A Master Space for Moduli Spaces of Gieseker-Stable Sheaves
aa r X i v : . [ m a t h . AG ] M a y A MASTER SPACE FOR MODULI SPACESOF GIESEKER-STABLE SHEAVES
DANIEL GREB, JULIUS ROSS, AND MATEI TOMA
Abstract.
We consider a notion of stability for sheaves, which we call multi-Gieseker stabil-ity , that depends on several ample polarisations L , . . . , L N and on an additional parameter σ ∈ Q N ≥ \ { } . The set of semi stable sheaves admits a projective moduli space M σ . Weprove that given a finite collection of parameters σ , there exists a sheaf- and representation-theoretically defined master space Y such that each corresponding moduli space is obtainedfrom Y as a Geometric Invariant Theory (GIT) quotient. In particular, any two such spacesare related by a finite number of “Thaddeus-flips”. As a corollary, we deduce that any twoGieseker-moduli space of sheaves (with respect to different polarisations L and L ) are re-lated via a GIT-master space. This confirms an old expectation and generalises results fromthe surface case to arbitrary dimension. This paper continues previous work [GRT14] of the authors in which, building on work of´Alvarez-C´onsul and King [ACK07], we re-examine the construction of the moduli space M L of Gieseker-semistable sheaves on a given projective scheme X , with an interest in how thesespaces change as the polarisation L varies. As a main result, we will prove that for any choice ofample line bundles L , L on X , there exists a sheaf- and representation-theoretically defined“master space” Y with the property that the moduli spaces M L and M L can be obtainedfrom Y as a Geometric Invariant Theory (GIT) quotient with respect to different stabilityparameters. In particular, any two such spaces fall into the Variation of Geometric InvariantTheory (VGIT) framework considered by Thaddeus [Tha96] and Dolgachev-Hu [DH98].The variation question for moduli of sheaves has attracted quite some attention in thepast, and a precise picture has emerged in the case of surfaces (see for example [MW97]),which has recently also been reconsidered from the point of view of Bridgeland stabilityconditions [BM15]. However, while a general master space result has been expected by manyexperts, the approaches used in the surface case fail, and the higher-dimensional situationuntil recently appeared to be quite mysterious; cf. the discussion of irrational walls in theintroduction of [Sch00].Before stating our results precisely, it is worth commenting on why such a statement doesnot follow immediately from Simpson’s GIT-construction of M L given in [Sim94]. To do so,we first recall this construction in the case of a single ample line bundle L , which startsby finding a space with a group action, whose orbits correspond to isomorphism classes ofsemistable sheaves. To this end one argues as follows: the family of sheaves of a given Date : November 5, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Gieseker stability, variation of moduli spaces, chamber structures, boundedness,moduli of quiver representations.During the preparation of this paper, DG was partially supported by the DFG-Collaborative ResearchCenter SFB/TR 45 ”Periods, moduli spaces and arithmetic of algebraic varieties“. Moreover, he wants tothank Sidney Sussex College, Cambridge, for hospitality and perfect working conditions during a research visitin October 2014. JR is supported by an EPSRC Career Acceleration Fellowship (EP/J002062/1). topological type that are Gieseker-semistable with respect to L is bounded, and thus for n sufficiently large, the evaluation map realises every such sheaf E as a quotient H ( E ⊗ L n ) ⊗ L − n → E → . Picking an identification between H ( E ⊗ L n ) and a fixed vector space of the appropriatedimension, i.e., picking a basis in H ( E ⊗ L n ), we can thus think of E as a point in a certainQuot scheme Q L . This space has a group action, coming from the possible choices of basis,and the tools of GIT are then applied to an appropriate subscheme R L ⊂ Q L to constructthe desired moduli space M L . Doing so requires a choice of linearisation for the group actionon R L , and it is a theorem [HL10, Theorem 4.3.3] that, for n sufficiently large, there is sucha choice for which GIT-stability agrees with Gieseker-stability.From this description, it is clear that there is a problem in using VGIT to relate two modulispaces M L and M L , as the moduli space M L is constructed as the quotient of a differentspace R L and not from the space R L . It would be nice if one could also produce M L as aquotient of R L with respect to some other linearisation, but it is not clear whether this is infact possible.Our approach is to consider a new kind of stability of sheaves that is a convex combinationof Gieseker-stability with respect to different polarisations. Precisely, let X be a projectivemanifold and fix a finite collection of ample line bundles L j on X for 1 ≤ j ≤ j . Furthermore,suppose that σ = ( σ , . . . , σ j ) is a non-zero vector of non-negative rational numbers. We shallsay a torsion-free coherent sheaf E on X is multi-Gieseker-semistable (or simply semistable )with respect to this data if for all non-trivial proper subsheaves F ⊂ E the inequality P j σ j χ ( F ⊗ L mj )rank( F ) ≤ P j σ j χ ( E ⊗ L mj )rank( E )holds for all m sufficiently large. On the one hand, note that by setting all but one of thecomponents of σ to zero we recover the classical notion of Gieseker-semistability. Hence,the notion of multi-Gieseker-stability formally allows us to interpolate between Gieseker-stability with respect to several polarisations. On the other hand, in [GRT14] we prove,under a boundedness hypothesis that we show to hold in many natural setups, that there isa projective moduli space M σ of sheaves of a given topological type that are semistable withrespect to σ . Analogous to the classical case, this moduli space parametrises S -equivalenceclasses of semistable sheaves.The goal of the present paper is to show that any two such spaces can be constructedas a GIT-quotient from the same master space. In the following, suppose L , . . . , L j are afixed collection of ample line bundles on X and that σ ( i ) = ( σ ( i )1 , . . . , σ ( i ) j ) for i = 1 , . . . , i is a finite collection of non-zero vectors in Q j ≥ . We work always under the assumption thatthe set of sheaves of the given topological type that are semistable with respect to σ ( i ) forsome i is bounded. We refer the reader to Section 1.4 for a brief discussion of some relatedboundedness results that are strong enough for most applications, including the variationquestion for Gieseker-moduli spaces considered above. Theorem (Master-Space Construction, Theorem 4.2) . There exists an affine master space Y with an action by a product G of general linear groups such that the following holds: for any i = 1 , . . . , i the moduli space M σ ( i ) is given as the GIT-quotient M σ ( i ) = Y// θ ( i ) G, where θ ( i ) is certain character of G used to define GIT-stability. MASTER SPACE FOR MODULI SPACES OF SHEAVES 3
Thus, we see that the M σ ( i ) are all related by so-called Thaddeus-flips that arise fromVGIT, see [Tha96]. In fact, the master space Y itself has a natural sheaf-theoretic meaningand connects the moduli problems considered to the representation theory of a certain quiver.For more details on this modular interpretation of Y the reader is referred to Sections 1.5,1.7, and 4. By setting j = 2 and considering two special parameters σ , the preceding resultimmediately implies the following: Corollary (Mumford-Thaddeus principle for Gieseker-moduli spaces, Corollary 4.4) . Let L and L be ample line bundles on X . Then, the moduli spaces M L and M L of sheaves of agiven topological type that are Gieseker-semistable with respect to L and L , respectively, arerelated by a finite number of Thaddeus-flips. Again, this is only a weak formulation of the outcome of our investigation, and the masterspace used is closely connected to the collection of moduli problems at hand.It is worth emphasising that the above theorem was proved in [GRT14] under the additionalassumption that all the σ ( i ) j are strictly positive. This suffices to conclude the above corollaryunder the additional assumption that we are dealing with torsion-free sheaves on a smooththreefold, see [GRT14, Theorem 12.1], or that L and L are “general” polarisations in asuitable sense, see [GRT15, Theorem 4.4], although the argument for these results is somewhatindirect. Thus, the real purpose of this paper is to address the degenerate case when someof the σ ( i ) j vanish, thus implementing the naive idea behind the introduction of the notionof multi-Gieseker-stability and allowing us to conclude the corollary in the stated generality.We also emphasise that the assumption that X be smooth and that E be torsion-free is notnecessary, and will be relaxed below.In [GRT15], we developed a general approach in order to realise the intermediate spacesthat appear in a VGIT-passage from one Gieseker-moduli space to another as moduli ofsheaves, and executed it in dimension three. In principle, given the main theorem of thepresent paper, this approach could be pushed through also in higher dimensions; however,the required Riemann-Roch computations seem to be growing in complexity rather quickly.For more information, the reader is refereed to Section 5 as well as to said paper [GRT15].It remains an interesting open problem to derive analogous more refined results concerningthe intermediate steps in the VGIT-passage even for special base manifolds X , low rank, andspecial choice of Chern-classes. Global Assumption.
We always work over an algebraically closed field k of characteristic zero.1. Preliminaries on multi-Gieseker stability and quiver representations
For convenience of the reader, and to make this paper reasonably self-contained, we recallsome preliminaries and definitions we need from [GRT14] and sketch the construction of themoduli space of multi-Gieseker-stable sheaves.1.1.
Geometric Invariant Theory.
For standard concepts from Geometric Invariant The-ory the reader is referred to [MFK94] as well as [Sch08]. When a group G acts on a scheme Y we shall write Y//G for the good quotient (when it exists). We shall use the following result(which is [Sch08, Exercise 1.5.3.3]) concerning quotients by product group actions.
Lemma 1.1.
Suppose G ′ and G ′′ are reductive linear algebraic groups. Let G ′ × G ′′ act on ascheme Z of finite type over k and suppose this action admits a good quotient Z// ( G ′ × G ′′ ) .Then, the induced action of G ′′ on Z admits a good quotient Z//G ′′ , which is endowed with D. GREB, J. ROSS, AND M. TOMA an induced G ′ -action, and Z// ( G ′ × G ′′ ) is a good quotient of Z//G ′′ by G ′ ; i.e., we obtain anisomorphism Z// ( G ′ × G ′′ ) ∼ = ( Z//G ′′ ) //G ′ . Proof.
First we recall the following well-known statement: suppose that G ′′ acts on schemes Y and Y ′ of finite type, and assume that Y ′ admits a good quotient Y ′ //G ′′ . Then if Y → Y ′ is a G ′′ -equivariant affine morphism, then Y also admits a good quotient Y//G ′′ , and theinduced map Y//G ′′ → Y ′ //G ′′ is affine (this statement is proved for example in [Ram96,Lemma 5.1] by first working affine locally and then patching). We apply this to the morphism Z → Z// ( G × G ′′ ), which is possible as this map is affine and G ′′ -invariant, and as Z// ( G × G ′′ )certainly admits a good quotient by the (trivial) action of G ′′ . This gives an affine Z//G ′′ → Z// ( G ′ × G ′′ ), which is easily checked to satisfy the requirements of being the good quotientof Z//G ′′ by G ′ . (cid:3) Preliminaries on sheaves.
Throughout this paper, X shall be a projective schemeover an algebraically closed field of characteristic zero. The dimension of a coherent sheaf E on X is the dimension of its support { x ∈ X : E x = 0 } , and we say that E is pure ofdimension d if all non-trivial coherent subsheaves F ⊂ E have dimension d . Let τ be anelement of B ( X ) Q := B ( X ) ⊗ Z Q , where B ( X ) is the group of cycles on X modulo algebraicequivalence, see [Ful98, Def. 10.3]. We say that E is of topological type τ if its homologicalTodd class τ X ( E ) equals τ . Given a very ample line bundle L on X we say E is n-regular with respect to L if H i ( E ⊗ L n − i ) = 0 for all i >
0. When dealing with several line bundles,the following definition is convenient:
Definition 1.2.
Suppose that L = ( L , . . . , L j ), where each L j is a very ample line bundleon X . We say that a coherent sheaf E is ( n, L ) -regular if E is n -regular with respect to L j for all j ∈ { , . . . , j } .1.3. Multi-Gieseker stability. A stability parameter σ = ( L, σ , . . . , σ j ) consists of L =( L , . . . , L j ) for some ample line bundles L j on X , and σ j ∈ R ≥ such that not all the σ j arezero. We say that σ is rational if all the σ j are rational. We say σ is positive if σ j > j ; otherwise σ is said to be degenerate .In the subsequent discussion the vector L will be fixed, so by abuse of notation we willsometimes confuse σ and the vector ( σ , . . . , σ j ). Thus, we allow σ to vary in a subset of( R ≥ ) j \ { } . We emphasise that whereas we allow the σ j to be irrational, we will alwaysassume that the L j are genuine (integral) line bundles. The multi-Hilbert polynomial of acoherent sheaf E of dimension d with respect to such a stability parameter σ is P σE ( m ) := X j σ j χ ( E ⊗ L mj ) = X di =0 α σi ( E ) m i i ! , where α σi ( E ) are uniquely defined coefficients, cf. [HL10, p. 10]. We denote the leading oneby r σE := α σd ( E ) and note that it is strictly positive by the hypothesis on σ . Definition 1.3 (Multi-Gieseker-stability) . The reduced multi-Hilbert polynomial of a coherentsheaf of dimension d is defined to be p σE ( m ) := P σE ( m ) r σE . We say that a coherent sheaf E is multi-Gieseker-(semi)stable with respect to this data if itis pure and if for all non-trivial proper subsheaves F ⊂ E we have p σF ( ≤ ) p σE . MASTER SPACE FOR MODULI SPACES OF SHEAVES 5
When convenient, we will refer to this notion as being σ -semistable or merely semistable in case no confusion may arise. Of course, if σ = e i = (0 , . . . , , . . . ,
0) is the standard basisvector, then E is (semi)stable with respect to σ if and only if it is Gieseker-(semi)stablewith respect to L i . The notion of multi-Gieseker-stability comes with the standard notion of Jordan-H¨older filtration , and we say two semistable sheaves E are S-equivalent if the gradedmodules associated with the respective filtrations are isomorphic.1.4.
Boundedness.
A set S of isomorphism classes of coherent sheaves on X is said to be bounded if there exists a scheme S of finite type and a coherent O S × X -sheaf E such that every E ∈ S is isomorphic to E { s }× X for some closed point s ∈ S . From [HL10, Lemma 1.7.6], weknow that the set of ( n, L )-regular sheaves of a given topological type is bounded. Conversely,it follows from Serre Vanishing that, if S is a bounded family of sheaves, then for n ≫ E ∈ S is ( n, L )-regular.Throughout this paper, we will work under the hypothesis that the set of sheaves of a giventopological type that are semistable with respect to any of the stability parameters underconsideration at that time is bounded. If X is smooth, and if the sheaves in question aretorsion-free, we proved in [GRT14, Corollary 6.12] that this hypothesis holds if either (a) therank of the sheaves in question is 2 or (b) the Picard rank of X is at most 2 or (c) the dimensionof X is at most 3. We also prove [GRT14, Theorem 6.8] that this boundedness hypothesisholds for torsion-free sheaves on a manifold of dimension d as long as the stability parameters σ = ( L , . . . , L j , σ , . . . , σ j ) in question have the property that P j σ j c ( L ) d − lies in a givencompact subset of the image of the ample cone under the map N ( X ) R → N ( X ) R , x x d − .This latter statement is usually adequate to ensure that the desired boundedness holds forstudying the problem of variation of moduli spaces of Gieseker-semistable sheaves, see alsothe discussion preceding the proof of the Mumford-Thaddeus principle for Gieseker-modulispaces in Section 4.1.5. Quivers and their representations.
Following the ideas of ´Alvarez-C´onsul–King pre-sented in [ACK07] we now discuss how to embed the category of sufficiently regular sheavesinto the category of representations for certain quivers. We will use the standard notationsused in representation theory of quivers, as fixed for example in [Kin94, Section 3]. We denoteby
Vect k the category of vector spaces over a field k . Given a j ∈ N + we define a labelledquiver Q = ( Q , Q , h, t : Q → Q , H : Q → Vect k ) as follows. Let Q := { v i , w j | i, j =1 , . . . , j } be a set of pairwise distinct vertices, and Q := { α ij | i, j = 1 , . . . , j } the set ofarrows, whose heads and tails are given by h ( α ij ) = w j and t ( α ij ) = v i . The arrows will eachbe labelled by a vector space, encoded by a function H : Q → Vect k written as H ( α ij ) = H ij ,which will be fixed later. This quiver can be pictured as follows (where for better readabilitywe restrict to the case j = 3): • H / / H ❘❘❘❘❘❘ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ H ❊❊❊❊❊❊❊ " " ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ •• H ❧❧❧❧❧❧ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ H / / H ❘❘❘❘❘❘ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ •• H ②②②②②②② < < ②②②②②②②②②②②②②②②②②②②②②②②②②②②②②② H / / H ❧❧❧❧❧❧ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ • . D. GREB, J. ROSS, AND M. TOMA A representation M of Q over a field k is a collection V i , W j , i, j = 1 , . . . , n of k -vectorspaces together with k -linear maps φ ij : V i ⊗ H ij → W j . By abuse of notation we write thisas M = L j V j ⊕ W j , where it is understood that V j (resp. W j ) is associated with the vertex v j (resp. w j ) of Q . A subrepresentation is a collection of subspaces V ′ i ⊂ V i and W ′ i ⊂ W i such that φ ij ( V ′ i ⊗ H ij ) ⊂ W ′ j . We call the vector d := (dim V , dim W , . . . , dim V j , dim W j ) =: ( d , d , . . . , d j , d j )the dimension vector of M . The abelian category of representation of Q is equivalent tothe abelian category of A -modules, where A is the (finite-dimensional) path algebra of Q ,cf. [Kin94]. Henceforth, will refer to a representation of Q as an A -module .1.6. Families.
Let S be a scheme. A flat family M of A -modules over S is a sheaf M ofmodules over the sheaf of algebras A := O S ⊗ A on S that is locally free as a sheaf of O S -modules. In other words, we have a decomposition M = L j V j ⊕ W j , where V j and W j arevector bundles on S associated with the vertices v j , w j of Q , together with sheaf morphisms V i ⊗ H ij → W j associated with the arrows of Q . By a trivialisation of such an M we mean atrivialisation of each V j and W j . The representation space R := Rep( Q , d ) := M j i,j =1 Hom k ( k d i ⊗ H ij , k d j ) (1.1)parametrises representations of Q . The reductive group G := Q j j =1 (cid:0) GL k ( d j ) × GL k ( d j ) (cid:1) acts linearly on R by conjugation, and the orbits correspond to isomorphism classes of A -modules of dimension d . We denote the tautological flat family of A -modules over R by U , which we observe comes with a canonical trivialisation. Note that a trivialisation of aflat family M of A -modules over a scheme S gives a canonical morphism σ : S → R andisomorphism σ ∗ U ∼ = M , and a different choice of trivialisation changes σ by the action of aunique morphism S → G .1.7. Embedding categories of sheaves into categories of quiver representations.
Toconnect the preceding discussion with the moduli theory of sheaves, given line bundles L j on X for j = 1 , . . . , j and integers m > n , we let H ij := Hom( L − mj , L − ni ). Given a coherentsheaf E we consider the representation of Q in which V i = H ( E ⊗ L ni ) and W j = H ( E ⊗ L mj )and φ ij : H ( E ⊗ L ni ) ⊗ H ij → H ( E ⊗ L mj ) are the natural multiplication maps. Defining asheaf T := L j j =1 L − nj ⊕ L − mj , we denote this representation byHom( T, E ) = M j H ( E ⊗ L nj ) ⊕ H ( E ⊗ L mj ) . In the set-up at hand, the path algebra is isomorphic to the finite-dimensional k -algebra A = L ⊕ H ⊂ End X ( T ) generated by the projection operators onto the summands L − ni and L − mj of T (collected in the subalgebra L ) and H = L i,j H ij (see [GRT14, Section 5.1] for moredetails). Then, the Q -representation Hom( T, E ) has the obvious structure of an A -module.The next theorem says that this way of associating an A -representation to a sheaf does notlose any information if the sheaf in question is n -regular and m is big enough. Theorem 1.4 (Embedding regular sheaves into the category of representations of Q , [GRT14,Theorem 5.7]) . Suppose that L = ( L , . . . , L j ) , where each L j is a very ample line bundle on X . Given n ∈ N , for all m ≫ n , the functor E Hom(
T, E ) MASTER SPACE FOR MODULI SPACES OF SHEAVES 7 is a fully faithful embedding from the category of ( n, L ) -regular sheaves of topological type τ to the category of representations of Q . In other words, if E is an ( n, L ) -regular sheaf oftopological type τ , the natural evaluation map ε E : Hom( T, E ) ⊗ A T → E is an isomorphism. We remark that part of the proof of the above theorem is the construction of the adjointfunctor of E Hom(
T, E ), which above and henceforth is denoted M M ⊗ A T . Inthe following, for a sheaf E on X × S and a sheaf M of A -modules on S we shall use theabbreviations H om X ( T, E ) := p S ∗ (cid:0) H om X × S ( p ∗ X T, E ) (cid:1) ∼ = p S ∗ (cid:0) E ⊗ O X × S T ∨ (cid:1) and M ⊗ A T := p ∗ S M ⊗ A p ∗ X T, where A := O S ⊗ A , and p S : X × S → S resp. p X : X × S → X are the canonical projections.The previous result, Theorem 1.4, extends in a natural way to flat families of sheaves andmodules (see [GRT14, Proposition 5.8] for the precise statement). In the next step, we identifythe image of the embedding functor. For this, we let P j ( k ) = χ ( E ⊗ L kj ) be the Hilbert-polynomial for a (resp. any) coherent sheaf E of topological type τ with respect to L j . Proposition 1.5 (Identifying the image of the embedding functor, [GRT14, Proposition 5.9]) . For m ≫ n as in Theorem 1.4 the following holds: If B is any Noetherian scheme and M a B -flat family of A -modules of dimension vector d = (cid:0) P ( n ) , P ( m ) , . . . , P j ( n ) , P j ( m ) (cid:1) , then there exists a (unique) locally closed subscheme ι : B [ reg ] τ ֒ → B with the following proper-ties. (a) ι ∗ M ⊗ A T is a B [ reg ] τ -flat family of ( n, L ) -regular sheaves on X of topological type τ ,and the unit map η ι ∗ M : ι ∗ M → H om X ( T, ι ∗ M ⊗ A T ) is an isomorphism. (b) If σ : S → B is such that σ ∗ M ∼ = H om X ( T, E ) for an S -flat family E of ( n, L ) -regular sheaves on X of topological type τ , then σ factors through ι : B [ reg ] τ ֒ → B and E ∼ = σ ∗ M ⊗ A T . Preservation of stability.
We now explain how the functor T Hom(
T, E ) preservesstability. For this we need King’s notion of stability for A -modules M of dimension vector d = ( d , d , . . . , d j , d j ). Fix σ = ( σ , . . . , σ j ) with σ j ∈ Q ≥ not all equal to zero. Definea vector θ σ = ( θ , θ , . . . , θ j , θ j ) by θ j := σ j P i σ i d i and θ j := − σ j P i σ i d i for j = 1 , . . . , j . For an A -module M ′ = L V ′ j ⊕ W ′ j , we set θ σ ( M ′ ) := X j θ j dim V ′ j + X j θ j dim W ′ j , which makes θ σ an additive function from the set N j of possible dimension vectors to R .Note that if M is an A -module of dimension vector d , then θ σ ( M ) = P j ( θ j d j + θ j d j ) = 0. D. GREB, J. ROSS, AND M. TOMA
Definition 1.6 (Semistability for A -modules) . Let M be an A -module with dimension vector d . We say that M is (semi)stable (with respect to σ ) if for all non-trivial proper submodules M ′ ⊂ M we have θ σ ( M ′ )( ≤ )0. We let R σ -ss denote the open subset of R = Rep( Q , d )consisting of modules that are semistable with respect to σ .Every σ -semistable A -module has a Jordan-H¨older filtration with respect to θ σ , cf. [Kin94,p. 521/522], and we call two modules S -equivalent if the graded modules associated with therespective filtrations are isomorphic. The following is the main technical result of [GRT14], forwhich, as always, we presume that the set of sheaves of topological type τ that are semistablewith respect to σ is bounded. Theorem 1.7 (Comparison of semistability and JH filtrations, [GRT14, Theorem 8.1]) . Forall integers m ≫ n ≫ p ≫ the following holds for any sheaf E on X of topological type τ : (1) E is semistable if and only if it is pure, ( p, L ) -regular, and Hom(
T, E ) is semistable. (2) Suppose σ is positive. If E is semistable, then Hom(
T, grE ) ∼ = gr Hom(
T, E ) , where gr denotes the graded object coming from a Jordan-H¨older filtration of E or Hom(
T, E ) , respectively. In particular, two semistable sheaves E and E ′ are S -equivalent if and only if Hom(
T, E ) and Hom(
T, E ′ ) are S -equivalent. Construction of the moduli space.
We sketch now how the above result is usedto construct the moduli space we are looking for. Assume σ is a positive rational stabilityparameter, and that the set of sheaves of topological type τ that are semistable with respectto σ is bounded. We choose natural numbers p, n, m ∈ N such that Theorem 1.7 holds.Moreover, by increasing m if necessary, we may assume that the conclusion of Theorem 1.4holds as well. Note that by assumption, every semistable sheaf E of topological type τ is( p, L )-regular, and therefore also ( n, L )- and ( m, L )-regular.Recall P j denotes the Hilbert polynomial of E with respect to L j , where E is a (any) sheafof topological type τ . We consider the dimension vector d = (cid:0) P ( n ) , P ( m ) , . . . , P j ( n ) , P j ( m ) (cid:1) , and as above let R := Rep( Q , d ) = M j i,j =1 Hom k ( k P i ( n ) ⊗ H ij , k P j ( m ) ) (1.2)be the representation space of the quiver Q corresponding to the dimension vector d . Here,as before we have used the notation H ij = H ( L − ni ⊗ L mj ).Let R [ n -reg] τ ⊂ R be the locally closed subscheme parametrising those modules that are inthe image of the Hom( T, − )-functor on the category of ( n, L )-regular sheaves, whose exis-tence is guaranteed by Proposition 1.5. Inside this, there is an open subscheme Q ⊂ R [ n -reg] τ parametrising those modules whose associated sheaf is not only ( n, L )- but actually ( p, L )-regular, and a further open subscheme Q [ σ -ss] ⊂ Q ⊂ R [ n -reg] τ parametrising representations Hom( T, E ), where E is σ -semistable and of topological type τ .The reductive group G := Y j j =1 (cid:0) GL k ( P j ( n )) × GL k ( P j ( m )) (cid:1) (1.3)acts linearly on R by conjugation, and R [ n - reg ] τ and Q [ σ - ss ] are preserved by the G -action. MASTER SPACE FOR MODULI SPACES OF SHEAVES 9
Theorem 1.8 (Existence of moduli spaces for σ -semistable sheaves, [GRT14, Theorems 9.4and 10.1]) . Assume that σ is a positive stability parameter such that the set of sheaves oftopological type τ that are semistable with respect to σ is bounded, and let Z := Q [ σ -ss ] be thescheme-theoretic closure in R . Then, Z σ -ss := Z ∩ R σ -ss = Q [ σ -ss ] . Moreover, M σ := Z σ -ss //G exists as a good quotient, and is the moduli space of sheaves of topological type τ that aresemistable with respect to σ . The proof of Theorem 1.8 rests on our comparison between stability of a sheaf E andstability of the corresponding A -module Hom( T, E ). Since we are assuming σ to be positive,Theorem 1.7(2) also allows us to compare S -equivalence classes before and after the categoricalembedding. This gives us enough control on the closure of G -orbits to ensure that the goodquotient Z σ -ss //G exists. We think of the quotient Z σ -ss //G as being embedded inside thegood quotient R σ -ss //G of semistable A -modules constructed by King. We refer the readerto [GRT14, Section 9] for a more detailed discussion, in particular, for a description of themoduli functor that M σ corepresents. Remark 1.9.
The above discussion requires that each σ j is rational. However, we showin [GRT14, Corollary 4.4] that, in the case of torsion-free sheaves on an integral scheme, thisis enough to also prove the existence of M σ even if some of the σ j are irrational.2. Degenerate stability parameters
Setup.
Let σ = ( σ , . . . , σ j ) ∈ Q j ≥ \ { } . In the previous section we sketched theconstruction of the moduli space M σ when σ is positive. When σ is degenerate, in orderto show the existence of the desired moduli space, suppose we ignore all those j such that σ j = 0. Since this does not change the (multi-Gieseker-)stability condition on sheaves weobtain in this way the moduli space M σ , but at a cost of modifying the parameter spaceused in the GIT construction. In particular, this approach does not directly yield the desiredmaster space that can be used for both positive and degenerate stability conditions at thesame time. For this reason we instead take a closely related but different approach.As above, X is a projective scheme over k , and L , . . . , L j are very ample line bundles on X . We also fix a topological type τ . Renaming indices if necessary, we may suppose there isa j ′ < j such that σ j > j ≤ j ′ and σ j = 0 for all j > j ′ . Consider the truncation σ ′ := ( L , . . . , L j ′ , σ , . . . , σ j ′ ) , which is a positive stability parameter, and set L ′ = ( L , . . . , L j ′ ). Our aim is to compare theGIT setup for σ with that for σ ′ .To this end, let Q be the quiver with vertices { v i , w i | i, j = 1 , . . . , j } discussed in Section1.5, and let Q ′ be the full subquiver with vertices { v i , w i | i, j = 1 , . . . , j ′ } . Pictorially, wethink of Q ′ as being the top j ′ rows of Q together with the induced arrows. As always,we assume the set of sheaves of topological type τ that are semistable with respect to σ isbounded, and we also choose integers m ≫ n ≫ p ≫ σ ′ . Now let T , A , d , R , Q , G be the objects associated with Q defined in Section 1, and let T ′ , A ′ , d ′ , R ′ , Q ′ , G ′ be the corresponding objects associated with Q ′ . Thus, for a sheaf E , the A -module Hom( T, E ) is a representation of Q and the A ′ -module Hom( T ′ , E ′ ) is a representationof Q ′ . If E is ( n, L )-regular of topological type τ , then Hom( T, E ) and Hom( T ′ , E ) havedimension vector d and d ′ respectively, and so yield elements of R = Rep( Q , d ) and R ′ =Rep( Q ′ , d ′ ), respectively. Definition 2.1 (Projection induced by subquiver) . We let π : R → R ′ be the projection induced from the inclusion Q ′ ⊂ Q .Pictorially, π can be thought of as taking a representation of Q and throwing away all butthe top j ′ rows; so, when j = 3 and j ′ = 2, this looks as follows: V H / / H ◗◗◗◗◗◗ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ H ❉❉❉❉❉❉❉ ! ! ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ W V H ♠♠♠♠♠♠ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ H / / H ◗◗◗◗◗◗ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ W V H ③③③③③③③ = = ③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③③ H / / H ♠♠♠♠♠♠ ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ W π V H / / H ❏❏❏❏❏ $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ W V H tttttt : : ttttttttttttt H / / W . Remark 2.2 (Kronecker quivers as subquivers of Q ) . Note that by setting all but one of therows to zero we obtain the Kronecker quiver used by ´Alvarez-C´onsul and King [ACK07] intheir construction of the moduli space of Gieseker-semistable sheaves. This observation is ouroriginal motivation to study degenerate stability parameters.As before, let θ j and θ j be given by θ j := σ j P j i =1 σ i d i and θ j := − σ j P j i =1 σ i d i for j = 1 , . . . , j , (2.1)and for an A -module M = L j V j ⊕ W j set θ σ ( M ) := X j θ j dim V j + X j θ j dim W j . (2.2)So, if M is an A -module of dimension vector d , then θ σ ( M ) = 0. Observe that if we define θ ′ j and θ ′ j in an analogous way, with σ replaced by σ ′ and j replaced by j ′ , then θ ′ ji = θ ji forall j ≤ j ′ and i = 1 ,
2. Thus, if M is an A ′ -module of dimension vector d ′ , then θ σ ′ ( M ) = 0.Write G := Q j j =1 (cid:0) GL k ( P j ( n )) × GL k ( P j ( m )) (cid:1) and observe that G acts on R as G = G ′ × G ′′ ,where G ′ is the product over those j up to j ′ , and G ′′ is the product over the remaining indices.The projection π : R → R ′ is G -equivariant, where G ′′ < G acts trivially on R ′ . We now let Z := Q [ σ -ss] ⊂ R and Z σ -ss := Z ∩ R σ -ss , and similarly Z ′ := Q ′ [ σ ′ -ss] ⊂ R ′ and Z ′ σ ′ -ss := Z ′ ∩ R ′ σ ′ -ss . MASTER SPACE FOR MODULI SPACES OF SHEAVES 11
The main technical result.
By Lemma 1.1, the existence of a good quotient for the G -action on Z σ -ss implies the existence of a good quotient Z σ -ss //G ′′ for the induced action of G ′′ , and our main technical result is to identify this quotient. Proposition 2.3 (Compatibility of the projection with GIT and embedding functors) . Theprojection π : R → R ′ induces a G ′ -equivariant, G ′′ -invariant map π : Z σ -ss → Z ′ σ ′ -ss . Moreover, the induced map ˆ π : Z σ -ss //G ′′ → Z ′ σ ′ -ss is a G ′ -equivariant isomorphism. As a consequence of the previous result, we can show the moduli space M σ of σ -multi-Gieseker semistable sheaves can be obtained as a GIT quotient from the G -space Z . Thisstatement will eventually lead to the desired master space construction for any finite numberof stability parameters. Corollary 2.4 (Obtaining moduli of multi-Gieseker semistable sheaves from Z ) . With theabove notations, we have M σ ∼ = Z σ -ss //G. Proof.
Observe first that (semi)stability of a sheaf E with respect to σ is precisely the same as(semi)stability with respect to σ ′ ; i.e., the moduli functors coincide and for the correspondingmoduli spaces we have M σ = M σ ′ . The construction of M σ ′ described in the Section 1.9,which applies as σ ′ is positive, says M σ ′ = Z ′ σ ′ -ss //G ′ . Thus, taking the quotient of Z σ -ss by G = G ′ × G ′′ in stages, by Lemma 1.1 and Proposition 2.3 leads to the following sequence ofisomorphisms Z σ -ss //G ∼ = ( Z σ -ss //G ′′ ) //G ′ ∼ = Z ′ σ ′ − ss //G ′ ∼ = M σ ′ = M σ and therefore establishes the desired result. (cid:3) Proof of the main technical result
In order to get a preliminary idea from the GIT-part of the theory as to why the resultwe are claiming is true, observe that taking the classical invariant-theoretic quotient of R by G ′′ yields R ′ . Moreover, as the character θ σ ′ of G computed from σ ′ is really a character of G ′ , i.e., θ σ ′ is trivial on G ′′ , the semistable set R σ -ss is saturated with respect to the goodquotient π : R → R//G ′′ ∼ = R ′ . Hence, the situation is quite clear for the relevant quiver-representations. What remains is proving that this geometric picture restricts well to thesubschemes parametrising representations coming from (semistable regular) sheaves.The proof of Proposition 2.3 is subdivided into two steps: in Section 3.1 we prove the firstassertion, in Section 3.2 we establish the second.3.1. Proving the first statement of Proposition 2.3.
In this section, we investigate therelation between the equivariant geometry of π and the categorical embedding provided byTheorem 1.4.We recall from Section 1.9 the definition of the subschemes Q ⊂ R and Q ′ ⊂ R ′ : the firstone parametrises those A -modules that are in the image of the Hom( T, − )-functor on thecategory of ( p, L )-regular sheaves, the second one those A ′ -modules that are in the image ofthe Hom( T ′ , − )-functor on the category of ( p, L ′ )-regular sheaves. Inside these there are opensubschemes Q [ σ -ss] and Q ′ [ σ ′ -ss] parametrising σ - and σ ′ -semistable sheaves, respectively. Thefollowing result shows that these subschemes are preserved by the projection π : R → R ′ . Lemma 3.1.
The restriction of π : R → R ′ to Q and Q [ σ -ss ] , respectively, induces morphisms π : Q → Q ′ and π : Q [ σ -ss ] → Q ′ [ σ ′ -ss ] . Proof.
We start by introducing some terminology. Given a flat family M of A -modules over S we write π ( M ) to mean the induced family of A ′ -modules over S obtained by ignoring allbut the top j ′ rows of the quiver Q . If u : S ′ → S is a morphism then u ∗ M is a family of A -modules over S ′ and clearly π ( u ∗ M ) = u ∗ π ( M ) . (3.1)Now let U be the tautological family of A -modules parametrised by R and U ′ the tauto-logical family of A ′ -modules parametrised by R ′ . Then, π ( U ) = π ∗ U ′ (3.2)as families of A ′ -modules parametrised by R .We first show that the restriction of π induces π : R [ n -reg] → R ′ [ n -reg] . By definition of ι : R [ n -reg] → R (Proposition 1.5 applied with B, M replaced by R, U ) we have that ι ∗ U ⊗ A T is an R [ n -reg] -flat family of ( n, L )-regular sheaves on X of topological type τ and the unit map η ι ∗ U : ι ∗ U → H om X ( T, ι ∗ U ⊗ A T )is an isomorphism. Observe that since each sheaf in the family ι ∗ U ⊗ A T is ( n, L )-regular, itis also ( n, L ′ )-regular. Moreover, using (3.2) we have ι ∗ π ∗ U ′ = ι ∗ π ( U ) = π ( ι ∗ U ) ∼ = π ( H om X ( T, ι ∗ U ⊗ A T )) = H om X ( T ′ , ι ∗ U ⊗ A T ) . So, from the defining property of R ′ [ n -reg] (the second statement in Proposition 1.5 with B, M replaced by R ′ , U ′ ) this precisely says that π ◦ ι : R [ n -reg] → R ′ factors through R ′ [ n -reg] , andmoreover ι ∗ U ⊗ A T ∼ = ι ∗ π ∗ U ′ ⊗ A ′ T ′ (3.3)as R [ n -reg] -flat families of sheaves on X of topological type τ .Now Q (resp. Q [ σ -ss] ) is the open subset of R [ n -reg] consisting of those points over which U ⊗ A T is a sheaf on X that is ( p, L )-regular (resp. σ -semistable), and similarly for Q ′ and Q ′ [ σ -ss] . But a sheaf being being ( p, L )-regular (resp. σ -semistable) clearly implies that it isalso ( p, L ′ )-regular (resp. σ ′ -semistable). Thus, (3.3) implies that π : Q → Q ′ and π : Q [ σ -ss] → Q ′ [ σ -ss] as claimed. (cid:3) Next, we consider the compatibility of π with the GIT-stability conditions σ and σ ′ . Thiswill establish the first statement of Proposition 2.3. Lemma 3.2.
As before, let Z σ -ss = Z ∩ R σ -ss and Z ′ σ ′ -ss = Z ′ ∩ R ′ σ ′ -ss . Then, the restrictionof π induces a morphism π : Z σ -ss → Z ′ σ ′ -ss . Proof.
Recall that Z = Q [ σ -ss] is the scheme-theoretic closure of Q [ σ -ss] in R and similarly Z ′ = Q ′ [ σ ′ -ss] . We claim that since the restriction of π : R → R ′ to Q [ σ -ss] induces π : Q [ σ -ss] → Q ′ [ σ ′ -ss] , the restriction of π to Z induces a morphism π : Z → Z ′ between the correspondingscheme-theoretic closures. To see this, recall that the scheme-theoretic image of a morphism f : X → Y between schemes is the smallest closed subscheme V of Y such that f factorsthrough V [SP16, Tag 01R6]. Let f : Q [ σ -ss] → R ′ be the composition of the inclusion andthe projection π : R → R ′ , and let V be the scheme-theoretic closure of f . Then, as f factors through Q ′ [ σ ′ -ss] ⊂ R ′ (by Lemma 3.1) it also factors through the closed subscheme MASTER SPACE FOR MODULI SPACES OF SHEAVES 13 Q ′ [ σ ′ -ss] ⊂ R ′ . Hence, V ֒ → Q ′ [ σ ′ -ss] = Z ′ . On the other hand, by [SP16, Tag 01R9] thecomposition f factors through a commutative diagram Q [ σ -ss] " " ❊❊❊❊❊❊❊❊❊ (cid:31) (cid:127) / / Z (cid:31) (cid:127) / / (cid:15) (cid:15) R π (cid:15) (cid:15) V (cid:31) (cid:127) / / R ′ . Thus, the restriction of π to Z induces π : Z → Z ′ , as claimed.Next, we claim that π maps R σ -ss to R ′ σ ′ -ss , which along with the claim proven in theprevious paragraph immediately establishes the statement of the lemma. But this is clear, forsuppose that M ∈ R σ -ss and M ′ is a proper A ′ -submodule of π ( M ). Then, we may extend M ′ to an A -module M by associating the zero vector space for all vertices of Q that are notin the first j ′ rows and also associating the zero morphism to any arrow that starts or endsoutside the first j ′ rows. Clearly, M is a proper A -submodule of M , and hence θ σ ′ ( M ′ ) = θ σ ( M ) ≤ , where the inequality uses semistability of M . Thus, π ( M ) ∈ R ′ σ ′ -ss , as required. (cid:3) Now, as Z σ -ss admits a good quotient by the G -action, by Lemma 1.1 it also admits agood quotient by G ′′ . As good quotients are categorical, the G ′′ -invariance of π together withLemma 3.2 yields an induced morphismˆ π : Z σ -ss //G ′′ → Z ′ σ ′ -ss . By construction, ˆ π is G ′ -equivariant.3.2. Proving the second statement of Proposition 2.3.
In order to prove the secondstatement of Proposition 2.3, we now construct an inverse to the morphism ˆ π . First observethat since σ ′ is a positive stability parameter, Theorem 1.8 gives Z ′ σ ′ -ss = Q ′ [ σ ′ -ss] . So, our aim is the construction of a natural morphismˆ s : Q ′ [ σ ′ -ss] → Z σ -ss //G ′′ . To this end, as before let U ′ be the tautological family of A ′ -modules parametrised by R ′ andlet ι : Q ′ [ σ ′ -ss] → R ′ be the inclusion. So, by construction E ′ := ι ∗ U ′ ⊗ A ′ T ′ (3.4)is a Q ′ [ σ ′ -ss] -flat family of sheaves on X of topological type τ that are σ ′ -semistable, and theunit map η : ι ∗ U ′ → H om X ( T ′ , E ′ ) (3.5)is an isomorphism.Consider now the family of A -modules N := H om X ( T, E ′ ) (3.6)over Q ′ [ σ ′ -ss] . Since each sheaf in the family E ′ is σ ′ -semistable, it is also σ -semistable andhence ( p, L )-regular by our choice of integers m ≫ n ≫ p . Thus, N is a flat family of A -modules of dimension d (flatness of N follows from flatness of E ′ and ( n, L )-regularity asexplained in the proof of [GRT14, Proposition 5.8]). Moreover, by preservation of semistabil-ity, Theorem 1.7(1), N is in fact a flat family of σ -semistable A -modules. We now define our desired map ˆ s , first over a sufficiently small affine open subset Ω ⊂ Q ′ [ σ ′ -ss] . To do this, recall that U ′ is canonically trivialised, and hence by (3.5) we have atrivialisation of H om X ( T ′ , E ′ ). Said another way, the vector bundles in N that are associatedwith the vertices of the subquiver Q ′ are trivialised. If Ω is sufficiently small, we may pick atrivialisation of the remaining part of N (so trivialisations for the vector bundles associatedwith the vertices outside of Q ′ ). Denote this trivialisation by φ . Then, N is trivialised overΩ, and so induces a morphism s φ : Ω → R fulfilling s ∗ φ U = N (3.7)(see the discussion at the end of Section 1.6). From the defining properties of Q [ σ -ss] and R σ -ss we infer that s φ factors through a morphism s φ : Ω → Q [ σ -ss] ∩ R σ -ss = Z σ -ss . We letˆ s φ : Ω → Z σ -ss //G ′′ be the composition of s φ with the good quotient Z σ -ss → Z σ -ss //G ′′ . Observe that, essentiallyby construction, ˆ s φ is independent of the chosen trivialisation φ , since any other choice canbe obtained by composing φ with the action of a morphism Ω → G ′′ , and since the quotientmap Z σ -ss → Z σ -ss //G ′′ is constant on G ′′ -orbits. Thus, we may glue the locally defined s φ toobtain the desired global morphismˆ s : Q ′ [ σ ′ -ss] → Z σ -ss //G ′′ . Now, recall that from Lemma 3.2 and the subsequent discussion that we have alreadyobtained morphisms fitting into the following diagram Z σ -ss π / / (cid:15) (cid:15) (cid:15) (cid:15) Z ′ σ ′ -ss Z σ -ss //G ′′ . ˆ π rrrrrrrrrrr In the final step, we prove compatibility of ˆ π with the morphism ˆ s just constructed. Lemma 3.3.
There is a commutative diagram Q ′ [ σ ′ -ss ] ˆ s / / ˆ π ◦ ˆ s % % ❑❑❑❑❑❑❑❑❑❑ Z σ -ss //G ′′ ˆ π (cid:15) (cid:15) π ( Z σ -ss ) (cid:31) (cid:127) ι / / Z ′ σ ′ -ss = Q ′ [ σ ′ -ss ] , where π ( Z σ -ss ) denotes scheme-theoretic image of π , and ι is the natural inclusion. Moreover,we have ι ◦ ˆ π ◦ ˆ s = id : Q ′ [ σ ′ -ss ] → Q ′ [ σ ′ -ss ] . Proof.
That ˆ π factors this way is clear from the definition. It is also clear from the aboveconstruction that ι ◦ ˆ π ◦ ˆ s is the identity pointwise. This also holds in families, as using theabove discussion (in particular (3.1), (3.2), (3.4), (3.5), (3.6), (3.7)), over a sufficiently smallaffine subset Ω of Q ′ [ σ ′ -ss] allowing for a trivialisation φ of N as above, we have canonicalidentifications( πs φ ) ∗ U ′ = s ∗ φ π ∗ U ′ = s ∗ φ π ( U ) = π ( s ∗ φ U ) = π ( N ) = π ( H om X ( T, E ′ )) = H om X ( T ′ , E ′ ) = U ′ , which shows that π ◦ s φ = id on Ω. Hence, ˆ π ◦ ˆ s is the identity of Q ′ [ σ ′ -ss] , as claimed. (cid:3) Lemma 3.4.
The scheme-theoretic image π ( Z σ -ss ) coincides with Q ′ [ σ ′ -ss ] = Z ′ σ ′ -ss . MASTER SPACE FOR MODULI SPACES OF SHEAVES 15
Proof.
Consider the sequence of maps Q ′ [ σ ′ -ss] = Z ′ σ ′ -ss ˆ π ◦ ˆ s −−−→ π ( Z σ -ss ) ι ֒ → Z ′ σ ′ -ss . The fact that ι ◦ ˆ π ◦ ˆ s = id in a first step gives equality π ( Z σ -ss ) red = ( Z ′ σ ′ -ss ) red on the levelof reduced spaces. But this implies in a second step that the scheme structures also agree:indeed, since these schemes have the same support we may work affine locally, so the schemestructures are related by ring morphisms A ι ∗ → B → A that compose to the identity. As atthe same time ι ∗ is surjective, it has to be an isomorphism. (cid:3) Consequently, the map ˆ π : Z σ -ss //G ′′ → Z ′ σ ′ -ss is a G ′ -equivariant isomorphism, establishing the second assertion of Proposition 2.3.4. Master space construction
We are now ready to prove our main variation result. We continue to use the notationintroduced above, so L , . . . , L j are fixed very ample line bundles on X and we have fixed atopological type τ . Suppose Σ ⊂ Q j ≥ { } is a finite collection of rational stability parameters,and we assume that the family consisting of those sheaves of topological type τ that aresemistable with respect to some σ ∈ Σ is bounded. We choose m ≫ n ≫ p so the conclusionsof Theorem 1.4 and Theorem 1.7(1) hold for each σ ∈ Σ. By enlarging these integers ifnecessary, we may assume Theorems 1.4 and 1.7 hold also for the truncation of each σ ( i ) inwhich all zero entries are ignored; cf. the organisation of our setup in Section 2.1.Let Q and d be as before, and consider R = Rep( Q , d ) with the group action of G = Y j j =1 (cid:0) GL k ( P j ( n )) × GL k ( P j ( m )) (cid:1) . Definition 4.1 (The master space) . Define Y := [ σ ∈ Σ Q [ σ -ss] ⊂ R, where Q [ σ -ss] denotes the scheme-theoretic closure of Q [ σ -ss] in R .The following is our main result. Theorem 4.2 (A master space for moduli of multi-Gieseker semistable sheaves) . The affinescheme Y is a master space for the moduli spaces M σ as σ varies in Σ . More precisely, foreach σ ∈ Σ , we have M σ = Y// θ σ G := Y σ -ss //G. In particular, any two such moduli spaces are related by a finite number of Thaddeus-flips.Proof.
Fix σ ∈ Σ. Given the work from the previous section, the only issue is in gainingcontrol over those components of Y that do not lie in Q [ σ -ss] . To this end, let Z := Q [ σ -ss ].We claim that Y σ -ss = Z σ -ss . (4.1)Clearly, we have a scheme-theoretic inclusion Z σ -ss ⊂ Y σ -ss , since Z ⊂ Y is a closed embedding.To prove the other inclusion, it is enough to check that for any σ ′ ∈ Σ we have a scheme-theoretic inclusion Q [ σ ′ -ss] ∩ R σ -ss ⊂ Z σ -ss . (4.2) Observe first that by our choice of integers m , n , and p , Theorem 1.7(1) implies Q [ σ -ss] ⊂ R σ -ss .We claim that (4.2) is implied by the inclusion Q [ σ ′ -ss] ∩ R σ -ss ⊂ Q [ σ -ss] . (4.3)To see this, observe that (4.2) is not affected by any component of Q [ σ ′ -ss] that does not meetthe Zariski open set R σ -ss , and thus we may assume that Q [ σ ′ -ss] ∩ R σ -ss is Zariski dense in Q [ σ ′ -ss] , at which point (4.2) follows formally from (4.3).Now, the definition of Q [ σ ′ -ss] and Proposition 1.5(a) show that Q [ σ ′ -ss] is the base of a flatfamily of σ ′ -semistable ( p, L )-regular sheaves of topological type τ on X . Each sheaf in thisfamily that lies over a point of the intersection Q [ σ ′ -ss] ∩ R σ -ss corresponds to a σ -semistable A -module and thus by Theorem 1.7(1) is σ -semistable as a sheaf on X . Therefore, Proposition1.5(b) and the definition of Q [ σ -ss] imply that the inclusion Q [ σ ′ -ss] ∩ R σ -ss → R σ -ss factorsthrough Q [ σ -ss] , which proves (4.3) and as consequence also (4.2) and (4.1).Finally, from (4.1) and Corollary 2.4 we obtain Y// θ σ G ∼ = Y σ -ss //G ∼ = Z σ -ss //G ∼ = M σ , which concludes the proof of Theorem 4.2. (cid:3) Remark 4.3 (Sheaf-semistability vs. quiver-semistability) . Note that in contrast to [GRT14,Thm. 10.1] we do not claim that Y σ -ss = Q [ σ -ss] . As the stability parameter σ is degenerate,the embedding of Q [ σ -ss] into R σ -ss is not saturated with respect to the GIT-quotient R σ -ss → R σ -ss //G . This is intimately related to the fact that one needs the relevant stability parameterto be positive for Theorem 1.7(2) to hold.As discussed in the Introduction, as a consequence of Theorem 4.2 we obtain the desiredVGIT statement about the relation between Gieseker-moduli spaces of semistable sheaveswith respect to two choices of ample line bundles on a fixed base scheme X . Corollary 4.4 (Mumford-Thaddeus principle for Gieseker-moduli spaces) . Let L and L be ample line bundles on X . Then, the moduli spaces M L and M L of sheaves of a giventopological type that are Gieseker-semistable with respect to L and L , respectively, are relatedby a finite number of Thaddeus-flips.Proof. Without loss of generality we may assume L and L are very ample. Then, apply theprevious theorem to the stability parameters ( L , L , ,
0) and ( L , L , ,
1) for the quiver Q we obtain setting j = 2. Note that the required boundedness requirements are fulfilled dueto classical results about Gieseker-semistability, see for example [HL10, Theorem 3.3.7]. (cid:3) Remark 4.5 (VGIT for a finite number of moduli spaces) . It is obvious from the above thatwe can discuss any number j of ample polarisations and their induced Gieseker moduli spacessimultaneously using the quiver Q with the appropriate number j of rows.5. Intermediate Spaces and Uniformity
The master space construction above works for any finite collection of stability parameters.However it has a deficiency in that it does not identify the intermediate spaces that appearin the Thaddeus-flips occuring between different moduli spaces. This is discussed in detailin our previous work [GRT15], the upshot being that given a family ( σ ( t )) t ∈ [0 , of stabilityparameters (all taken with respect to the same vector L of ample line bundles), one can onlyreasonably expect to be able to control these intermediate spaces if one makes some additional MASTER SPACE FOR MODULI SPACES OF SHEAVES 17 assumptions that we discuss next. For this section we assume that X is smooth of dimension d and all the sheaves in question are torsion-free.Fix ample line bundles L j for j = 1 , . . . , j on X. We extend the notion of multi-Giesekerstability to allow twisting by a fixed collection of line bundles B j , j = 1 , . . . , j . Given thesedata, the multi-Hilbert polynomial with respect to σ ∈ ( R ≥ ) j \ { } of a coherent sheaf E isnow taken to be P σE ( m ) := X j σ j χ ( E ⊗ L mj ⊗ B j ) . (5.1)This twisting adds no new difficulties (see [GRT15]). We say a path σ : [0 , → R j ≥ \ { } , σ ( t ) = ( σ ( t ) , . . . , σ j ( t )) is a stability segment if each σ j is a linear function of t and X j vol( L j ) σ j ( t ) = 1 for all t ∈ [0 , , where vol( L j ) := R X c ( L j ) d . We say a stability segment ( σ ( t )) t ∈ [0 , is bounded if the set ofsheaves of a given topological type that are semistable with respect to σ ( t ) for some t ∈ [0 , Definition 5.1 (Uniform stability) . We say a stability segment ( σ ( t )) t ∈ [0 , is uniform if forevery torsion-free sheaf E and every t ∈ [0 ,
1] we have P j ( t ) σ j χ ( E ⊗ L kj ⊗ B j )rank( E ) = k d d ! + a d − ( E ) k d − + · · · + a ( E ) k + a ( E, t ) , where a d − ( E ) , . . . , a ( E ) are independent of t and a ( E, t ) is linear in t .Key to the notion of uniform stability segment is the following semicontinuity property : if E is a sheaf that is semistable with respect to σ ( t ) for all t < t then it is also semistable withrespect to σ ( t ) (this is easy to see, and discussed further in [GRT15, Remark 2.5]). Theorem 5.2 (Thaddeus-flips through moduli spaces of sheaves) . Let X be smooth andprojective, let τ be a topological type and ( σ ( t )) t ∈ [0 , be a bounded uniform stability segment.For t ∈ [0 , let M σ ( t ) be the moduli space of torsion-free sheaves on X of topological type τ that are σ ( t ) -semistable.Then given any t ′ , t ′′ in [0 , the moduli spaces M σ ( t ′ ) and M σ ( t ′′ ) are connected by a finitecollection of Thaddeus-flips of the form M σ ( t i ) $ $ ❏❏❏❏❏❏❏❏❏ M σ ( t i +1 ) y y ssssssssss M σ ( t ′ i ) . for some t i , t ′ i ∈ [0 , .Proof. We remark that the only difference between the statements of Theorem 5.2 and[GRT15, Theorem 2.6] is that here we allow the parameters t ′ and t ′′ to be among the endpoints t = 0 or t = 1. The crucial point is that the uniformity assumption on ( σ ( t )) t ∈ [0 , allows usto choose the integers m ≫ n ≫ p appearing in the proof of Theorem 4.2 in such a way thatthey work uniformly over all t ∈ [0 , σ (0) and σ (1) may not be positive stability parameters. (cid:3) References [ACK07] Luis ´Alvarez-C´onsul and Alastair King,
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E-mail address : [email protected] JR: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge,Wilberforce Road, Cambridge, CB3 0WB, UK
E-mail address : [email protected] MT: Institut de Math´ematiques ´Elie Cartan, Universit´e de Lorraine, B.P. 70239,54506 Vandoeuvre-l`es-Nancy Cedex, France
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