( t ) = ( t , , . . . , t ,l , t , t , , . . . , t ,ν ) its image in G .Let q t : U ⊗ O Y ( − n ) → E be the locally free quotient sheaf corresponding to t . Theweighted filtration ( U • , m ) induces a filtration of E defined by E u := q ( U u ⊗ O Y ( − n )) ⊂ E .Assume that h ( Y, E u ( n )) = 0 and l u := dim( U u ) = h ( Y, E u ( n )). Then, the semistabilityfunction is given by (see Section 2.4) µ ( λ, Gies( t )) = l X i =1 b i µ G ( λ, t ,i ) + cµ G ( λ, t ) + ν X i =1 k i µ G r ( λ, t ,i ) == l X i =1 b i s X u =1 m u (rk( E iu ) p − rh ( Y, E u ( n )))++ c s X u =1 m u ( ν ( I , l u ) p − ah ( Y, E u ( n )))++ X i ∈ I ( e ) k i s X u =1 m u ( p dim( q i ( E u ( y i ) ⊕ E u ( y i ))) − e i h ( Y, E u ( n ))) . (9)We fix now a concrete polarization, defined as follows (recall h i = deg( O Y | Y i )), b i := bh i , b := p − b ′ , b ′ := b ′ + b ′ , b ′ := aδ , b ′ := r P j ∈ I ( e ) κ j ,c := δrh = P li =1 δrh i k i := re i κ i α. (10)11hen, Equation (9) becomes, µ ( λ, Gies( t )) = s X u =1 m u ( bα u p − h ( Y, E u ( n )) αp ++ cν ( I , l u ) p + X i ∈ I ( e ) ακ i re i p dim( q i ( E u ( y i ) ⊕ E u ( y i )) . Again, since b = p − b ′ − b ′ , b ′ = aδ and α u = P li =1 h i rk( E iu ), we get µ ( λ, Gies( t )) p = s X u =1 m u ( pα u − αh ( Y, E u ( n )) + δ l X i =1 h i ( rν ( I , l u ) − a rk( E iu ))++ X i ∈ I ( e ) ακ i re i dim( q i ( E u ( y i ) ⊕ E u ( y i )) − b ′ α u . Since the first cohomology groups are assumed to be 0, we find pα u − αh ( E u ( n )) = α u P E ( n ) − αP E u ( n ) = α u deg( E ) − α deg( E u ) . We also know that κ -pardeg( E u ) = deg( E u ) − P i ∈ I ( e ) κ i re i dim( q i ( E u ( y i ) ⊕ E u ( y i )) and κ -pardeg( E ) = deg( E ) − r ( P i ∈ I ( e ) κ i ). Then, we finally get µ ( λ, Gies( t )) p = s X u =1 m u (cid:26) ( α u κ -pardeg( E ) − ακ -pardeg( E u )) + δ ( αν ( I , l u ) − aα u ) (cid:27) . Given a swamp with generalized parabolic structure,( E , q, φ ) rank r , degree d and type tp = ( a, b, c, L , e ), we will use the following nota-tion, par χ ( E ( n )) := χ ( E ( n )) − P i ∈ I ( e ) κ i re i dim q i ( F ( y i ) ⊕ F ( y i )) , par h ( E ( n )) := h ( Y, E ( n )) − P i ∈ I ( e ) κ i re i dim q i ( F ( y i ) ⊕ F ( y i )) , par µ ( E ) := κ -pardeg( E ) α . In the next theorem we adapt the result [16, Theorem 2.12] to our case.
Theorem 3.5.
There exists n ∈ N such that for very n > n and every ( κ, δ ) -(semi)stable generalized parabolic swamp, ( E , q, φ ) , the following inequality s X i =1 m i (par χ ( E ( n )) α i − par h ( E i ( n )) α ) + δµ ( E • , m, φ )( ≥ )0 holds true for every weighted filtration ( E • , m ) .Proof. Let ( E • , m ) be a weighted filtration. Assume that each E i satifies that E i ( n ) isglobally generated and h ( Y, E i ( n )) = 0 for each i = 1 , . . . , s . Then, for each i we havepar χ ( E ( n )) α i − par h ( E i ( n )) α = κ -pardeg( E ) α i − κ -pardeg( E i ) α, and we are done. Let C be the constant given in Proposition 3.2 and let C be another constant. Consider thebounded family of isomorphism classes of locally free sheaves E ′ satisfying a) µ ( E ′ ) ≥ dα − C , b) 1 ≤ α ′ ≤ α − µ max ( E ′ ) ≤ dα + C . Let E be a locally free sheafappearing in a a ( κ, δ )-(semi)stable swamp of rank r and degree d , and let E ′ ⊂ E be12 locally free subsheaf that do not belongs to the above family. Applying Le Potier-Simpson Estimate to the factors of the Harder-Narashimham filtration of E ′ (see [11,Corollary 3.3.8]), we get h ( E ′ ( n )) ≤ α ′ ( α ′ − α ′ [ dα + C + n + B ] + + 1 α ′ [ dα − C + n + B ] + ) , where B := − α ( α + 1) /
2. Assume n is large enough so that dα + C + n + B and dα − C + n + B are positive. Then, h ( E ′ ( n )) ≤ α ′ ( dα + n + B − C α + C ( α − χ ( E ( n )) α ′ − h ( E ′ ( n )) α ≥ − [ B ′ ] + α + C − C α ( α − , where B ′ := B + dα .Since B depends only on α , we can define the constant K = K ( C , C , α, l, κ, d ) := − [ B ′ ] + α + C − C α ( α − − rα ( P j ∈ I ( e ) κ j ). Then, par χ ( E ( n )) α i − par h ( E i ( n )) α ≥− [ B ′ ] + α + C − C α ( α − − rα ( P j ∈ I ( e ) κ j ) . Let C be large enough so that K >δa ( α −
1) and let n be large enough so that, for every E ′ satisfying a), b) and c), h ( Y, E ′ ( n )) = 0 and E ′ is globally generated. Let ( E • , m ) be a weighted filtrationwith E • ≡ (0) ⊂ E ⊂ . . . ⊂ E s ⊂ E and m = ( m , . . . , m s ). We make a partitionof this filtration as follows. Let j , . . . , j t be the indices such that µ ( E j i ) ≥ dα − C , E j i ( n ) is globally generated and h ( Y, E j i ( n )) = 0 for i = 1 , . . . , t . Let l , . . . , l s − t theset of indices { , , . . . , s }\{ j , . . . , j t } in increasing order. Define the weighted filtrations( E , • , m ) and ( E , • , m ) as E • , ≡ (0) ⊂ E j ⊂ . . . ⊂ E j t ⊂ E , m = ( m j , . . . , m j t ) , E • , ≡ (0) ⊂ E l ⊂ . . . ⊂ E l s − t ⊂ E , m = ( m l , . . . , m l s − t ) . From Equation (5) we find that µ ( E • , m, φ ) ≥ µ ( E • , , m , φ ) − ( P tq =1 m j q ) a ( α − . Thus s X i =1 m i (par χ ( E ( n )) α i − par h ( E i ( n )) α ) + δµ ( E • , m, φ ) ≥≥ t X q =1 m j q (par χ ( E ( n )) α j q − par h ( E j q ( n )) α ) + δµ ( E • , , m , φ )++ ( s − t X q =1 m l q ) K − δ ( s − t X q =1 m l q ) a ( α − ≥ , and the result is proved. ( κ, δ ) -semistability and Hilbert-Mumford semistability The goal now is to proveTheorem 3.7, which shows that ( κ, δ )-(semi)stability is equivalent to GIT (semi) stabilityin the Gieseker space under some conditions.Let B := − α ( α + 1) / K ′ be a constant such that d + K ′ > αK ′ > max (cid:26) d ( w − α ) + αrν + aδ ( α −
1) + Bα ( α − | w = 1 . . . α − (cid:27) , (11) Proposition 3.6.
There exists n ∈ N and a constant C such that for every n ≥ n and for any triple t = ( q : U ⊗ O Y ( − n ) → E , q, φ ) of degree d and multiplicity α whoseinduced map U → H ( Y, E ( n )) is injective and giving a semistable point in the Giesekerspace, G ( s ) s , µ max ( E ) ≤ µ ( E ) + C .Proof. It is enough to show that deg( E ′ ) < d + K ′ for the maximal destabilizing subsheaf,since in such case we would have µ ( E ′′ ) ≤ µ ( E ′ ) < d + K ′ α ( E ′ ) ≤ d + K ′ ≤ µ ( E ) + C forevery subsheaf E ′′ ⊂ E , where C := µ ( E )( α −
1) + K ′ .13et Q := E / E ′ be the (semistable) quotient locally free sheaf. Let us use the notation α ′ := α ( E ′ ) , α ′′ := α ( Q ) , d ′ := deg( E ′ ) , d ′′ := deg( Q ) , µ ′ := µ ( E ′ ) and µ ′′ := µ ( Q ).Assume that d ′ ≥ d + K ′ and and let us show that we get a contradiction. For all n ∈ N we have h ( Y, Q ( n )) ≤ α ′′ [ µ ′′ + n + B ] + . Then we have to study two different cases.Consider the first case, h ( Y, Q ( n )) ≤ α ′′ ( µ ′′ + n + B ). Set U ′ := H ( Y, E ′ ( n )) ∩ U . Thenwe have, dim( U ′ ) ≥ p − h ( Y, Q ( n )) ≥ α ( 1 − gh ) + d + αn − α ′′ ( µ ′′ + n + B ) ≥≥ α ( 1 − gh + n ) + d − d ′′ − α ′′ ( 1 − gh + n ) − α ′′ B ≥≥ α ′ ( 1 − gh + n ) + d + K ′ − B ( α − . Consider the locally free sheaf b E := Im( U ′ ⊗ O Y ( − n ) → E t ). Thus, we have U ′ ⊂ H ( Y, b E ( n )) ∩ U (see [8, Lemma 3.3 ], which also holds true in our case), rk( b E | Y i ) ≤ rk( E ′ | Y i ) and b E is generically generated by global sections. Let { u , . . . , u i } be a basisfor U ′ and complete it to a basis u = { u , . . . , u p } of U . Let λ = λ ( u, γ ( i ) p ) be the asso-ciated one parameter subgroup. Then we clearly have that µ G i ( λ, i ,i ( t )) = p rk( b E | Y i ) − r dim( U ′ ) ≤ p rk( E ′ | Y i ) − r dim( U ′ ) . Since ν ( I, i ) ≤ a , we also have µ G ( λ, i ( t )) ≤ a ( p − dim( U ′ )). Therefore, µ G ( λ, Gies( t )) = l X i =1 b i µ G i ( λ, i ,i ( t )) + cµ G ( λ, i ( t ))++ ν X i =1 k i ( p dim( q i ( b E ( y i ) ⊕ b E ( y i ))) − e i dim( U ′ )) ≤≤ l X i =1 d i ( p − aδ − r ( X i ∈ I ( e ) κ i ))( p rk( E ′ | Y i ) − r dim( U ′ ))++ l X i =1 d i δra ( p − dim( U ′ ))++ X i ∈ I ( e ) κ i re i α ( p dim( q i ( E ′ ( y i ) ⊕ E ′ ( y i )))) − rh dim( U ′ )) . An easy calculation give us µ G ( λ, Gies( t )) p ≤ α ′ ( p − r ( X i ∈ I ( e ) κ i )) − α { dim( U ′ ) −− X i ∈ I ( e ) κ i re i α (dim( q i ( E ′ ( y i ) ⊕ E ′ ( y i )))) } + aδ ( α − α ′ ) . (12)Since p = α ( n + − gh ) + d and dim( U ′ ) ≥ d + K ′ + α ′ ( n + gh ) − B ( α − µ G ( λ, Gies( t )) p ≤ aδ ( α − α ′ ) − αK ′ + Bα ( α − − rα ′ ( X i ∈ I ( e ) κ i )++ α ( X i ∈ I ( e ) κ i re i dim( q i ( E ′ ( y i ) ⊕ E ′ ( y i )))) + d ( α ′ − α ) . Since α ′ r ( P i ∈ I ( e ) κ i ) > α P i ∈ I ( e ) κ i re i dim( q i ( E ′ ( y i ) ⊕ E ′ ( y i ))) < ανr (because κ i 1, we get µ G ( λ, Gies( t )) < 0. However Gies( t ) is semistable so we14et a contradiction.Consider now the second case, h ( Y, Q ( n )) = 0. Assuming n > g − h , we have dim( U ′ ) = p . The same calculation as before (see Equation (12)) shows that µ G ( λ, Gies( t )) p ≤ α ′ ( p − r ( X i ∈ I ( e ) κ i )) − α { dim( U ′ ) −− X i ∈ I ( e ) κ i re i dim( q i ( E ′ ( y i ) ⊕ E ′ ( y i ))) } + aδ ( α − α ′ ) ≤≤ ( α ′ − α )( p − aδ ) + ανr. Assume n is large enough so that p − aδ > − ανrα ′ − α (recall that p = rχ ( O Y ) + d + αn ).Then, µ G ( λ, Gies( t )) < Theorem 3.7. There exists n ∈ N such that for every n ≥ n , ( E t , q t , τ t ) is ( κ, δ ) -(semi)stable if and only if t ∈ Gies − ( G ( s ) s ) .Proof. 1) From the construction of the parameter space, we know that q t induces anisomorphism U ≃ H ( Y, E t ( n )). Then, by Proposition 3.6, Gies( t ) ∈ G δ − ( s ) s implies µ max ( E t ) ≤ deg( E ) α + C . We also know, by Proposition 3.1, that ( E t , q t , φ t ) is ( κ,δ )-(semi)stable if and only if P κ ( E • , m )+ δµ ( E • , m, φ )( ≥ )0 for every ( E • , m ) with par µ ( E j ) ≥ par µ ( E ) − C . Observe that, in this case, µ ( E j ) > par µ ( E j ) ≥ par µ ( E ) − C ≥ µ ( E ) − νh − C . Denote C = νh + C . Consider the family of locally free sheaves satisfyinga) µ max ( E ′ ) ≤ deg( E ) α + C , b) par µ ( E ′ ) ≥ par µ ( E ) − C and c) 1 ≤ α ′ ≤ α − . Thisfamily is clearly bounded. Therefore, there is a natural number, n ∈ N , large enoughsuch that E ′ ( n ) is globally generated and h ( Y, E ′ ( n )) = 0 for any E ′ of this family.Now, fix a weighted filtration ( E • , m ) of E t satisfying conditions a), b) and c). Let u = { u , . . . , u p } be a basis of U , such that there are indices l , . . . , l s with U ( l j ) := h u , . . . , u l j i ≃ H ( Y, E j ( n )) for each j . Define γ = P sj =1 α j γ ( l j ) p and consider theone parameter subgroup, λ ( u, γ ( l j ) p ) . Let I be a multiindex giving the minimum in µ G ( λ ( u, γ )). Then µ G ( λ ( u, γ ) , Gies( t ))( ≥ 0) if and only if µ G ( λ ( u, γ ) , Gies( t )) /p ( ≥ ≤ ) µ G ( λ ( u, γ ) , Gies( t )) p == s X u =1 m u { ( b α u κ -pardeg( E ) − ακ -pardeg( b E u )) + δ ( αν ( I , l u ) − a b α u ) } , b E i being the saturated subsheaf generated by E i . Finally, since b α i := α ( b E i ) = α i and κ -pardeg( b E i ) ≥ κ -pardeg( E i ), we get0( ≤ ) µ G ( λ ( u, γ ) , Gies( t )) p == s X u =1 m u { ( α E u κ -pardeg( E ) − ακ -pardeg( b E u )) + δ ( αν ( I , l u ) − a b α u ) } ≤≤ s X u =1 m u { ( α u κ -pardeg( E ) − ακ -pardeg( E u )) + δ ( αν ( I , l u ) − aα u ) } == P κ ( E • , m ) + δµ ( E • , m, τ ) . Thus, the swamp is ( κ, δ )-semistable.2) By Theorem 3.5 we deduce that s X i =1 m i (par χ ( E ( n )) α i − par h ( E i ( n )) α ) + δµ ( E • , m, φ )( ≥ )0 (13)15or any weighted filtration ( E • , m ) of E t . Let λ be a one parameter subgroup and ( U • , m ′ )a weighted filtration such that λ = λ ( U • , m ′ ). This filtration together with the quotient q t : U ⊗ O Y t ( − n ) → E t induces a chain(0) ⊆ E ′ ⊆ . . . ⊆ E ′ s ′ ⊆ E t (14)and, therefore, a filtration E • ≡ (0) ⊂ E ⊂ . . . ⊂ E s ⊂ E t , formed by the differentsubsheaves collected in the above chain. Let J = ( i , . . . , i s ) be the multiindex definedby the following condition: i j ∈ { , . . . , s ′ } is the maximum index among those k ∈{ , . . . , s ′ } such that E j = E ′ k . Denote by m j the sum of the numbers m ′ k correspondingto those sheaves in the chain (14) which are equal to E i , i.e., m j = m k + m k +1 + . . . + m i j , ( k, k + 1 , . . . , i j ) being the indices such that E ′ k = E ′ k +1 = . . . = E ′ i j = E j . We get in thisway a weighted filtration ( E • , m ). Multiplying by p in Equation (13) we get0 ≤ s X i =1 m i (cid:26) p α i − ph ( Y, E i ( n )) α + δp ( αν i ( I ) − aα i )++ p X j ∈ I ( e ) κ j re j dim( q j ( E i ( y j ) ⊕ E i ( y j ))) α − rp ( X j ∈ I ( e ) κ j ) α i (cid:27) . The inverse calculation presented in Subsection 3.3.1 gives0 ≤ l X u =1 b u s X i =1 m i (rk( E ui ) p − rh ( Y, E i ( n )))++ c s X i =1 m i ( ν i ( I ) p − ah ( Y, E i ( n )))++ X j ∈ I ( e ) k j s X i =1 m i ( p dim( q j ( E i ( y j ) ⊕ E i ( y j ))) − e j h ( Y, E i ( n ))) . (15)Since l i := dim U i ≤ h ( Y, E i ( n )), Equation (15) turns into0 ≤ l X u =1 b u s ′ X i =1 m ′ i (rk( E ui ) p − rl i )++ c s ′ X i =1 m ′ i ( ν i ( I ) p − al i )++ X j ∈ I ( e ) k j s ′ X i =1 m ′ i ( p dim( q j ( E i ( y j ) ⊕ E i ( y j ))) − e j l i ) == µ G ( λ ( U • , m ′ ) , Gies( t )) , and the proposition is proved. The moduli space The last step before proving the existence of the moduli space consists in showing thatthe restriction of the Gieseker map to the ( κ, δ )-semiststable locus is proper. Proposition 3.8. There exists n large enough such that the Gieseker morphism, Gies : I ( κ,δ )-(s)s d → G s(s) , is proper for any d ∈ I r,d,δ .Proof. For the sake of notation we drop the subindex d . We use the the valuative criterionfor properness. Let ( O , m , k ) be a DVR, K being its field of fractions and assume we16ave a conmutative diagram Spec( K ) h K / / (cid:15) (cid:15) I ( κ,δ )-(s)s d (cid:15) (cid:15) { , η } = S : Spec( O ) h / / G ( s ) s . The morphism h K is given by a family ( q K , q K , φ K ) over Y K := Y × Spec( K ), where q K : U ⊗ O Y K ( − n ) ։ E K φ K :( E ⊗ aK ) ⊕ b → det( E K ) ⊗ c ⊗ L K q iK :Γ( E K | y i ,y i ) → R K (16)Let us see that h K can be extended to a family, b h = ( q S , φ S , q S ), over Y × S . Thequotient q K defines a point in the Quot scheme of quotients of U ⊗ O Y ( − n ) with thefixed Hilbert polynomial P ( n ). Therefore, there exists a (unique) flat extension q S : U ⊗ π ∗ O Y ( − n ) ։ E S (17)over Y × S . Define now the sheaves M := π S ∗ (det( E S ) ⊗ c ⊗ π ∗ Y L ⊗ π ∗ Y O Y ( an )) and G = π S ∗ (( U ⊗ a ) ⊕ b ⊗ π ∗ Y O Y ). Both sheaves are locally free, so we can form the projectivespace over S , pr S : P := P (Hom O ( G , M ) ∨ ) → S, which carries a tautological morphismover P × Y ,pr ∗ P pr P ∗ (( U ⊗ a ) ⊕ b ⊗ pr ∗ Y O Y ) → (id Y × pr S ) ∗ det( E S ) ⊗ c ⊗ pr ∗ Y O Y ( an ) ⊗ pr ∗ Y L ⊗ pr ∗ P O P (1)Now, the canonical morphism ∆ : pr ∗ P pr P ∗ (( U ⊗ a ) ⊕ b ⊗ π ∗ Y O Y ) → ( U ⊗ a ) ⊕ b ⊗ π ∗ Y O Y inducesa diagram K / / g ( ( pr ∗ P pr P ∗ (( U ⊗ a ) ⊕ b ⊗ pr ∗ Y O Y ) / / (cid:15) (cid:15) (id Y × pr S ) ∗ ( E S ( n ) ⊗ a ) ⊕ b H ′ , where H ′ = (id Y × pr S ) ∗ det( E S ) ⊗ c ⊗ pr ∗ Y O Y ( an ) ⊗ pr Y L ⊗ pr ∗ P O P (1). Let S ⊂ P be theclosed subscheme over which g is the zero morphism, i.e., over which the tautologicalmorphism factorizes through (id Y × pr S ) ∗ ( E S ( n ) ⊗ a ) ⊕ b . Thus, we have over S × Y amorphism (id Y × pr S ) ∗ ( E ⊗ aS ) ⊕ b → (id Y × pr S ) ∗ det( E S ) ⊗ c ⊗ pr Y L ⊗ pr ∗ P O P (1) . Note nowthat the morphism φ K : ( E ⊗ aK ) ⊕ b → det( E K ) ⊗ c ⊗ L K defines a point Spec( K ) → S .Since S is projective this point extends (uniquely) to a point Spec( O ) → S , i.e., to amorphism φ S : ( E ⊗ aS ) ⊗ b → det( E S ) ⊗ c ⊗ π ∗ Y L ⊗ N (18)Let us extend now the parabolic structure. Since E S,η ≃ E K we have an isomorphism π K ∗ ( E S,η | D i ) ≃ π K ∗ ( E K | D i ). Thus composing with π K ∗ ( E K | D i ) ։ R K , we get a surjec-tion π K ∗ ( E S,η | D i ) ։ R K . Observe that the morphism π S : D i × S → S is finite, thus affineand proper. By flat base change, we know that π K ∗ ( E S,η | D i ) = j ∗ π S ∗ ( E S | D i ) , j beingthe open embedding j : η ֒ → S . Now, taking the push-forward and composing with thecanonical map π S ∗ ( E S | D i ) → j ∗ j ∗ π S ∗ ( E S | D i ), we get a morphism π S ∗ ( E S | D i ) → j ∗ R K .Let R S ⊂ j ∗ R K be its image. Then by [9, Proposition 2.8.1], R S is S -flat (thus a free O -module) and the quotient q iS : π S ∗ ( E S | D i ) ։ R S (19)extends q iS : π K ∗ ( E K | D i ) ։ R K (thus rk( R S ) = e i ). Then the family given in Equations(17), (18), (19), b h = ( q S , φ S , q S ), extends the family given in Equation (16) to S . Clearly,17he family ( q S , φ S , q S ) defines an S -valued point t : S → G in the Gieseker space. Since t ( η ) = h ( η ) we deduce that t (0) = h (0), thus it defines a semistable point in G . Let usshow that q (0) induces an isomorphism U ≃ H ( Y, E (0) ( n )). To show that it is injective,we consider the kernel, H ⊂ U , of H ( q (0) ( n )) : U → H ( Y, E (0) ( n )). Since t (0) issemistable we have, µ G ( λ, t (0)) = l X i =1 b i µ G i ( λ, t ,i (0)) + cµ G ( λ, t (0))++ X i ∈ I ( e ) k i µ G r ( λ, t ,i (0)) == l X i =1 b i ( − r dim( H )) + ca ( − dim( H ))++ X i ∈ I ( e ) k i ( p dim( t i ( H ⊕ H ) − e i dim( H )) == l X i =1 d i ( p − aδ − r X j ∈ I ( e ) κ j )( − r dim( H )) + l X i =1 d i δra ( − dim( H ))++ X i ∈ I ( e ) κ i α ( − r dim( H )) = − αp dim( H ) ≥ H ) = 0, i.e, U → H ( Y, E (0) ( n )) is injective. Let us show that it isin fact an isomorphism. For that we just need to show that h ( Y, E (0) ( n )) = 0. Supposeit does not. Then, by Serre duality, there is a non trivial morphism E (0) ( n ) → ω Y . Let G be its image, and consider the linear map Ω : U ֒ → H ( Y, E (0) ( n )) → H ( Y, G ) . Let H ⊂ U be the kernel of Ω, let λ be the corresponding one parameter subgroup and F ⊂ E (0) the subsheaf generated by H . Since t (0) is semistable, we get:0 ≤ µ ( λ, Gies( t )) p = pα F − α dim( H ) + δ l X i =1 d i ( rν ( I , dim( H )) − a rk( F i ))++ X i ∈ I ( e ) ακ i re i dim( q i ( F ( y i ) ⊕ F ( y i )) − b ′ α F . Since h ( Y, G ) ≥ p − dim( H ), we get0 ≤ − pα G + αh ( Y, G ) + δ l X i =1 d i ( rν ( I , dim( H )) − a rk( F i ))++ X i ∈ I ( e ) ακ i re i dim( q i ( F ( y i ) ⊕ F ( y i )) − b ′ α F . and therefore h ( Y, G ) ≥ pα + M, M being a constant not depending on G . Note that p = αn + d + rχ ( O Y ) and that we can assume h ( Y, ω Y ) ≥ h ( Y, G ). Then, if n is largeenough we get a contradiction, so h ( Y, E (0) ( n )) = 0.Let us show now that E (0) has no torsion. Assume it has torsion, T ⊂ E (0) ( n ),supported on the divisors D i , and let T = H ( Y, T ). Let now H := H ( q (0) ( n )) − ( T ) ⊂ . Again, since t (0) is semistable, we have0 ≤ µ G ( λ, t (0)) = l X i =1 b i µ G i ( λ, t ,i (0)) + cµ G ( λ, t (0))++ X i ∈ I ( e ) k i µ G r ( λ, t ,i (0)) == l X i =1 b i ( − r dim( H )) + ca ( − dim( H ))++ X i ∈ I ( e ) k i ( p dim( t i ( H ⊕ H ) − e i dim( H )) == X i ∈ I ( e ) κ i re i α ( p dim( t i ( H ⊕ H )) − αp dim( H ) ≤≤ X i ∈ I ( e ) κ i re i αp dim( T D i ) − ν X i =1 αp dim( T D i )Since κ i < e i r we must have dim( T D i ) = 0, that is T = 0, so E (0) has no torsion supportedon the divisors D i . Furthermore, from the last calculation it is clear that there can notbe any torsion subsheaf supported outside the divisors D i , therefore E (0) is locally free.Thus, the extended family defines a point in I d . Since the corresponding point in G liesin the semistable locus we deduce that the extended family lies in the semistable locus, G s ( s ) , as well and by Theorem 3.7 we are done.Let d ∈ I r,d,δ be as in Section 3.2.1, Equation (7), and let I d be the parameter spaceconstructed in Section 3.2.1. Over Y × I d there is a universal family satisfying the localuniversal property (follows as in [18, Proposition 2.8]). Note also that the natural SL( U )action on Q , h and G r i determines an action on the space I d , Γ : SL( U ) × I d → I d , andthat the universal family satisfies the glueing property as well (again it follows as in [18,Proposition 2.10]). Finally, we have Theorem 3.9. There exist a projective scheme SGPS ( κ,δ ) -ss r,d, tp and an open subscheme SGPS ( κ,δ ) -s r,d, tp together with natural transformation α ( s ) s : SGPS ( κ,δ ) − ( s ) sr,d, tp → h SGPS ( κ,δ ) -(s)s r,d, tp with the following propoerties:1) For every scheme S and every natural transformation SGPS ( κ,δ ) − ( s ) sr,d, tp → h N , there exists a unique morphism ϕ : SGPS ( κ,δ ) -(s)s r,d, tp → S with α ′ = h ( ϕ ) ◦ α ( s ) s .2) The scheme SGPS ( κ,δ ) -s r,d, tp is a coarse moduli space for SGPS ( κ,δ ) − sr,d, tp .Proof. We may assume without lost of generality that e i = 0 for each i = 1 , . . . , ν .Consider the Gieseker map Gies : I d ֒ → G , which is injective and SL( U )-equivariant (seeProposition 3.4). Consider on G the polarization given in Section 3.3.1, and let L :=Gies ∗ O ( b , . . . , b l , c, k i , . . . , k i ν ′ ) . From ([14, Chap.2, § − ( G ( s ) s ) = I (s)s d , and therefore Theorem 3.7 implies that I (s)s d = I ( κ,δ )-(s)s d . By Proposition 3.8, wededuce that the restriction of the Gieseker map to the semistable locus is a SL(U)-equivariant injective and proper morphism. Thus1) the good quotient SGPS ( κ,δ )-ss r,d, tp := I ( κ,δ )-ss d // SL( U ) exists and is projective,2) the geometric quotient SGPS ( κ,δ )-s r,d, tp := I ( κ,δ )-s d / SL( U ) exists and is an open sub-scheme of SGPS ( κ,δ )-ss r,d, tp .Define SGPS ( κ,δ )-(s)s r,d, tp := ` d ∈ I ( r,d,δ ) SGPS ( κ,δ )-(s)s r,d, tp . Now, 1) and 2) follow from this con-struction, the local universal property and the glueing property.19 The parameter space Let r ∈ N , d ∈ Z , e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r , and δ ∈ Q > . In order to prove theexistence of a coarse projective moduli space for the moduli functor given in Equation(2) we need to rigidify the moduli problem. Let n ∈ N and U := C P ( n ) . Consider thefunctor rig SPBGPS ( ρ ) nr,d,e ( S ) = isomorphism classes of tuples ( E S , q S , τ S , g S )where ( E S , τ S ) is a family of singular principal G -bundles parametrized by S with rank r and degree d , ( E S , q S ) is a family of generalizedparabolic locally free sheaves of type e and g S : U ⊗ O S → π S ∗ E S ( n ) is a morphism suchthat the induced morphism U ⊗ O Y × S ( − n ) → E S is surjective . (20)and let us show that there is a representative for it.We may assume without loss of generality that e i = 0 for all i = 1 , . . . , ν . Recallfrom Proposition 3.2 that the family of locally free sheaves E of rank r and degree d thatappear in ( κ, δ )-(semi)stable swamps with generalized parabolic structure is bounded. Inconsequence, there is a natural number n ∈ N such that for n ≥ n , E ( n ) is globallygenerated and H ( Y, E ( n )) = 0. Fix n > max { n , n } and d = ( d , . . . , d l ) ∈ N l with d = P li =1 d i , and let p = rχ ( O Y ) + d + αn . Let U be the vector space C ⊕ p . Denoteby Q the quasi-projective scheme parametrizing equivalence classes of quotients q : U ⊗ π ∗ Y O Y ( − n ) → E where E is a locally free sheaf of uniform multirank r and multidegree( d , . . . , d l ) on Y , and such that the induced map U → H ( Y, E ( n )) is an isomorphism.On Q × Y , we have the morphism, h : S • ( V ⊗ U ⊗ π ∗ Y O Y ( − n )) → S • ( V ⊗ E Q ) → S • ( V ⊗ E Q ) G . Let s ∈ N be as in [13, Theorem 4.2, Remark 4.3]. Then h ( L si =1 S i ( V ⊗ U ⊗ π Y O Y ( − n ))), contains a set of generators of S • ( V ⊗ E Q ) G . Observe that everymorphism k : ⊕ si =1 S i ( V ⊗ U ⊗ O Y ( − n )) → O Y breaks into a family of morphisms k i : S i ( V ⊗ U ) ⊗ O Y ( − in ) ≃ S i ( V ⊗ U ⊗ O Y ( − n )) → O Y and therefore into morphisms k i : S i ( V ⊗ U ) ∆ ֒ → S i ( V ⊗ U ) ⊗ C ⊕ l → H ( Y, O Y ( in )), ∆ being the diagonal morphism.From this point onwards we can proceed as in [13, § D ⊂ Q ∗ together with a universal family ( E D , τ D ) of singular principal G -bundles of uniform multirank r and multidegree ( d , . . . , d l ). In order to include theparabolic structure as well we need to consider the Grassmannians G r i := Grass e i ( U ⊕ )of e i dimensional quotients of U ⊕ . Define Z := D × G r × . . . × G r ν , and denote by c i : Z → G r i the projection onto the i th Grassmannian. Consider the pullback of theuniversal quotient of the i th Grassmannian to Z , q iZ : U ⊕ ⊗ O Z → R Z , and take the directsum q Z : U ⊕ ν ⊗ O Z → L ν R Z . Denote by q Z , E Z and τ Z the pullbacks to Z × Y of thecorresponding objects over D . Consider the morphism π i : Z × { y i , y i } → Z × { x i } ≃ Z .and look at the following commutative diagram For each i , there are quotients f i : U ⊕ × O Z → π i ∗ ( E Z | y i ,y i ) and we can form f := ⊕ ( f i ) : U ⊕ ν × O Z → L π i ∗ ( E Z | y i ,y i ).Consider the following diagram,0 / / Ker( f ) / / q ′ % % U ⊕ ν × O Z f / / q Z (cid:15) (cid:15) L π i ∗ ( E Z | y i ,y i ) / / L R Z . Denote by M d ( G ) ⊂ Z the closed subscheme given by the zero locus of the morphism q ′ q Z to M d ( G ) factorizes L π i ∗ ( E Z | y i ,y i ) | M d ( G ) L R M d ( G ) L π i M d ( G ) ∗ ( E M d ( G ) | y i ,y i ) q M d ( G ) / / L R Z | M d ( G ) . Since f and q Z are diagonal morphisms we deduce that q M d ( G ) is also diagonal. Therefore q M d ( G ) is determined by ν morphisms q i M d ( G ) : π i M d ( G ) ∗ ( E M d ( G ) | y i ,y i ) → R M d ( G ) . Denoteby ( E M d ( G ) , τ M d ( G ) ) the restriction of ( E Z , τ Z ) to M d ( G ). Then ( E M d ( G ) , q M d ( G ) , τ M d ( G ) )is a universal family of singular principal G -bundles with generalized parabolic structure. Theorem 4.1. The functor rig SPBGPS ( ρ ) nr,d,e is representable.Proof. Follows from the construction of M d ( G ) and taking the disjoint union over all thepossible multidegrees as in Theorem 3.9, which we denote by M ( G ). The moduli space Recall from Proposition 3.2 that the family of locally free sheaves E of fixed degree andrank which appears in a ( κ, δ )-(semi)stable swamp with generalized parabolic structureis bounded. As a consequence, there is a natural number n ∈ N such that for n ≥ n , E ( n ) is globally generated and h ( Y, E ( n )) = 0. Fix such natural number n and considerthe functors rig SGPS nr,d, tp and rig SPBGPS ( ρ ) nr,d,e given in Equation (4) and Equation(20) respectively. Note that there is a natural GL( U ) action on the space M ( G ), Γ :GL( U ) × M ( G ) → M ( G ). We can view this GL( U )-action as a ( C ∗ × SL( U ))-action.Thus, we will construct the quotient of M ( G ) by GL( U ) in two steps, considering theactions of C ∗ and SL( U ) separately. Consider the action of C ∗ on rig SPBGPS ( ρ ) nr,d,e .Let tp = ( a, b, , O Y , e ), where a and b are as in [13, Theorem 5.5]. The map given inEquation (1) induces an injective C ∗ -invariant natural transformation rig SPBGPS ( ρ ) nr,d,e ֒ → rig SGPS nr,d, tp , which in turn induces a SL( U )-equivariant injective and proper morphism, β : M ( G ) // C ∗ ֒ → I r,d, tp = a d ∈ I I d . Furthermore, the universal family on M ( G ) satisfies the local universal property as wellas the glueing property. We finally have Theorem 4.2. There is a projective scheme SPBGPS( ρ ) ( κ,δ ) − ss r,d,e and an open subscheme SPBGPS( ρ ) ( κ,δ ) − s r,d,e ⊂ SPBGPS( ρ ) ( κ,δ ) − ss r,d,e together with a natural tranformation α (s)s : SPBGPS ( ρ ) ( κ,δ )-(s)s r,d,e → h SPBGPS( ρ ) ( κ,δ )-(s)s r,d,e with the following properties:1) For every scheme S and every natural transformation α ′ : SPBGPS ( ρ ) ( κ,δ )-(s)s r,d,e → h S , there exists a unique morphism ϕ : SPBGPS ( κ,δ )-(s)s r,d,e ( ρ ) → S with α ′ = h ( ϕ ) ◦ α (s)s .2) The scheme SPBGPS ( κ,δ )-s r,d,e ( ρ ) is a coarse quasi-projective moduli space for themoduli functor SPBGPS ( κ,δ )-s r,d,e ( ρ ) .Proof. Considering the linearized invertible sheaf L given in the proof of Theorem 3.9and defining L ′ := β ∗ L , it follows as in the connected case (see [18]).21 Let X be a projective nodal curve with nodes x , . . . , x ν and l irreducible components,and π : Y = ` li =1 Y i → X its normalization. Let O X (1) be an ample invertible sheaf on X and denote by O Y (1) the ample invertible sheaf obtained by pulling O X (1) back to Y . As usual, h is the degree of O Y (1), y i , y i are the points in the preimage of the i thnodal point x i , D i = y i + y i are the corresponding divisor on Y and D = P D i is thetotal divisor. Torsion free sheaves over a reducible nodal curve Let F be a torsion free sheaf on X of rank r , that is, of uniform multirank r . C. S.Seshadri showed (see [20, Chapter 8]) that for each nodal point x (regardless of howmany components this point lies on), there is a natural number 0 ≤ l ≤ r such that F x ≃ O lX,x ⊕ m r − lx . Then, it is said that a torsion free sheaf of rank r is of type l =( l , . . . , l ν ) if F x i ≃ O l i X,x i ⊕ m r − l i x i at the i th nodal point.If F be a torsion free sheaf on X of rank r and of type l , then the canonical map α : F → π ∗ π ∗ ( F ) is injective, and T := Coker( α ) is a torsion sheaf supported on thenodes. A short calculation shows that length( T ) = P νi =1 (2 r − l i ). anddeg( π ∗ F ) = deg( F ) + rν − X l i , deg( T ( F )) = 2( rν − X l i ) , (21) T ( F ) being the torsion subsheaf of π ∗ ( F ) (see [1] for the irreducible case). Proposition 5.1. If F is a torsion free sheaf of rank r and type l = ( l , . . . , l ν ) on X ,then the natural morphism β : F ֒ → π ∗ ( E ) , where E := π ∗ ( F ) /T ( F ) , is injective and length(Coker( β )) = l := P l i . Furthermore, Coker( β ) = L νi =1 C l i x i .Proof. Let F be a torsion free sheaf on the nodal curve X and let T ( F ) be the torsionsubsheaf of π ∗ ( F ). Consider the natural morphism β : F → π ∗ ( π ∗ ( F ) /T ( F )). This isinjective at every smooth point so it is injective since F is torsion free. Consider nowthe exact sequence 0 → F ֒ → π ∗ ( π ∗ ( F ) /T ( F )) → Coker( β ) → . (22)Then, we have χ ( π ∗ ( F ) /T ( F )) = χ ( F ) + length(Coker( β )) and, therefore, rχ ( O Y ) +deg( π ∗ ( F ) /T ( F )) = rχ ( O X ) + deg( F ) + length(Coker( β )). However χ ( O Y ) − χ ( O X ) = ν , so length(Coker( β )) = rν + deg( π ∗ ( F ) /T ( F )) − deg( F ) and applying Equation (21)we get the result. Corollary 5.2. Let F be a torsion free sheaf of rank r and type l = ( l , . . . , l ν ) on X . Suppose there exists a locally free sheaf E on Y of the same rank and an injection i : F ֒ → π ∗ E . Then length(Coker( i )) = e if and only if length(Coker( π ∗ ( λ ))) = e − l ,where l = P l i .Proof. Let F be a torsion free sheaf of rank r on X and suppose there exists a locally freesheaf of rank r , E , on the normalization and an injection i : F ֒ → π ∗ ( E ). Then, there isan injection λ : E ֒ → E such that π ∗ ( λ ) ◦ β = i . From the above observation, it followsthat Coker( i ) / Coker( β ) ≃ Coker( π ∗ ( λ )). Hence, we deduce that length(Coker( π ∗ ( λ ))) =length(Coker( i )) − length(Coker( β )). Since length(Coker( i )) = e and length(Coker( β )) = l , we can conclude using Proposition 5.1. 22 .2. — Descending singular principal G -bundles Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . Let ( E , q, τ ) be a singularprincipal G -bundle with generalized parabolic structure on Y with rank r , degree d and type e . Consider the natural surjection ev D = ⊕ ev i : E → E | D = L E | D i andtake the push-forward, π ∗ (ev D ) : π ∗ ( E ) → π ∗ ( E | D ). Since π ∗ ( E | D ) is precisely thevector space L ( E ( y i ) ⊕ E ( y i )) supported on the nodes, we can consider R = L R i as askycraper sheaf supported on the nodes and compose π ∗ (ev D ) with q to get the morphism q ◦ π ∗ (ev D ) : π ∗ ( E ) → R → 0. Defining F = Ker( q ◦ π ∗ (ev D )), we get an exact sequence0 → F ֒ → π ∗ ( E ) p → R → F is a torsion free sheaf of rank r and degree d + P νi =1 ( r − e i ), and R has lengthlength( R ) := e + . . . + e ν .It remains to construct τ ′ : Spec( F ⊗ V ) G → O X from the data ( E , q, τ ). Consider thecanonical isomorphism, π ∗ (Spec( F ⊗ V ) G ) ≃ Spec( π ∗ ( F ) ⊗ V ) G . Now, the identity map π ∗ E → π ∗ E induces a morphism π ∗ π ∗ E → E by adjunction and therefore a morphism ofalgebras π ∗ S • ( V ⊗ π ∗ E ) G → S • ( V ⊗ E ) G which, in turn, induces a morphism of algebras S • ( V ⊗ π ∗ E ) G → π ∗ S • ( V ⊗ E ) G again by adjunction. This induces a diagram S • ( V ⊗ F ) G / / τ ′ ( ( PPPPPPPPPPPPP S • ( V ⊗ π ∗ E ) G ˆ τ (cid:15) (cid:15) / / O X (cid:31) (cid:127) / / π ∗ O Y / / L νi =1 C x i / / Remark 5.3. Let ( E , q ) be a generalized parabolic locally free sheaf of rank r , degree d and type e ′ = ( e ′ , . . . , e ′ ν ). For each i = 1 , . . . , ν , denote by K i the kernel of the i th parabolic structure E ( y i ) ⊕ E ( y i ) → R i and by C i (resp. C i ) the kernel of theinduced linear map K i → E ( y i ) (resp. K i → E ( y i )). From [3, Proposition 3.7], itfollows that the associated torsion free sheaf F satisfies F x i ≃ O e i X ⊕ m r − e i x i , where e i = 2 r − e ′ i − dim( C i ) − dim( C i ). Definition 5.4. Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . A descending G -bundle of rank r , degree d and type e on Y is a singular principal G -bundle withgeneralized parabolic structure of rank r , degree d and type e , ( E , q, τ ), such that τ ′ takes values in O X ⊂ π ∗ ( O Y ). Definition 5.5. Let r ∈ N , d ∈ Z , e := ( e , . . . , e ν ) ∈ N ν , and let δ ∈ Q > . For each i ∈ I ( e ) fix κ i ∈ (0 , e i r ) ∩ Q . A descending G -bundle is ( κ, δ )-(semi)stable if it is assingular principal G -bundle with generalized parabolic structure.A family of descending G -bundles parametrized by a scheme S is defined in theobvious way, and we can consider the moduli functor, D ( ρ ) ( κ,δ )-(s)s r,d,e ( S ) = isomorphism classes of families of( κ, δ )-(semi)stable descending G -bundles on Y parametrized by S with rank r degree d and type e . Then one can show the next theorem following a similar argument as given for provingTheorem 4.2 and [18, Main Theorem]. Theorem 5.6. There exist a projective scheme D( ρ ) ( κ,δ ) -ss r,d,e and an open subscheme D( ρ ) ( κ,δ ) -s r,d,e ⊂ D( ρ ) ( κ,δ ) -ss r,d,e together with a natural tranformation α ( s ) s : D ( ρ ) ( κ,δ ) -(s)s r,d,e → h D( ρ ) ( κ,δ ) -(s)s with the following properties:1) For any scheme S and any natural transformation α ′ : D ( ρ ) ( κ,δ )-(s)s r,d,e → h S , thereexists a unique morphism ϕ : D( ρ ) ( κ,δ ) -(s)s r,d,e → S with α ′ = h ( ϕ ) ◦ α ( s ) s .2) The scheme D( ρ ) ( κ,δ ) -s r,d,e is a coarse moduli space for the moduli functor D ( ρ ) ( κ,δ ) -s r,d,e . .3. — Relation to the moduli space of principal G -bundles over a reduciblenodal curve. Specializations Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . Let ( E , q, τ ) be a descending G -bundle of rank r , degree d and type e , and ( F , τ ′ ) the induced singular principal G -bundle. Recall that both sheaves, E and F , are related through the exact sequence givenin Equation (23) where the morphism p factorizes over the surjection q : π ∗ ( E | D ) → R . For any subsheaf G ⊂ E , the image of p restricted to π ∗ ( G ) ⊂ π ∗ ( E ) is precisely L νi =1 q i ( G ( y i ) ⊕ G ( y i )). Therefore we can construct the following diagram0 / / F (cid:31) (cid:127) / / π ∗ ( E ) / / R / / / / Ker( p ′ ) (cid:31) (cid:127) / / ?(cid:31) O O ✤✤✤ π ∗ ( G ) / / ?(cid:31) O O L νi =1 q i ( G ( y i ) ⊕ G ( y i )) / / ?(cid:31) O O S ( G ) := Ker( p ′ ). If G is saturated then S ( G ) is clearly saturated. Thisconstruction allows us to attach to any weighted filtration ( E • , m ) of E by saturatedsheaves a weighted filtration ( S ( E • ) , m ) of F by saturated sheaves. Moreover, any sat-urated subsheaf can be constructed from a saturated subsheaf of E (follows as in theconnected case [18]).In what follows, we will use the notation κ ( e ) for ( e i r , . . . , e i ν ′ r ), where i , . . . , i ν ′ arethe indices in I ( e ). Proposition 5.7. Let ( E , q, τ ) be a descending G -bundle of rank r degree d and type e and ( F , τ ′ ) the induced singular principal G -bundle on X . Then, ( F , τ ′ ) is δ -(semi)stableif and only if ( E , q, τ ) is a ( κ ( e ) , δ ) -(semi)stable G -bundle with a generalized parabolicstructure.Proof. This follows as in the irreducible case [18, Proposition 5.2.2] Proposition 5.8. Let r ∈ N , d ∈ Z and e := ( e , . . . , e ν ) ∈ N ν with e i ≤ r . There exists ǫ ∈ R ∩ (0 , , such that for any κ with e i r − ǫ < κ i < e i r , any integral parameter δ , andany singular principal G -bundle ( E , q, τ ) with a generalized parabolic structure of rank r ,degree d and type e , we have1) if ( E , q, τ ) is ( κ, δ ) -semistable, then it is ( κ ( e ) , δ ) -semistable,2) if ( E , q, τ ) is ( κ ( e ) , δ ) -stable, then it is ( κ, δ ) -stable.Proof. Recall that the ( κ, δ )-(semi)stability condition for a singular principal G -bundlewith a generalized parabolic structure has to be checked just for the weighted filtrations( E • , m ) of E for which m i < A for suitable constant A depending only on the numericalinput data (see Remark 2.7). This implies that we can find a natural number n such that P κ ( e ) ( E • , m ) + δµ ( E • , m, τ ) ∈ Z [ 1 n ] for all such weighted filtrations. A short calculationshows that for every generalized parabolic bundle ( E , q ) and every weighted filtration( E • , m ) we have P κ ( e ) ( E • , m ) − P κ ( E • , m ) ≤ νrǫAα . In fact we can also show that P κ ( e ) ( E • , m ) − P κ ( E • , m ) ≥ − νrǫAα . Take ǫ so that the inequality νrǫAα < n holds.Now 1) and 2) follow by a similar argument as given in [18, Proposition 5.2.3.].Let r ∈ N , d ∈ Z and e ∈ J ( r ) := { ( e , . . . , e ν ) ∈ N ν | e i ≤ r } . Denote by D r,d ( e,r ) ,e the set of isomorphism classes of descending G -bundles over Y with rank r type e anddegree d ( e, r ) = d − P νi =1 ( r − e i ), and by SPB r,d,e the set of isomorphism classes ofsingular principal G -bundles over X of rank r degree d and type e . From Corollary 5.2,it follows that there is a map Θ e : D r,d ( e,r ) ,e −→ S e ′ ≤ e SPB r,d,e ′ Theorem 5.9. Θ e induces a bijection Θ − e ( SPB r,d,e ) → SPB r,d,e . emark 5.10. From Remark 5.3 it follows that Θ − e ( SPB r,d,e ) consists of descendingsingular principal G -bundles ( E , q, τ ) ∈ D r,d ( e,r ) ,e satisfying dim( C i ) + dim( C i ) = 2( r − e i ) for i = 1 , . . . , ν . Proof. 1. Let ( F , τ ) be a singular principal G -bundle of rank r , degree d and type e ,and consider the exact sequence0 / / T ( F ) / / π ∗ ( F ) / / E = π ∗ F /T ( F ) / / . (25)Since S • ( V ⊗ π ∗ F ) G → S • ( V ⊗ E ) G → S • ( V ⊗ E ) G ) ֒ → Spec( S • ( V ⊗ π ∗ F ) G ). We have the followingdiagramSpec( S • ( V ⊗ E ) G ) (cid:31) (cid:127) / / Spec( S • ( V ⊗ π ∗ F ) G ) / / (cid:15) (cid:15) Spec( S • ( V ⊗ F ) G ) (cid:15) (cid:15) Y π / / π ∗ ( τ ) C C ✺✤ ✠ X , τ C C ✺✤ ✠ The morphism π ∗ ( τ ) : π ∗ ( S • ( V ⊗ F ) G ) = S • ( V ⊗ π ∗ F ) G → π ∗ O X = O Y is the onethat we obtain by adjunction when we take the composition of S • ( V ⊗ F ) G → O X with the natural inclusion of rings O X ⊂ π ∗ O Y . Let us denote by W the opensubset Y \ π − (Sing( X )). Restricting the exact sequence (25) to this open subsetwe get π ∗ F | W = E | W so Spec( S • ( V ⊗ E | W ) G ) = Spec( S • ( V ⊗ π ∗ F | W ) G ) whichmeans that the restriction π ∗ ( τ | W ) takes values in Spec( S • ( V ⊗ E | W )). From thechain of immersionsSpec( S • ( V ⊗ E | V ) G ) ֒ → Spec( S • ( V ⊗ E ) G ) closed ֒ → Spec( S • ( V ⊗ π ∗ F ) G )it follows that π ∗ ( τ ) must then take values in Spec S • ( V ⊗ E ) G , that is, the mor-phism S • ( V ⊗ π ∗ F ) G → O Y factorizes through the surjection S • ( V ⊗ π ∗ F ) G → S • ( V ⊗ E ) G → τ the morphism of algebras S • ( V ⊗ E ) G → O Y . On the otherhand, given a node x ∈ X , π ∗ ( E ) x ⊗ O X,x O X,x / m x ≃ E ( y ) ⊕ E ( y ). Therefore,the surjection π ∗ ( E ) → Coker( β ) defined in Proposition 5.1 induces a surjection q i : E ( y i ) ⊕ E ( y i ) → Coker( β ) x i of dimension e i for each i = 1 , . . . , ν , which,in turn, induce a generalized parabolic structure of type e = ( e , . . . , e ν ). Fromthis construction, it follows that the singular principal G -bundle with general-ized parabolic structure ( E , τ , q ) of rank r , degree d − P νi =1 ( r − e i ) and type e = ( e , . . . , e ν ) is a descending principal G -bundle and it descends to ( F , τ ).This shows surjectivity. On the oder hand, if ( E , τ , q ) ∈ Θ − e ( SPB r,d,e ) is an-other singular principal G -bundle with generalized parabolic structure descendingto ( F , τ ), then we have two exact sequences0 / / F / / ψ ! ! π ∗ E / / R / / / / F / / π ∗ E / / R / / E is locally free, the morphism ψ induces a morphism ι : E → E byadjunction, and therefore a morphism ψ ′ : π ∗ E → π ∗ E making the left squarecommutative. This in turn implies that ψ ′ induces a morphism ψ ′′ : R → R making the right square commutative, and by the Short-Five lemma, Ker( ψ ′ ) =Ker( ψ ′′ ) and Coker( ψ ′ ) = Coker( ψ ′′ ). However, Ker( ψ ′′ ) must be a torsion sheafwhile π ∗ E is torsion free, so we deduce that ψ ′′ is an isomorphism and, therefore, ψ ′ 25s an isomorphism as well. From [20, Huitime partie, II, Proposition 10], it followsthat ι : E ≃ E and that this isomorphism induces an isomorphism between theparabolic structures. Now, since E ≃ E and both, ( E , τ , q ) and ( E , τ , q ),descend to ( F , τ ), we deduce that the diagramSpec( S • ( V ⊗ E )) Spec( S • ( V ⊗ E )) Y τ f f ◆◆◆◆◆◆◆◆◆◆◆ τ ♣♣♣♣♣♣♣♣♣♣♣ commutes when is restricted to W := Y \ π − (Sing( X )). Since τ and τ areseparated morphisms, we finally deduce that the diagram commutes and, therefore, ι : E → E induces an isomorphism of singular principal G -bundles with generalizedparabolic structures. This shows injectivity.Let r ∈ N , d ∈ Z , δ ∈ Z > and define J ( r ) := { e = ( e , . . . , e ν ) ∈ N ν | e i ≤ r } . For each e ∈ J ( r ) fix ǫ = ǫ ( e ) and κ as in Proposition 5.8. Let SPB( ρ ) δ -(s)s r,d be the moduli space of δ -(semi)stable singular principal G -bundles of rank r and degree d on the nodal curve X (see [13]). Then Proposition 5.7 and Proposition 5.8 imply that, for each e ∈ J ( r ), thereis a well defined functor D ( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e → SPB ( ρ ) δ − ( s ) sr,d , where d ( e, r ) = d − P νi =1 ( r − e i ),and thus a proper morphismΘ : D( ρ ) ( κ,δ )-(s)s r,d := a e ∈ J ( r ) D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e −→ SPB( ρ ) δ -(s)s r,d (26)between the moduli spaces. Let e ∈ J ( r ) and let SPB( ρ ) δ -(s)s r,d,e be the subscheme thatparametrizes singular principal G -bundles, ( F , τ ), with F a torsion free sheaf of type e .Then, by Corollary 5.2, Θ induces a proper morphismΘ e : D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e −→ [ e ′ ≤ e SPB( ρ ) δ -(s)s r,d,e ′ . Let us denote by SPB( ρ ) δ -s r,d,e the schematic closure in SPB( ρ ) δ -(s)s r,d , which lies in the closedsubscheme S e ′ ≤ e SPB( ρ ) δ -(s)s r,d,e ′ . Obviously Θ e maps Θ − e (SPB( ρ ) δ -s r,d,e ) to SPB( ρ ) δ -s r,d,e . Theorem 5.11. If the open subscheme D( ρ ) ( κ,δ )-s r,d ( e,r ) ,e ⊂ D( ρ ) ( κ,δ )-ss r,d ( e,r ) ,e is dense, then Θ e induces a birational, proper and surjective morphism Θ e : D( ρ ) ( κ,δ )-ss r,d ( e,r ) ,e −→ SPB( ρ ) δ -s r,d,e .Proof. From Proposition 5.8 and Theorem 5.9 it follows that Θ e induces an isomor-phism Θ − e (SPB( ρ ) δ -s r,d,e ) ≃ SPB( ρ ) δ -s r,d,e . Let us denote by W e the dense open subschemeof D( ρ ) ( κ,δ )-(s)s r,d ( e,r ) ,e parametrizing descending principal bundles with generalized parabolicstructure such that dim( C i )+dim( C i ) = 2( r − e i ) for i = 1 , . . . , ν (see Remark 5.3). FromProposition 5.7 and Remark 5.10 it follows that Θ − e (SPB( ρ ) δ -s r,d,e ) = W e ∩ D( ρ ) ( κ,δ )-s r,d ( e,r ) ,e .Therefore, it is a dense open subscheme. 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