On some isomorphism of compactifications of moduli scheme of vector bundles

A morphism of the reduced Gieseker -- Maruyama moduli functor (of semistable coherent torsion-free sheaves) to the reduced moduli functor of admissible semistable pairs with the same Hilbert polynomial, is constructed. It is shown that main components of reduced moduli scheme for semistable admissible pairs ((\tilde S, \tilde L), \tilde E) are isomorphic to main components of reduced Gieseker -- Maruyama moduli scheme.


Introduction
In the present article S is a smooth irreducible projective algebraic surface over an algebraically closed field k of characteristic 0, O S is its structure sheaf, E coherent torsion-free O S -module, E ∨ := Hom OS (E, O S ) dual O S -module. In this case E ∨ is reflexive and, consequently, locally free. In the sequel we make no difference between locally free sheaf and the corresponding vector bundle, and both terms are used as synonyms. Let L be very ample invertible sheaf on S; it is fixed and will be referred to as the polarization. The symbol χ(·) denotes the Euler characteristic, c i (·) i-th Chern class. Also if Y ⊂ X be the locally closed subscheme of the scheme X, then Y be its scheme-theoretic closure in X. Remark 0.2. In the further considerations, if necessary, we replace L by its big enough tensor power. As shown in [7], this power can be chosen constant and fixed. All Hilbert polynomials are compute with respect to these new L and L correspondingly.
Definition 0.3. [7,8] S-(semi)stable pair (( S, L), E) is the following data: • S = i≥0 S i admissible scheme, σ : S → S morphism which is called canonical, σ i : S i → S its restrictions onto components S i , i ≥ 0; • E vector bundle on the scheme S; • L ∈ P ic S distinguished polarization; such that • χ( E ⊗ L m ) = rp(m), the polynomial p(m) and the rank r of the sheaf E are fixed; • the sheaf E on the scheme S is stable (respectively, semistable) in the sense of Gieseker, i.e. for any proper subsheaf F ⊂ E for m ≫ 0 • for each additional component S i , i > 0, the sheaf E i := E| Si is quasi-ideal, namely it has a description E i = σ * i ker q 0 /tors. (0.1) In the series of papers of the author [3] - [8] the projective algebraic scheme M is constructed as reduced moduli scheme for S-semistable pairs.
The scheme M contains an open subscheme M 0 which is isomorphic to the subscheme M 0 of Gieseker-semistable vector bundles, in Gieseker -Maruyama moduli scheme M for semistable torsion-free sheaves with Hilbert polynomial equal to χ(E ⊗ L m ) = rp(m). We make use of the Gieseker's definition of semistability.
if for any proper subsheaf F ⊂ E of rank r ′ = rank F for m ≫ 0 the following holds: .
Let E be a semistable locally free coherent sheaf. Then, obviously, the sheaf I = F itt 0 Ext 1 (E, O S ) is trivial and S ∼ = S. Consequently, (( S, L), E) ∼ = ((S, L), E) and there is a bijection M 0 ∼ = M 0 .
Let E be a semistable nonlocally free sheaf, then the scheme S contains reduced irreducible component S 0 such that σ 0 := σ| S0 : S 0 → S is a morphism of blowing up of the scheme S in the sheaf of ideals I = F itt 0 Ext 1 (E, O S ). Formation of the sheaf I is a way to characterize singularities of the sheaf E, i.e. its difference from local freeness. Indeed, the quotient sheaf κ := E ∨∨ /E is an Artinian sheaf and its length is not greater then c 2 (E), and is a sheaf of ideals of (in general case nonreduced) subscheme Z of bounded length [9] supported in a finite number of points on the surface S. Hence, as it is shown in [6], the rest irreducible components S i , i > 0 of the scheme S in general case can carry nonreduced scheme structure.
We assign to each semistable torsion-free coherent sheaf E a pair (( S, L), E) with ( S, L) defined as described before.
Let U be Zariski-open subset in one of components S i , i ≥ 0, and σ * E| Si (U ) be the corresponding group of sections which is a O Si (U )-module. Sections s ∈ σ * E| Si (U ) which are annihilated by prime ideals of positive codimension in O Si (U ), form a submodule in σ * E| Si (U ). This submodule will be denoted as tors i (U ). The correspondence U → tors i (U ) defines a subsheaf tors i ⊂ σ * E| Si . Note that associated primes of positive codimension which annihilate sections s ∈ σ * E| Si (U ), correspond to subschemes supported in the preimage σ −1 (Supp κ) = i>0 S i . Since by the construction the scheme S = i≥0 S i is connected, then subsheaves tors i , i ≥ 0, allow to form a subsheaf tors ⊂ σ * E. This subsheaf is defined in the following way. A section s ∈ σ * E| Si (U ) satisfies the condition s ∈ tors| Si (U ) if and only if • there exists a section y ∈ O Si (U ) such that ys = 0, • at least one of following two conditions is satisfied: either y ∈ p, where p is prime ideal of positive codimension; or there exist a Zariski-open subset V ⊂ S and a section s ′ ∈ σ * E(V ) such that V ⊃ U , s ′ | U = s, and s ′ | V ∩ S0 ∈ tors(σ * E| S0 )(V ∩ S 0 ). The torsion subsheaf tors(σ * E| S0 ) in the recent expression is understood in the usual sense.
In our construction the subsheaf tors ⊂ σ * E plays the role which is completely analogous to the role of the subsheaf of torsion for the case of reduced and irreducible base scheme. Since no ambiguity occur, the symbol tors anywhere in this article is understood as mentioned above. The subsheaf tors is called a torsion subsheaf.
In [7] it is proven that sheaves σ * E/tors are locally free. The sheaf E in the pair (( S, L), E) is defined by the formula E = σ * E/tors. In this case there is an isomorphism In [8] it is shown that the restriction of the sheaf E on each component S i , i > 0, is given by the condition of quasi-ideality (0.1) where q 0 : O ⊕r S ։ κ is an epimorphism defined by the exact triple 0 → E → E ∨∨ → κ → 0 and by local freeness of the sheaf E ∨∨ .
Resolution of singularities of a semistable sheaf E can be globalized in a flat family by the procedure developed in articles [4,5,7] in different settings. Let T be reduced irreducible quasiprojective scheme, E be a sheaf of O T ×S -modules, L invertible O T ×S -sheaf very ample relatively to T and such that L| t×S = L and χ(E⊗ L m | t×S ) = rp(m) for all closed points t ∈ T . Also suppose that T contains nonempty open subset T 0 such that E| T0×S is locally free O T0×S -module. Then the following data is defined: • T integral normal scheme obtained by some blowing up φ : T → T of the scheme T , • π : Σ → T flat family of admissible schemes with invertible O Σ -module L such that L| t×S is the distinguished polarization of the scheme π −1 (t), • E locally free O Σ -module and ((π −1 (t), L| π −1 (t) ), E| π −1 (t) ) is S-semistable admissible pair.
In this case there is a blowup morphism Φ : Σ → T × S, and (Φ * E) ∨∨ = φ * E. This follows from the coincidence of sheaves from the left hand side and from the right hand side, on the open subset out of a subscheme of codimension 3. Both sheaves are reflexive. The scheme T × S is integral and normal. The mechanism described is called in [7] as standard resolution.
In section 1 we recall definitions of reduced functors (f GM / ∼) of moduli of coherent semistable torsion-free sheaves ("the Gieseker -Maruyama moduli functor") and (f/ ∼) of moduli of admissible semistable pairs. The rank r and the polynomial p(m) are fixed and equal for both moduli functors.
In the present article we prove following results. 1 Morphism of moduli functors: proof of theorem 0.5 Following [2, ch. 2, sect. 2.2] we recall some definitions. Let C be a category, C o its dual, C ′ = F unct(C o , Sets) the category of functors to the category of sets. By Yoneda's lemma the functor C → C ′ : F → (F : X → Hom C (X, F )) includes C in C ′ as a full subcategory.
Let T be a scheme over the field k. Consider families of semistable pairs and a functor f : (Schemes k ) o → (Sets) from the category of k-schemes to the category of sets, assigning to a scheme T the set {F T }. The moduli functor (f/ ∼) assigns to a scheme T the set of equivalence classes ({F T }/ ∼).
The equivalence relation ∼ is defined as follows. Families (( π : In this paper we restrict ourselves by the consideration of the full subcategory RSch k of reduced schemes and of the reduced moduli functor (f red / ∼) = (f| (RSch k ) o / ∼) [9]. Since no ambiguity occur, we use the symbol (f/ ∼) for the reduced moduli functor.
In the general case results of the paper [7] provide existence of a coarse moduli space for any maximal under inclusion irreducible substack in T ∈Ob(RSch . Such pairs will be referred to as Spairs. We mention under M namely the moduli space of a substack containing semistable S-pairs and emphasize this speaking about main components of the moduli scheme.
Analogously, we mention the Gieseker -Maruyama scheme M as union of those components of reduced moduli scheme of semistable torsion-free sheaves, that contain locally free sheaves.
The Gieseker -Maruyama functor f GM : (Schemes k ) o → Sets is defined as follows: are said to be equivalent if there are invertible O T -sheaves L ′ and L ′′ such that for the projection p : T × S → T one has For this functor we use the convention which is analogous to the convention 1.3. The functor morphism t :  (respectively, F T ′ ) with some connected base T ′ and containing locally free sheaves (respectively, S-pairs) according to the fibres diagram In particular, this restriction excludes embedded components of moduli scheme from our consideration whenever these components do not contain locally free sheaves (respectively, S-pairs). Then it is enough to construct diagrams (1.1) only for families which contain locally free sheaves (respectively, S-pairs), where T is reduced scheme.
Let p : Σ T → T be a flat family of schemes such that its fibre is isomorphic to S, L T be a family of very ample invertible sheaves on fibres of the family p, E T be a flat family of coherent torsion-free sheaves on fibres of p. The sheaves are mentioned to have rank r and Hilbert polynomial rp(m) and to be semistable with respect to polarizations induced by the family L T . The application of the standard resolution leads to the collection of data ( π : Σ T → T , L T , E T ). Let Σ T := Σ T × T T where φ : T → T is birational morphism also provided by the standard resolution, and (φ, id S ) : Σ T → Σ T is the induced morphism.
Further, due to the considerations of [4,5,7] there is a partial morphism of functors t −1 : (f/ ∼) → (f GM / ∼) defined by the morphism σ σ : Σ T → Σ T and the operation (σ σ * −) ∨∨ on those families which are obtained by standard resolution from families of coherent semistable torsion-free sheaves. Then t −1 • t = id Sets . Since t −1 is defined only partially, it is impossible to claim that t is an isomorphism. Remark 1.5. Also, as it is shown in [7], there is a birational morphism of moduli schemes κ : M → M which are mentioned to be reduced [8]. The scheme M can be not normal. Hence, although κ is a bijective morphism and becomes a morphism of integral schemes if restricted on each of irreducible components, this does not imply that κ is an isomorphism.
In the further text we will show that there is a morphism of reduced Gieseker -Maruyama moduli functor to the reduced moduli functor of admissible semistable pairs. Namely, for any reduced scheme T the correspondence E T → (( Σ T , L T ), E T ) will be constructed. This correspondence defines the map of sets ({E T }/ ∼) → ({(( Σ T , L T ), E T )}/ ∼). This all means that for any family of semistable coherent sheaves E T which is flat over its base T one can build up a family (( Σ T , L T ), E T ) of admissible semistable pairs with the same base T .
The procedure of standard resolution yields in the family of admissible schemes π : Σ T → T which is flat over T , in the locally free O Σ T -sheaf E T and in the invertible O Σ T -sheaf L T , which is very ample with respect to the morphism π. Proposition 1.6. There exist • π : Σ T → T flat morphism, The proposition formulated implies the functor morphism of interest t : (f GM / ∼) → (f/ ∼). It is defined for any reduced scheme T ∈ Ob RSch k by the commutative diagram (1.1) T ' ' y y y y y y y y y y y y where the right vertical arrow is a morphism (mapping) in the category of sets. This mapping is defined by the proposition 1.6. The horizontal and the skew arrows are defined by functorial correspondences (f GM / ∼) and (f/ ∼) respectively.
Proof of proposition 1.6. For the construction of the scheme Σ T we assume that m ≫ 0 is as big as the sheaf of O T -modules π * ( E T ⊗ L m T ) is locally free, the canonically defined morphism π * π * ( . Also there is a (relative) Plücker immersion of the Grassmannian bundle into the (relative) projective space

Besides consider the isomorphism of
Consequently, formation of exterior powers and passing to projectivizations and to Grassmannian bundles induces the fibred diagram It is compatible with Plücker embeddings in the sense that the square The immersion here is induced by the sheaf π * ( E T ⊗ L m T ) ⊗ L ′ ; the morphism g comes from the previous diagram. To convince that the scheme Σ ′ T is flat over T , it is enough to verify the coincidence of images for fibres of the scheme Σ T over closed points of the fibre φ −1 (t) for each closed point t ∈ T . But by the construction, images of all fibres of the scheme Σ T over points of the subscheme φ −1 (t), have equal collections of local equations in fibres of the projective bundle P ( r p * (E T ⊗ L m T )). Consequently, they are also given by equal local equations in fibres of the Grassmannian bundle G(p * (E T ⊗ L m T ), r). This completes the proof of flatness of Σ ′ T over T . Also it is clear that the morphism φ ′ : Σ T → Σ ′ T of the scheme Σ T onto its image becomes an isomorphism when restricted to the open subset π −1 (T 0 ). T 0 is open subscheme of the scheme T where φ is an isomorphism. This implies that the morphism φ ′ is birational. Now turn to closed immersions j T : Σ T ֒→ G( π * ( E T ⊗ L m T ) ⊗ L ′ ), r) and j : Σ ′ T ֒→ G(p * (E T ⊗ L m T ), r). We denote by the symbol S T the universal (locally free) quotient sheaf of rank r on G(p * (E T ⊗ L m T ), r). Also the symbol S T denotes the universal (locally free) quotient sheaf of rank r on G( π * ( E T ⊗ L m T ) ⊗ L ′ ), r). It is clear that by the construction (1.2, 1.3) of Grassmannian bundles of our interest we can write S T = g * S T . Then j * T S T = E T ⊗ L m T ⊗ π * L ′ , and j * T S T is a locally free sheaf on the scheme Σ ′ T . Consider the invertible O Σ T -sheaf L T providing the distinguished polarization on fibres of the morphism π : Σ T → T , and its direct image π * L T . The recent sheaf is locally free by the choice of the sheaf L T . Also take (any) invertible O ΣT -sheaf L T which is very ample relatively to the projection p : Σ T → T and such that sheaves L T and L T induce equal polarizations on fibres over points of open subscheme T 0 . The sheaf p * L T is also locally free. Since the Hilbert polynomial on fibres of morphisms π and p is constant over the base and the morphism Σ ′ T → Σ T is birational, ranks of locally free sheaves π * L T and p * L T are equal. Moreover, their restrictions on the open subscheme T 0 ⊂ T are isomorphic. Then there exists an invertible O T -sheaf L ′′ such that π * L T ⊗ L ′′ = φ * p * L T = p * (φ, id S ) * L T . Now consider the relative projective space P ( π * L T ⊗ L ′′ ) together with closed immersion i : Σ T ֒→ P ( π * L T ⊗ L ′′ ). Also take the relative projective space of same dimension P (p * L T ). Then the square is Cartesian. Denote by Σ ′′ T the image of the composite map Σ T ֒→ P ( π * L T ⊗ L ′′ ) → P (p * L T ). By the construction the scheme Σ ′′ T is flat over T . Denote by j : Σ ′′ T ֒→ P (p * L T ) the corresponding closed immersion of this subscheme; let O P (1) be the canonical invertible sheaf on the projective bundle P (p * L T ). Then define L ′′ T := j * O P (1). Now form the fibred product G(p * (E T ⊗ L m T ), r) ⊗ T P (p * L T ) and consider the mapping . This map is induced by mappings into each factor constructed earlier. Let Σ T be the scheme image of this map. Then there is the following commutative diagram: Birational morphisms p ′ , p ′′ are defined by projections of the product G(p * (E T ⊗L m T ), r)× T P (p * L T ) to each factor.
It rests to define sheaves L T and E T on the scheme Σ T by formulas: If T is an integral and normal scheme then the functorial correspondence which has been constructed is invertible. Then the class of schemes where the partial functor t −1 is defined, contains all integral and normal schemes. The immersion j is defined non-uniquely but up to isomorphy class H 0 ( S, E ⊗ L m ) ∼ −→ V modulo multiplication by nonzero scalars ϑ ∈ k * . Let P (m) = χ(j * O G(V,r) (m)) be the Hilbert polynomial of the subscheme j( S) ⊂ G(V, r). Hence the point corresponding to the subscheme j( S) ⊂ G(V, r), is defined in the Hilbert scheme Hilb P (t) G(V, r) up to the action of the group P GL(V ).
In [7] it is shown that the data Q, L, E Q defines the morphism µ : Q → Hilb P (m) G(V, r). Also it is proven there that there is a morphism of projective schemes κ : M → M .
For the further consideration we also need the notion of M-equivalence for semistable admissible pairs. This notion is introduced and investigated in [7].
We construct for any reduced scheme B and for a natural transformation ψ ′ : f → F ′ the unique natural transformation ω : M → F ′ such that ψ ′ = ω • ψ. The natural transformation ψ ′ corresponds to a flat family π : Σ → B with fibrewise polarization L and equipped with the family of locally free sheaves E. It is enough to restrict ourselves by the case when B is integral scheme. Indeed, any reduced scheme can be represented as union of irreducible components. Also we take into account the remark 1. 4 Restriction on any fibre π −1 (b) of the morphism π yields in an object ((π −1 (b), L| π −1 (b) ), E| π −1 (b) ) of the class F. Then there is a morphism into the Grassmannian bundle Σ → G(π * ( E⊗ L m ), r). This morphism becomes an immersion j b : π −1 (b) ֒→ G(H 0 (π −1 (b), E ⊗ L m | π −1 (b) ), r) when restricted to fibres of the morphism π.
The sheaf π * ( E ⊗ L m ) is locally free and then the Grassmannian bundle G(π * ( E ⊗ L m ), r) is locally trivial over B. Let i B i = B be Zariski-open trivializing cover. Subfamilies Σ i are given by fibred squares where horizontal arrows are open immersions. Fix isomorphisms of trivialization τ i : defines a morphism of the base into Hilbert scheme µ i : B i → Hilb P (t) G(V, r). The morphism µ i factors through the subscheme µ( Q). For universal scheme Univ P (m) G(V, r) over Hilbert scheme Hilb P (m) G(V, r) we have the fibred diagram Univ P (m) G(V, r)| µ( Q) ' ' y y y y y y y y y y y y y y / / Hilb P (m) G(V, r) µ( Q) Define schemes B iQ and Σ iQ as fibred products B iQ := B i × µ( Q) Q and Σ iQ := Σ i × Q B iQ . Let β : Σ iQ → Σ i be the projection onto the factor. Define E iQ := β * E i . Then there is a morphism Φ i : Σ iQ → B iQ ×S obtained by the fibred product from the morphism Φ : Σ Q → Q×S. By Serre's theorem and by choice of the invertible sheaf L one has an epimorphism π * π * ( E iQ ⊗ L m ) ⊗ L −m ։ E iQ . Let B 0 be the open subset of the scheme B, where Σ is locally trivial and a fibre is isomorphic to S. Refining the cover {B i } if necessary, we come to epimorphisms V ⊗ L −m ⊠ O Bi0 ։ E iQ | Bi0 on overlaps B i0 = B 0 ∩ B i . Then there is a morphism q i0 : B i0 → Quot rp(m) (V ⊗ L −m ). Since E iQ | Bi0 = E iQ | Bi0 are families of semistable locally free sheaves, the morphism q i0 factors through the subscheme Q ′ . Form a closure q i0 B i0 ⊃ q i0 B i0 in the scheme Q ′ , and a product B i ×q i0 B i0 . Let B i be a closure of the subset of points of the view (b, q i0 (b)), b ∈ B i0 in the product B i × q i0 B i0 . The projection ρ : B i → B i onto the first factor is a birational morphism of integral schemes. Introduce the notation for the composite map q i :