Moduli spaces of stable sheaves on schemes : Restriction theorems, boundedness and the GIT construction.

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Contents 1

Stable sheaves 1 1 What is the moduli of vector bundles? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Definition of stable sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Generalities on boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Openness of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 1 2 3 4 5 6 7

Restriction theorems and boundedness Restriction of Harder-Narasimhan filtration . . . . . . . . . . . . . . . . . . . . . . . . . Theorem of Grauert-M¨ ulich-Spindler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statements of boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundedness of a family of torsion free sheaves . . . . . . . . . . . . . . . . . . . . . Tensor product of semi-stable sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem of Mumford-Mehta-Ramanathan . . . . . . . . . . . . . . . . . . . . . . . . . . Simpson’s stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 4 5 6 7 8 9

Construction of Moduli Spaces 81 Construction of Quot-schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Geometric invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 S-equivalence and e-semi-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 General setting and a fundamental lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Moduli spaces of stable sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Moduli spaces of semi-stable sheaves : char 0 . . . . . . . . . . . . . . . . . . . . . . . 120 Closed orbits of a Grassmann variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Langton’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Moduli spaces of semi-stable sheaves – general case . . . . . . . . . . . . . . . . . 142

3

24 24 31 37 42 49 61 72

Appendix A On Langer’s work 147 1 Boundedness of semi-stable sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2 Dimension estimate of global sections of torsion free sheaves . . . . . . . . 148 Appendix B

Some properties of the moduli

149

Bibliography

153

Glossary of Notations

154

xi

Chapter 1

Stable sheaves First we shall show that if we collect all the vector bundles on a projective variety and require a weak universal property of a moduli space, then there does not exist the moduli space. This motivates us to introduce the notion of stability and semi-stability. The idea of Harder-Narasimhan filtration plays a crucial role sometimes behind strong results and sometimes very explicitly. Two of basic results on boundedness are proved in the section 3. The formulation of the first is due to L. S. Kleiman [K2] and the second is a theorem of Grothendieck [G]. We shall show a beautiful application of the second result in the proof of the openness of stability.

Definition of quot-schemes We define quot-schemes. The construction will be given in Chapter III. Let f : X → S be a morphism of noetherian schemes and F a coherent sheaf on X. For an S-scheme T , we set QuotF/X/S (T ) = {FT → E → 0 | E is flat over T }/ ∼, where FT = F ⊗OS OT is the pull-back of F to X × T and where two members are equivalent under the relation ∼ if they are isomorphic as quotient sheaves of FT . This defines a contravariant functor QuotF/X/S of the category (Sch/S) of locally noetherian S-schemes to that of sets. Assuming f to be projective, we shall fix an f -ample invertible sheaf OX (1). If FT → E is a member of QuotF/X/S (T ), then the flatness of E over T and the invariance of Hilbert polynomials of a flat family tell us that T is the direct sum of subschemes T P such that the Hilbert polynomial of E(t) is P (m) on every fiber of XT over T P . Especially if the functor QuotF/X/S is represented by a pair (Q, G) of a locally noetherian S-scheme Q and the universal quotient sheaf FQ → G, then Q is the direct sum of QP . For a numerical polynomial P , we define QuotP F/X/S (T ) to be the subset of QuotF/X/S (T ) consisting of quotient sheaves of FT with Hilbert polynomial P on each fiber of XT over T . Then we have a subfunctor QuotP F/X/S of QuotF/X/S . What we have seen in the above is that QuotF/X/S is representable if and only if so is QuotP F/X/S is representable for all P . Moreover, in this case the ` ` pair Q = P QP and G = P GP represents QuotF/X/S , where (QP , GP ) does QuotP F/X/S . We shall prove in Theorem III 1.5 that if f : X → S is a projective morphism of noetherian schemes and F is a coherent sheaf on X, then QuotP F/X/S is representable by a projective S-scheme, which we denote by QuotP F/X/S . 1

2

Chapter 1

STABLE SHEAVES

By studying infinitesimal deformations of a quotient sheaf, we can obtain a description of the Zariski tangent space of a quot-scheme. We here give the description without the proof. Let X be a projective scheme over S = Spec k with k a field, and let F a coherent sheaf on X. Consider a point x ∈ Q = QuotF/X/S and let K be the residue field of OQ,x . The point x corresponds to a surjective homomorphism α : FK → E of coherent sheaves on XK . Put K = ker(α). Then the Zariski tangent space HomK (mx /m2x , K) of Q at x is isomorphic to HomOXK (K, E).

1 What is the moduli of vector bundles? Let X be a non-singular projective variety over an algebraically closed field k and VB(X) be the set of isomorphism classes of algebraic vector bundles on X. The problem of moduli of algebraic vector bundles is intuitively to endow VB(X) with a natural structure V BX of scheme. What is the meaning of a natural structure then? We require at least V BX to have the following property: (1.1) Let S be an algebraic k-scheme and E a vector bundle on X ×k S. Then the map S(k) 3 s 7→ [E(s)] = [E ⊗OS k(s)] ∈ VB(X) is induced by a morphism of S to V BX , where [ ] denotes the isomorphism class. Fix an ample line bundle OX (1) on X. Let H be a numerical polynomial of degree = dim X. Pick two vector bundles E1 and E2 on X with χ(E1 (m)) = χ(E2 (m)) = H(m). If m is a sufficiently large integer, then Ei (m) has the following property: (1.2) Ei (m) is generated by its global sections and H q (X, Ei (m)) = 0 for all q ≥ 1. By replacing Ei by Ei (m), we may assume that Ei itself has the property. Let us take a look at the quot-scheme Q = QuotH V ⊗k OX /X/k , where V is a vector space of dimension N = H(0) = dim H 0 (X, Ei ). If θ : V ⊗k OX×Q → F is the universal quotient sheaf on X ×k Q, then there exists an open set U of Q such that U (k) is exactly the set {y ∈ Q(k) | F (y) = F ⊗OQ k(y) has the following properties (a), (b) and (c) }: (a) F (y) is locally free, (b) θ(y) : V ⊗k OX → F (y) induces an isomorphism Γ(θ(y)) : V = H 0 (X, V ⊗k OX ) → H 0 (X, F (y)), (c) H q (X, F (y)) = 0 for all q ≥ 1. By abuse of notation we denote the restriction of the universal quotient sheaf to X ×k U by θ : V ⊗k OX×U → F . Through the natural action on V , G = GL(V ) acts on Q and U is a G-invariant open subscheme. It is easy to see that the center Gm of G acts trivially on Q and hence G = G/Gm acts on U . For a point y ∈ U (k), an automorphism σ of F (y) induces an element σ 0 of G by the property (b) :

1

WHAT IS THE MODULI OF VECTOR BUNDLES?

3

Γ(θ(y))

V −−−−−→ H 0 (X, F (y))   Γ(σ)  σ0 y y Γ(θ(y))

V −−−−−→ H 0 (X, F (y)) Obviously, σ 0 is an element of the stabilizer group StG (y) of G at y. Conversely, τ is an element of StG (y) if and only if there is an automorphism τ 0 of F (y) which makes the following diagram commutative θ(y)

V ⊗k OX −−−−→ F (y)     0 τy yτ θ(y)

V ⊗k OX −−−−→ F (y) Thus τ gives rise to an automorphism of F (y). Thanks to (b) again, τ 0 is not ∼ the identity unless τ = id. We see therefore that StG (y) → Aut(F (y)) for every y ∈ U (k). The image of Gm by this isomorphism is the multiplications by elements of k × on F (y), that is, (1.3)

∼

StG (y) −→ Aut(F (y))/k × .

An observation similar to the above shows that (1.4) for y1 , y2 ∈ U (k), both y1 and y2 belong to the same orbit of G if and only if F (y1 ) ∼ = F (y2 ). Now, by (1.2) for Ei with m = 0, we have a surjective ηi : V ⊗k OX → Ei such that Γ(ηi ) : V → H 0 (X, Ei ) is bijective. The universal property of Q provides us with points z1 , z2 of Q(k) such that F (zi ) ∼ = Ei and θ(zi ) ∼ = ηi . Both z1 and z2 are in U (k) because of (1.2). Assume that V BX exists. Then, by the property (1.1) for V BX , we obtain a morphism f : U → V BX . For each k-rational point y of U , f −1 f (y) is the G-orbit of y by (1.4). On one hand, dimy f −1 f (y) is upper semi-continuous on U by a theorem of Chevalley. On the other hand, dimy f −1 f (y) = dim o(y) = dim G−dim StG (y) = N 2 −1−dim StG (y) is lower semi-continuous. Thus dim StG (y) is constant over each connected component of U . By this and (1.3) we see that dimk EndOX (F (y)) = dim Aut(F (y)) depends only on the connected component containing y. We have therefore (1.5) if dimk EndOX (E1 ) 6= dimk EndOX (E2 ), then z1 and z2 belong to different connected components of U . Let us give an example of a family of vector bundles which are parameterized by an irreducible curve but whose spaces of endomorphisms jump at a special point.

4

Chapter 1

STABLE SHEAVES

Example 1.6. Assume that dim X ≥ 2. Let T be a non-singular subvariety of codimension 1 in X and D a very ample divisor on T . Assume (1.6.1) H 0 (T, OX (T ) ⊗OX OT (−D)) = 0, (1.6.2) dim H 0 (T, OT (D)) ≥ dim X + 1 = n + 1. Choose an (n + 1)-ple (s0 , . . . , sn ) of elements of H 0 (T, OT (D)) so generally that Tn−1 s0 , . . . , sn are linearly independent over k and i=0 (si )0 = ∅. Set Z = X ×k A1 and S = T ×k A1 . On S we have a homomorphism α : OSn+1 → p∗ (OT (D)) such that on Su = T × u, u ∈ A1 , α(u) is defined by Γ(α(u))(ei ) = si (0 ≤ i ≤ n − 1) and Γ(α(u))(en ) = usn , where p : S → T is the projection and {e0 , . . . , en } is a fixed Tn−1 free basis of OSn+1 . By the fact that i=0 (si )0 = ∅ we see that α(u) is surjective for all u and hence α is surjective by Nakayama’s lemma. Put K = ker(α). Then K is a vector bundle of rank n on S. Dualizing K ,→ OSn+1 and composing it with the n+1 n+1 natural OZ → OSn+1 , we have a surjective homomorphism β : OZ → K ∨ . Put n+1 0∗ E = ker(β)⊗OZ p OX (T ), so called, an elementary transform of OZ along K ∨ and it is well-known that E is a vector bundle of rank n+1 on Z, where p0 : Z → X is the projection. At the origin 0 ∈ A1 , E(0) ∼ = E 0 ⊕ OX , whence dim EndOX (E(0)) ≥ 2. 1 Let us show that if u ∈ A is not zero, then EndOX (E(u)) ∼ = k. By a property of elementary transform, there is a short exact sequence ϕ

n+1 0 → OX − → E(u) → OX (T ) ⊗OX OT (−D) → 0.

By (1.6.1), the map Γ(ϕ) : k n+1 → H0 (X, E(u)) is bijective. It follows that if γ ∈ EndOX (E(u)), then we have a commutative diagram ϕ

n+1 OX −−−−→ E(u)    γ Γ(γ)y y ϕ

n+1 OX −−−−→ E(u).

From this we obtain a commutative diagram OT (−D) −−−−→ OTn+1     Γ(γ)y y OT (−D) −−−−→ OTn+1 , where the left vertical arrow is the multiplication of some a ∈ k. Since s0 , . . . , sn−1 , usn are linearly independent, Γ(γ) is also the multiplication of a. This implies that γ is the multiplication of a. Let us consider the E in the above example. If m is sufficiently large, then for E(m) = E ⊗OX OX (m) and for all u in A1 , E(m)(u) is generated by its global sections and H i (X, E(m)(u)) = 0 for all positive i. By virtue of the base change theorem of cohomologies, q∗ (E(m)) is locally free, the natural map ψ : q ∗ q∗ (E(m)) → E(m)

2

DEFINITION OF STABLE SHEAVES

5

is surjective and N ∼ k ⊕N ∼ ) = H 0 (X, q ∗ q∗ (E(m))(u)) → H 0 (X, E(m)(u)) = H 0 (X, OX

is bijective, where q is the projection of Z to A1 . Since every vector bundle on A1 is trivial, q∗ (E(m)) ∼ = V ⊗k OA1 with V a k-vector space. We obtain therefore a surjective ψ : V ⊗k OZ → E(m). What we have shown implies that there is ∗ a morphism g : A1 → QuotH V ⊗k OX /X/k such that (1X × g) (θ) is isomorphic to ψ : V ⊗k OZ → E(m), where H is the Hilbert polynomial of an E(m)(u). g(A1 ) is a subset of the open set U for QuotH V ⊗k OX /X/k . Set E2 = E0 and E1 = E(u) with u 6= 0. Then z1 = g(u) and z2 = g(0) belong to the same connected component of U . This violates (1.5) because dimk EndOX (E1 ) = 1 < 2 ≤ dimk End OX (E2 ). For every pair (X, T ), there are D’s which satisfy the conditions (1.6.1) and (1.6.2). Moreover, when dim X = 1, the construction of the family of vector bundles on X as in Example 1.6 is much easier. We see therefore the following. Theorem 1.7. If X is a non-singular projective variety with positive dimension, then there does not exist V BX with the property (1.1). The above theorem tells us that to get a moduli space of vector bundles on a projective variety X we have to restrict ourselves to a good subfamily of VB(X). What is the good family then?

2 Definition of stable sheaves Throughout this section we shall work on a field k which may not be algebraically closed. We fix an algebraic closure k¯ of k. An algebraic scheme X over k is said to be geometrically integral if X ⊗k k¯ is integral. It seems more natural to study our moduli problem in a category wider than that of vector bundles. Definition 2.1. Let X be an integral algebraic scheme over k and E a coherent sheaf on X. (1) There is a non-empty open set U of X over which E is locally free. The rank of E|U is said to be the rank of E and denoted by r(E). (2) E is said to be torsion free if the natural map of E to E ⊗OX K(X) is injective, where K(X) is the sheaf of function field of X. If X is quasi-projective, then the definition of torsion free sheaves is simpler and clearer. Lemma 2.2. Assume that X is a quasi-projective, integral scheme. A coherent sheaf E on X is torsion free if and only if E is a subsheaf of a coherent locally free sheaf F (that is, a vector bundle) on X. In this case F can be chosen to be of rank r(E). Proof. Assume that there is a vector bundle F on X and an injection α of E to F . Since the sheaf K(X) contains OX , F is a subsheaf of F ⊗OXK(X). The composition α of E → E ⊗OX K(X) with α ⊗OX K(X) is the same as E −→ F → F ⊗OX K(X).

6

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STABLE SHEAVES

Since the latter is injective, we see that the natural map E → E ⊗OX K(X) is injective and hence E is torsion free. Conversely, if E is torsion free, then the natural map of E to its double dual (E ∨ )∨ is injective because E has no local sections supported by proper closed sets. Taking an ample line bundle OX (1) on X, E ∨ (m) = E ∨ ⊗OX OX (m) is generated by its global sections. Then, if we pick sufficiently general members s1 , . . . , sr of H 0 (X, E ∨ (m)) with r = r(E) = r(E ∨ ), then they define a generically surjective map β : OV⊕r → E ∨ (m). Dualizing β and tensoring OX (m), we have an injection of (E ∨ )∨ to OX (m)⊕r because (E ∨ )∨ is torsion free. As a simple application of the above lemma we see that the restriction of a torsion free coherent sheaf to a general hyperplane section is again torsion free. Indeed, we have the following result. Corollary 2.3. Let E be a torsion free coherent sheaf on a quasi-projective integral scheme X over an infinite field k, L a line bundle on X and let W be a k-vector subspace of H 0 (X, L) which generates L. Assume that for a general member s of W , the zero scheme Z(s) of s is integral. Then, for a sufficiently general member s of W , the restriction E|Z(s) = E ⊗ OZ(s) of E to Z(s) is torsion free. Proof. Since X is quasi-projective and E is torsion free, E is a subsheaf of a vector bundle F by the above lemma. Let C be the quotient sheaf F/E. If s is a sufficiently general member of W , then Z(s) is integral and does not contain any point of Ass(C) because W generates L and k is infinite. Using the section s, we have the following exact commutative diagram 0   y

0   y ×s

E −−−−→ E|Z(s) −−−−→ 0    γ y y

×s

F −−−−→ F |Z(s) −−−−→ 0     y y

×s

C −−−−→ C|Z(s) −−−−→ 0     y y

0 −−−−→ E ⊗ L∨ −−−−→   y 0 −−−−→ F ⊗ L∨ −−−−→   y 0 −−−−→ D −−−−→ C ⊗ L∨ −−−−→   y 0

0

0

Thanks to the way of choice of s, D must be zero and then the snake lemma implies that γ is injective. Since Z(s) is integral, Lemma 2.2 implies that E|Z(s) is torsion free. Let us fix a pair (X, OX (1)) of a geometrically integral projective scheme X over k and an ample line bundle OX (1) on X. Assume that X is of dimension n.

2

DEFINITION OF STABLE SHEAVES

7

For a coherent sheaf E of rank r on X, there are integers a1 (E), a2 (E), . . . , an (E) Pn such that the Hilbert polynomial χ(E(m)) = i=0 (−1)i dim H i (X, E(m)) can be written in the form   n−1   X m+n m+i χ(E(m)) = rd + an−i (E) (2.4) n i i=0 where d is the degree of X with respect to OX (1). We introduce the following notation which plays a key role in the sequel. Definition 2.5. Let E be a coherent sheaf on a geometrically integral projective scheme X over k and let OX (1) be an ample line bundle on X. Assume that r = r(E) is positive. (1) For the integer a1 (E) in (2.4), we define µ0 (E) to be the rational number a1 (E)/r. (2) We denote the polynomial χ(E(m))/r by PE (m). By Riemann-Roch theorem we have Lemma 2.6. If X is non-singular, then we have a1 (E) = d(E, OX (1)) −

rd(KX , OX (1)) rd(n + 1) − , 2 2

where for a coherent sheaf F on X, d(F, OX (1)) is the degree of the first Chern class of F with respect to OX (1) and where KX is the canonical sheaf of X. Now a usual definition of µ is as follows. Definition 2.7. Let (X, OX (1)) and E be as in Definition 2.5. If X is, in addition, smooth, then we define a rational number µ(E) to be µ(E) =

d(E, OX (1)) . r

The above lemma shows that the difference µ(E) − µ0 (E) is d(KX , OX (1))/2 + d(n + 1)/2, which depends only on the pair (X, OX (1)). Now we are ready to introduce the notion of stability of coherent sheaves. Definition 2.8. Let (X, OX (1)) be a pair of a geometrically integral projective scheme X over k and an ample line bundle OX (1) on X and let E be a coherent OX -module. (1) E is said to be µ-stable (or, µ-semi-stable) if E is torsion free and for every coherent subsheaf F of E ⊗k k¯ with 0 < r(F ) < r(E), we have the inequality µ0 (F ) < µ0 (E) (or, µ0 (F ) ≤ µ0 (E), resp.). (2) E is said to be stable (or, semi-stable) if E is torsion free and for every coherent subsheaf F of E ⊗k k¯ with 0 < r(F ) < r(E) and for all sufficiently large integers m, we have the inequalities PF (m) < PE (m) (or, PF (m) ≤ PE (m), resp.).

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STABLE SHEAVES

¯ Remark 2.9. (1) In Definition 2.8 we may take only F with E ⊗k k/F is torsion ¯ , then for the inverse image F 0 of free. Indeed, if T is the torsion part of E ⊗k k/F ¯ we obtain χ(F 0 (m)) = χ(F (m)) + χ(T (m)), deg χ(T (m)) < n and the T to E ⊗k k, leading coefficient of χ(T (m)) is positive. Thus the inequality for F 0 implies that of F . (2) By Lemma 2.6 we see that if X is smooth, then the above definition of the µ-stability and µ-semi-stability is the same as the usual, that is, we can use µ instead of µ0 to define the µ-(semi-)stability. (3) Directly from our definition we have the following implications: µ-stability w w 

=⇒

stability w w 

µ-semi-stability

⇐=

semi-stability

One of the main aims of this book is to construct moduli spaces of stable sheaves and then, from the viewpoint of (1.5), (5) in the following is an evidence that the stable sheaves must form a good family to construct their moduli space. Proposition 2.10. Let (X, OX (1)) be as in Definition 2.8 and let E and F be coherent OX -modules. (1) If both E and F are µ-semi-stable and if µ0 (E) > µ0 (F ), then every homomorphism of E to F is 0. (2) If E is µ-stable, F is µ-semi-stable and if µ0 (E) = µ0 (F ), then every nonzero homomorphism of E to F is injective. (3) If both E and F are semi-stable and if PE (m) > PF (m) for all sufficiently large m, then every homomorphism of E to F is 0. (4) If E is stable, F is semi-stable and if PE (m) = PF (m), then every non-zero homomorphism of E to F is injective and F/E is semi-stable. (5) If E is stable, then E is simple, that is, EndOX (E) is isomorphic to k which is naturally embedded in EndOX (E) as the multiplication by constants. (6) Assume that E is an extension of torsion free coherent sheaves 0 −→ E1 −→ E −→ E2 −→ 0 such that PE (m) = PE1 (m) = PE2 (m) (or, µ0 (E) = µ0 (E1 ) = µ0 (E2 )). Then E is semi-stable (or, µ-semi-stable, resp.) if and only if so are E1 and E2 . Proof. We shall prove only (3), (4) and (5). The proofs of (1) and (2) are obtained from those of (3) and (4) by replacing the polynomials PE (m), PF (m) with µ0 (E), µ0 (F ). (6) is obvious by the definition. (3) Let α be a non-zero homomorphism of E to F and G the image of α. The semi-stability of E implies that for K = ker(α) and all sufficiently large m, PK (m) ≤ PE (m) and hence PG (m) =

χ(E(m)) − χ(K(m)) r(E)PE (m) − r(K)PE (m) ≥ = PE (m) r(G) r(G)

2

DEFINITION OF STABLE SHEAVES

9

for all sufficiently large m. On the other hand, we have that PG (m) ≤ PF (m) for all sufficiently large m because G is a subsheaf of F . Then, by our assumption, PG (m) < PE (m) for all sufficiently large m. This is a contradiction. Thus α must be 0. (4) Let α, G and K be as in the proof of (3). By the semi-stability of F and our assumption we get PG (m) ≤ PF (m). Then a computation similar to the above shows that PK (m) ≥ PE (m). Then the stability of E implies that K is 0 and hence α is injective. If F/E is not torsion free, then we obtain a coherent subsheaf E 0 of F with PE 0 (m) > PF (m) as in Remark 2.9. Thus F/E must be torsion free and then F/E is semi-stable by (6). ¯ ∼ (5) Since EndO (E ⊗k k) = EndO (E) ⊗k k¯ and since we have only to prove X

X

dimk EndOX (E) = 1, we may assume that k is algebraically closed. Pick a non-zero element α of EndOX (E). Then, by virtue of (4) α is injective. Since χ(coker(α)(m)) = 0, we see that coker(α) = 0 or α is surjective. Thus every non-zero endomorphism of E is an isomorphism. Take a k-rational point x of X. α induces a linear transformation α(x) of the vector space E(x). Let λ be an eigen value of α(x). Then β = α − λ id cannot be an isomorphism because β(x) is not surjective. Therefore, β must be 0, which implies that α is the multiplication by λ. Harder-Narasimhan filtration affords us a tool to measure how far a torsion free coherent sheaf is from the semi-stability. Proposition 2.11. Let (X, OX (1)) be as in Definition 2.8 and let E be a torsion free coherent OX -module. (1) There is a unique filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα = E by coherent subsheaves such that (a) Ei /Ei−1 is µ-semi-stable for 0 < i ≤ α and (b) µ0 (Ei /Ei−1 ) > µ0 (Ei+1 /Ei ) for 0 < i < α. (2) There is a unique filtration 0 = E00 ⊂ E10 ⊂ · · · ⊂ Eβ0 = E by coherent sub0 0 sheaves such that (c) Ei0 /Ei−1 is semi-stable for 0 < i ≤ α and (d) PEi0 /Ei−1 (m) > 0 0 PEi+1 /Ei (m) for all sufficiently large integer m and for 0 < i < β. Definition 2.12. The filtration in (1) (or, (2)) of Proposition 2.11 is called the Harder-Narasimhan filtration by µ-semi-stability (or, semi-stability, resp.). Proof of Proposition 2.11. We shall prove only (1) because the idea of the proofs of (1) and (2) is common. Let us prove the uniqueness by induction on r(E). If E is µ-semi-stable, especially if r(E) = 1, then there is nothing to prove. Assume that E is not µ-semi-stable and hence we have a filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα = E with the properties (a) and (b) and with α > 1. Suppose that we have another filtration 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fβ = E with the properties (a) and (b). We may assume, without losing any generality, that µ0 (F1 ) ≥ µ0 (E1 ). Let i be the smallest integer such that Ei contains F1 . If i > 1, then the composition F1 ,→ Ei → Ei /Ei−1 is a non-zero homomorphism. On the other hand, since µ0 (F1 ) ≥ µ0 (E1 ) > µ0 (Ei /Ei−1 ), Proposition 2.10, (1) implies that the composition

10

Chapter 1

STABLE SHEAVES

must be 0. This is a contradiction. Hence i = 1, which means that F1 is a subsheaf of E1 . By the µ-semi-stability of E1 we have the equality µ0 (F1 ) = µ0 (E1 ). Then, changing the role of the two filtrations, we know that E1 ⊂ F1 . Thus E1 = F1 . By our induction hypothesis we see that α = β and Fi /E1 = Ei /E1 for 1 < i ≤ α. This completes the proof of the uniqueness. To prove the existence, assume first that k is algebraically closed. If E is µsemi-stable, then our assertion is obvious. Assuming that E is not µ-semi-stable, set µ = max{ µ0 (F ) | 0 6= F ⊂ E }. Then µ is finite by Proposition 3.12 which will be proved later. Let E1 be a maximal coherent subsheaf of E with µ0 (E1 ) = µ. By the maximality of E1 , E/E1 is torsion free and by the maximality of µ, E1 is µ-semi-stable. By using induction on r(E), we may assume that E/E1 has a filtration with the properties (a) and (b). Let E1 ⊂ E2 ⊂ · · · ⊂ Eα = E be the lift of the filtration of E/E1 to E. We have to prove µ0 (E1 ) > µ0 (E2 /E1 ). Assuming the contrary, we shall lead a contradiction. Since µ0 (E1 ) ≤ µ0 (E2 /E1 ), we get r(E2 )µ0 (E2 ) = r(E1 )µ0 (E1 ) + r(E2 /E1 )µ0 (E2 /E1 ) ≥ r(E1 )µ0 (E1 ) + r(E2 /E1 )µ0 (E1 ) = r(E2 )µ0 (E1 ). We have therefore that µ0 (E2 ) ≥ µ0 (E1 ). If µ0 (E2 ) = µ0 (E1 ), then it contradicts the maximality of E1 . The inequality µ0 (E2 ) > µ0 (E1 ) violates the maximality of µ. If k is not algebraically closed, the above proof shows that there is a Harder¯ What we have ¯0 ⊂ E ¯1 ⊂ · · · E ¯α = E ¯ of E ¯ = E ⊗k k. Narasimhan filtration 0 = E ¯ to show now is all the Ei ’s are defined over k. Let ϕ be a homomorphism of E¯1 to ¯ E ¯1 . Take the smallest integer i such that im ϕ ⊂ E¯i /E¯1 . If i > 1, then the same E/ argument as in the proof of the uniqueness leads us to a contradiction. Thus i = 1 ¯1 , E/ ¯ E ¯1 ) = 0. Let us consider the and then ϕ = 0. We see therefore that HomOX¯ (E k ¯ Quot-scheme Q = QuotE/X/k . Let x ¯ be the k-valued point of Q corresponding to ¯ → E/ ¯ E ¯1 → 0 and x be the scheme point which is the image of x E ¯. The residue field ˜1 of E ˜ = E ⊗k K K of OQ,x is a finite algebraic extension and there is a subsheaf E ¯ ˜ ¯ ˜ ˜ such that E1 ⊗K k = E1 . Applying the induction on r(E) to E/E1 , we may assume ¯i is defined over K. If K is not purely inseparable over k, then it that every E ˜ Since K is now contradicts the uniqueness of the Harder-Narasimhan filtration of E. ˜1 , E/ ˜ E ˜1 )∨ purely inseparable extension, OQ,x ⊗k K is a local ring. Since HomOXK (E is isomorphic to the Zariski tangent space of Q at x, we see that for the maximal ˜1 , E/ ˜ E ˜1 )∨ ⊗K k¯ ∼ ¯1 , E/ ¯ E ¯1 )∨ = 0. ideal m, m/m2 ⊗K k¯ ∼ = HomOXK (E = HomOXk¯ (E Thus m = 0 by Nakayama’s lemma and hence OQ,x ⊗k K is a field. This means that OQ,x = K = k because K is a purely inseparable extension of k. By induction on r(E) we get our assertion. The above proposition shows that for the definition of the semi-stability, we do not need to extend the base field to the algebraic closure. Corollary 2.13. Let (X, OX (1)) and E be as in Definition 2.8.

2

DEFINITION OF STABLE SHEAVES

11

(1) E is µ-semi-stable if E is torsion free and for every coherent subsheaf F of E with 0 < r(F ) < r(E), we have the inequality µ0 (F ) ≤ µ0 (E). (2) E is semi-stable if E is torsion free and for every coherent subsheaf F of E with 0 < r(F ) < r(E) and for all sufficiently large integers m, we have the inequalities PF (m) ≤ PE (m). Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα = E be the Harder-Narasimhan filtration of a torsion free coherent sheaf on (X, OX (1)). Connecting successively the points x(Ei ) = (r(Ei ), a1 (Ei )) by segments in a two dimensional real plane, we obtain the Harder-Narasimhan polygon. a1 .....

.......... ... ... ... ... 3 ... ...... ....... ....... ... ....... . . . . ... . . . ....... ... ....... ... ..... ..... ... ..... . 2 . . ... ... . .. ... ... ..... ... ..... ... ..... ..... . . ... . . ..... α−1 .... . . . ..... ... . . . . ..... ..... . . ..... . ... . . . ..... ... . . . .. . 1 ... . ..... . . .... . . . . .. . . ... . . . . ..... . ....... ..... ... .... ....... .. ... ... ....... ... ... ............ . ... ....... . 0 ................ ............................................................................................................................................................................................................................................................ .. ... .. ...

x(E ) •

• x(E )

• x(E )

x(E )

• x(F )

• x(E

)

•x(E) r

One of the main results on Harder-Narasimhan filtration is ¯ the point x(F ) = Proposition 2.14. For every coherent subsheaf F of E ⊗k k, (r(F ), a1 (F )) is below or on the Harder-Narasimhan polygon. ¯ We are going to Proof. Using the same notation as above, we set Fi = F ∩Ei ⊗k k. show by induction on i that x(Fi ) is below or on the Harder-Narasimhan polygon. Since E1 is µ-semi-stable, µ0 (F1 ) ≤ µ0 (E1 ) or x(F1 ) is the origin according as F1 6= 0 or 0. This and the fact that r(F1 ) ≤ r(E1 ) imply our assertion for i = 1. Suppose that the assertion is established up to i − 1. Let j and ` be the integers such that r(Ej−1 ) < r(Fi−1 ) ≤ r(Ej ) and r(E`−1 ) < r(Fi ) ≤ r(E` ). We have obviously j ≤ ` ≤ i. By the definition of the Harder-Narasimhan filtration we know that for all k ≤ i, µ0 (Ek /Ek−1 ) ≥ µ0 (Ei /Ei−1 ) ≥ µ0 (Fi /Fi−1 ) = ν. This and our induction hypothesis show that for all j ≤ m ≤ ` − 1, the point xm is equal to or above the point ym = (r(Em ), a1 (Fi−1 ) + ν{r(Em ) − r(Fi−1 )}). Then, by the same argument, we see that x(Fi ) = (r(Fi ), b + ν{r(Fi ) − r(E`−1 )}) is below or on the Harder-Narasimhan polygon, where b is the second coordinate of y`−1 . The notion of the type is another way to measure the distance of a coherent torsion free sheaf from the µ-stability and it is going to play a key role in studying the boundedness of stable sheaves. Definition 2.15. Let (X, OX (1)) and E be the same as in Definition 2.8. Assume that r = r(E) and let (α) = (α1 , α2 , . . . , αr−1 ) be a sequence of rational numbers. E is said to be of type (α) if E is torsion free and for every coherent quotient sheaf F of E ⊗k k¯ with 0 < r(F ) < r, we have inequalities µ0 (F ) ≥ µ0 (E) − αr(F ) .

12

Chapter 1

STABLE SHEAVES

As a direct consequence of the definition we have Lemma 2.16. (1) In the definition of type (α) we may further assume that F is torsion free. (2) Under the situation of Definition 2.15, E is of type (α) = (α1 , . . . , αr−1 ) if and only if E is torsion free and for all coherent subsheaves F of E ⊗k k¯ with 0 < r(F ) < r, we have inequalities µ0 (F ) ≤ µ0 (E) + αr−r(F )

r − r(F ) . r(F )

(3) For every integer m, E is of type (α) if and only if so is E(m). (4) If µ0 (E) + αr−1 (r − 1) < µ0 (OX ), then H 0 (X, E) = 0. Hence there is an integer m0 determined by (X, OX (1)), r, a1 (E) and αr−1 such that for every integer m with m0 ≥ m, we have H 0 (X, E(m)) = 0. Proof. The first assertion is the same as in Remark 2.9. For the second let F 0 be ¯ . then we have the quotient sheaf E ⊗k k/F a1 (E) = a1 (F ) + a1 (F 0 ). It is immediate to see that the inequality for F 0 in the definition is equivalent to the inequality in our lemma. If one notes that for a coherent OX -module F , a1 (F (m)) = a1 (F ) + r(F )dm, the assertion (3) is obvious. If H 0 (X, E(m)) 6= 0, then E(m) contains OX as a subsheaf. This, (2) and (3) show our assertion (4). A relationship between the Harder-Narasimhan filtration and the type of E is stated as follows. Lemma 2.17. Let E be a torsion free coherent sheaf on (X, OX (1)) and 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eν = E be the Harder-Narasimhan filtration of E. Then E is of type (α, α, . . . , α) for a rational number α if and only if for all 0 < i < ν, µ0 (Ei ) ≤ µ0 (E) + α

r(E) − r(Ei ) . r(Ei )

Proof. The ’only if’ part is obvious by Lemma 2.16. To prove the converse, pick a coherent subsheaf F of E ⊗k k¯ with 0 < r(F ) < r(E). Assume that rj = r(Ej ) ≤ s = r(F ) < rj+1 = r(Ej+1 ). Then, by Proposition 2.14 and our assumption, we have (s − rj ){a1 (Ej+1 ) − a1 (Ej )} (rj+1 − rj ) (s − rj )rj+1 µ0 (Ej+1 ) + (rj+1 − s)rj µ0 (Ej ) = (rj+1 − rj )

a1 (F ) ≤ a1 (Ej ) +

≤ sµ0 (E) + α(r − s), which completes our proof.

3

GENERALITIES ON BOUNDEDNESS

13

3 Generalities on boundedness A set of coherent sheaves on a scheme over a field k is, roughly speaking, said to be bounded if it is parameterized by an algebraic k-scheme. We shall define a class of coherent sheaves and then introduce the precise idea of the boundedness. Definition 3.1. Let X be a scheme of finite type over a noetherian scheme S. (1) For a point s of S and extension fields K, K 0 of k(s), we are given a coherent sheaf FK and FK 0 on XK = X ⊗S K and XK 0 , respectively. FK and FK 0 are said to be equivalent or belong to the same class of coherent sheaves on the fibers of X over S if there are k(s)-homomorphisms of K and K 0 into another extension field L such that FK ⊗K L is isomorphic to FK 0 ⊗K 0 L on XL . (2) A family F of classes of coherent sheaves on the fibers of X over S is said to be bounded if there is a scheme T of finite type over S and a coherent sheaf E on X ×S T such that a subset of the set {E(t) | t ∈ T } represents all the classes in F. In this case, we say that E bounds F. Remark 3.2. In (2) of the above definition we may replace T by Tred and hence may assume that T is reduced. If T is reduced, then there is a dense open set U ` of T , over which E is flat. Replacing T by the disjoint union U (T \ U ), applying the same argument to T \ U and using an induction on dim T , we may assume that E is flat over T . One of the main benefits of the boundedness is that all the members of a bounded family behave rather uniformly from a cohomological point of view. In fact we have the following. Proposition 3.3. Let f : X → S be a projective morphism of noetherian schemes and OX (1) an f -ample invertible sheaf on X. If F is a bounded family of classes of coherent sheaves on the fibers of X over S, then there is an integer N such that for all integers n ≥ N and all representatives F of classes in F, F (n) is generated by global sections and H i (XK , F (n)) = 0 for all i > 0, where F is a sheaf on the fiber XK . Proof. By the definition of the boundedness and Remark 3.2 there is a scheme T of finite type over S and a T -flat coherent sheaf E on X ×S T such that a subset of the set {E(t) | t ∈ T } represents all the classes in F. Our assertion is independent of the choice of representatives of classes of F and hence we have to find N for the set {E(t) | t ∈ T }. Pick a point t of T . Then there is an integer N (t) such that for all n ≥ N (t) and i > 0, E(t)(n) is generated by its global sections and H i (Xt , E(t)) = 0. By the base change theorem we can find an open neighborhood U (t) of t such that for all u ∈ U (t), f ∗ f∗ (E(n))(u) is canonically isomorphic to H 0 (Xu , E(n)(u)) ⊗ OXu and H i (Xu , E(n)(u)) = 0. Then the natural map ρ : f ∗ f∗ (E(n)) → E(n) is surjective on Xt and hence on Xu , by shrinking U (t) if necessary. This means that N (t) is an integer we want for {E(u) | u ∈ U (t)}.

14

Chapter 1

STABLE SHEAVES

Since T is quasi-compact, there are a finite number of points t1 , . . . , tr such that S T = U (ti ). Thus N = max{N (ti )} meets our requirement. Our main task in this section is to prove a converse of the above proposition. Definition 3.4. Let X be a projective scheme over an algebraically closed field k and OX (1) an ample line bundle on X which is generated by global sections. For an integer m, a coherent sheaf F on X is said to be m-regular if for every positive integer p, we have H p (X, F (m − p)) = 0. Since OX (1) is generated by its global sections and k is an infinite field, there is a global section s of OX (1) passing through no associated points of F . Then the multiplication by s defines an injection of F (−1) into F . Lemma 3.5. Set G to be the quotient sheaf F/F (−1). If F is m-regular, then so is G. Proof. The exact sequence 0 −→ F (m − p − 1) −→ F (m − p) −→ G(m − p) −→ 0 gives rise to another exact sequence H p (X, F (m − p)) −→ H p (X, G(m − p)) −→ H p+1 (X, F (m − p − 1)). The m-regularity of F means the vanishing of the first and third terms of the above and hence the second should be 0 as required. The following two propositions are the main results on the m-regularity. Proposition 3.6. Assume that a coherent sheaf F on X is m-regular. Then for every integer n ≥ m, we have the following: (1) F is n-regular. (2) The natural map H 0 (X, F (n)) ⊗k H 0 (X, OX (1)) → H 0 (X, F (n + 1)) is surjective. (3) F (n) is generated by global sections. Proof. Our proof of (1) and (2 )is by induction on dim Supp(F ). If dim Supp(F ) = 0, then our assertions are obvious. Let us look at the exact sequence we used in Lemma 3.5 0 −→ F (−1) −→ F −→ G −→ 0. Since dim Supp(G) < dim Supp(F ) and since G is m-regular by Lemma 3.5, our induction hypothesis implies that G is n-regular and the natural map H 0 (X, G(n))⊗k H 0 (X, OX (1)) → H 0 (X, G(n + 1)) is surjective. The above sequence provides us with an exact sequence of cohomologies H p (X, F (n − p − 1)) −→ H p (X, F (n − p)) −→ H p (X, G(n − p)).

3

GENERALITIES ON BOUNDEDNESS

15

Since H p (X, G(n − p)) = 0, H p (X, F (n − p − 1)) is mapped surjectively to H p (X, F (n − p)). By the m-regularity of F we know that H p (X, F (m − p)) = 0. Then, using the surjectivity repeatedly, we see that H p (X, F (n − p)) must be zero, which shows (1). (2) can be easily proved by chasing the following exact commutative diagram: H 0 (F (n)) ⊗k H 0 (OX (1)) → H 0 (G(n)) ⊗k H 0 (OX (1)) → 0   *    u   y y  0 0 0 0 → H (F (n)) H (F (n + 1)) H (G(n + 1)) ↓ 0 where the top row is exact by (1) proved in the above and where u is the map that sends a to a ⊗ s. To prove (3) we shall use the following commutative diagram: v

H 0 (F (n)) ⊗k H 0 (OX (1)) ⊗k OX −−−n−→ H 0 (F (n + 1)) ⊗k OX    wn+1 y y w ⊗id

H 0 (F (n)) ⊗k OX (1)

−−n−−→

F (n + 1)

By the above (2), vn is surjective and a theorem of Serre implies that for sufficiently large n, wn+1 is surjective. For this n, wn is surjective by the above diagram. Thus, using a descending induction on n, we see that wn is surjective for all n ≥ m. Proposition 3.7. Let 0 −→ F (−1) −→ F −→ G −→ 0 be an exact sequence as in Lemma 3.5. Assume that G is m-regular. Then, we have (1) for all p ≥ 2 and all n ≥ m − p, H p (X, F (n)) = 0, (2) for n ≥ m − 1, dim H 1 (X, F (n − 1)) ≥ dim H 1 (X, F (n)), (3) for all n ≥ m − 1 + dim H 1 (X, F (m − 1)), H 1 (X, F (n)) = 0. In particular, F is [m + dim H 1 (X, F (m − 1))]-regular. Proof. We have an exact sequence of the cohomologies H p−1 (X, G(n)) −→ H p (X, F (n − 1)) −→ H p (X, F (n)) −→ H p (X, G(n)). If n ≥ m − p, then H p (X, G(n)) = 0 by the m-regularity of G and Proposition 3.6. This shows our assertion (2). Assuming p ≥ 2 and n ≥ m − p + 1, we have furthermore H p−1 (X, G(n)) = 0 and hence H p (X, F (n − 1)) ∼ = H p (X, F (n + a)) for all a ≥ 0. Combining this with a theorem of Serre, we get (1). To prove (3) let us consider the following commutative diagram u ⊗id

H 0 (F (n)) ⊗k H 0 (OX (1)) −−n−−→ H 0 (G(n)) ⊗k H 0 (OX (1))    v y yn H 0 (F (n + 1))

un+1

−−−−→

H 0 (G(n + 1))

16

Chapter 1

STABLE SHEAVES

If H 1 (X, F (n − 1)) → H 1 (X, F (n)) is injective, or un is surjective, then by Proposition 3.6 vn (un ⊗ id) is surjective and hence so is un+1 . Repeating this procedure, we see that for all q ≥ n, uq is surjective or equivalently H 1 (X, F (q − 1))is a subspace of H 1 (X, F (q)). This and Serre’s Theorem imply that H 1 (X, F (n − 1)) = 0. Assume contrarily that if un is not surjective, then dim H 1 (X, F (n)) is strictly less than dim H 1 (X, F (n − 1)). We see therefore that if H 1 (X, F (m − 1)) 6= 0, then dim H 1 (X, F (n − 1)) is strictly decreasing while n is increasing until dim H 1 (X, F (n − 1)) becomes 0. This shows (3) A key ingredient of criteria for a family of classes of coherent sheaves to be bounded is given in the following way. Definition 3.8. Let X and OX (1) be as in Definition 3.4, F be a coherent sheaf on X and let (b) = (b0 , . . . , br ) be a sequence of integers with r ≥ dim Supp(F ). F is said to be a (b)-sheaf if the following two conditions are satisfied: (1) dim H 0 (X, F (−1)) ≤ b0 . (2) If r ≥ 1, then there is a section s in H 0 (X, OX (1)) which defines an exact sequence 0 −→ F (−1) −→ F −→ G −→ 0 such that G is a (b1 , . . . , br )-sheaf. Lemma 3.9. If F is a (b)-sheaf, then F is m-regular for an integer m which is determined by the sequence (b) and the Hilbert polynomial χ(F (n)) of F . Proof. By induction on dim Supp(F ), we shall prove our assertion and the fact that for every integer a ≥ 0, dim H 0 (X, F (a)) is bounded above by an integer determined by a, (b) and the polynomial χ(F (n)). If dim Supp(F ) = 0, then all of our assertion is obvious. We have an exact sequence 0 −→ F (−1) −→ F −→ G −→ 0. Since G is a (b1 , . . . , br )-sheaf and χ(G(n)) = ∆χ(F (n)) = χ(F (n))−χ(F (n−1)), we can find an integer m1 determined by (b) and the polynomial χ(F (n)) such that G is m1 -regular. Moreover, there is a sequence of integers c0 , c1 , . . . such that ci depends only on i, (b) and the polynomial χ(F (n)) and such that dim H 0 (X, G(i)) ≤ ci . We may assume that m1 > 0. The above exact sequence gives rise to inequalities dim H 0 (X, F (i))

≤ dim H 0 (X, F (i − 1)) + dim H 0 (X, G(i)) ≤ dim H 0 (X, F (i − 1)) + ci .

Summing up these from i = 0 to a and using the condition dim H 0 (X, F (−1)) ≤ b0 , we obtain a X dim H 0 (X, F (a)) ≤ b0 + ci i=0

3

GENERALITIES ON BOUNDEDNESS

17

which is the second assertion of our induction. Since G is m1 -regular, Proposition 3.7 implies that H q (X, F (a)) = 0 for all a ≥ m1 − 1 and q ≥ 2. Thus, setting M to Pm1 −1 be b0 + i=0 ci , we get dim H 1 (X, F (m1 − 1)) = dim H 0 (X, F (m1 − 1)) − χ(F (m1 − 1)) ≤ M − χ(F (m1 − 1)). Applying Proposition 3.7 again we see that F is [m1 + M − χ(F (m1 − 1))]-regular.

Let us mention, without proof, a quite useful criterion for flatness. Lemma 3.10. Let A be a noetherian local ring, B be a noetherian A-algebra and let I be an ideal of A such that the Jacobson radical of B contains IB . Assume that an exact sequence of finite B-modules u

v

M 0 −→ M −→ M 00 −→ 0 enjoys the following properties: (1) M is A-flat and M 00 ⊗A A/I is A/I-flat, (2) the map u ⊗A id : M 0 ⊗A A/I → M ⊗A A/I is injective. Then, M 00 is A-flat and u is injective. Now we come to the main result on criteria for a family of classes of coherent sheaves to be bounded, which plays always a key role when we handle problems of boundedness. Theorem 3.11. Let f : X → S be a projective morphism of noetherian schemes and let OX (1) be an f -ample invertible sheaf on X such that for every point s of S, OX (1)(s) = OX (1) ⊗OS k(s) is generated by global sections. For a family F of classes of coherent sheaves of the fibers of X over S, the set {χ(F (n)) | F ∈ F} of Hilbert polynomial of members of F is denoted by χF . Then the following conditions are equivalent to each other. (1) The family F is bounded. (2) The set χF is finite and there is a sequence (b) = (b0 , . . . , br ) of integers such that every class in F is represented by a (b)-sheaf FK with K algebraically closed. (3) The set χF is finite and there is an integer m such that every member FK ∈ F is m-regular. (4) The set χF is finite and there is a scheme T of finite type over S and a coherent sheaf E on X ×S T such that the family of equivalence classes of quotient sheaves of E(t) contains F, where t runs over field valued points of T . (5) There is a scheme T of finite type over S and there are coherent sheaves E and E 0 on X ×S T such that the family of the equivalence classes of the cokernels of homomorphisms E 0 (t) → E(t) contains F, where t runs over field valued points of T .

18

Chapter 1

STABLE SHEAVES

Proof. Assuming (1), we have a scheme T of finite type over S and a coherent sheaf E on X ×S T such that every class in F is represented by an E(t) with t a field valued point of T . We may assume that E is flat over T (see Remark 3.2). By the invariance of Hilbert polynomials of a flat family, ]χF is bounded by the number of connected components of T . To show the implication (1) =⇒ (2) we have only to prove that there is a sequence (b) = (b0 , . . . , br ) such that for every algebraically closed field K and every K-valued point t of T , E(t) is a (b)-sheaf. Set e = max{dim Supp(E(t)) | t ∈ T }. If e = 0, then our assertion is obvious by the upper semi-continuity of the dimensions of cohomologies. Assume that the implication is true if e < a. On the union of the connected components of T where dim Supp(E(t)) < a, we have, by our induction hypothesis, a sequence (b0 ) which meets our requirement. Thus we may assume that dim Supp(E(t)) is constant over T . Since T can be chosen to be reduced, the base change theorem implies that after splitting T into a direct sum of locally closed subsets, we may assume that fT ∗ (OX (1)T ) is locally free. Pick a point t of T . There is a finite extension field K of k(t) and a section s in H 0 (XK , OX (1)K ) = H 0 (Xt , OX (1)(t)) ⊗k(t) K which defines an injection νs of E(t)(−1)K to E(t)K . Then, we have a finite cover Ut of an affine open neighborhood of t in T , a point u of Ut over t and a section s˜ in H 0 (Ut , fUt ∗ (OX (1)Ut )) such that s˜(u) = s. Since the map νs˜ of E(−1)Ut to EUt defined by the section s˜ induces the injection νs at u, we can apply Lemma 3.10 and see that over Spec(OUt ,u ), νs˜ is injective and coker(νs˜) is flat. Shrinking Ut if necessary, we may assume that νs˜ is injective and coker(νs˜) is flat over Ut . There are a finite number of Ut ’s whose images are a covering of T . Replacing T by the direct sum of these Ut ’s, we may assume that there exists a section s˜ of H 0 (XT , OX (1)T ) such that at every geometric point t of T , νs˜(t) is injective. Setting G = coker(νs˜) and applying our induction hypothesis to G we can find a sequence (b1 , . . . , br ) such that for every geometric point t of T , G(t) is a (b1 , . . . , br )-sheaf. By the upper semi-continuity of the dimensions of cohomologies, there is an integer b0 such that for every geometric point t of T , dim H 0 (X ⊗S k(t), E(t)(−1)) ≤ b0 . Then, (b) = (b0 , b1 , . . . , br ) meets our requirement. The implication (2) =⇒ (3) is a direct consequence of Lemma 3.9. If we assume (3), then we see by Proposition 3.6 that every member of F is represented by such a coherent sheaf F on a fiber XK that F (m) is generated by global sections. Thus, setting M = max{χ(F (m)) | F ∈ F }, T = S and E = OX (−m)⊕M satisfy the conditions of (4). Assuming the condition (4), let us pick up a representative F of a member of F on a fiber Xt with t a field valued point of T . Then we have an exact sequence 0 −→ F 0 −→ E(t) −→ F −→ 0 Since the family of classes of E(t) is bounded by the definition, the implication (1) =⇒ (2) shows that ]{χ(E(t)(n))} is finite and there is a sequence (b) = (b0 , . . . , br ) such that the family of the classes of E(t) is a (b)-family. This and the above exact sequence imply that ]{χ(F 0 (n))} is finite and the family of the classes of F 0 is a (b)-family because χF is finite. Then, by the implication (2) =⇒ (3), we can

3

GENERALITIES ON BOUNDEDNESS

19

find an integer m such that every F 0 is m-regular. The above proof of (3) =⇒ (4) shows us that there is a positive integer M 0 such that every F 0 is a quotient sheaf 0 of OX (m)⊕M , which gives rise to (5). K Finally let us prove that (5) implies (1). Assuming (5), we have T , E and E 0 which satisfy the condition of (5). We may assume that E and E 0 are flat over T . By (1) =⇒ (3) we have integers m and M such that for every point y of T , there is a surjection ψ of OX (−m)⊕M (y) to E 0 (y). We may assume that m is so large that for every point y of T and every integer i with i > 0, we have H i (Xy , E(m)(y)) = 0. For a member of F, there is a geometric point t of T and a homomorphism ϕ of E 0 (t) to E(t) such that the member is represented by F = coker(ϕ). Composing ϕ with ψ(t), we see that F is isomorphic to coker(ϕψ(t)). Thus we may assume that E 0 is OX (−m)⊕M . Let H0 be HomOXT (E 0 , E). Then we see that H0 ∼ = E(m)⊕M and ⊕M 0 ∼ ∼ hence for every point y of T , H0 (y) = E(m) (y) = HomOXy (E (y), E(y)). Moreover, our choice of m implies that for all positive integers i, we have H i (Xy , H0 (y)) = 0. Thus H = fT ∗ (H0 ) is a locally free OT -module and for every point y of T , the natural map H(y) → HomOXy (E 0 (y), E(y)) is isomorphic. Then, on the space H = Spec(S(H ∨ )) we have the universal section Φ of HH , where S(∗) stands for the symmetric algebra over OT . g

XH −−−−→   fH y

XT  f yT

H −−−−→ T Since the base change theorem implies H 0 (H, HH ) ∼ = H 0 (H, fH ∗ g ∗ (H0 )) ∼ = H 0 (XH , g ∗ (H0 )) ∼ = H 0 (XH , EH (m)⊕M ) ∼ = HomO (E 0 , EH ), X

H

H

0 to EH . If F˜ is the cokernel Φ can be regarded as the universal homomorphism of EH ˜ of Φ, then F obviously bounds our F.

Let F be a coherent sheaf on an algebraic scheme Y . For an integer r, let Tr be the subsheaf of F such that on each open set U of Y , Γ(U, Tr ) = {s ∈ Γ(U, F ) | dim Supp(s) < r}. Then Tr is a coherent subsheaf of F . If r > dim Y , then Tr = F . Let us denote F/Tr by F(r) . The following is due to Grothendieck, which is another key result on boundedness. Proposition 3.12. Let f : X → S be a projective morphism of noetherian schemes, OX (1) be an f -very ample invertible sheaf on X and let E be a coherent sheaf on X. Let F be a family of the classes of quotient coherent sheaves F of sheaves E(s), where s runs over geometric points of S. Assume that for every geometric point s

20

Chapter 1

STABLE SHEAVES

of S, dim Supp(E(s)) is less than or equal to r and put χ(F (n)) = a(F )

nr nr−1 + b(F ) + terms of degree < r − 1. r! (r − 1)!

Then {a(F ) | F ∈ F} is bounded and {b(F ) | F ∈ F} is bounded below. Moreover, if {b(F ) | F ∈ F} is bounded, then the family F(r) of classes of F(r) is bounded. Proof. Taking the closed subscheme of X defined by the ideal of annihilators of E instead of X and replacing S by the disjoint union of suitable subschemes of S, we may assume that dim Xs ≤ r for every s ∈ S, X is a closed subscheme of a projective (1). After space PN S over S and OX (1) is isomorphic to the inverse image of OPN S replacing S by the disjoint union of schemes which are finite over open subschemes r r of S, a general linear projection of PN S to PS induces a finite morphism g : X → PS such that OX (1) ∼ = g ∗ (OPrS (1)). For a geometric point s of S and a coherent sheaf F on Xs , Leray’s spectral sequence and the projection formula imply that each cohomology H i (Xs , F (m)) is isomorphic to H i (Prk(s) , (gs )∗ (F )(m)). Hence we see that χ(F (n)) = χ((gs )∗ (F )(n)) and that F is m-regular if and only if so is (gs )∗ (F ). If F is a quotient sheaf of E(s), then (gs )∗ (F ) is a quotient of (gs )∗ (E(s)). Moreover, by the finiteness of gs we have that (gs )∗ (F )(r) = (gs )∗ (F(r) ). Combining these and the equivalence of (1) and (3) of Theorem 3.11, we can reduce our problem to the case X = PrS . We may further assume that E = OPrS (d)⊕M . Suppose now that F is a representative of a member of F and hence a quotient coherent sheaf of E(s). Then, for the kernel G of the quotient homomorphism, we know that a(F ) + a(G) = M and a(G) is non-negative. Thus we see 0 ≤ a(F ) ≤ M. The coefficient of nr−1 in χ(F (n)) is bounded below (or, bounded) if and only if so ⊕M is for χ(F (n − d)). Thus, to prove the remaining we may assume that E = OP r . S Taking the torsion part Tr of F there is an exact sequence 0 −→ Tr −→ F −→ F(r) −→ 0. Since the Hilbert polynomial of Tr is of degree less than r and the leading coefficient of a Hilbert polynomial is positive, we obtain an inequality b(F(r) ) ≤ b(F ). This shows that to prove our assertion we may assume that F is torsion free. Then there is a closed set Y of codimension at least 2 in Prk(s) such that on U = Prk(s) \Y , F is a locally free sheaf of rank a = a(F ). Since codim Y ≥ 2, there is a unique line bundle Va Le = OPrk(s) (e) which is isomorphic to F on U . On the other hand, we have a Va Va Va surjection of the trivial bundle E(s) to F and hence F is generated by its global sections. Thus Le |U is generated by its global sections. Since codim Y ≥ 2, the global sections of Le |U extend to those of Le , which means that e ≥ 0. By the definition of e we see that c1 (F ) = e and then Riemann-Roch implies that b(F ) = a(F )

(r + 1) + e. 2

4

OPENNESS OF STABILITY

21

We see therefore that b(F ) ≥ 0 and hence b(F ) is bounded below. Now, assuming that {b(F ) | F ∈ F} is bounded, we shall prove that F(r) is bounded. Pick a representative of a member F of F(r) . Since b(F ) is bounded, the integer e defined in the above is bounded. We may assume that a = a(F ), b = b(F ) and hence e are Va fixed. The natural surjection E(s)|U → Le |U extends uniquely to a generically Va surjective homomorphism θ : E(s) → Le . Let η be the homomorphism of E(s) Va−1 Va to Hom( E(s), E(s)) provided by the exterior product and let θ∗ be the Va−1 Va Va−1 homomorphism of Hom( E(s), E(s)) to Hom( E(s), Le ) defined by the composition of θ. Composing η with θ∗ we obtain a homomorphism δ : E(s) −→ HomOPr

k(s)

a−1 ^

(

E(s), Le ).

It is easy to see that im(δ) coincides with F on the open set U . Then the following lemma shows that F = im(δ). Lemma 3.13. Let Z be an integral scheme and let Q1 and Q2 be quotient quasicoherent sheaves of a quasi-coherent OZ -module P . Suppose that Q1 and Q2 coincide on a non-empty open set U and both are torsion free. Then Q1 = Q2 as quotient sheaves of P . Proof. Let Ki be the kernel of the quotient homomorphism of P to Qi and let K be the quasi-coherent submodule of P generated by K1 and K2 . Our assumption means that K = Ki on the open set U . Thus K/Ki is, on one hand, a torsion sheaf and, on the other hand, a subsheaf of Qi . Since Qi is torsion free, we see that K1 = K = K2 . Let us come back to the proof of Proposition 3.12. What we have seen is that F is the image of a homomorphism of E(s) to (OPrS (e)⊕A )(s), where A is the rank Va−1 of E. Since giving the image of a homomorphism is equivalent to giving the cokernel, the equivalence of (1) and (5) of Theorem 3.11 completes our proof. Remark 3.14. Let (X, OX (1)) and E be as in Definition 2.8. The above proposition shows that there is a sequence (α) = (α1 , . . . , αr(E)−1 ) of rational numbers such that E is of type (α).

4 Openness of stability To fix our idea let us introduce the notion of the openness property on coherent sheaves. Definition 4.1. Let P be a property of coherent sheaves on pairs of a geometrically integral projective scheme and an ample line bundle on it. Let f : Y → S be a flat, geometrically integral, projective morphism of locally noetherian schemes and let OY (1) be an f -ample line bundle on Y . The property P is said to be open if for every (Y /S, OY (1)) and S-flat coherent sheaf F on Y , there is an open set U of S

22

Chapter 1

STABLE SHEAVES

such that for every algebraically closed field k, the set U (k) of k-valued geometric points1 of U is exactly { s ∈ S(k) | F (s) has the property P on the geometric fiber (Ys , OY (1)(s)) }. Our theorem on the openness is stated as follows. Theorem 4.2. (1) For every sequence (α) = (α1 , . . . , αr−1 ), the property that a coherent sheaf is of type (α) is open. (2) The property that a coherent sheaf is stable is open. (3) The property that a coherent sheaf is semi-stable is open. Proof. We are given (Y /S, OY (1)) and an S-flat, coherent OY -module E as in Definition 4.1. We may assume that S is connected. Since Y and E are flat over S, we may also assume that for every geometric point s of S, Supp(E(s)) = Ys . Then, the torsion freeness of E(s) is equivalent to the property (S1 ) of E(s), that is, depth E(s)y ≥ min{1, dim E(s)y } for every point y ∈ Ys . (In fact E(s) satisfies (S1 ) if and only if Ass(E(s)) consists only of the generic point of Ys .) Thus the property that a coherent sheaf is torsion free is open by virtue of [EGA, IV Theorem 12.2.1] (for the proof of the projective case, see Proposition II.7.15). We may now assume that for every geometric point s of S, E(s) is torsion free. (1) Let us consider the family B of the classes of the coherent sheaves F on the fibers of Y over S with the following properties: (a)

F is a coherent quotient sheaf of an E(s) with s a geometric point of S.

(b)

F is torsion free.

(c)

µ0 (F ) < µ0 (E(s)) − αr(F ) .

Set Q = QuotE/Y /S . By virtue of the above properties and Proposition 3.12. we see that B is bounded and hence there are only a finite number of components Q1 , . . . , Qt of Q such that Qi (k) ∩ B(k) 6= ∅ for some algebraically closed field k, where B(k) = {F ∈ B | F is represented by a sheaf on the fiber over a k-valued point of S}. It is obvious that t [

Qi (k) ⊇ B(k).

i=1

Since Q0 =

t [

Qi

i=1

is projective over S, T = g(Q0 ) is closed in S, where g is the structure morphism of Q0 over S. We claim that the open set U = S \ T is the required one for type (α). In fact, if for a k-valued point s of S, E(s) is not of type (α), then there exists a coherent quotient sheaf F of E(s) with properties (b) and (c). F defines 1 A geometric point of a scheme U is a morphism Spec k → U with k an algebraically closed field.

4

OPENNESS OF STABILITY

23

a k-valued point s0 of Q0 whose image by g is the s. Thus E(s) is of type (α) if s is a geometric point of U . Conversely, take a k-valued point s of T . Then there is a k-valued geometric point s0 of Q0 with g(s0 ) = s. For the universal quotient sheaf F˜ on X ×S Q0 , F = F˜ (s0 ) is a coherent quotient sheaf of E(s). Since for the connected component containing s0 , say Qi , and for a suitable algebraically closed field k 0 , Qi (k 0 ) ∩ B(k 0 ) 6= ∅, we see that µ0 (F ) < µ0 (E(s)) − αr(F ) . Therefore, E(s) is not of type (α). For the openness of stability or semi-stability, we can apply the same argument as above if we replace (c) by the following (c0 ) or (c00 ), respectively: (c0 )

PF (m) ≤ PE(s) (m) for all sufficiently large integers m.

(c00 )

PF (m) < PE(s) (m) for all sufficiently large integers m.

As a direct consequences of (1) of the above theorem we obtain Corollary 4.3. (1) The property that a coherent sheaf is µ-stable is open. (2) The property that a coherent sheaf is µ-semi-stable is open.

Chapter 2

Restriction theorems and boundedness Is the semi-stability preserved when we make the restriction of a semi-stable sheaf to a hyperplane section? We have to, of course, put some conditions on semistable sheaves and hyperplanes. What kind of conditions then? We shall show here several types of restriction theorems and apply them to proving results on boundedness of semi-stable sheaves.

1 Restriction of Harder-Narasimhan filtration Throughout this section we shall assume that k is an algebraically closed field and (X, OX (1)) is a pair of a projective integral scheme X over k and an ample line bundle OX (1) on X. Let V be a vector subspace of H 0 (X, OX (1)) which defines a very ample linear system L, that is, the natural homomorphism of V ⊗k OX to OX (1) is surjective and the induced morphism i : X = P(OX (1)) → P(V ⊗k OX ) → P(V ) is a closed immersion. For the dual vector space V ∨ of V , we set P = P(V ∨ ) and then we have an exact sequence of locally free sheaves 0 −→ A −→ V ∨ ⊗k OP −→ OP (1) −→ 0. On the projective bundle P(A) there is another exact sequence ∗ 0 −→ B −→ πA (A) −→ OP(A) (1) −→ 0,

where πA is the projection of P(A) to P . Since B is a vector subbundle of V ∨ ⊗OP(A) and r(B) = dim V − 2, the universal property of a Grassmann variety provides us with a morphism π of P(A) to the Grassmann G of 2-dimensional subspaces of V or equivalently (dim V − 2)-dimensional subspaces of V ∨ . The surjection of V ⊗k OP to A∨ defines a closed immersion of P(A∨ ) to P(V ) ×k P . Another surjection ∗ πA (A∨ ) → B ∨ gives rise to a closed immersion of P(B ∨ ) to P(A∨ ) ×P P(A). Thus we obtain a sequence of subschemes P(B ∨ ) ⊂ P(A∨ ) ×P P(A) ⊂ P(V ) ×k P(A). This sequence and the projections to P(A) parameterize pairs of linear subspaces of codimension 1 and 2 in P(V ) such that the first contains the second. For every closed point y of P(A), X ×{y} is integral and no hyperplane of P(V )×{y} contains it. Thus the scheme theoretic intersection X 0 of X ×k P(A) and P(A∨ ) ×P P(A) in P(V ) ×k P(A) is a relative Cartier divisor of X ×k P(A) over P(A). Let Y 0 be the scheme theoretic intersection of X ×k P(A) and P(B ∨ ) in P(V ) ×k P(A). Take the maximal open set T of P(A) over which Y 0 is flat. T is exactly the set of points t of 24

1

RESTRICTION OF HARDER-NARASIMHAN FILTRATION

25

P(A) such that Yt0 is locally of complete intersection of codimension 2 in X × {t}. ˜ = X 0 ×P(A) T Then there is an open set G0 of G such that T = π −1 (G0 ). Setting X and Y˜ = Y 0 ×P(A) T , we get the following diagram p - X

˜ Y˜ ⊂ X q ? T π1

P

A π2 U A G0

Now, for a point u of G0 , π2−1 (u) = P1k(u) and π1 induces a linear embedding of P1k(u) to P . For every point v of π2−1 (u), Y˜v is the subscheme of X which is the base locus of the linear pencil in L ⊗k k(u) defined by π1 (P1k(u) ). From this view point, G0 can be regarded as a parameter space of linear pencils in L which have base loci of codimension 2. ˜ and Y˜ are geometrically U0 denotes the maximal subset of T over which both X integral. Then U0 is a non-empty open set of T . Let E be a coherent torsion free ˜ be the pull-back p∗ (E). Shrinking U0 if necessary, we may sheaf on X and E U0 ˜ ˜ assume that E is flat over U0 and for every geometric point t of U0 , E(t) is torsion free (Corollary I.2.3 and [EGA, IV Theorem 12.2.1]). Let u be the generic point of ˜ U0 and F be E(u). Take the Harder-Narasimhan filtration of F 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fα = F. By virtue of Proposition I.2.11, the above filtration is defined over k(u). Thus the quotient coherent sheaf F/F1 of F defines a morphism s0 of Spec(k(u)) to Q, ˜ X ˜ U /U0 . Since U0 is an integral scheme, there where Q is the Quot-scheme of E/ 0 is a non-empty open set U1 of U0 and a morphism s of U1 to Q with πs = idU1 and s ⊗ k(u) = s0 , where π is the structure morphism of Q as a U0 -scheme. Let ϕ1 ˜U −→ ˜ 0 → 0 be the pull-back of the universal quotient on XQ by s. Applying E E 1 1 ˜ 0 and F2 /F1 , we obtain a non-empty open set U2 of U1 the above argument to E 1 ϕ2 ˜ 0 )U −→ ˜ 0 of U2 -flat coherent sheaves on X ˜U and a surjective homomorphism (E E 1 2 2 2 such that ϕ2 (u) is the quotient homomorphism F/F1 → F/F2 . Repeating these procedures, we have a non-empty open set Uα of U0 and surjective homomorphisms ˜0 → E ˜ 0 (0 ≤ i ≤ α − 1) of Uα -flat coherent sheaves on X ˜ U such that ϕi : E α i i+1 0 ˜U and ϕi (u) is the quotient homomorphism F/Fi → F/Fi+1 . Set E ˜ = E ˜i = E 0 α ˜0 = 0. Then each E ˜i is Uα -flat and E ˜i (u) = Fi . In this ker(ϕi−1 · · · ϕ0 ) and E situation, we obtain a non-empty open subset U 0 of Uα such that for all geometric ˜i /E ˜i−1 (t) is µ-semi-stable (Theorem I.4.2). Clearly, for all points t of U 0 , every E 0 ˜ i /E ˜i−1 )(t)) = µ0 (Fi /Fi−1 ) and hence points t of U , µ0 ((E ˜0 (t) ⊂ E ˜1 (t) ⊂ · · · ⊂ E ˜α (t) = E(t) ˜ 0=E

26

Chapter 2

RESTRICTION THEOREMS AND BOUNDEDNESS

˜ is the Harder-Narasimhan filtration of E(t). For the generic point u of U 0 , Fi |Y˜u is torsion free by Corollary I.2.3 and hence Fi |Y˜u is a subsheaf of F |Y˜u . Thus there exists a non-empty open subset U of U 0 such ˜i )U ) is a subsheaf of J˜ = j ∗ ((E ˜α )U ) and that J/ ˜ J˜i is U -flat that each J˜i = j ∗ ((E ˜ ˜ for 0 ≤ i ≤ α, where j : YU → XU is the closed immersion. Then we can find an open subset W0 of U , which may be empty, such that W0 = {t ∈ U | (J˜i /J˜i−1 )(t) are µ-semi-stable, 1 ≤ i ≤ α}. Since ˜ i /E ˜i−1 )(t)) µ0 ((J˜i /J˜i−1 )(t)) = µ0 ((E for all point t of W0 , ˜ ˜ 0 = J˜0 (t) ⊂ J˜1 (t) ⊂ · · · ⊂ J˜α = E| Yt ˜ ˜. is the Harder-Narasimhan filtration of E| Yt ˜i }, we obtain On the other hand, by the same procedure as the construction of {E ˜0 ⊂ H ˜1 ⊂ · · · ⊂ H ˜ β = J˜W on a non-empty open set W of U such a filtration 0 = H ˜ β /H ˜ i is W -flat and that for every point t of W , that each H ˜ 0 (t) ⊂ H ˜ 1 (t) ⊂ · · · ⊂ H ˜ β (t) = E| ˜ ˜ 0=H Yt ˜ ˜ . Set `i = min{` | (J˜i )W → J˜W /H ˜ ` is is the Harder-Narasimhan filtration of E| Yt zero on the generic fiber of Y˜ over W }. Then there is a non-empty open set Wi0 of ˜ ` )W 0 . We let Ci = Supp(coker((J˜i )W 0 → W such that (J˜i )Wi0 is a subsheaf of (H i i i 0 ˜ (H`i )Wi0 )). Since q(Ci ) is closed in Wi , Wi = Wi0 \ q(Ci ) is an open subset of T . ˜ ` . Furthermore, Wi is non-empty if and only if for the On Wi , J˜i coincides with H i ˜ generic point u of T , Ji (u) = Fi |Y˜u is a filter of the Harder-Narasimhan filtration ˜ ˜ . of E| Yu Summarizing the above results, we have Lemma 1.1. Let E be a torsion free coherent sheaf on X. ˜U = p∗ (E)U has a (1) There exists a non-empty open set U of T such that E ˜0 ⊂ E ˜1 ⊂ · · · ⊂ E ˜α = E ˜ with the following properties (a), (b), filtration Φ : 0 = E (c) and (d):

(b)

˜ U and Y˜U are geometrically integral over U , X ˜ E ˜i is flat over U , every E/

(c)

for every point t of U ,

(a)

˜0 (t) ⊂ E ˜1 (t) ⊂ · · · ⊂ E ˜α (t) = E| ˜ Φ(t) : 0 = E Xt

(d)

is the Harder-Narasimhan filtration of E|X˜ t , ˜ U , each J˜i = j ∗ (E ˜i ) is a subsheaf of for the closed immersion j : Y˜U → X ˜ ˜ ˜ ˜ J = E and each J/Ji is flat over U (1 ≤ i ≤ α).

1

RESTRICTION OF HARDER-NARASIMHAN FILTRATION

27

(2) There exists an open set W0 of U such that for every point t of W0 , the filtration Φ|Y˜t has the following property (e0 ) and that W0 is non-empty if and only if Φ|Y˜t has the property (e0 ) when t is the generic point of U : ˜ (e0 ) Φ|Y˜t is the Harder-Narasimhan filtration of J(t). (3) There exists an open set Wi (1 ≤ i < α) of U such that for every point t of Wi , J˜i (t) has the following property (ei ) and that Wi is non-empty if and only if for the generic point t of U , J˜i (t) has the property (ei ): ˜ (ei ) J˜i (t) is a filter of the Harder-Narasimhan filtration of J(t). Remark 1.2. (1) Let f : X → S be a projective, geometrically integral morphism of noetherian schemes, OX (1) be an f -very ample invertible sheaf on X and let E be a coherent sheaf on X. Assume that S is integral. Let F1 and F2 be coherent subsheaves of E with the following properties: (a)

E/F1 and E/F2 are flat over S,

(b)

r(F1 ) = r(F2 ),

(c)

for the generic point s of S, E(s) is torsion free and both F1 (s) and F2 (s) are filters of the Harder-Narasimhan filtration of E(s).

Then we have F1 = F2 . (2) By virtue of (1), the filtration Φ in (1) of Lemma 1.1 is unique. Proof of (1). Let Q is the Quot-scheme of E/X/S. By (a), E/Fi defines a section gi of S to Q. Thanks to (b), (c) and the uniqueness of the Harder-Narasimhan filtration, g1 (s) = g2 (s) as morphism of Spec(k(s)) to Q. Since Q is separated over S and since S is integral, we see g1 = g2 As an application of the above we get a gluing theorem. Theorem 1.3. Let (X, OX (1)) be a pair of a projective integral scheme X over an algebraically closed field k and an ample line bundle OX (1) on X and let V be a vector subspace of H 0 (X, OX (1)) which defines a very ample linear system L. Let E be a torsion free coherent sheaf on X. Assume that dim X ≥ 3 and that for some 0 < i < α, Wi for E in Lemma 1.1 is not empty. Then there is a coherent subsheaf ˜i (t)) and r(Ei ) = r(E ˜i (t)) for a point t of Wi . Ei of E such that µ0 (Ei ) = µ0 (E Proof. Our proof consists of several steps. Step I. Since π2 : T → G0 is flat, π2 (Wi ) is a non-empty open set of G0 . Pick a sufficiently general k-rational point v of π2 (Wi ). Let Y be the base locus of the linear pencil π2−1 (v) ∼ = P1k . Then we may assume that the singular locus Sing(Y ) of Y is exactly Y ∩Sing(X). Take a k-rational point t of P1k . Since Y is a hyperplane cut of ˜ t and of codimension 2 in X, we see that Sing(X ˜ t ) ∩ Y ⊂ Sing(Y ). Conversely, if y X ˜ t ⊂ Sing(X ˜ t ). is in Sing(Y ), then y is a point of Sing(X) and hence it is in Sing(X)∩X ˜ ˜ Thus we have Sing(Y ) = Y ∩ Sing(Xt ). If Xt is reducible, then Y must be in the

28

Chapter 2

RESTRICTION THEOREMS AND BOUNDEDNESS

˜ t and intersection of irreducible components because Y is a hyperplane cut of X ˜ t is Y is irreducible. This violates the generically smoothness of Y and hence X −1 ˜ irreducible. Moreover, it is easy to see that f : Z = X ×T π2 (v) → X is the blowing-up of X with center Y , D = Y˜ ×T π2−1 (v) is the exceptional divisor of f and D ∼ = Y ×k π2−1 (v). f

Z S g

P1k

D∼ = Y ×k P1k

  h  

-X S fD Y



Step II. For each point t of P1k , Zt = g −1 (t) is the member of the linear pencil corresponding to t and h−1 (t) = Dt ∼ = Y as subschemes of Zt ⊂ X. Set ˜R and E ˜i,R = (E ˜i )R are flat over R and this is a subsheaf R = Wi ∩ π −1 (v). Then E

π2−1 (v)

2

of that because of the property (b) of Lemma 1.1. Let B be the subsheaf of g0 ˜R . =E torsions of f ∗ (E) and let E 0 = f ∗ (E)/B. Then E 0 is g-flat and clearly ER 0 0 0 0 0 It is easy to construct a coherent subsheaf Ei of E such that F = E /Ei is g-flat ˜ R /E ˜i,R as quotient sheaves of E 0 . Consider the exact sequence and FR0 = E R E ⊗ IY −→ E −→ E|Y −→ 0, where IY is the ideal sheaf of Y . Since IY OZ = OX˜ (−1)|Z = OZ (−1), the pull-back of the above sequence gives rise to an exact commutative diagram ∗ (E|Y ) −−−−→ 0 f ∗ (E)(−1) −−−−→ f ∗ (E) −−−−→ fD      γ y y y

E 0 (−1)   y 0

−−−−→

E0   y

−−−−→

E 0 |D

−−−−→ 0

0

By five lemma γ is surjective and obviously it is isomorphic over R. Thus the kernel ∗ of γ is a torsion sheaf. On the other hand, fD (E|Y ) is torsion free and hence γ is ∗ 0 injective. We see therefore that fD (E|Y ) = E |D . Step III. Pick a k-rational point t of P1k \ R and put ∆ to be the open set of smooth points of Zt . As we have seen in Step I, ∆ ∩ Dt is exactly the open set of smooth points of Dt . Let N be the subsheaf of F 0 (t) formed by local sections supported by a set of codimension ≥ 1 (cf. the paragraph before Proposition I.3.12). Suppose that Supp(N ) contains Dt . Let us denote dim X by n + 1 and F 0 (t)/N by F¯ 0 . Take a k-rational point t0 of R. Using the notation in section 2 of Ch. I, we

1

RESTRICTION OF HARDER-NARASIMHAN FILTRATION

29

obtain the following equalities a1 (F¯ 0 ) = a1 (F 0 (t)) − a1 (N ) = a1 (F 0 (t0 )) − a1 (N ). By our assumption Supp(N ) ⊃ Dt , we know that a1 (N ) is a positive integer. Let us consider the sheaf H = F¯ 0 |Dt . Since F¯ 0 |∆ is torsion free, the map δ : F¯ 0 (−1) → F¯ 0 of the multiplication by the local equation of Dt is injective on ∆. This means that ker(δ) is supported by Zt \∆ which is of dimension at most n−1 by the irreducibility of Zt . Thus, by the definition of F¯ 0 , we see that ker(δ) = 0. We get therefore an exact sequence 0 −→ F¯ 0 (−1) −→ F¯ 0 −→ H −→ 0. This gives rise to an equality χ(H(m)) = χ(F¯ 0 (m)) − χ(F¯ 0 (m − 1)) and hence a1 (H) = a1 (F¯ 0 ). Since F¯ 0 is torsion free on ∆, it is locally free on ∆ outside a set of codimension at least 2. On the other hand, ∆ ∩ Dt is of codimension 1 in ∆. Thus F¯ 0 is locally free at general points of Dt and hence we have r(H) = r(F¯ 0 ) = r(F 0 (t0 )). We obtain therefore a1 (H) µ0 (H) = r(H) a1 (F¯ 0 ) = r(F 0 (t0 )) a1 (F 0 (t0 )) − a1 (N ) = < µ0 (F 0 (t0 )). r(F 0 (t0 )) On the other hand, E 0 |D ∼ = j ∗ (E|Y ) and H is a quotient sheaf of E 0 |D with t

t

t

r(H) = r(F 0 (t0 )) = r(F 0 (t0 )|Dt0 ), where jt is the isomorphism of Dt to Y . These and Proposition I.2.14 imply that µ0 (H) ≥ µ0 (F 0 (t0 )|Dt0 ) = µ0 (F 0 (t0 )). This contradicts the above inequality and hence Dt \ Supp(N ) 6= ∅. The set of pinching points of F 0 |∆ in ∆ \ Supp(N ) is of codimension at least 2, F 0 |∆ is locally free at general points of Dt . Thanks to the g-flatness of F 0 , we see that (1.3.1) if C is the set of pinching points of F 0 , then for all t ∈ P1k , Ct does not contain Dt . Step IV. Let 0 = H0 ⊂ H1 ⊂ · · · ⊂ Hβ be the Harder-Narasimhan filtration of ˜ j = f ∗ (Hj ). Let K be the torsion part of F 0 |D and let I be (F 0 |D )/K. E|Y and let H D Then we see that Supp(K) ⊂ C ∩ D, I is flat over P1k and IR = F 0 |DR . Moreover, I ˜ ` for an ` by virtue of the property (ei ) for Wi and Remark coincides with (E 0 |D )/H 1.2, (1). Thus Ct ∩ Dt is jt−1 (S) if t ∈ R, where S is the set of pinching points of (E|Y )/H` . From this we deduce that   [ [  fD (C ∩ D) = S jt (Ct ∩ Dt ) . t∈P1 \R

30

Chapter 2

RESTRICTION THEOREMS AND BOUNDEDNESS

Since P1k \ R is a finite set, (1.3.1) shows that fD (C ∩ D) 6= Y . we have therefore ∗ (1.3.2) On D \ (C ∩ D), F 0 |D is isomorphic to fD ((E|Y )/H` ) as quotient sheaves 0 of E |D and f (C ∩ D) 6= Y .

Step V. By replacing E by E(m) with m big enough, we may assume that E is generated by its global sections ⊕M ψ : OX −→ E −→ 0.

Pulling back ψ to Z and composing it with the homomorphism f ∗ (E) → E 0 , we get surjective homomorphisms τ1 and τ2 as follows τ

τ

1 2 ⊕M OZ −→ E 0 −→ F 0.

Let C 0 be the set of pinching points of E. Then C 0 \ Sing(X) ∩ C 0 is of codimension at least 2 and hence for all points of P1k , Zt 6⊂ f −1 (C 0 ). We may assume also that Y 6⊂ C 0 . Set Z0 = Z \ f −1 (C 0 ∪ f (C)), X0 = X \ f (C) ∪ C 0 and f0 = f |Z0 . The restrictions of τ1 and τ2 to Z0 define a morphism u of Z0 to the flag variety Flag(M, r(E 0 ), r(F 0 )) of pairs of quotient vector spaces (V1 , V2 ) of an M -dimensional vector space such that dim V1 = r(E 0 ), dim V2 = r(F 0 ) and that V2 is a quotient ∗ of V1 . Since fD ((E|Y )/H` ) = F 0 |D on Z0 ∩ D as quotient bundles of E 0 |Z0 ∩D by (1.3.2), u(f −1 (y)) is one point for all Y ∩ X0 . Thus u factors through X0 , that is, u = u0 f0 for a u0 : X0 → Flag(M, r(E 0 ), r(F 0 )). This means that there ⊕M are locally free quotient sheaves η1 : OX → E0 and η2 : E0 → F0 such that 0 ∗ 0 ∗ ∗ 0 ∼ ∼ ∼ f0 (E0 ) = E |Z0 = f0 (E|X0 ), f0 (F0 ) = F |Z0 , f0∗ (η1 ) ∼ = τ1 and f0∗ (η2 ) ∼ = τ2 . Since ∗ ∗ ∼ ∼ ∼ E0 = f0 ∗ f0 (E0 ) = f0 ∗ f0 (E|X0 ) = E|X0 , η2 induces a surjection of E|X0 to F0 . Then we can extend this to a surjection ζ : E −→ F with F torsion free. Step VI. Set Ei = ker(ζ). Then Ei meets our requirement. In fact, for a krational point t of P1k , we shall identify Zt with f (Zt ). F can be embedded in a vector bundle F˜ so that F˜ /F is locally free at a point x if and only if so is F . F |Zt has torsions if and only if Zt passes through an associated point of F˜ /F (see the proof of Corollary I.2.3). Moreover, if Zt contains exactly x1 , . . . , xa of Ass(F˜ /F ), then the torsion part of F |Zt is supported by a subset of the closure of the set {x1 , . . . , xa }. On the other hand, Y is the base locus of the pencil {Zt }. Thus for general t, Zt can avoid the associated points outside Y and hence F |Zt has at most torsions supported by a closed subset of Y . Since the set of pinching points of F is a subset of C 0 ∪ f (C), this shows that the torsion part of F |Zt is supported by a proper closed subset of Y , which is of codimension at least 2 in Zt . Fix a general k-rational point t of P1k and set Q to be the maximal open subset where F |Zt is torsion free. Then F 0 (t) and F |Zt are the same quotient sheaf of E|Zt on a non-empty open set of Zt and torsion free on Q. Thanks to Lemma I.3.13, we see

2

¨ THEOREM OF GRAUERT-MULICH-SPINDLER

31

that F 0 (t) = F |Zt on Q. This and the fact that codim(Zt \ Q, Zt ) ≥ 2 imply that a1 (F 0 (t)) = a1 (F |Zt ) and r(F 0 (t)) = r(F |Zt ). Thus we have ˜i (t)) = a1 (E|Z ) − a1 (F 0 (t)) a1 (E t = a1 (E|Zt ) − a1 (F |Zt ) = a1 (Ei |Zt ) = a1 (Ei ) ˜i (t)) = r(Ei ). and r(E The above theorem can be used to prove a restriction theorem of semi-stable sheaves under a condition on their rank. Theorem 1.4. Let (X, OX (1)), V and L be the same as in Theorem 1.3. Let E be a µ-semi-stable sheaf on (X, OX (1)). Assume that dim X ≥ 2 and r(E) < dim X. Then for general hyperplane sections Y in L, E|Y is µ-semi-stable with respect to OY (1) = OX (1)|Y . Proof. We shall prove our theorem by induction on r(E). If r(E) = 1, then it is enough to show that E|Y is torsion free and it follows from Corollary I.2.3. Assume now that r(E) > 1. Take the open set U of T and the filtration ˜0 ⊂ E ˜1 ⊂ · · · ⊂ E ˜α = E ˜ Φ:0=E of Lemma 1.1. Pick a k-rational point t in U . E|X˜ t is not µ-semi-stable if and only if α > 1. π1−1 π1 (t) ∩ U can be regarded as an open set of the linear system TrX˜ t (L) ˜ t which is defined by the image of V in H 0 (X ˜ t , O ˜ (1)). Assume that E| ˜ on X Xt Xt ˜ ˜ ˜ t . Thus, by is not µ-semi-stable. Then, we have r(Ei /Ei−1 (t)) < r(E) ≤ dim X ˜i /E ˜i−1 (t)|D is our induction hypothesis, for general member D of TrX˜ t (L), every E ˜ µ-semi-stable. We may assume, therefore, that our filtration Φ of E and open set U have the property (e0 ) of Lemma 1.1. Since dim X > r(E) ≥ 2, the above and Theorem 1.3 provide us with a coherent sheaf E1 of E with µ0 (E1 ) > µ0 (E). This contradicts the assumption that E is µ-semi-stable. Hence E|X˜ t is µ-semi-stable.

2 Theorem of Grauert-M¨ ulich-Spindler Let k, (X, OX (1)) and V be the same as in the preceding section. We assume furthermore that X is smooth in codimension 1. Fix a positive integer n0 which is smaller than n = dim X. If G is the Grassmann variety of n0 -dimensional linear subspaces of V . We have the following diagram of incidence correspondence: p0

P −−−−→ P(V )   q0 y G

32

Chapter 2

RESTRICTION THEOREMS AND BOUNDEDNESS

Both p0 and q0 are fiber bundles in Zariski topology and a fiber q0−1 (x) can be identified with the linear subspace of codimension n0 in P(V ) that is defined by the subspace of V corresponding to x. ˜ = p−1 (X). There exists a non-empty open X is embedded in P(V ). Set X 0 ˜ Z is flat, geometrically integral and for every point z of subset Z of G such that X ˜ z is smooth in codimension 1. Put X ˜0 = X ˜ Z , p = p0 | ˜ and q = q0 | ˜ 0 . Z, the fiber X X

X

p

˜0 ⊂ X ˜ −−−−→ X X     qy y Z ⊂ G ˜ 0 , Z, q) parameterizes the geometrically integral subschemes that are The triple (X smooth in codimension 1 and complete intersections of n0 members in the linear system L defined by V . Thus the fiber q −1 (x) can be regarded as a subvariety in X and hence p|q−1 (x) is a closed immersion. ˜ be the pull-back Let E be a coherent sheaf of type (α) on (X, OX (1)) and let E ∗ ¯ ˜ p (E). Then, for the generic point z¯ of Z, E = E|q−1 (¯z) is torsion free on the fiber ˜ z¯. The Harder-Narasimhan filtration of E ¯ is defined over k(¯ X z ) and hence, as in the preceding section, we have a non-empty open set Z0 of Z and a filtration of ˜ q−1 (Z ) E| 0 ˜0 ⊂ E ˜1 ⊂ · · · ⊂ E ˜ν = E| ˜ q−1 (Z ) 0=E 0 ˜ i /E ˜i−1 are flat over Z0 and that for all z in Z0 , such that all the E ˜0 (z) ⊂ E ˜1 (z) ⊂ · · · ⊂ E ˜ν (z) = E(z) ˜ 0=E ˜ q−1 (z) . Replacing Z by Z0 , we may assume is the Harder-Narasimhan filtration of E| ˜ ˜ ˜ E ˜λ . Let W be the maximal open set of Z = Z0 . Pick a filter Eλ and set F = E/ 0 ˜ ˜ ˜ ˜ Then X over which E is locally free and P(F ) is a projective subbundle of P(E). ˜ z \ Wz , X ˜ z ) ≥ 2. We shall consider the closure Ψ of P(F˜ )W for all z ∈ Z, codim(X ˜ in P(E). ˜ −−−g−→ P(E) Ψ⊂P(E)    π π ˜y y ˜ X

p

−−−−→ X ˜λ ) − 1. It is clear that dim g(Ψ) ≥ dim X + r(E) − r(E ˜λ ) − 1, then there exists a coherent Lemma 2.1. If dim g(Ψ) = dim X + r(E) − r(E ˜λ (z) as subsheaves of subsheaf Eλ of E such that for general z in Z, Eλ |X˜ z ∼ = E ˜ E|X˜ z ∼ = E(z). Proof. Let U be a non-empty open set of X such that E|U is locally free and for ˜λ ) − 1. Set U ˜ = p−1 (U ) ∩ W . Then p(U ˜) all points x of U , dim g(Ψ)x = r(E) − r(E ˜ is open in X because p is flat. Thus, replacing U by p(U ), we may assume that ˜ ). Since Ψ ˜ is a projective subbundle of P(E) ˜ ˜ = P(E) ×X U ˜ , we obtain U = p(U U U

2

¨ THEOREM OF GRAUERT-MULICH-SPINDLER

33

˜ to the Grassmannian Grassr (E|U ) of r-dimensional quotient a morphism γ of U ˜x ) ≥ 1 spaces of the fibers of E|U , where r = r(F˜ ). Pick a point x of U . If dim γx (U ⊕r(E) in Grassr (E|U )x = Grassr (k(x) ), then dim g(Ψ)x ≥ r, which is not the case. ˜x ) is one point. This implies that γ = γ 0 · p with γ 0 a morphism of U Thus γx (U to Grassr (E|U ). Therefore, there is a quotient vector bundle F0 of E|U such that F˜ |U˜ = p∗U˜ (F0 ). We can so extend ker(E|U → F0 ) to a coherent subsheaf Eλ of E that E/Eλ is torsion free. If we take a general k-rational point z of Z, then ˜ E˜λ )(z) and (E/Eλ )| ˜ (E/Eλ )|X˜ z is torsion free by Corollary I.2.3. Moreover, (E/ Xz ˜z as quotient sheaves of E(z). ˜ coincide on the open set U Then, by Lemma I.3.13, we have that Eλ |X˜ z = E˜λ (z). ˜λ ) − 1, then for every z ∈ Z, we Corollary 2.2. If dim g(Ψ) = dim X + r(E) − r(E have r − rλ µ0 (Eλ (z)) ≤ µ0 (E) + αr−rλ , rλ ˜λ ). where r = r(E) and rλ = r(E ˜λ ) − 1 and zero is the characNow, assume that dim g(Ψ) > dim X + r(E) − r(E teristic of k. Then there is a non-empty open set ∆ of ΨZ such that g|Ψ : Ψ → g(Ψ) ˜ z is irreis smooth on ∆. Pick a k-rational point z of Z in q(˜ π (∆)). Y = X ducible and can be regarded as a complete intersection subvariety of codimension n0 in X. Put Y0 = Y ∩ W . Recall that codim(Y \ Y0 , Y ) ≥ 2. A = ΨY0 ˜ Y . g induces an isomorphism of P(E) ˜ Y to is a projective subbundle of P(E) 0 0 P(E)Y0 and hence B = g(A) is a projective subbundle of P(E)Y0 . Note that ˜λ ) + n − n0 − 1. dim A = dim B = r(E) − r(E Let V 0 be the linear subspace of V that corresponds to the point z of G. Since the tangent space of G at z is canonically isomorphic to Homk (V 0 , V /V 0 ), we obtain the following exact commutative diagram: NY /X˜   y

−−−−→

δ

−−−−→ 0

NY /X   y 0

0 −−−−→ HY −−−−→ Homk (V 0 , V¯ ) ⊗k OY −−−−→ OY (1)⊕n −−−−→ 0 where V¯ = V /V 0 . Y is the complete intersection of Y1 , . . . , Yn0 with Yj ∈ L. If {e1 , . . . , en0 } is a basis of V 0 corresponding to {Y1 , . . . , Yn0 }, then δ can be written in a direct sum of δj ’s by using this basis: V ⊗k O X   y δj : V¯ ⊗k OY x  

−−−−→

H 0 (X, OX (1)) ⊗k OX   y

,−→

H 0 (Y, OY (1)) ⊗k OY x  

−−−−→

OY (1) x  

V¯ ⊗k OP(V¯ ) −−−−→ H 0 (P(V¯ ), OP(V¯ ) (1)) ⊗k OP(V¯ ) −−−−→ OP(V¯ ) (1)

34

Chapter 2

RESTRICTION THEOREMS AND BOUNDEDNESS

Thus the kernel HY of δ is the direct sum of n0 -copies of ΩP(V¯ ) (1)|Y , where Y is naturally embedded in the linear subspace P(V¯ ) of P(V ): 0 HY ∼ = (ΩP(V¯ ) (1)|Y )⊕n .

Let θ be the projection of A to Y0 that is identified with the projection of B to Y0 . 0 Using the natural isomorphisms of θ∗ (NY0 /X ) to θ∗ (OY0 (1)⊕n ) and of θ∗ (NY0 /X˜ ) to NA/ΨW , we obtain the following exact commutative diagram θ ∗ (dp)

0 ←−−−− θ∗ (HY |Y0 ) −−−−→ θ∗ (NY0 /X˜ ) −−−−→





yo

θ∗ (NY0 /X )   yo

−−−−→ 0

0

−−−−→ θ∗ (OY0 (1)⊕n ) −−−−→ 0   yo

0 −−−−→ θ∗ (HY |Y0 ) −−−−→ NA/ΨW     ωy d(g|Ψ )y

0 −−−−→ NB/P(E)Y0 −−−−→ NB/P(E) −−−−→ NP(E)Y0 /P(E) −−−−→ 0 On the other hand, it is not hard to see that ˜λ (z)0 )∨ ⊗ M, NB/P(E)Y0 ∼ = θ∗ (E ˜λ (z)0 = E ˜λ (z)|Y and where M is the tautological line bundle of E|Y on where E 0 0 P(E)Y0 . Since z is taken from q(˜ π (∆)), we have that r(im(d(g|Ψ ))) = dim g(Ψ)−dim B ≥ n +1. Moreover, codim(P(E)Y0 , P(E)) = n0 and hence r(NP(E)Y0 /P(E) ) = n0 . Thus we see that 0

r(im(ω))

= r(im(d(g|Ψ )) ∩ NB/P(E)Y0 ) ≥

r(im(d(g|Ψ ))) + r(NB/P(E)Y0 ) − r(NB/P(E) )

≥

n0 + 1 − n0

=

1.

This implies that ω cannot be zero. What we have seen is

(2.3)

0 6= ω ∈ HomOA (θ∗ (HY |Y0 ), NB/P(E)Y0 ) ∼ ˜λ (z)0 )∨ ⊗ M ) = HomO (θ∗ (HY |Y ), θ∗ (E

Y0

where E0 = E|Y0 .

0

A

∼ ˜λ (z)∨ ⊗ (E0 /E ˜λ (z)0 )) = HomOY0 (HY |Y0 , E 0 ∼ ˜λ (z)0 , (E0 /E ˜λ (z)0 )), = HomO (HY |Y ⊗ E 0

2

¨ THEOREM OF GRAUERT-MULICH-SPINDLER

35

Lemma 2.4. There exists an open set Y 0 of Y with codim(Y \ Y 0 , Y ) ≥ 2 and a filtration 0 = G0 ⊂ G1 ⊂ · · · ⊂ Gt = ΩP(V¯ ) (1)|Y by coherent subsheaves of ΩP(V¯ ) (1)|Y such that (1) G1 ∼ = OY (−1), (2) Gi /Gi−1 |Y 0 ∼ = OY 0 (2 ≤ i ≤ t) and (3) Gt /Gi ’s are torsion free. Proof. Let us look at Euler sequence 0 −→ OP(V¯ ) (−1) −→ V¯ ∨ ⊗k OP(V¯ ) −→ TP(V¯ ) (−1) −→ 0, where TP(V¯ ) is the tangent bundle of P(V¯ ). Let H be a codimension 2 linear subspace of P(V¯ ). H is given by a codimension 2 vector subspace VH of V¯ ∨ . It is easy to see that VH ⊗k OP(V¯ ) is a subsheaf of TP(V¯ ) (−1) and provides us with an exact sequence 0 −→ VH ⊗k OP(V¯ ) −→ TP(V¯ ) (−1) −→ IH (1) −→ 0, where IH is the ideal sheaf of H. Taking the dual of this we get another exact sequence w

0 −−−−→ OP(V¯ ) (−1) −−−−→ ΩP(V¯ ) (1) −−−−→ VH∨ ⊗k OP(V¯ ) −−−−→ Ext1O Since Ext1O

P(V¯ )

P(V¯ )

(IH (1), OP(V¯ ) ) −−−−→ 0.

(IH (1), OP(V¯ ) ) ∼ = ωH (t), the above gives rise to an exact sequence 0 −→ G −→ VH∨ ⊗k OP(V¯ ) −→ ωH (t) −→ 0,

where G is the image of w. Let H1 and H2 be hyperplanes of P(V¯ ) such that H = H1 ∩ H2 . The last exact sequence shows that if H1 does not pass through the generic point of Y and if H2 contains no associated points of H1 ∩ Y , then G|Y is a subsheaf of VH∨ ⊗k OY and hence it is torsion free. A filtration of VH∨ ⊗k OY by trivial line bundles induces that of G|Y . Combining this and the subsheaf OY (−1), we obtain a filtration of ΩP(V¯ ) (1)|Y . It is now easy to see that this and Y 0 = Y \Y ∩H meet our requirement. Now we come to the main result of this section. Theorem 2.5. Let (X, OX (1)) be a pair of a projective integral scheme X over an algebraically closed field k and an ample line bundle OX (1) on X and let V be a vector subspace of H 0 (X, OX (1)) which defines a very ample linear system L. Assume that k is of characteristic 0 and X is smooth in codimension one. If E is a coherent sheaf of type (α) on X, then for general members Y1 , . . . , Yn0 of L (n0 < n), the restriction of E to the subvariety Y = Y1 · . . . · Yn0 of dimension n − n0 has the Harder-Narasimhan filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eν−1 ⊂ Eν = E|Y with the following property

36

Chapter 2

(a) or

RESTRICTION THEOREMS AND BOUNDEDNESS

µ0 (Eλ ) ≤ µ0 (E) + (r − rλ )αr−rλ /rλ

(b) µ0 (Eλ /Eλ−1 ) − µ0 (Eλ+1 /Eλ ) ≤ d,

where r = r(E) and d is the degree of X with respect to OX (1). Proof. Assume that (a) does not hold. Then by virtue of Corollary 2.2 we have the ˜λ ) − 1. We may assume that Y is smooth inequality dim g(Ψ) > dim X + r(E) − r(E in codimension one and E|Y is torsion free. Let Y0 be the largest open set of Y over which both E|Y and Eλ are locally free and put E0 = E|Y0 and F = Eλ |Y0 . If Y is sufficiently general, then (2.3) provides us with a non-zero homomorphism ω of HY |Y0 ⊗F to E0 /F . Thus we obtain a non-zero homomorphism ω 0 of ΩP(V¯ ) (1)|Y0 ⊗F 0 to E0 /F because HY ∼ = (ΩP(V¯ ) (1)|Y )⊕n . Now consider the filtration of ΩP(V¯ ) (1)|Y given in Lemma 2.4. Since codim(Y \Y0 , Y ) ≥ 2, we may assume that Y0 contains Y 0 in the lemma. Let j be the maximum among the integers with ω 0 ((Gi |Y0 ) ⊗ F ) = 0. ω 0 induces a non-zero homomorphism ω ¯ of (F ⊗ (Gj /Gj−1 ))|Y 0 to (E0 /F )|Y 0 . For 0 the immersion v : Y → Y , we have the homomorphism δ : Eλ ⊗ OY (β) ⊂ v∗ ( (Eλ ⊗ OY (β))|Y 0 ) ∼

v∗ (¯ ω)

− → v∗ ((F ⊗ (Gj /Gj−1 ))|Y 0 ) −−−→ v∗ ((E0 /F )|Y 0 ), where β = −1 or 0 according as j = 1 or j 6= 1. There is a torsion free coherent subsheaf Q of v∗ ((E0 /F )|Y 0 ) such that Q|Y 0 = (E0 /F )|Y 0 and δ(Eλ ⊗ OY (β)) ⊂ Q. We denote the induced map of Eλ ⊗ OY (β) to Q by the same δ. Since δ|Y 0 = ω ¯, δ is not zero. If 0 = Q0 ⊂ Q1 ⊂ · · · ⊂ Qν−λ = Q is the Harder-Narasimhan filtration of Q, then (Qi /Qi−1 )|Y 0 = (Ei+λ /Ei+λ−1 )|Y 0 and µ0 (Qi /Qi−1 ) = µ0 (Ei+λ /Ei+λ−1 ) because codim(Y \ Y 0 , Y ) ≥ 2. Set h = max{i | δ(Ei−1 ⊗ OY (β)) = 0} and m = min{i | δ(Eh ⊗ OY (β)) ⊂ Qi } for these h and m, δ defines a non-zero homomorphism δ¯ : (Eh /Eh−1 ) ⊗ OY (β) −→ Qm /Qm−1 . Since h ≤ λ and m ≥ 1, we see µ0 (Eh /Eh−1 ⊗ OY (β)) ≥ µ0 (Eλ /Eλ−1 ) + βd ≥ µ0 (Eλ /Eλ−1 ) − d and µ0 (Qm /Qm−1 ) = µ0 (Em+λ /Em+λ−1 ) ≤ µ0 (Eλ+1 /Eλ ). If µ0 (Eλ /Eλ−1 ) − d > µ0 (Eλ+1 /Eλ ), then the above shows that µ0 (Eh /Eh−1 ⊗ OY (β)) > µ0 (Qm /Qm−1 ). Then, δ¯ must be zero because both Eh /Eh−1 and Qm /Qm−1 are µ-semi-stable. This is a contradiction and hence we see that µ0 (Eλ /Eλ−1 ) − µ0 (Eλ+1 /Eλ ) ≤ d. As for the type of E|Y , the theorem leads us to the following corollary.

3

STATEMENTS OF BOUNDEDNESS

37

Corollary 2.6. Under the same situation as in Theorem 2.5, we denote the rational number max{α1 , . . . , αr−1 , 0} by β and let βs be max{(r − 1)β + (r − 2)d/2, (r − 1)d}(r − s)/s. Then, E|Y is of type (β1 , . . . , βr−1 ). Proof. If E|Y happens to be µ-semi-stable, there is nothing to prove. Suppose that E|Y is not µ-semi-stable and let 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eν−1 ⊂ Eν = E|Y be the Harder-Narasimhan filtration of E|Y . Set δi = a1 (Ei ), ri = r(Ei ), δ = a1 (E) and r = r(E). Assume that the set C = {λ | (a) of Theorem 2.5 holds for λ} is not empty and put τ = min C. Then for all λ with λ < τ , we have δλ+1 − δλ . rλ+1 − rλ

µ0 (E1 ) − λd ≤ µ0 (Eλ+1 /Eλ ) = This is equivalent to

(rλ+1 − rλ ){µ0 (E1 ) − λd} + rλ µ0 (Eλ ) ≤ rλ+1 µ0 (Eλ+1 ). Summing up these inequalities from λ = 1 to λ = τ − 1, we obtain rτ µ0 (E1 ) ≤ rτ µ0 (Eτ ) +

τ −1 X

(rτ − rλ )d.

λ=1

Since µ0 (Eτ ) ≤ µ0 (E) + (r − rτ )β/rτ and inequality

Pτ −1

λ=1 (rτ

− rλ ) ≤ rτ (rτ − 1)/2, we get an

(r − 2)d . 2 On the other hand, if C = ∅, then Theorem 2.5 implies that µ0 (E1 ) ≤ µ0 (E) + (r − 1)β +

µ0 (E1 ) − µ0 (Eν /Eν−1 ) ≤ (r − 1)d. Now, by Proposition I.2.14, we see that for all coherent subsheaves F of E|Y with F 6= 0, we have µ0 (F ) ≤ µ0 (E1 ), µ0 (E) − µ0 (Eν /Eν−1 ) ≥ 0 and therefore µ0 (F ) ≤ µ0 (E1 ) ≤ µ0 (E1 ) + µ0 (E) − µ0 (Eν /Eν−1 ). Combining these inequalities, the following is obtained µ0 (F ) ≤ µ0 (E) + max{(r − 1)β + = µ0 (E) +

(r − 2)d , (r − 1)d} 2

(r − r(F ))βr−r(F ) . r(F )

3 Statements of boundedness In this section we shall formulate our statements of boundedness of families of torsion free coherent sheaves. Let us fix a noetherian ring Λ and let f : X → S be a smooth, geometrically integral, projective morphism of noetherian schemes over Λ. We shall first work on this situation and then show that most results under much

38

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more general setting will be deduced from those under that restrictive situation (see section 7). We shall further fix an f -very ample invertible sheaf OX (1). Assume that the dimensions of the fibers of f are constant n. For integers r, a1 , . . . , an and for a sequence (α) = (α1 , . . . , αr−1 ) of rational numbers, we define TX/S (n, r; a1 , . . . , an ; (α)) to be the family of the classes of coherent sheaves on the fibers of X over S such that E on a geometric fiber Xs is a member of the family if and only if E has the following two properties: (3.1.1) E is of type (α) with respect to OX (1)(s), (3.1.2) r(E) = r, a1 (E) = a1 and ai (E) ≥ ai (2 ≤ i ≤ n). 0 TX/S (n, r; a1 , a2 ; (α)) is the family of the classes of coherent sheaves on the fibers of X over S such that E on a geometric fiber Xs is a member of the family if and only if E has the properties (3.1.1) and

(3.1.3) E satisfies Serre’s condition (S2 ), (3.1.4) r(E) = r, a1 (E) = a1 and a2 (E) ≥ a2 . Here we say that a coherent sheaf F on a noetherian scheme Z satisfies the condition (Sk ) if depth Fz ≥ min{k, dim Fz } for every point z ∈ Z. To define the third family, let H(x) be a numerical polynomial of degree n . We 00 set TX/S (n, r; H(m); (α)) to be the family of the classes of coherent sheaves on the fibers of X over S such that E on a geometric fiber Xs is a member of the family if and only if E has the properties (3.1.1) and (3.1.5) r(E) = r and χ(E(m)) = H(m). Our formulation of the boundedness is stated as follows: (3.2.1) Ln,r (Λ) : TX/S (n, r; a1 , . . . , an ; (α)) is bounded for all f : X → S, OX (1), a1 , . . . , an and (α), whenever n, r and Λ are fixed. 0 (3.2.2) L0n,r (Λ) : TX/S (n, r; a1 , a2 ; (α)) is bounded for all f : X → S, OX (1), a1 , a2 and (α), whenever n, r and Λ are fixed. 00 (3.2.3) L00n,r (Λ) : TX/S (n, r; H(m); (α)) is bounded for all f : X → S, OX (1), H(x) and (α), whenever n, r and Λ are fixed.

One of main tasks of this chapter is to examine to what extent the above statements are true and the author believes that all the statements should hold good for every n, r and Λ.1 A relationship among the three statements is as follows. Lemma 3.3. (1) Ln,r (Λ) implies L00n,r (Λ). (2) L0n,r (Λ) implies L00n,r (Λ). Proof. (1) is obvious. Take a coherent sheaf E on a geometric fiber Xs whose 00 class is a member of TX/S (n, r; H(m); (α)). It is enough to show that E is a (b)sheaf with (b) = (b0 , . . . , bn ) a sequence of integers which depends only on the 1 See

Appendix A.

3

STATEMENTS OF BOUNDEDNESS

39

00 family TX/S (n, r; H(m); (α)) (see Theorem I.3.11). Let E 0 be the double dual of 0 E. Then E satisfies the condition (S2 ), E is naturally a subsheaf of E 0 and T = E 0 /E is a torsion sheaf such that codim(Supp(T ), Xs ) ≥ 2. Thus we see that χ(T (m)) = cmn−2 + terms of degree < n − 2 with c ≥ 0 and hence a0 (E 0 ) = a0 (E), a1 (E 0 ) = a1 (E) and a2 (E 0 ) = a2 (E) + c ≥ a2 (E). Therefore, E 0 is a member of 0 TX/S (n, r; a1 , a2 ; (α)), where

H(m) =

n X i=0

ai

  m+n−i . n−i

L0n,r (Λ) implies that there exists a sequence (b) = (b0 , . . . , bn ) of integers such that 0 every member of TX/S (n, r; a1 , a2 ; (α)) is a (b)-sheaf. For elements σ1 , . . . , σn of 0 H (Xs , OX (1)(s)), the zero scheme of σ1 , . . . , σi is denoted by Yi . Since for general member σ1 , . . . , σn of H 0 (Xs , OX (1)(s)), E|Yi is a subsheaf of E 0 |Yi , E is also a (b)-sheaf. For the given f : X → S and OX (1), we can construct a smooth, projective, geometrically integral morphism f 0 : X 0 → S 0 of Λ-schemes and a commutative diagram X 0 ,−→ X ×S S 0 −−−−→ X    f f 0y y S 0 −−−−−−−−−−−− −−−−→ S such that for every geometric point s of S, f 0 ⊗S k(s) : X 0 ⊗S k(s) → S 0 ⊗S k(s) parameterizes all the non-singular member of the complete linear system |OX (1)(s)| defined by H 0 (Xs , OX (1)(s)) and that the pull back OX 0 (1) of OX (1) to X 0 is f 0 very ample. Proposition 3.4. Let F (or, F 0 ) be a subfamily of TX/S (n, r; a1 , . . . , an ; (α)) (or, 0 TX/S (n, r; a1 , a2 ; (α)), resp.). Assume that there exists a sequence (α0 ) = (α10 , . . . , 0 αr−1 ) of rational numbers such that if a coherent sheaf E on Xs is a member of F (or, F 0 , resp.), then for general member Y of |OX (1)(s)| E|Y is of type (α0 ). If Ln−1,r (Λ) (or, L0n−1,r (Λ), resp.) holds good, then F (or, F 0 , resp.) is bounded. For the proof of the proposition, we need the following lemma. Lemma 3.5. Let Z be a smooth projective scheme of dimension n and let OZ (1) be a very ample line bundle on Z. Let E be a coherent sheaf on Z, and let fn

fn−1

f1

f0

0 −−−−→ Fn −−−−→ Fn−1 −−−−→ · · · −−−−→ F0 −−−−→ E −−−−→ 0 be a locally free resolution of E. Let Bi be the set of pinch points of Ki = ker(fi ). (We understand that K−1 = E.) Assume that each irreducible component of Supp(E) has dimension d. (1) E satisfies the condition (Sk ) if and only if dim Bi ≤ n − k − i − 2 for i ≥ n − d − 1.

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(2) If E satisfies the condition (Sk ), then for a general member Y of |OZ (1)|, E|Y satisfies (Sk ). (3) If E satisfies the condition (Sk ) for k ≥ d, then H j (Z, E(−`)) = 0 for 0 ≤ j < k and sufficiently large `. Proof of Lemma 3.5. (1) Assume that E satisfies (Sk ). For i ≥ n − d − 1, consider a point z ∈ Z with dim {z} = n − k − i − 1. If z ∈ / Supp(E), then Ki is locally free at z. If z ∈ Supp(E), then dim Ez = d − n + k + i + 1. We have depth Ez ≥ min{k, dim Ez } = d − n + k + i + 1. Hence projdim Ez ≤ n − d. It follows that Bi is locally free at z. Conversely, assume that dim Bi ≤ n − k − i − 2 for i ≥ n − d − 1. Then for a point z ∈ Z with dim Ez = e, we have projdim Ez ≤ n − d + e − min{k, e}. Since dim Oz = n − d + e, we have depth Ez ≥ min{k, e}. (2) By (1) we have dim Bi ≤ n − k − i − 2 for i ≥ n − d − 1. For a general Y ∈ |OZ (1)|, the sequence 0 −→ Fn |Y −→ Fn−1 |Y −→ · · · −→ F0 |Y −→ E|Y −→ 0 is exact and dim(Bi ∩ Y ) ≤ n − k − i − 3. Note also that each component of E|Y has dimension d − 1. Since the set of pinch points of ker(fi |Y ) is Bi ∩ Y , we see that for the above Y , E|Y satisfies (Sk ). (3) This is a well-known theorem of Serre. By (1) there is a locally free resolution fn−k−1

f1

f0

0 −−−−→ Kn−k−1 −−−−→ Fn−k−1 −−−−−→ · · · −−−−→ F0 −−−−→ E −−−−→ 0 of length n − k. Since Fi and Kn−k−1 are locally free and coherent, Serre duality implies that there is an integer `0 such that for all ` ≥ `0 , H j (Z, Fi (−`)) = 0 (0 ≤ j < n). By induction on i and exact sequences 0 −→ Ki −→ Fi −→ Ki−1 −→ 0, we have H j (Z, Ki (−`)) = 0 for 0 ≤ j < i + k + 1 and ` ≥ `0 . For i = −1, this means that H j (Z, E(−`)) = 0 for 0 ≤ j < k and ` ≥ `0 . Proof of Proposition 3.4. Let E ∈ F (or, F 0 ) and σ be a general element of H 0 (Xs , OX (1)(s)). We have an exact sequence (3.4.1)

0 −→ E(−1) −→ E −→ E|Y −→ 0,

where Y is the zero scheme of σ. The exact sequence (3.4.1) implies that χ(E|Y (m)) = ∆χ(E(m)) = χ(E(m)) − χ(E(m − 1)). This and the above lemma shows that if E|Y is of type (α0 ), then E|Y is a member of TX 0 /S 0 (n − 1, r; a1 , . . . , an−1 ; (α0 )) or TX0 0 /S 0 (n − 1, r; a1 , a2 ; (α0 )) according as E ∈ F or E ∈ F 0 . Replacing F (or, F 0 ) by F(m) = {E(m) | E ∈ F} (or, F 0 (m) = {E(m) | E ∈ F 0 }, resp.) with m sufficiently large, we may assume that (i) H i (Y, (E|Y )(`)) = 0 for all i > 0, all ` ≥ 0 and all E ∈ F (or, E ∈ F 0 , resp.),

3

STATEMENTS OF BOUNDEDNESS

41

(ii) dim H 0 (Y, (E|Y )(`)) ≤ c` , dim H 1 (Y, (E|Y )(`)) ≤ c0` for all E ∈ F (or, E ∈ F 0 , resp.), (iii) ]{∆χ(E(m)) | E ∈ F (or, F 0 , resp.)} < ∞ and (iv) there is a sequence (b) = (b1 , . . . , bn ) of integers such that E|Y is a (b)-sheaf for all E ∈ F (or, E ∈ F 0 , resp.) because Ln−1,r (Λ) (or, L0n−1,r (Λ), resp.) holds good. In the case of F 0 , we may assume moreover that (v) there is an integer `0 such that H 0 (Y, (E|Y )(−`)) = 0 and H 1 (Y, (E|Y )(−`)) = 0 for all ` ≥ `0 and all E ∈ F 0 . The exact sequence (3.4.1) and (i) imply that for all i ≥ 2 and ` ≥ 0, H i (Xs , E) ∼ = H i (Xs , E(`)) and hence i H (Xs , E) = 0 for all i ≥ 2. Therefore, we get (3.4.2)

dim H 0 (Xs , E) − dim H 1 (Xs , E) = χ(E) = rd + a1 + d2 + · · · + dn ,

where     n−2   X m+n m+n−1 m+i χ(E(m)) = rd + a1 dn−i + . n n−1 i i=0 Since E is of type (α) and since µ0 (E(t)) = a1 /r + dt, H 0 (Xs , E(t)) must vanish for all t ≤ t0 , where t0 is the largest among the integers less than  −

a1 (r − 1)αr−1 d(KXs , OXs (1)) n + 1 + + + rd d 2d 2d

 .

On the other hand, dim H 0 (Xs , E(`))−dim H 0 (Xs , E(`−1)) ≤ dim H 0 (Y, E|Y (`)) ≤ c` . Thus dim H 0 (Xs , E) ≤ c0 + c−1 + · · · + ct0 = b0 . Note that the t0 is independent of the choice of E. Combining this inequality with (3.4.2), we get (3.4.3)

b0 ≥ dim H 0 (Xs , E) ≥ rd + a1 + d2 + · · · + dn .

By virtue of (iii) above, d2 , . . . , dn−1 range over a finite set of integers and hence, dn is bounded from above. In the case of F, dn is bounded because dn ≥ an . Thus we see that ]{χ(E(m)) | E ∈ F} < ∞. Assume that E is a member of F 0 . The exact sequence (3.4.1) shows that dim H 1 (Xs , E(`)) − dim H 1 (Xs , E(` − 1)) ≤ dim H 1 (Y, E|Y (`)) ≤ c0` . (v) implies that for all ` ≥ `0 and all E ∈ F 0 , H 1 (Xs , E(−`)) ∼ = H 1 (Xs , E(−`0 )). By this and Lemma 3.5, we have that for all ` ≥ `0 and all E ∈ F 0 , H 1 (Xs , E(−`)) = 0. We see therefore 0 ≤ dim H 1 (Xs , E) ≤ c00 + c0−1 + · · · + c01−`0 = c This and (3.4.2) imply that −c ≤ rd + rd + a1 + d2 + · · · + dn . Since d2 , . . . , dn−1 range over a finite set, dn is bounded from below. Thus dn is bounded in the case of F 0 , too. This proves ]{χ(E(m)) | E ∈ F 0 } < ∞. On the other hand, (iv) and (3.4.3) show that every member of F or F 0 is (b0 , b1 , . . . , bn )sheaf. Then the boundedness of F or F 0 follows from Theorem I.3.11.

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4 Boundedness of a family of torsion free sheaves The aim of this section is to prove various results on boundedness of families of torsion free coherent sheaves. Throughout this section Λ and f : X → S are the same as in the previous section. (A) First we shall treat the case of dim X/S = 1. Since a torsion free coherent sheaf is locally free on a smooth curve, all the statements Ln,r (Λ), L0n,r (Λ) and L00n,r (Λ) are the same. We have to prove that for a fixed Hilbert polynomial H(x) and a sequence (α) of rational numbers, there is a sequence of (b) = (b0 , b1 ) of integers and an integer m0 such that if a locally free sheaf E is of type (α) and χ(E(m)) = H(m), then E(m0 ) is a (b)-sheaf. By virtue of Lemma I.2.16 we can find an integer m0 which depends on (X, OX (1)) and H(x) such that H 0 (X, E(m0 )(−1)) = 0 if E is locally free, of type (α) and χ(E(m)) = H(m). For a non-zero global section s of OX (1), the zero scheme (s)0 of s is artinian and dim H 0 (X, O(s)0 ) = d, where d is the degree of X with respect to OX (1). Since E is locally free, we see that dim H 0 (X, E(m0 )|(s)0 (−1)) = r(E)d. Thus E(m0 ) is a (0, r(E)d)-sheaf. Thus we get the following. Proposition 4.1. For all r and Λ, all the L1,r (λ), L01,r (λ) and L001,r (λ) hold good. (B) If E is a torsion free coherent sheaf of rank 1, then the type of E puts no condition on E. This fact and Proposition 3.4 show the boundedness of torsion free coherent sheaves of rank 1. Proposition 4.2. Ln,1 (Λ) and L0n,1 (Λ) are true and hence so is L00n,1 (Λ). (C) Let us consider the case where dim X/S = 2. Assume that L02,r (Λ) is true. Take a member E of TX/S (2, r; a1 , a2 ; (α)). The double dual E 0 of E is now locally free because dim X/S = 2 and X is smooth over S. Since E 0 /E is supported by a 0-dimensional subscheme, a2 (E 0 ) ≥ a2 (E) ≥ a2 . On the other hand, E 0 is 0 obviously of type (α). Thus we see that E 0 is a member of TX/S (2, r; a1 , a2 ; (α)). Our assumption says that the last family is bounded and hence a2 (E 0 ) is bounded above by an integer a02 . This implies that a2 ≤ a2 (E) ≤ a02 , which means that ]{χ(E(m)) | E ∈ TX/S (2, r; a1 , a2 ; (α))} < ∞. Moreover, there is a sequence of integers (b) = (b0 , b1 , b2 ) such that all the E 0 ’s are (b)-sheaves and hence so are E’s. Then by Theorem I.3.11, TX/S (2, r; a1 , a2 ; (α)) is bounded. Thus we obtain a lemma. Lemma 4.3. L02,r (Λ) implies L2,r (Λ). 0 Pick a member E of TX/S (2, r; a1 , a2 ; (α)). Since X is smooth over S, dim X/S = 2 and since E satisfies the condition (S2 ), E is locally free. If F is a coherent subsheaf of E ∨ with E ∨ /F torsion free, then the natural map of E to F ∨ is surjective in codimension 1. Since E is of type (α), we see that µ(F ∨ ) ≥ µ(E) − αr(F ) or equivalently µ(F ) ≤ µ(E ∨ ) + αr(F ) . This and Lemma I.2.16, (2) imply that

4

BOUNDEDNESS OF A FAMILY OF TORSION FREE SHEAVES

43

E ∨ is of type (β) = (β1 , . . . , βr−1 ) with βt = (r − t)αr−t /t. Let KXs be the canonical sheaf of Xs , where E is on the fiber Xs . By Serre duality H 2 (Xs , E(m)) is the dual space of H 0 (Xs , E ∨ (−m) ⊗ KXs ). Since E ∨ ⊗ KXs is of type (β) and a1 (E ∨ ⊗KXs ) is determined by a1 , Lemma I.2.16, (4) implies that there is an integer 0 m0 such that for every m ≥ m0 and every member E of TX/S (2, r; a1 , a2 ; (α)), H 0 (Xs , E ∨ (−m)⊗KXs ) and hence H 2 (Xs , E(m)) vanish. For every positive integer m, the integer χ(E(m)) is greater than or equal to the integer     m+2 m+1 rd + a1 + a2 . 2 1 0 Thus there is an integer m1 determined by the family TX/S (2, r; a1 , a2 ; (α)) such that for every m ≥ m1 , χ(E(m)) is positive. Fix an integer ` ≥ max{m0 , m1 } and then the above choice of m0 and m1 implies that

dim H 0 (Xs , E(`)) = χ(E(`)) + dim H 1 (Xs , E(`)) ≥ χ(E(`)) > 0. Pick a non-zero member σ of H 0 (Xs , E(`)), which gives rise to an exact sequence 0 −→ OXs −→ E(`) −→ E10 −→ 0. Lemma 4.4. Let L be a torsion free coherent sheaf of rank 1 on a locally factorial scheme Z. If L satisfies the condition (S2 ), then it is invertible. Proof. Since L is torsion free and Z is normal, there is a closed subset Z 0 with codim Z 0 ≥ 2 such that L is invertible on the open set U = Z \ Z 0 . The local factoriality of Z provides us with such a Weil divisor DU on U that OU (DU ) ∼ = L|U . Extending DU to a divisor D on the whole space Z, we obtain an invertible sheaf OZ (D). L and OZ (D) are isomorphic on U and both satisfy the condition (S2 ). Then they are isomorphic to i∗ (L|U ) ∼ = i∗ (OU (DU )), where i is the open immersion of U into Z. Let us go back to the exact sequence right before the above lemma. The inverse image L of the torsion part of E10 to E(`) is of rank 1 and satisfies the condition (S2 ) and has a non-zero global section. Thus the above lemma gives us an effective Cartier divisor D on Xs such that L is isomorphic to OXs (D). We have therefore an exact sequence 0 −→ OXs (D) −→ E(`) −→ E1 −→ 0 with D effective and E1 torsion free. Since the degree of E(`) is fixed, the condition of type on E(`) implies that the degree of D is bounded above by a constant depending only on a1 , ` and (α). It is known that a family of effective cycles of bounded dimension and degree on a smooth projective variety is bounded. We shall give here a proof of the fact for the case of divisors.

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Proposition 4.5. Let D be a family of effective divisors on the fibers of a smooth projective scheme (Z, OZ (1)) over a noetherian scheme S. Assume that all the members of D have a fixed degree d with respect to OZ (1). Then the family D is bounded. Proof. We may replace OZ (1) by a high multiple OZ (m) and S by a suitable scheme of finite type over S. Then a general projection provides us with a finite, flat, surjective morphism π : Z → PnS , where n is the dimension of Z. The support of any member D of D is mapped to a divisor of degree ≤ d in a fiber Pns . Thus there is a hypersurface HD in Pns such that deg HD ≤ d and π ∗ (HD ) contains D as a subscheme. Note that HD may not be reduced. The family of HD ’s is parameterized by a scheme of finite type over S, or more precisely there is a scheme T of finite type over S and an effective relative Cartier divisor H of PnT over T which contains the family of HD ’s. The pull-back F = πT∗ (H) of H to X ×S T is an effective relative Cartier divisor of X ×S T over T . For every member D of D, there is a point t of T such that D is equivalent to a subscheme of Ft . Let U0 be a maximal open set of T over which F is a family of geometrically integral Cartier divisors (U0 can be empty). For the generic point t of an irreducible component T1 of T \ U0 , a geometrically integral Cartier subdivisor D1 of F ⊗T k(t) is defined over a finite algebraic extension of k(t). Thus if P be the Hilbert polynomial of D1 , then HilbP F1 /T1 is finite over T1 , where F1 = F ×T T1 . There is a non-empty open set U1 of T1 which parameterizes all the members of geometrically integral Cartier divisors of the universal family of the Hilbert scheme. Repeating this procedure, we obtain a finite number of schemes U0 , . . . , Uh of finite type over S and effective relative Cartier divisors Ei of XUi over Ui such that E0 , . . . , Eh parameterize all the integral Cartier subdivisors of F. Combining these suitably, we obtain a parameterization of D over a scheme of finite type over S. This proposition shows that the set {χ(OXs (D)(m))} of Hilbert polynomials of D’s appearing before the proposition is finite. A quotient coherent sheaf F of E1 is that of E(`) and hence µ(F ) is bounded below by µ(E(`)) − αr(F ) . On the other hand, {µ(E1 )} is a finite set. Thus there is a sequence (β) = (β1 , . . . , βr−2 ) such that every E1 constructed in the above is of type (β). Now we shall prove L2,r (Λ) by induction on r. L2,1 (Λ) holds by Proposition 4.2. 0 Assuming that L2,r−1 (Λ) is true, pick a member E of TX/S (2, r; a1 , a2 ; (α)). Then the above argument shows that we obtain `, OXs (D) and E1 and that E1 is an element of a finite union of TX/S (2, r − 1; b1 , b0 ; (β)). By our induction hypothesis 0 the set {E1 | E ∈ TX/S (2, r; a1 , a2 ; (α))} is bounded. We proved moreover that the set {OXs (D)} is bonded, too. Therefore, E is in the family of extensions of two coherent sheaves both of which are in bounded families. This implies that 0 TX/S (2, r; a1 , a2 ; (α)) is bonded and hence L02,r (Λ) holds good. Then Lemma 4.3 tells us that L2,r (Λ) is true. We completed our proof of the following. Proposition 4.6. For all r and Λ, all the L2,r (Λ), L02,r (Λ) and L002,r (Λ) hold good.

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BOUNDEDNESS OF A FAMILY OF TORSION FREE SHEAVES

45

(D) By virtue of Proposition 4.6 both L2,2 (Λ) and L02,2 (Λ) are true. Then, Theorem 1.4 and Proposition 3.4 provide us with a proof of Ln,2 (Λ) and L0n,2 (Λ) by induction on n. Proposition 4.7. For all n and Λ, all the Ln,2 (Λ), L0n,2 (Λ) are true and hence so is L00n,2 (Λ). (E) Our next task is to treat the case of rank 3. By Theorem 1.4 and Proposition 3.4 we have only to prove L3,3 (Λ), L03,3 (Λ). Let us begin with a general remark on Harder-Narasimhan filtrations. Lemma 4.8. Let Y be an integral projective scheme over an algebraically closed field and OY (1) a very ample invertible sheaf on Y . Let E be a µ-semi-stable sheaf on (Y, OY (1)) and a very ample linear subsystem L of the complete linear system |OY (1)|. For a general integral member Z of L, E|Z is torsion free and then we can consider the Harder-Narasimhan filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eν = E|Z of E|Z . If Z is sufficiently general, then r(Ei /Ei−1 ) ≥ dim Y − 1 for some i (1 ≤ i ≤ ν). Proof. By Theorem 1.4, all the (Ei /Ei−1 )|W are µ-semi-stable if r(Ei /Ei−1 ) < dim Y − 1 = dim Z for all i and if W is a sufficiently general member of the trace of L to Z. Then the Harder-Narasimhan filtration of E|W is the restriction of the given filtration to W . Using the argument of the proof of Theorem 1.4, we see that this contradicts the assumption that E is µ-semi-stable. Let us assume for a while that S = Spec(k) with k an algebraically closed field and that dim X = 3. Pick a coherent sheaf E of rank 3 on (X, OX (1)) such that a1 (E) = a1 , a2 (E) ≥ a2 and E is of type (α) = (α2 , α2 ). Let L be a very ample linear subsystem of the complete linear system |OX (1)|. We shall prove that there is a sequence (α0 ) = (α20 , α20 ), which depends only on ai , (α) and (X, OX (1)), such that for every general member Y of L, E|Y is of type (α0 ). If E is not µ-semi-stable, then Theorem 1.4 shows that for every general Y in L, the Harder-Narasimhan filtration of E|Y is just the restriction of that of E to Y . In this case, Lemma I.2.17 implies that E|Y is of type (α0 , α0 ) with α0 = max{α2 , α2 }. Thus we may assume that E is µ-semi-stable. If E|Y is µ-semi-stable, there is nothing to prove. When E|Y is not µ-semi-stable, Lemma 4.8 tells us that two is the length of the Harder-Narasimhan filtration of E|Y : 0 = E0 ⊂ E1 ⊂ E2 = E|Y . Set E2 /E1 = F . Then a1 (E1 ) + a1 (F ) = a1 . Suppose that r(E1 ) = 1. Since F is µsemi-stable and of rank 2 and since H 2 (Y, F (m)) is the dual space of H 0 (Y, F (m)∨ ⊗ KY ), H 2 (Y, F (m)) must be 0 if d(F (m)∨ ⊗KY , OY (1)) = −a1 (F )+d(KY , OY (1))−

46

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3d − 2md < 0, where KY is the canonical line bundle and d is the degree of X with respect to OX (1). (4.9.1)

κ − a1 (F ) − 1 with κ = d(KX , OX (1)), we have that for 2d 2 every m > m1 , H (Y, F (m)) = 0. setting m1 =

The definition of ai (F ) and Riemann-Roch Theorem provide us with

(4.9.2)

χ(F (m)) = dm2 + {a1 (F ) + 3d}m + a1 (F ) + a2 (F ) + 2d a1 (F ) = d(F, OY (1)) − κ − 4d

Set (4.9.3)

p   −a1 (F ) − 3d + a1 (F )2 + 2a1 (F )d − 4a2 (F )d + d2 or m2 = 2d  one of the roots of χ(F (m)) = 0

according as the roots of χ(F (m)) = 0 are real or not. In any case χ(F (m2 )) = 0. We put the terms in the square root to be A(F ): (4.9.4) A(F ) = a1 (F )2 + 2a1 (F )d − 4a2 (F )d + d2 . (EI) Assume that m2 is real and m2 > m1 . Let m be the integer [m2 ] + 1. Then we see that χ(F (m)) > 0 and H 2 (Y, F (m)) = 0 and hence that H 0 (Y, F (m)) 6= 0. A non-zero global section of F (m) gives rise to an exact sequence 0 −→ M1 (m) −→ F (m) −→ M2 (m) −→ 0 such that M1 and M2 are torsion free and of rank one and that d(M1 (m), OY (1)) ≥ 0. By (4.9.3) and µ-semi-stability of F we have p a1 (F ) + d − A(F ) a1 (F ) + κ + 4d < d(M1 , OY (1)) ≤ = µ(F ) 2 2 (4.9.5) p a1 (F ) + 2κ + 7d + A(F ) d(M2 , OY (1)) < 2 Pick a general member Z of the trace TrY (L) of the linear system L to Y . Then, for every coherent subsheaf M of F |Z with r(M ) = 1, we obtain d(M, OZ (1)) ≤ max{d(M1 , OY (1)), d(M2 , OY (1))}. The µ-semi-stability of F implies that d(M1 , OY (1)) ≤ d(M2 , OY (1)). Combining these with (4.9.5), we come to p a1 (F ) + 2κ + 7d + A(F ) d(M, OZ (1)) < . 2

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BOUNDEDNESS OF A FAMILY OF TORSION FREE SHEAVES

47

If d(E1 , OY (1)) > d(M, OZ (1)), then E1 |Z is a filter of the Harder-Narasimhan filtration of E|Z . By Theorem 1.3, we get a coherent subsheaf E 0 of E such that µ(E 0 ) = µ(E1 ) > µ(E). This is not the case because of µ-semi-stability of E. Therefore, we can find a coherent subsheaf M of F |Z with r(M ) = 1 such that 2a1 (E1 ) + κ + 4d = d(E1 , OY (1)) ≤ d(M, OZ (1)) 2 p a1 (F ) + 2κ + 7d + A(F ) < 2 It follows from this and the equality a1 (F ) = a1 − a1 (E1 ) that a2 (E1 )d > 2a1 (E1 )2 − (a1 + c1 )a1 (E1 ) + c2 a1 + da2 + c3 or (4.9.6) a1 (E1 )
1. By the uniqueness tration of E. ˜ of the Harder-Narasimhan filtration, we have that for all σ ∈ G = Gal(k(X)/k(X)), σ −1 ˜ ˜ F1 = F1 . For X0 = X \ (Sing(X) ∪ f (Sing(X))), set X0 = f (X0 ). Then ˜ 0 /G = X0 because X0 is normal and k(X) ˜ is a Galois extension of k(X). Since X ˜ ˜ 0 and X0 are smooth, f | ˜ is flat. f | ˜ : X0 → X0 is finite surjective and both X X0

X0

The separability of f implies that there is a non-empty open set U of X0 such that f |V : V = f −1 (U ) → U is ´etale. Then the action of G on V is free. Thanks to the f.p.q.c-descent theory of quasi-coherent sheaves, F1 |V descends to a coher˜ 1 |V is torsion free, so is (E|U )/H. Let E1 be ent subsheaf H of E|U . Since E/F the coherent subsheaf of E such that E1 |U = H and E/E1 is torsion free. Since ˜ ˜ and (E| ˜ ˜ )/f ∗ (E1 )| ˜ is torsion free. f |X˜ 0 is flat, f ∗ (E1 )|X˜ 0 is a subsheaf of E| X0 X0 X0 Lemma I.3.13 shows that these and the fact that F1 |V = f ∗ (E1 )|V imply that F1 |X˜ 0 = f ∗ (E1 )|X˜ 0 . Since ˜ X) ≥ 2, codim(Sing(X) ∪ f (Sing(X)),

52

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we have µ(f ∗ (E1 ), f ∗ (OX (1))) ˜ : X] [X ∗ µ(F1 , f (OX (1))) = ˜ : X] [X ˜ f ∗ (OX (1))) µ(E, > ˜ : X] [X

µ(E1 , OX (1)) =

= µ(E, OX (1)). ˜ is µ-semi-stable. This contradicts the assumption that E is µ-semi-stable. Hence E

M We shall now introduce a key tool for our proof. Let g : PN k → Pk be the Veronese embedding by the degree d polynomials, where   N +d − 1. M= d N Take a linear subspace B in PM k of dimension M − N − 1 with g(Pk ) ∩ B = N N ∅. Then the projection PM k → Pk with center B induces a morphism of g(Pk ) N N to Pk . Composing it with g, we have a morphism fB of Pk to itself. Since every morphism of PN k to a projective variety is finite or constant, fB is finite and hence surjective. It is clear that fB is represented by a homomorphism ϕ : k[Z0 , . . . , ZN ] → k[Y0 , . . . , YN ] such that (i) ϕ(Zi ) is a homogeneous form of degree d for all i and (ii) only (Y0 , . . . , YN ) = (0, . . . , 0) is the common root of ϕ(Z0 ) = · · · = ϕ(ZN ) = 0. Conversely, if a homomorphism ϕ of k[Z0 , . . . , ZN ] to k[Y0 , . . . , YN ] with the properties (i) and (ii) is given, then we can construct a morphism fB . This correspondence between fB ’s and ϕ’s is bijective up to multiplying ϕ by non-zero constants. Note that all the fB ’s are flat and that fB∗ (OPN (m)) = OPN (md).

Let (X, OX (1)) be a pair of a projective variety X over k and a very ample invertible sheaf OX (1) on X. There is a closed immersion j : X → PN k such that OX (1) ∼ = j ∗ (OPN (1)). Taking the base change of the fB , we obtain a flat finite ˜ → X. morphism f : X Proposition 5.4. Assume that the characteristic of k is not a divisor of d and X is integral. Then we have ˜ → X such that (1) If X is smooth in codimension one, then we can find f : X ˜ is integral and smooth in codimension one. X ˜ → X such that X ˜ is smooth and (2) If X is smooth, then there is an f : X integral. ˜ is connected. The direct image E = Proof. Let us prove first of all that X fB∗ (OPN ) is locally free and of finite rank because fB is flat and finite. Since E(m) = fB∗ fB∗ (OPN (m)) = fB∗ (OPN (md)) and since fB is a finite morphism,

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53

N i Leray’s spectral sequence implies that H i (PN k , E(m)) = H (Pk , OPN (md)). Thus N i we see that for all 1 ≤ i < N and m, H (Pk , E(m)) = 0. A theorem of Horrocks [Ho] then tells us that E is a direct sum of line bundles. This and the fact ∼ that dim H 0 (PN k , E) = 1 imply that E = OPN ⊕ (⊕OPN (ai )) with ai < 0. Since X is integral, we see therefore that dim H 0 (X, E|X ) = 1. On the other hand, ˜ O ˜) = the base change theorem gives rise to E|X ∼ = f∗ (OX˜ ). Thus dim H 0 (X, X 0 ˜ dim H (X, E|X ) = 1 and hence X is connected.

To show (1) let us choose a sequence of hyperplanes H0 , . . . , HN as follows: (a) for every i, Hi ∩ X is an irreducible divisor and Hi · X is reduced as a divisor on X, (b) for every i 6= j, Hi ∩ Hj ∩ X is of codimension two in X and (c) H0 , . . . , HN are linearly independent. By the property (c) if one takes a system of coordinates suitably, then PN k = Proj(k[Z0 , . . . , ZN ]) and each Hi is defined by Zi = 0. Set Fi = Yid . Then (F0 , . . . , FN ) has the property (i) and (ii) explained right before this proposition and hence it defines a morphism fB of PN k to itself and a flat ˜ ˜ finite covering f : X → X. Let D be the ramification locus of the morphism fB . ˜ is exactly H0 ∪· · ·∪HN . By virtue of the properties Obviously the image D = fB (D) (a) and (b), Hi ∩ X \ ∪j6=i Hj ∩ X contains a smooth point x of Hi · X. We claim ˜ is smooth at each point of f −1 (x). Pick a j with j 6= i, say j = 0, then here that X Spec(k[z1 , . . . , zN ]) is an affine open neighborhood of x, where z` = Z` /Z0 . Since y`d = z` for y` = Y` /Y0 , we see that k[y1 , . . . , yN ] = k[z1 , . . . , zN ][T1 , . . . , TN ]/(T1d − z1 , . . . , TNd − zN ). Hi and X intersect transversely at x because x is a smooth point of Hi · X. This means that there are elements t2 , . . . , tn of the maximal ideal mx of OX,x such that {¯ zi , t2 , . . . , tn } form a regular system of parameters of OX,x , where z¯i is the image of zi in OX,x . Since fB is finite, A = k[y1 , . . . , yN ] ⊗k[z1 ,...,zN ] OX,x is s finite OX,x -algebra. Thus the completion Aˆ of the semi-local ring A is isomorphic to ˆX,x [T1 , . . . , TN ]/(T d − z¯1 , . . . , T d − z¯N ). O 1 N On the other hand, z¯` is unit in OX,x if ` 6= i. This and the assumption that the characteristic of k is coprime to d imply that T`d

− z¯` =

d Y

(T` − ζ α u` )

α=1

ˆX,x , where ζ is a primitive d-th root of unity. Applying the Chiwith u` ∈ O N −1 nese remainder theorem repeatedly, we see therefore that Aˆ ∼ with B = = B ⊕d ˆX,x [T ]/(T d − z¯i ). As is easily seen, B is a local ring of dimension n whose maximal O ideal is generated by {T¯, t2 , . . . , tn }, where T¯ is the class of T in B. Thus Aˆ is a ˜ is smooth at regular ring and hence so is A. This proves, as we claimed, that X −1 each point of f (x). Since f is ´etale outside X ∩ D, we complete the proof that

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˜ is smooth in codimension one. Let g : X 0 → X be the normalization of X and X consider the cartesian product 0

g ˜ 0 −−− X −→   f 0y

˜ X  f y

g

X 0 −−−−→ X Since f is flat and finite, the base change theorem and the computation in the very beginning of this proof provide us with an isomorphism f∗0 (OX˜ 0 ) ∼ = g ∗ (OX ) ⊕ ∗ 0 ˜0 ˜0 (⊕g (OX (ai ))) with ai < 0, Thus we see that dim H (X , OX˜ 0 ) = 1 and hence X 0 ˜ and X ˜ are isomorphic outside closed is connected. On the other hand, since X 0 ˜ ˜ 0 satisfies set of codimension at least two, X is smooth in codimension one. X 0 0 moreover the condition (S2 ) because so is X and f is flat. By Serre’s criterion for ˜ 0 is normal. Thus X ˜ 0 is irreducible and hence so is X. ˜ the normality, we see that X ˜ satisfies (S1 ), too. This means Since X satisfies the condition (S1 ) and f is flat, X ˜ is reduced. that X To prove (2) let us choose a sequence H0 , . . . , HN of hyperplanes so generally that for every 0 ≤ i1 < i2 < · · · < iα ≤ N , X, Hi1 , . . . , Hiα intersect transversely at each point of X∩Hi1 ∩· · ·∩Hiα . By using these H0 , . . . , HN we can construct the flat finite morphism fB of PN k to itself. Take a point x in, say X ∩ Hi ∩ · · · ∩ HN \ ∪j 0 such that for every n ≥ n0 , F ⊗OX S n (E) is generated by its global sections.

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55

For some of basic properties of ample vector bundles the author refers the reader to [Har1, Propositions 2.2, 3.2 and Corollary 3.4]. Proposition 5.7. Let E be a vector bundle on an algebraic scheme X. (1) E is ample if and only if the tautological line bundle OP(E) (1) of E on P(E) is ample. (2) Assume that E is a direct sum of two vector bundles E1 and E2 . E is ample if and only if so are both E1 and E2 . (3) Assume that X is proper and there is an exact sequence 0 −→ E1 −→ E −→ E2 −→ 0. If E1 and E2 are ample, then so is E. Following Barton’s work [B], we shall present a proof of the fact that the tensor product of ample vector bundle is ample. Let X be a proper scheme over an algebraically closed field. A1 (X) (or, A1 (X)) denotes the R-vector space generated by the numerical equivalence classes of line bundles (or, curves, resp.) on X. The intersection pairing between divisors and curves induces a perfect paring ( , ) : A1 (X) × A1 (X) → R. Let us fix a norm k k on the space A1 (X). Barton’s proof is based on the following criterion for ampleness of a divisor due to Kleiman [K1, p. 327, Proposition 2]. Proposition 5.8. A line bundle L on X is ample if and only if there is a positive real number  such that for all integral curve C on X, we have (L, C) ≥ kCk. If we fix a basis {z1 , . . . , zρ } of A1 (X), then the perfect pairing between A1 (X) and A1 (X) defines a norm of A1 (X) by setting for C ∈ A1 (X) kCk =

ρ X

|(zi , C)|.

i=1

Let E be a vector bundle on a non-singular projective curve Y . δ(E) denotes the minimum of the degrees of quotient line bundles of E. Applying Proposition 5.8 to the tautological line bundle of a vector bundle, we have Proposition 5.9. Let X be a projective scheme over an algebraically closed field and fix a basis of A1 (X) to define a norm of A1 (X) as in the above. A vector bundle E on X is ample if and only if there is a positive real number  such that for every non-singular curve Y and every finite morphism g : Y → X, we have δ(g ∗ (E)) ≥ kg∗ (Y )k. Proof. Put P = P(E). We have to prove that the tautological line bundle OP (1) of E on P is ample if and only if the condition of the proposition is satisfied. Let {z1 , . . . , zρ } be the fixed basis of A1 (X), z0 be the numerical equivalence class of OP (1) and let π be the projection of P to X. Identifying A1 (X) with a vector subspace of A1 (P) by the natural injection π ∗ , {z0 , z1 , . . . , zρ } forms a basis of

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A1 (P). Pick an integral curve C in P. If C is in a fiber of Py , then (z0 , C) = deg C and (zi , C) = 0 for all i ≥ 1, where deg C is the degree of C as a curve in the projective space Py . Thus kCk = deg C = (z0 , C). Assume next that no fiber of P over X contains C. Then π induces a finite morphism g0 : C → X. Let Y be the normalization of C and g : Y → X the composition of the natural morphism of Y to C and g0 . Then g is finite and [Y : g(Y )] = [C : g(Y )]. The inclusion of C into P gives rise to a section σ of the projection p : P(g ∗ (E)) = P ×X Y → Y . The section corresponds to a quotient line bundle L of g ∗ (E) such that q ∗ (OP (1))|σ(Y ) ∼ = L, where q is the projection of P ×X Y to P and σ(Y ) is identified with Y . Thus we get (z0 , C) = (q ∗ (OP (1)), σ(Y )) = deg L ≥ δ(g ∗ (E)). On the other hand, since g∗ (Y ) = π∗ (C), we get kg∗ (Y )k = kπ∗ (C)k =

ρ X

|(zi , π∗ (C))| =

i=1

ρ X

|(π ∗ (zi ), C)| = kCk − |(z0 , C)|

i=1

Suppose that the inequality δ(g ∗ (E)) ≥ kg∗ (Y )k holds. We may assume that  < 1. Then the above computations implies (z0 , C) ≥ kg∗ (Y )k = kCk − |(z0 , C)|. Thus we have

 kCk, 1+ which proves the ampleness of OP (1) by Proposition 5.8. Conversely, assume that OP (1) is ample. There is a quotient line bundle M of g ∗ (E) with δ(g ∗ (E)) = deg M . M provides us with a section σ 0 of p. Let C 0 be the integral curve qσ 0 (Y ). Then π(C 0 ) = g(Y ). As the computation in the above, we have (z0 , C) ≥

δ(g ∗ (E)) = deg M = (q ∗ (z0 ), σ 0 (Y )) = [σ 0 (Y ) : C 0 ](z0 , C 0 ). If OP (1) is ample, then there is a positive real number 0 < 1 such that (z0 , C 0 ) ≥ 0 kC 0 k for all C 0 . Combining this with the above equality, we come to δ(g ∗ (E)) ≥ 0 [σ 0 (Y ) : C 0 ]kC 0 k = 0 (z0 , q∗ (σ 0 (Y ))) + 0

ρ X

|(π ∗ (zi ), q∗ (σ 0 (Y )))|

i=1

= 0 δ(g ∗ (E)) + 0

ρ X

|(zi , π∗ q∗ (σ 0 (Y )))|

i=1

= 0 δ(g ∗ (E)) + 0

ρ X

|(zi , g∗ (Y ))|

i=1

= 0 δ(g ∗ (E)) + 0 kg∗ (Y )k and hence δ(g ∗ (E)) ≥ kg∗ (Y )k

with  =

0 . 1 − 0

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TENSOR PRODUCT OF SEMI-STABLE SHEAVES

57

Assume now that characteristic of k is p > 0. For q = ps with s a positive integer, Fq denotes the s-th power of the Frobenius morphism Fp : X → X. If E is a vector bundle on X, then E (q) = Fq∗ (E) is the vector bundle defined by the transition matrices whose entries are q-th power of those of E. Definition 5.10. A vector bundle E on X is said to be p-ample if for every coherent s sheaf F on X, there is a positive integer s0 such that for all integers s ≥ s0 , F ⊗E (p ) is generated by its global sections. For a proof of the following basic result we refer to [Har1, Propositions 6.3 and 7.3]. Proposition 5.11. Let X be a projective scheme over a field of characteristic p > 0 and E a vector bundle on X. (1) If E is p-ample, then E is ample. (2) If X is a non-singular curve and if E is ample, then E is p-ample. Using Proposition 5.9 we obtain a tricky result. Proposition 5.12. Let E be an ample vector bundle on a projective scheme over a field of characteristic p. For every vector bundle E 0 on X, there is a positive integer s s0 such that for all s ≥ s0 , E 0 ⊗ E (p ) is ample. Proof. As easily seen, a quotient vector bundle of an ample vector bundle is ample. Since X is projective, E 0 is a quotient bundle of a direct sum of line bundles. Thus we may replace E 0 by a line bundle L. Take a basis of A1 (X) and define the norm of A1 (X) by using the basis as in Proposition 5.9. Consider the continuous map y → (L, y)kyk−1 of A1 (X) \ {0} to R. Since the map is constant on every positive ray and bounded on the compact {y ∈ A1 (X) | kyk = 1}, there is a positive integer N such that (L, y) ≥ −N kyk. Let g be a finite morphism of a non-singular curve Y to X. Then we have δ(g ∗ (L)) = deg g ∗ (L) = (L, g∗ (Y )) ≥ −N kg∗ (Y )k. Since E is ample, Proposition 5.9 provides us with a positive real number  such that for all g : Y → X, δ(g ∗ (E)) ≥ kg∗ (Y )k. Applying this to Fq ◦ g, we get an inequality δ(g ∗ (E (q) )) ≥ qkg∗ (Y )k. Pick a positive integer s0 such that ps0  > N . Then for every q = ps with s ≥ s0 and every finite g : Y → X with Y a non-singular curve, we get δ(g ∗ (L ⊗ E (q) )) = δ(g ∗ (E (q) )) + deg g ∗ (L) ≥ (q − N )kg∗ (Y )k. By Proposition 5.9 L ⊗ E (q) is ample. To state Barton’s result on tensor products of ample vector bundles we need the notion of positive tensors.

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Definition 5.13. Let r and m be positive integers. A positive tensor T is a rule which, to every vector bundle E of rank r on a scheme X, assigns another vector bundle T (E) of rank m on X, subject to the conditions: (1) For every morphism f : X 0 → X, T (f ∗ (E)) ∼ = f ∗ (T (E)). (2) If L1 , . . . , Lt are line bundles on X such that E is a quotient bundle of the direct sum L1 ⊕ · · · ⊕ Lt , then T (E) is a quotient bundle of a finite direct sum of t 1 ⊗ · · · ⊗ L⊗n , where the ni ’s are non-negative integers and line bundles L⊗n t 1 n1 + · · · + nt > 0. Note that E ⊗α , E (q) , S α (E) and ∧α E are all positive. Theorem 5.14. Let E be a vector bundle on a projective scheme X over an algebraically closed field k. If E is ample, then so is every positive tensor bundle T (E). In particular, the tensor product of two ample vector bundle is ample. Proof. Assume first that characteristic of k is zero. There is a ring R of algebraic integers, a projective scheme Z over an open subscheme U of Spec(R) and a vector ˜ on Z such that the quotient field K of R is a subfield of k, Z ×U Spec(k) ∼ bundle E = ˜ ⊗O k ∼ ˜ X, E E and Z is flat over U . Since the tautological line bundle L of E = U ˜ on P(E) is ample on the generic fiber over U , we may assume that it is ample on ˜ over U . If we can prove our assertion on a special fiber, then we every fiber of P(E) get the ampleness of the tensor bundle on the generic fiber by the openness of the ampleness and by the functorial property of positive tensors. Thus we may assume that the characteristic of k is p > 0. If E and F is ample, then so is the direct sum G = E ⊕F . Since E ⊗F is a direct summand of the positive tensor G ⊗ G, the second assertion follows from the first. Fix a norm k k of A1 (X) as in Proposition 5.9. Take an ample line bundle M on X and then there exists a positive real number 0 such that for all finite morphism g of a non-singular curve Y to X, we have the inequality δ(g ∗ (M )) ≥ 0 kg∗ (Y )k. On the other hand, by Proposition 5.12 there is a positive power q of p such that M ∨ ⊗ E (q) is ample. We are going to prove that the tensor bundle T (E) satisfies the inequality in Proposition 5.9 with  = 0 /q and hence T (E) is ample. Note first that if h : Y 0 → Y is a finite morphism of degree d and the required inequality holds for g 0 = gh, then we have the inequality for g. In fact, we see easily ∗ that dδ(g ∗ (T (E))) ≥ δ(g 0 (T (E))) ≥ kg∗0 (Y 0 )k = dkg∗ (Y )k. Since g ∗ (M ∨ ⊗ E (q) ) is ample, it is p-ample by Proposition 5.11. Replacing g by gFq0 with a large q 0 , we may assume that g ∗ (M ∨ ⊗E (q) ) is generated by its global sections, or it is a quotient of a trivial bundle OY⊕N . Thus. for gq = gFq , gq∗ (E) = g ∗ (E (q) ) is a quotient of g ∗ (M )⊕N . Replacing g by gq , we have only to prove δ(gq∗ (T (E))) ≥ k(gq )∗ (Y )k. By the positivity of T and the fact we proved in the above, gq∗ (T (E)) = T (gq∗ (E)) is a quotient of a finite direct sum of line bundles g ∗ (M )⊗α with α > 0. Thus we

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have that δ(gq∗ (T (E))) ≥ δ(g ∗ (M )) ≥ 0 kg∗ (Y )k. On the other hand, 0 kg∗ (Y )k = qkg∗ (Y )k = k(gq )∗ (Y )k. Combining these two, we obtain the inequality as required. Another application of Proposition 5.9 is an algebraic proof of a theorem of Hartshorne ([Har2]). Theorem 5.15. Let X be a non-singular projective curve over an algebraically closed field of characteristic zero. If E is a semi-stable vector bundle on X with deg E > 0, then E is ample. Proof. A divisor of degree 1 defines a norm k k in A1 (X). Let g : Y → X be a finite morphism of non-singular projective curve. By Lemma 5.3 g ∗ (E) is semistable and hence δ(g ∗ (E)) ≥ µ(g ∗ (E)) = d · deg E/r, where d is the degree of g and r is the rank of E. On the other hand, we have kg∗ (Y )k = d. Thus if we set  = deg E/r, then we get the required inequality of Proposition 5.9. We completed now our preparation to prove the main theorem of this section. Theorem 5.16. Let (X, OX (1)) be a pair of a geometrically integral projective scheme X over a field k of characteristic zero and an ample invertible sheaf OX (1) on X. If E1 and E2 are µ-semi-stable sheaves on (X, OX (1)) and if X is smooth ¯ 2 = (E1 ⊗ E2 )/torsion is µ-semi-stable. in codimension one, then E1 ⊗E Proof. First of all. replacing OX (1) by OX (m) with m large, we may assume that OX (1) is very ample. We also may assume that k is algebraically closed Let e be the degree of X with respect to OX (1), ei be d(Ei , OX (1)) and ri be the ˜ → X for d = r1 r2 e which is given by Corollary 5.5 rank of Ei . Take a covering g : X ˜ : X] and n be the dimension and set OX˜ (1) = M . Let h be the covering degree [X of X. Then, for α1 = −r2 e1 and α2 = −r1 e2 , we have d(g ∗ (Ei ), g ∗ (OX (1))) dn−1 ∗ ri d(g (OX (αi )), g ∗ (OX (1))) + dn h(dei + ri αi e) = dn h(r1 r2 eei − r1 r2 eei ) = = 0 dn

d(g ∗ (Ei ) ⊗ OX˜ (αi ), OX˜ (1)) =

by virtue of Lemmas 5.1 and 5.2. Since Ei is µ-semi-stable, so is g ∗ (Ei ) ⊗ OX˜ (αi ) ¯ ∗ (E2 ) is µ-semi-stable if and only if so by Lemma 5.3. On the other hand, g ∗ (E1 )⊗g ∗ ∗ ∼ ¯ ¯ ∗ (E2 )⊗OX˜ (α1 +α2 ). Since g is (g (E1 )⊗OX˜ (α1 ))⊗(g (E2 )⊗OX˜ (α1 )) = g ∗ (E1 )⊗g

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¯ ∗ (E2 ) ∼ ¯ 2 ) and then E1 ⊗E ¯ 2 is µ-semi-stable is flat, we see that g ∗ (E1 )⊗g = g ∗ (E1 ⊗E ∗ ∗ ¯ (E2 ) by Lemma 5.3 again. Thus, replacing Ei by g ∗ (Ei ) ⊗ OX˜ (αi ) if so is g (E1 )⊗g and OX (1) by OX˜ (1), we may assume that d(Ei , OX (1)) = 0 for both i = 1 and 2. ¯ 2 is not µ-semi-stable. Then there is a torsion free, Now assume that E1 ⊗E ¯ 2 with µ(Q) < µ(E1 ⊗E ¯ 2 ) = µ(E1 ⊗ E2 ) = coherent, quotient sheaf Q of E1 ⊗E µ(E1 ) + µ(E2 ) = 0 (see Lemma 5.1). Pick a positive integer d0 such that (5.16.1) −µ(Q) > 2re/d0 with r = max{r1 , r2 }. ˜ → X and the very ample line bundle O ˜ (1) = M on X ˜ Take a covering g : X X 0 ∗ ˜i = g (Ei ) ⊗ L is given by Corollary 5.5 for d = d again. Set L = OX˜ (r). Then E ˜ µ-semi-stable on (X, OX˜ (1)) and we have surjective homomorphism ˜1 ⊗ ˜2 −→ Q ˜ −→ 0 ¯E E

˜ = g ∗ (Q) ⊗ L⊗2 . with Q

Since OX˜ (1) is very ample, for sufficiently general members Y1 , . . . , Yn−1 of the complete linear system |OX˜ (1)|, C = Y1 · . . . · Yn−1 is a non-singular curve and ˜i |C is locally free. Moreover, by Theorem 2.5 we may assume that for the HarderE ˜i |C , we have Narasimhan filtration 0 = E0i ⊂ E1i ⊂ · · · ⊂ Eλi i = E i i 0 < µ(Eji /Ej−1 ) − µ(Ej+1 /Eji ) ≤ e0 ,

˜ with respect to O ˜ (1). From this it follows that where e0 is the degree of X X i i d(Eji ) − d(Ej−1 ) ≤ {r(Eji ) − r(Ej−1 )}{ti + (λi − j)e0 }

for 1 ≤ j ≤ λi ,

where ti = µ(Eλi i /Eλi i −1 ). Summing up these inequalities from j = 1 to j = λi , we have ˜i , O ˜ (1)) = d(Eλi ) ≤ ri ti + e0 d(E X i

λX i −1

r(Eji ) ≤ ri ti +

j=1

ri (ri − 1)e0 . 2

˜i , O ˜ (1)) = rri d(O ˜ (1), O ˜ (1)) = rri e0 because the degree On the other hand, d(E X X X ∗ of g (Ei ) is 0. Therefore, we see that µ(Eλi i /Eλi i −1 ) > 0. By virtue of Theorem 5.15 i ˜1 |C and E ˜2 |C are ample and hence Theorem 5.14 all the Eji /Ej−1 are ample. Then E ∼ ˜ ˜ ˜ ˜ ¯ E2 )|C = (E1 |C ) ⊗ (E2 |C ) is ample. Furthermore, we assume that implies that (E1 ⊗ ˜ C is locally free. Then, since Q| ˜ C is a quotient bundle of an ample (E ˜1 ⊗ ˜2 )|C , ¯E Q| it is ample, a fortiori, ˜ O ˜ (1)) = µ(Q| ˜ C ) > 0. µ(Q, X On the other hand, we compute ˜ O ˜ (1)) = µ(g ∗ (Q), O ˜ (1)) + µ(L⊗2 , O ˜ (1)) µ(Q, X X X µ(g ∗ (Q), g ∗ (OX (1))) 2rµ(g ∗ (OX (1)), g ∗ (OX (1))) + d0 n d0 n−1 0 + 2re ˜ : X] d µ(Q, OX (1)) = [X . n 0 d =

By (5.16.1) the last term of the above is negative, which is a contradiction. We see ¯ 2 is µ-semi-stable. therefore that E1 ⊗E

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To state another result on the tensor bundles we need the notion of homogeneous tensor sheaves. Definition 5.17. Let r and m be positive integers and let w be an integer. A homogeneous tensor T of weight w is a rule which, to every torsion free, coherent sheaf E of rank r on an algebraic scheme X smooth in codimension one, assigns another torsion free, coherent sheaf T (E) of rank m on X, subject to the conditions: (1) For every morphism f : X 0 → X, T (f ∗ (E)) ∼ = f ∗ (T (E)), where X 0 is smooth in codimension one and means modulo “torsion”. (2) T (E) is a quotient sheaf of E ⊗α ⊗(E ∨ )⊗β with α and β non-negative integers. (3) For a line bundle L on X, T (E ⊗ L) ∼ = T (E) ⊗ Lw . (4) c1 (T (E)) = γc1 (E) for some integer γ. ¯ ¯ α E are all homogeneous, where ¯ “means modulo torsion”. E ⊗α , S¯α (E) and ∧

Theorem 5.18. Let (X, OX (1)) be as in Theorem 5.16. If E is µ-semi-stable sheaf on (X, OX (1)), then every homogeneous tensor sheaf T (E) is µ-semi-stable. ˜ → X is a flat finite covering with X ˜ smooth in codimension one, Proof. If g : X ∗ ⊗w ∼ ∗ ∗ ∼ then T (g (E) ⊗ L) = T (g (E)) ⊗ L = g (T (E)) ⊗ L⊗w by the conditions (1) and (3) of the homogeneous tensor, where w is the weight of T . Therefore, as in the proof of Theorem 5.16, we may assume that d(E, OX (1)) = 0. Let F be E ⊗α ⊗ (E ∨ )⊗β modulo torsion. Since T (E) is torsion free, it is a quotient sheaf of F by virtue of (2) of Definition 5.17. On the other hand, we see that F is µ-semistable by using Theorem 5.16 repeatedly. Moreover, (4) of Definition 5.17 implies that d(T (E), OX (1)) = γd(E, OX (1)) = 0. Combining these with Proposition I.2.10 we find that T (E) is µ-semi-stable. Corollary 5.19. Let (X, OX (1)) be as in Theorem 5.16. If E1 , . . . , Eh are µ-semistable sheaves on (X, OX (1)) and if T1 (E1 ), . . . , Th (Eh ) are homogeneous tensor ¯ · · · ⊗T ¯ h (Eh ) is µ-semi-stable. In particular, S¯m (Ei ), ∧ ¯ s Ei , sheaves, then T1 (E1 )⊗ s s m m ¯ · · · ⊗E ¯ h are µ-semi-stable, where ¯ ¯ Ei ), ∧ ¯ (S¯ (Ei )), Hom(Ei , Ej ) and E1 ⊗ S¯ (∧ means “modulo torsion”.

6 Theorem of Mumford-Mehta-Ramanathan As we have shown in Theorem 1.4, the restriction of a µ-semi-stable sheaf to a general hyperplane is again µ-semi-stable if the rank is less than the dimension. This does not hold, however, without assuming the condition on the rank. In fact, the restriction of the tangent bundle Tn of the projective space Pn to a hyperplane is isomorphic to Tn−1 ⊕OPn (1) which is obviously not µ-semi-stable. In this section we are going to show a beautiful result due to Mumford, Mehta and Ramanathan [MR] which overcomes this trouble by taking a positive multiple of the polarization. Let X be a non-singular projective variety of dimension n ≥ 2 over an algebraically closed field k and let OX (1) be a very ample line bundle on X which

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defines an embedding of X into PN k . For each positive integer m, we take a vector subspace Wm of H 0 (X, OX (m)) which gives rise to a very ample linear subsystem ∨ of |OX (m)|. Sm = P(Wm ) parameterizes the hypersurfaces of degree m of X. Let m be a vector (m1 , . . . , mt ) of positive integers with 1 ≤ t ≤ n − 1. For the product Sm = Sm1 ×k · · · ×k Smt , we can construct a diagram qm

X ×k Sm ⊃ Zm −−−−→ Sm   pm y X −1 (s) the closed where for a k-valued point s = (s1 , . . . , st ) ∈ Sm , pm makes qm subscheme of X defined by the ideal generated by {s1 , . . . , st }. Since Wm generates OX (m), there is an exact sequence 0 −→ Gm −→ Wm ⊗k OX −→ OX (m) −→ 0 with Gm a locally free sheaf of rank dim Wm − 1. We thus obtain a projective ∨ subbundle P(G∨ m ) of X ×k Sm = P(Wm ⊗k OX ). At a k-valued point x of X, the ∨ fiber P(Gm )x is the hyperplane in Sm formed by {s ∈ Wm | s(x) = 0}. It is not hard to see that ∨ Zm = P(G∨ m1 ) ×X · · · ×X P(Gmt ) and hence Zm is a fiber bundle whose fibers are product of projective spaces. This implies that Zm is a non-singular projective variety. We denote the function field of Sm by Km and the generic fiber of qm by Ym . qm

Zm −−−−→ x  ϕm 

Sm x  

Ym −−−−→ Spec(Km ) Ym is geometrically integral and smooth and we call it the generic complete intersection of type m. 0 00 Proposition 6.1. Let F be a coherent sheaf on X and let Sm and Sm be the open subschemes of Sm 0 −1 Sm = {z ∈ Sm | dim qm (z) = n − t} 00 0 Sm = {z ∈ Sm | F ⊗ Iz → F is injective}, −1 −1 0 0 where Iz is the ideal of qm (z) in X. Then qm (Sm ) is flat over Sm and p∗m (F ) is 00 flat over Sm . 0 Proof. As we have seen in the above Zm is smooth and equidimensional on Sm . 0 0 Then the smoothness of Sm implies the flatness of qm over Sm . Since F ⊗OX OX×Sm 00 is flat over Sm , our condition for Sm is exactly the condition (2) in Lemma I.3.10 ∗ 00 for the flatness of pm (F ) over Sm .

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We are going to prove Weil’s lemma which is a very basic result on the Picard group of the generic complete intersection. Proposition 6.2. Let X be a non-singular projective variety of dimension n ≥ 2 over an algebraically closed field k and let m = (m1 , . . . , mt ) be a vector of positive integers such that 1 ≤ t ≤ n−1 and there are positive integers mi1 , mi2 and mi3 with mi = mi1 + mi2 + mi3 . If the image of Wmi1 ⊗k Wmi2 ⊗k Wmi3 in H 0 (X, OX (mi )) is a subspace of Wmi , then the restriction map Pic(X) → Pic(Ym ) induced by the composition of ϕm and pm is an isomorphism. Proof. Take an affine open set Spec(A) of Sm . Then Km is the quotient field Q(A). For a subvariety D of Ym , let a1 , . . . , ad ∈ A be the denominators of generators of the ideal of D and let U be the open subscheme D(a1 ) ∩ · · · ∩ D(ad ) of Sm . Then D extends to a subscheme of Zm ×Sm U . If a member L of Pic(Ym ) corresponds to a P ˜i Weil divisor ni Di , then there is an open set U of Sm and are Cartier divisors D ˜ on Zm ×Sm U such that Di ∩ Ym = Di because Zm is smooth. Taking the closure of ˜ i in Zm , we get a Cartier divisor D ¯ on Z such that D ¯ i ∩ Ym = Di . Let L ¯ be the D Pi ¯ m ∗ ¯ ∼ invertible sheaf on Zm defined by ni Di . Then, obviously we have ϕ (L) = L. m

We claim ∗ (Pic(Sm )). (6.2.1) Pic(Zm ) = p∗m (Pic(X)) ⊕ qm

In fact, since each fiber of pm is the product of hyperplanes of Smi , for every invertible sheaf M on Zm , there is an invertible sheaf M 0 on Sm such that M ⊗ ∨ ∗ qm (M 0 ) is trivial on every fiber of pm . Then, by the see-saw theorem we can find ∨ ∗ (M 0 ) ∼ an invertible sheaf M 00 on X such that M ⊗ qm = p∗m (M 00 ), which implies ∗ ∗ ∗ (Pic(Sm )), that Pic(Zm ) = pm (Pic(X)) + qm (Pic(Sm )). On one hand, if L is in qm ∗ then L ∼ = qm (OSm (a1 , . . . , at )), where a1 , . . . , at are integers and OSm (a1 , . . . , at ) is the tensor product of the pull-backs of OSm1 (a1 ), . . . , OSmt (at ) to Sm . On the other hand, if L is a member of p∗m (Pic(X)), then it is trivial on every fiber of pm . Since each fiber of pm is the products of hyperplanes of Sm1 , . . . , Smt , for an element L ∗ of p∗m (Pic(X)) ∩ qm (Pic(Sm )), we have a1 = a2 = · · · = at = 0 which means that L is trivial. This completes the proof of our claim (6.2.1). ∗ Since qm (Pic(Sm )) is in the kernel of the map ϕ∗ : Pic(Zm ) → Pic(Ym ), ϕ∗ ∗ passes through Pic(Zm )/qm (Pic(Sm )) which is isomorphic to p∗m (Pic(X)) by the claim (6.2.1). This proves the part of the surjectivity in our proposition because ϕ∗ is surjective as we showed before the claim (6.2.1).

To prove the part of the injectivity we need lemmas. Lemma 6.3. Let q : Z → S be a proper flat morphism of noetherian integral schemes and L be an invertible sheaf on Z. If every fiber of q is integral, then the following are equivalent with each other. (1) L is trivial on the generic fiber. (2) There is a non-empty open set U of S such that for every geometric point x of U , L(x) is trivial.

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(3) L is trivial on every geometric fiber of q. (4) There is an invertible sheaf M on S such that L ∼ = q ∗ (M ). Proof. Since every fiber is proper and integral, for a point t of S, L(t) is trivial if and only if H 0 (Zt , L(t)) 6= 0 and H 0 (Zt , L(t)∨ ) 6= 0. Moreover, if L(t) is trivial, then so is L(s) for every geometric point of S over t. Since S is connected, these and the upper semi-continuity of the dimensions of cohomologies prove the equivalence of (1) and (3). Obviously, (3) implies (2) and (2) does (1). The equivalence of (3) and (4) is due to the see-saw theorem. The following is the reason why we need the condition on our linear system in the proposition. Lemma 6.4. Under the situation of proposition 6.2, let m be an mi . (1) For a closed point x of X, let V = {s ∈ Wm | s(x) = 0 and the divisor defined by s = 0 is singular at x}. Then the rational map defined by V induces an isomorphism of X \ {x} to its image. (2) There is a non-empty open set V0 of V such that for every s ∈ V0 , the divisor defined by s = 0 is integral. −1 (s) is not integral} is a closed set. Moreover, (3) The set A = {s ∈ Sm | qm −1 codim(A, Sm ) ≥ 2 and codim(qm (A), Zm ) ≥ 2.

Proof. By the condition on Wm there are positive integers α, β and γ such that m = α + β + γ and Wm contains the image of Wα ⊗k Wβ ⊗k Wγ . Let Lt be the linear system defined by Wt . For closed points x1 , x2 of X \ {x}, there is a member D1 or D2 of Lα or Lβ respectively such that neither passes through xi but both contain x. Then for every D of Lγ , D + D1 + D2 is in Lm and the corresponding section is in V . Thus the linear system LV defined by V separates two points and the tangent space at a point of X \ {x}. This proves our assertion (1). Since the linear system LV embeds X \ {x} to a projective space, we see that (2) holds by Bertini’s theorem (cf. [Jou, Theorem 6.10]). By [EGA, IV Theorem 12.2.4] A is closed. Pick a closed point y of X and a section s0 in Wm with s0 (y) 6= 0. By sending s in Wm to (s/s0 )y in OX,y , we obtain a k-linear map of Wm to OX,y and then composing this with the natural map of OX,y to OX,y /m2y , a k-linear map −1 θy : Wm → OX,y /m2y is defined. Let By be the set {s ∈ Sm | qm (s) is singular at y}. Since Lm is very ample, θy is surjective. Thus, noting that By = ker(θy )/k ∗ ˜ to be and dim OX,y /m2y = n + 1, we have that dim By = dim Sm − n − 1. Set B −1 ˜ {(y, s) ∈ X × Sm | qm (s) is singular at y}. The fiber of the projection B → X over ˜ is a projective bundle over X. These y is the projective space By and in fact B ˜ ˜ is imply that B is irreducible and of dimension dim Sm − 1. The image B = qm (B) then irreducible and a proper closed subset of Sm . The connectedness theorem of Zariski implies that every non-integral member of Lm is singular and hence A ⊂ B. We have only to prove now that A is different from B. Our assertion (2) provides −1 us with an s such that qm (s) is integral but singular. s is obviously in B but not −1 in A. Finally since qm is equidimensional, qm (A) is of codimension ≥ 2.

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Now we shall prove the injectivity of Pic(X) → Pic(Ym ). Our proof is by induction on t. Assume that t = 1 and take an L in Pic(X) such that ϕ∗m p∗m (L) is ] ] −1 ] trivial. Consider the morphism qm : Zm = Zm \ qm (A) → Sm = Sm \ A and the ] ∗ ] restriction L = pm (L)|Zm ] . Then, by Lemma 6.3, the triviality of L on the generic ] ] ] ∼ ] fiber provides us with an invertible sheaf M on Sm such that L = qm (M ] ). Since Sm is smooth, M ] extends to an invertible sheaf M on the whole space Sm . On ] −1 ∼ ∗ one hand, p∗m (L)|Zm ] = qm (M )| ] and on the other hand, Zm \ Zm = qm (A) is of Zm ∗ ∗ ∼ codimension ≥ 2. Thus we see that pm (L) = qm (M ). Then, by (6.2.1) p∗m (L) must be trivial and hence so is L. Assume that t > 1 and the injectivity holds for u = (m1 , . . . , mt−1 ). Let L be an invertible sheaf on X such that ϕ∗m p∗m (L) is trivial. By Lemma 6.3 L is −1 −1 trivial on qm (s) if qm (s) is integral. If s0 is a general point of Su , then qu−1 (s0 ) is −1 (s) embedded in qu−1 (s0 ), L|q−1 (s) is a smooth variety. For every integral fiber qm m trivial. Applying the case of t = 1 to the smooth variety qu−1 (s0 ), we see that L is trivial on qu−1 (s0 ). Then, by Lemma 6.3 L is trivial on the generic fiber of qu . Our induction hypothesis now implies that L is trivial on X. Proposition 6.5. Let A be a discrete valuation ring whose field of quotients Q(A) is K and whose residue field A/m is k. Let D be a projective flat scheme over S = Spec(A) such that D is smooth outside B with codim(B, D) ≥ 3, the generic fiber DK is smooth and irreducible and that the special fiber Dk is reduced and the sum of smooth irreducible Cartier divisors Dk1 , . . . , Dkr as a divisor of D. Fixing a line bundle OD (1) on D which is ample over S, for a reflexive sheaf V on D, we set µK = max{µ(W ) | W is a coherent subsheaf of VK } µik = max{µ(W ) | W is a coherent subsheaf of Vk |Dki }. Pr Then we have that µK ≤ i=1 µik . Proof. First of all note that D is normal and hence V satisfies the condition (S2 ). This implies that every Vki = V |Dki is torsion free. Let WK be a coherent subsheaf of VK such that µ(WK ) = µK and QK = VK /WK is torsion free. The properness of the Quot-scheme shows that the exact sequence 0 −→ WK −→ VK −→ QK −→ 0 extends to an exact sequence 0 −→ WA −→ VA −→ QA −→ 0 with QA torsion free. Set Wk = WA |Dk and Wki = WA |Dki . Since V satisfies the condition (S2 ), so is WK and hence there is an open set D0 of Dk such that ˜ be the disjoint codim(Dk \ D0 , Dk ) ≥ 2 and W0 = WA |D0 is locally free. Let D `r i ˜ to Dk . For the finite covering union i=1 Dk and π be the natural morphism of D −1 ˜ $ : D0 = π (D0 ) → D0 , we have an isomorphism $∗ $∗ (W0 (m)) ∼ = W0 (m) ⊗OD0 $∗ (OD˜ 0 ).

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Thus the Hilbert polynomials of π∗ π ∗ (Wk ) and Wk ⊗ π∗ (OD˜ ) are the same in the terms of degree dim Dk − 1. This and the equality χ(π∗ π ∗ (Wk ) ⊗ O(n)) = Pr i i=1 χ(Wk (n)) give rise to r X

a1 (Wki ) = a1 (Wk ⊗ π∗ (OD˜ )).

i=1

On the other hand, the exact sequence 0 −→ Wk (m) −→ Wk ⊗ π∗ (OD˜ )(m) −→ Wk (m) ⊗ (π∗ (OD˜ )/ODk ) −→ 0 provides us with a1 (Wk ⊗ π∗ (OD˜ )) = a1 (Wk ) + r(Wk )a0 (π∗ (OD˜ )/ODk ) . Moreover, we have the following obvious equalities r X

a1 (ODki )

= a1 (OD˜ )

i=1

= a1 (π∗ (OD˜ )) = a1 (ODk ) + a0 (π∗ (OD˜ )/ODk ) . Using these and the equalities µ(Wki )

=

a1 (Wki ) − r(Wk )a1 (ODki ) r(Wk )

our computation proceeds as follows Pr r r X a1 (Wki ) X µ(Wki ) = i=1 − a1 (ODki ) r(Wk ) i=1 i=1 a1 (Wk ) + a0 (π∗ (OD˜ )/ODk ) − a1 (ODk ) − a0 (π∗ (OD˜ )/ODk ) r(Wk ) a1 (Wk ) − a1 (ODk ). = r(Wk )

=

The last term of the above is equal to µ(WK ) because a1 (WK ) = a1 (Wk ), r(WK ) = Pr r(Wk ) and a1 (ODK ) = a1 (ODk ). By the very definition i=1 µik is greater than or equal to the above. We have to construct a degenerating family of divisors of X which has the properties in Proposition 6.5. From now on we are going to assume that for positive integers m, m1 , . . . , mt with m = m1 + · · · + mt , the image of Wm1 ⊗k · · · ⊗k Wmt in H 0 (X, OX (m)) is a subspace of Wm . This is possible, for example, if we set Wm to be the image of S m (W1 ) in H 0 (X, OX (m)). Corollary 6.6. Under the situation of Proposition 6.5, if Vki is µ-semi-stable for all i, then VK is µ-semi-stable.

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67

Proof. By the same argument as in the proof of Proposition 6.5 we see that Pr i i i=1 µ(Vk ) = µ(VK ). Since Vk is µ-semi-stable, we have µK ≤

r X i=1

µik ≤

r X

µ(Vki ) = µ(VK ),

i=1

which means the µ-semi-stability of VK . Lemma 6.7. Assume that for each positive integer m, a non-empty open set Um of Sm is given. Let m, m1 , . . . , mt be positive integers with m = m1 +· · ·+mt . Then there is a discrete valuation ring A with the field of quotients K and the residue field k and there is an embedding of Spec(A) into Sm such that the induced family −1 (Spec(A)) → Spec(A) has the following properties: D = qm −1 (¯ y ) with y¯ ∈ Um (K). (1) The generic fiber DK is smooth and DK = qm

(2) The special fiber Dk is reduced and Dk = Dk1 + · · · + Dkt such that every Dki −1 is smooth, Dki = qm (yi ) with yi ∈ Umi (k) and Dk1 , . . . , Dkt intersect transversely. i (3) D is smooth outside B with codim(B, D) ≥ 3. −1 (s), C1 = Proof. Take s ∈ Um (k) and si ∈ Umi (k) such that the divisors C = qm −1 −1 qm1 (s1 ), . . . , Ct = qmt (st ) are smooth and intersect transversely. Regarding the Qt points s, s1 , . . . , st as members of Wm , Wm1 , . . . , Wmt , the set {us + v i=1 si | (u : v) ∈ P1 } gives rise to a projective line ` in Sm . Let y be the point (0 : 1) on `. It −1 −1 ˜ = qm is easy to see that D (`) is smooth on the fiber qm (y) outside p−1 m (∪i6=j C ∩ ˜ Ci ∩ Cj ). Since D is smooth over the point (1 : 0) and the point is in Um , the local ring A = O`,y meets our requirement.

Our main theorem in this section is stated as follows. Theorem 6.8. Let (X, OX (1)) be a pair of a non-singular projective variety over an algebraically closed field and a very ample line bundle OX (1) on X. Let W1 be a vector subspace of H 0 (X, OX (1)) which defines a very ample linear subsystem of the complete linear system |OX (1)|, Wm be the image S m (W1 ) in H 0 (X, OX (m)) and let Lm be the linear system defined by Wm . For a µ-semi-stable sheaf E on (X, OX (1)), there is a positive integer m0 such that for every integer m ≥ m0 and general members D of Lm , E|D is µ-semi-stable. Proof. First of all we may assume that E is reflexive and d(E, OX (1)) > 0. Let Ym be the generic fiber of qm . Suppose that none of E|Ym is µ-semi-stable. Then, there is a non-empty open set Um with the following properties: −1 −1 (6.8.1) For all s ∈ Um , qm (s) is smooth and E|qm (s) is reflexive. −1 (6.8.2) Em = p∗m (E)|qm (Um ) contains a coherent subsheaf Fm such that Qm = Em /Fm is torsion free and flat over Um . −1 −1 (6.8.3) For all s ∈ Um , Qm |qm (s) is torsion free and Fm |qm (s) is the first filter of −1 (s). the Harder-Narasimhan filtration of E|qm

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Note that Em is flat over Um and hence so is Fm . Set rm = r(Fm ) and βm = rm −1 µ(Fm |qm Fm )∨∨ . If m ≥ 3, then Proposi(s) ). Let det Fm be the line bundle (∧ tion 6.2 provides us with a line bundle Mm on X such that Mm |Ym ∼ = det Fm |Ym . −1 Then we see by Lemma 6.3 that for every s ∈ Um , Mm |qm is isomorphic to (s) −1 det Fm |qm . We set d = d(M , O (1)). Writing m = 3t + a with 3 ≤ a ≤ 5, we m m X (s) consider a family D in Lemma 6.7 for Um , U3 and Ua such that Dk = Dk1 +· · ·+Dkt+1 with Dk1 , . . . , Dkt ∈ U3 and Dkt+1 ∈ Ua , where we identify Dki with the correspond˜ be the double dual of the pull-back of E to D. Then ing point in S3 or Sa . Let E ∼ ˜ E|Dki = E|Dki . Hence, by Proposition 6.5 we have βm ≤ tβ3 + βa . On the other hand, since βm = mdm /rm and β` > `µ(E) ≥ 0, we come to an inequality rm (tβ3 + βa ) dm ≤ ≤ r(β3 + βa ) m and hence dm is bounded. Then the rational numbers δm = βm /m = dm /rm are bounded and discrete. Let δ be min{δm | m ≥ 3} and r = min{rm | δm = δ}. We shall consider the set G = {m | m ≥ 3, δm = δ and rm = r}. If m1 and m2 are in G, we have a family D of divisors in Lemma 6.7 for Um , Um1 , Um2 and the reflexive ˜ on D as before. Applying Proposition 6.5 to this situation the coherent sheaf E inequality βm ≤ βm1 + βm2 is obtained. By our definition of δ we see mδ ≤ mδm = βm ≤ βm1 + βm2 = m1 δ + m2 δ = mδ. ˜ with Thus δm = δ. Moreover, if FA is the extension of Fm |DK to a subsheaf of E ˜ E/FA torsion free, then H = Fk |Dk1 is a subsheaf of E|Dk1 such that µ(H) = m1 δ = βm1 and r(H) = rm ≥ r. Since r = rm1 is the maximal among the ranks of the subsheaves H 0 of E|Dk1 with µ(H 0 ) = βm1 , rm must be equal to r. These prove that m is a member of G. Let b be the smallest integer in G and pick an integer c with 1 ≤ c ≤ b. The above result shows that once c + t0 b is in G, tb + c is in G for all integers t ≥ t0 . If tb + c is not in G for t > 1 and if δtb+c = δ, then rtb+c > r. Applying the same argument as above to m = tb + c, m1 = b and m2 = (t − 1)b + c, we come to the contradiction that rb > r. Thus if tb + c is not in G for t > 1, then we must have δtb+c > δ. Assume that δtb+c ≥ δ(t−1)b+c . The same argument as above gives rise to (tb + c)δtb+c = βtb+c ≤ βb + β(t−1)b+c = bδ + {(t − 1)b + c}δ(t−1)b+c < (tb + c)δtb+c , which is a contradiction. We obtain therefore that δtb+c < δ(t−1)b+c , that is, δtb+c is strictly monotone decreasing while tb + c is not in G. On the other hand, the set {δm | m ≥ 3} is bounded and discrete. Thus there is tc such that {tb + c | t ≥ tc } is in G. Then, for t0 = max{t1 , . . . tb−1 }, all the integers m ≥ t0 b are members of G. We set m0 = t0 b.

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69

Lemma 6.9. For m ≥ m0 , Mm are all isomorphic. Proof. Pick an integer m with m ≥ 2m0 or m = m0 + m1 with m1 ≥ m0 . Using Lemma 6.5 for Um , Um0 and Um1 , we have a family π : D → Spec(A) of divisors over −1 discrete valuation ring A such that the generic fiber is DK = qm (¯ s) with s¯ ∈ Um (K) 0 1 i −1 and the special fiber is Dk + Dk such that Dk = qmi (si ) with si ∈ Umi (k). Dk0 and Dk1 meet transversely in X and D is smooth outside B such that codim(B, D) ≥ 3 ˜ be (p∗m (E)|D )∨∨ . The first filter of the Harderand B is a subset of Dk0 ∩ Dk1 . Let E ˜K = E|D extends to a coherent subsheaf F˜ of E ˜ such Narasimhan filtration of E K i ˜ i ˜ ˜ ˜ ˜ that E/F is torsion free. Set F = F |Dki . We know that µ(F ) = βmi and r(F˜ i ) = rm = r = rmi . Thus F˜ i coincides with the first filter of the Harder-Narasimhan filtration of F˜ i outside B i with codim(B i , Dki ) ≥ 2. Put det F˜ = (∧r F˜ )∨∨ . Then det F˜ is reflexive and invertible on the set of smooth points of D. As we have seen in the above, det F˜ |DK is isomorphic to Mm |DK and det F˜ |Dki ∼ = Mmi |Dki on ∨ i i ∗ ˜ Ui = Dk \ B ∪ B. Set N = pm (Mm ) ⊗ det F . Then N is reflexive and hence flat over A. Since dim H 0 (DK , N |DK ) = 1, π∗ (N ) is a torsion free A-module of rank 1 and hence isomorphic to A. Pick the element g corresponding to 1 ∈ A. π ∗ (g) gives rise to a global section σ ∈ Γ(D, N ). A uniformanizing parameter u of A defines an exact sequence 0 −→ uN −→ N −→ N |Dk −→ 0 and σ induces a section σk of N |Dk . Assume that σk |U = 0 with U = U0 ∪ U1 . Since N |Dk satisfies the condition (S1 ), σk = 0 and then it is not hard to see that g cannot correspond to a unit. Thus σk |U 6= 0 and we can find i such that σ i = σ|Dki 6= 0. Since N |Dki is invertible, of degree 0 on Ui , has a non-zero section and since Dki \ Ui is of codimension at least 2, N |Ui is isomorphic to the trivial invertible sheaf OUi . Now σ i defines an element of Γ(Ui , OUi ) = k. Moreover, Ui ∩ Dk0 ∩ Dk1 6= ∅, σk does not vanish on Uj with j 6= i and hence it is a constant on U . We have proved therefore that σ : Mm → det F˜ is isomorphic on V = D \ B ∪ B 0 ∪ B 1 . Since det F˜ is reflexive and since B ∪ B 0 ∪ B 1 is of codimension at least 3, det F˜ is isomorphic to Mm and hence it is invertible. On the other hand, det F˜ |Ui ∼ = Mmi |Ui . This and that imply that det F˜ |Dki ∼ = Mmi |Dki . Therefore, for general s ∈ Sm0 , ∼ ∼ ∼ ˜ −1 −1 −1 Mm |qm (s) = det F |qm (s) = Mm0 |qm (s) and hence Mm = Mm0 by Lemma 6.3 and 0 0 0 ∼ Lemma 6.4. Similarly we have that Mm = Mm1 . We shall go back to the proof of Theorem 6.8. Take the line bundle M that is isomorphic to all Mm with m ≥ m0 and the maximal open set X0 where E is locally free. Let E0 be the locally free sheaf EX0 . The vector bundle ∧r E0 extends to a reflexive sheaf H on the whole space. The Grassmann bundle Grassr (E0 ) of rank r subbundles is embedded in P(∧r E0∨ ) = P(M ⊗ ∧r E0∨ ). Let Γ be the cone in the vector bundle V(M ⊗ ∧r E0∨ ) over the Grassmann bundle. A section σ ∈ H 0 (X, M ∨ ⊗H) defines a section σ ˜ of the structure morphism V(M ⊗∧r E0∨ ) → X0 . Γ(σ) denotes the closed set {x ∈ X0 | σ ˜ (x) ∈ Γ}. {Γ(σ) | σ ∈ H 0 (X, M ∨ ⊗ H)} form a bounded family of closed sets of X0 . Since the degrees of the closures in X of Γ(σ)’s are bounded, there is an integer m1 such that for all m ≥ m1 , Γ(σ) does not

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−1 contain any hypersurface qm (s) unless Γ(σ) = X0 . On the other hand, we can find an integer m2 such that for all m ≥ m2 and general s ∈ Sm , H 0 (X, M ∨ ⊗ H) → −1 −1 H 0 (qm (s), M ∨ ⊗ H|qm (s) ) is surjective. Let m3 be max{m0 , m1 , m2 }. For an −1 −1 m ≥ m3 and general s ∈ Sm , let σ ¯ be a member of H 0 (qm (s), M ∨ ⊗ H|qm (s) ) −1 corresponding to the first filter F¯m = Fm |qm of the Harder-Narasimhan filtration (s) −1 of E|qm ¯ to an element σ of H 0 (X, M ∨ ⊗H). By the definition of (s) . We can lift the σ −1 Γ(σ), Γ(σ) must contain qm (s) and then Γ(σ) = X0 by the choice of m. On the open 0 set X0 = {x ∈ X0 | σ(x) 6= 0}, σ gives rise to a vector subbundle F which coincides with F¯m on q −1 (s)∩X 0 . Now, extending F to a coherent subsheaf F˜ of E with E/F˜ m

0

−1 −1 torsion free, we come to a contradiction because codim(qm (s) \ X00 , qm (s)) ≥ 2 and ˜ ¯ −1 hence µ(F ) = µ(Fm )/m > µ(E|qm )/m = µ(E). (s)

Remark 6.10. Let f : X → T be a smooth projective morphism of noetherian schemes, OX (1) be an f -very ample invertible sheaf. Let W1 → f∗ (OX (1)) be a homomorphism with W1 a locally free OT -module such that at each point t ∈ T , the image of W1 (t) in H 0 (Xt , OXt (1)) via the natural map f∗ (OX (1))(t) → H 0 (Xt , OXt (1)) defines a very ample linear subsystem of |OXt (1)|. Let Lm (t) be the linear system defined by the image of S m (W1 (t)) in H 0 (Xt , OXt (m)). Let E be a coherent sheaf on X such that E is flat over T and for every point t of T , E(t) is µ-semi-stable. Then there exists a positive integer m0 such that for every m ≥ m0 , every t ∈ T and general member D of Lm (t), E(t)|D is µ-semi-stable. In fact we can construct Sm and Zm relatively over T . Pick a point t ∈ T . By Theorem 6.8 there is an integer m(t) such that for every m ≥ m(t) and general point s ∈ (Sm )t , E|qm (t)−1 (s) is µ-semi-stable. By openness of µ-semi-stability there is an open set Um (t) of Sm such that its image in T contains t and E is µ-semistable on every fiber of Zm over Um (t). Let U (t) be the intersection of the images of Um(t) (t), . . . , U2m(t)−1 (t) in T . By using the argument in the proof of the previous theorem and Corollary 6.6, we see that for every point y of U (t), every m ≥ m(t) and every general point s of (Sm )y , the restriction of E to the fiber of Zm over s is µ-semi-stable. We can cover T by a finite number of U (t)’s and hence we can find an integer m0 with the desired property. A generalization of the above theorem to the case of complete intersections is stated as follows. Corollary 6.11. Under the same situation as in Theorem 6.8, let t be an integer with 1 ≤ t < dim X. There is a positive integer m0 such that if m1 , . . . , mt are integers greater than or equal to m0 , then E|q−1 (s) is µ-semi-stable for general s ∈ m Sm with m = (m1 , . . . , mt ). Proof. Our proof is by induction on t. If t = 1, then our assertion is nothing but Theorem 6.8. Assume that t ≥ 2. By Theorem 6.8 we have N1 such that for all m ≥ −1 N1 , there is a non-empty open set Um such that E|qm (s) is µ-semi-stable if s ∈ Um . −1 Then, applying our induction hypothesis and Remark 6.10 to qm (Um ) → Um and ∗ −1 pm (E)|qm there is an integer N (m) such that for every m = (m, m2 , . . . , mt ) 2 (Um )

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THEOREM OF MUMFORD-MEHTA-RAMANATHAN

71

with mi ≥ N2 (m), there is an open set Um of Um × Sm2 × · · · × Smt such that for s ∈ Um , E|q−1 (s) is µ-semi-stable. Let N2 = max{N2 (m) | N1 ≤ m ≤ 2N1 − 1}. m Pick integers m1 , m2 , . . . , mt such that m1 ≥ N1 and mi ≥ N2 for all i ≥ 2. Writing m1 = aN1 + b with N1 ≤ b < 2N1 − 1, construct a degenerating family D → Spec(A) in Lemma 6.6 such that DK , Dk1 , . . . , Dka and Dka+1 correspond to general points of Sm1 , SN1 and Sb , respectively. If we take hypersurfaces H2 , . . . , Ht which correspond to general points of Sm2 , . . . , Smt , respectively and if we make intersections of D, H1 × Spec(A), . . . , Ht × Spec(A) in X × Spec(A), then we obtain a family Dm → Spec(A) which satisfies the conditions of Proposition 6.5 such that each irreducible component of the special fiber corresponds to a point of Um with m = (N1 , m2 , . . . , mt ) or (b, m2 , . . . , mt ). Then, by Corollary 6.6 the restriction of E to the generic fiber of Dm is µ-semi-stable. Now m0 = max{N1 , N2 } meets our requirement.

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7 Simpson’s stability C. Simpson [S] introduced a new notion of stable sheaves which is a natural generalization of ours. We are going to show here the boundedness of Simpson’s semi-stable sheaves in characteristic zero. Definition 7.1. Let (X, OX (1)) be a pair of a projective scheme X over a field k and an ample invertible sheaf OX (1). A coherent sheaf F on X is said to be of pure dimension d if dim Supp(F ) = d and if for every non-zero coherent subsheaf F 0 of F , we have dim Supp(F 0 ) = d. Note that F is of pure dimension d if and only if each irreducible component of Supp(F ) has dimension d and F satisfies the condition (S1 ). In fact F satisfies (S1 ) if and only if Ass(F ) consists only of the generic points of Supp(F ). Thus, for every over field K of k, F ⊗k K is of pure dimension d on XK if and only if so is F on X. It is obvious that if X is integral and Supp(F ) = X, then the pure dimensionality is nothing but the torsion freeness and that if a coherent sheaf is of pure dimension d, then so is every its coherent subsheaf. If a coherent sheaf F on X is of pure dimension d, then there are integers a0 (F ), a1 (F ), . . . , ad (F ) such that d X

  m+d−i χ(F (m)) = ai (F ) . d−i i=0 Since d is not necessarily dim X and since Ann(F ) may define a reducible or even non-reduced subscheme, we have no reasonable notion of the rank and hence we are going to use a0 (F ) instead of r(F ). Definition 7.2. Let (X, OX (1)) be as in Definition 7.1 and let F be a coherent sheaf on X of pure dimension d. (1) The rational number a1 (F )/a0 (F ) is denoted by µS (F ) and we denote the polynomial χ(F (m))/a0 (F ) by PFS (m). (2) F is said to be µ-stable (or, µ-semi-stable) if for every coherent subsheaf E of F ⊗k k¯ with 0 < a0 (E) < a0 (F ), we have µS (E) < µS (F ) (µS (E) ≤ µS (F ), resp.). (3) F is said to be stable (or, semi-stable) if for every coherent subsheaf E of F ⊗k k¯ with 0 < a0 (E) < a0 (F ) and for all sufficiently large integers m, we have PES (m) < PFS (m) (PES (m) ≤ PFS (m), resp.). As in Chapter I we have the following basic results whose proof is completely the same as that of Proposition I.2.10. Proposition 7.3. Let (X, OX (1)) be as in Definition 7.1 and let E and F be coherent sheaves on X of pure dimension d. (1) If both E and F are µ-semi-stable and if µS (E) > µS (F ), then every homomorphism of E to F is 0.

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73

(2) If E is µ-stable, F is µ-semi-stable and if µS (E) = µS (F ), then every nonzero homomorphism of E to F is injective. (3) If both E and F are semi-stable and if PES (m) > PFS (m) for all sufficiently large m, then every homomorphism of E to F is 0. (4) If E is stable, F is semi-stable and if PES (m) = PFS (m), then every non-zero homomorphism of E to F is injective and F/E is semi-stable. (5) Assume that E is stable, then E is simple. (6) Assume that E is an extension of coherent sheaves of pure dimension d 0 −→ E1 −→ E −→ E2 −→ 0 such that PES (m) = PES1 (m) = PES2 (m) (or, µS (E) = µS (E1 ) = µS (E2 )). Then E is semi-stable (µ-semi-stable, resp.) if and only if so are E1 and E2 . Depending on this we can also prove the existence and the uniqueness of the Harder-Narasimhan filtration of a coherent sheaf of pure dimension d. Proposition 7.4. Let (X, OX (1)) be as in Definition 7.1 and let E be a coherent OX -module of pure dimension d. (1) There is a unique filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα = E by coherent subsheaves such that (a) Ei /Ei−1 is µ-semi-stable for 0 < i ≤ α and (b) µS (Ei /Ei−1 ) > µS (Ei+1 /Ei ) for 0 < i < α. (2) There is a unique filtration 0 = E00 ⊂ E10 ⊂ · · · ⊂ Eβ0 = E by coherent sub0 sheaves such that (c) Ei0 /Ei−1 is semi-stable for 0 < i ≤ β and (d) PES0 /E 0 (m) > i

PES0

0 i+1 /Ei

i−1

(m) for all sufficiently large integer m and for 0 < i < β.

As in Chapter I, connecting successively the points xi (Ei ) = (a0 (Ei ), a1 (Ei )) by segments in a two dimensional real plane, we get the Harder-Narasimhan polygon of E and obtain the following. Proposition 7.5. Let E be as in Proposition 7.4. For every coherent subsheaf F ¯ the point x(F ) is below or on the Harder-Narasimhan polygon of E. of E ⊗k k, For convenience sake, let us introduce the notion of type of a coherent sheaf of pure dimension d. Definition 7.6. Let (X, OX (1)) be as in as in Definition 7.1 and let E be a coherent OX -module of pure dimension d. By setting r = a0 (E), let (α) = (α1 , α2 , . . . , αr−1 ) be a sequence of rational numbers. E is said to be of type (α) if for every coherent quotient sheaf F of E ⊗k k¯ of pure dimension d with 0 < a0 (F ) < r, we have the inequality µS (F ) ≥ µS (E) − αs , where s = a0 (F ). We shall prove the boundedness of semi-stable sheaves in the sense of Simpson along the idea of Simpson and Le Potier under the assumption of characteristic 0 or of pure dimension ≤ 2. To study our problem in a more general setting we consider the following families.

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Definition 7.7. Let f : X → S be a projective morphism of noetherian schemes and OX/S (1) an f -very ample invertible sheaf. Fix non-negative integers d and r, a numerical polynomial H(x) of degree d with the leading coefficient r/d!, a sequence (a1 , . . . , ad ) of integers and a sequence (α) = (α1 , . . . , αr−1 ) of rational numbers. (1) Let SX/S (d; r, a1 , . . . , ad ; (α)) be the family of the classes of coherent sheaves on the fibers of X over S such that E on a geometric fiber Xs is a member of the family if E has the properties (7.7.1) and (7.7.2). 0 (2) Let SX/S (d; r, a1 , a2 ; (α)) be the family of the classes of coherent sheaves on the fibers of X over S such that E on a geometric fiber Xs is a member of the family if E has the properties (7.7.1) and (7.7.3).

(3) Let SX/S (d; H(m), (α)) be the family of the classes of coherent sheaves on the fibers of X over S such that E on a geometric fiber Xs is a member of the family if E has the properties (7.7.1) and if χ(E(m)) = H(m). (7.7.1) E is of pure dimension d and of type (α). (7.7.2) a0 (E) = r, a1 (E) = a1 and ai (E) ≥ ai

(2 ≤ i ≤ d).

(7.7.3) E satisfies the condition (S2 ), a0 (E) = r, a1 (E) = a1 and a2 (E) ≥ a2 . We shall show first a relationship among the above three families. Lemma 7.8. Let the pair of (X, OX (1)) be the same as in Definition 7.7. (1) If SX/S (d; r, a1 , . . . , ad ; (α)) is bounded for all (a1 , . . . , ad ), then so is SX/S (d; H(m), (α)) for all H(x). 0 (2) If SX/S (d; r, a1 , a2 ; (α)) is bounded for all pairs (a1 , a2 ), then so is SX/S (d; H(m), (α)) for all H(x)

Proof. (1) is obvious by the very definition of the two families. To prove (2) take a representative E on a geometric fiber Xs of a member of SX/S (d; H(m), (α)). Since H(x) is a numerical polynomial, we can write d X

  m+d−i H(m) = ai (H) d−i i=0 with a0 (H) = r and ai (H) integers. Let Z be the closed subscheme of Xs defined by the ideal Ann(E) and let W be the set of points z of Z with dim OZ,z ≥ 2. 0 Let us look at the sheaf E 0 = HZ/W (E)3 . Since Z is embedded in a projective space over k(s), the condition (S1 ) for E implies that E 0 is coherent by [EGA IV, Proposition 5.11.1]. Then the natural homomorphism ρZ/W : E → E 0 is injective and isomorphic outside a set of codimension ≥ 2 and moreover E 0 satisfies the condition (S2 ) by [EGA IV, Corollaire 5.10.15]. Thus E 0 is of type (α), a0 (E 0 ) = a0 (E) = r, a1 (E 0 ) = a1 (E) = a1 (H) and a2 (E 0 ) ≥ a2 (E) = a2 (H). This means 3 For

a noetherian scheme Z, a subset W of Z which is a union of closed subsets Wα and an OZ -module F , there is a natural map (iα )∗ (F |Z\Wα ) → (iβ )∗ (F |Z\Wβ ) if Wα ⊂ Wβ , where 0 iα : Z \ Wα → Z is an inclusion map. The sheaf HZ/W (F ) is defined to be lim(iα )∗ (F |Z\Wα ). − → α

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75

0 that E 0 is a member of SX/S (d; r, a1 (H), a2 (H); (α)). The boundedness of the last family implies that there is a sequence of integers (b) = (b0 , . . . , bd ) such that E 0 ’s are (b)-sheaves and hence so are E’s.

We can reduce the boundedness of the above defined three families to the case of torsion free sheaves on smooth projective schemes. The main idea here is due to C. Simpson and J. Le Potier. Theorem 7.9. 4 Let the pair of (X, OX (1)) be the same as in Definition 7.7. Assume that S is over a field of characteristic 0 or that d ≤ 2. Then both 0 SX/S (d; r, a1 , . . . , ad ; (α)) and SX/S (d; r, a1 , a2 ; (α)) are bounded. Hence, in these cases, SX/S (d; H(m), (α)) is bounded, too. Proof. The final assertion is by Lemma 7.8. We are going to prove our theorem 0 only for SX/S (d; r, a1 , a2 ; (α)) because the proof of another family is similar and easier. For a coherent sheaf E on a geometric fiber Xs of pure dimension d, we set µmax (E) to be the µS (E1 ) with E1 the first filter of the Harder-Narasimhan filtration of E. Obviously if E is of type (α), then µmax (E) is bounded by an constant determined by (α). Conversely, by Proposition 7.5 we can bound the type 0 by bounding µmax (E). Thus we may replace the family SX/S (d; r, a1 , a2 ; (α)) by 0 SX/S (d; r, a1 , a2 ; C) which is obtained by bounding µmax from above by C instead of putting the condition on type. If we break up S into the disjoint union of suitable subschemes of S, then X is embedded in a projective space PnS over S. Hence we may assume that X = PnS and OX (1) is the invertible sheaf corresponding to hyperplanes. Fix a linear subspace Y ∼ = PSn−d−1 and set SY to be the subfamily 0 of SX/S (d; r, a1 , a2 ; C) consisting the classes of E on geometric fibers Pns such that Supp(E) does not meet Ys . The projective linear group P GL(n + 1) so acts on 0 SX/S (d; r, a1 , a2 ; C) that every orbit meets the subfamily SY . Thus it is enough to prove that SY is bounded. Pick a coherent sheaf E on a geometric fiber Pns which represents a member of SY . Let π : Pns \ Ys → Pds be the linear projection from Ys . Since Supp(E) is contained in Pns \ Ys and proper over k(s), the direct image F = π∗ (E) is a coherent sheaf whose Hilbert polynomial is the same as that of E. The pure d-dimensionality implies that F is torsion free. We can say more. Lemma 7.10. If E is a coherent sheaf on an algebraic scheme Z of pure dimension d over a field k with Supp(E) = Z, E satisfies the condition (S2 ) and if π : Z → Pdk is a finite morphism, then π∗ (E) satisfies the condition (S2 ). Proof. Since E satisfies the condition (S1 ) and since Z is pure dimension d, E is of pure dimension d. This and the fact that π is finite imply that π∗ (E) is torsion free. Then there is an open set W of Pdk such that π∗ (E)|W satisfies the condition (S2 ) and codim(Pdk \ W, Pdk ) ≥ 2. Setting W 0 = π −1 (W ), we shall consider the 4 See

Appendix A.

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commutative diagram: δ

W 0 −−−−→   π0 y

Z  π y

γ

W −−−−→ Pdk Since codim(Z \ W 0 , Z) ≥ 2, the condition (S2 ) on E implies that δ∗ δ ∗ (E) = E. Then we have simply γ∗ γ ∗ (π∗ (E)) = γ∗ (π∗0 δ ∗ (E)) = π∗ (δ∗ δ ∗ (E)) = π∗ (E). Thus π∗ (E) satisfies the condition (S2 ). By this lemma F satisfies the condition (S2 ). If we can prove that there is a constant C 0 such that for all F in the above, we have µmax (F ) ≤ C 0 , then F is 0 0 a member of the family SP d /S (d; r, a1 , a2 ; C ). Under our condition that S is over a filed of characteristic 0 or that d ≤ 2, this family is bounded by virtue of [M, Main Theorem]. Then, the set of Hilbert polynomials of F ’s is finite. Moreover, we can find an integer m such that every F is m-regular. On the other hand, Leray’s spectral sequence implies that H p (Pds , F (m − p)) ∼ = H p (Pns , E(m − p)). Thus every member of SY is m-regular. These complete our proof. We have only to prove, therefore, the following. Lemma 7.11. Assume that X is embedded in PnS . Fix positive integers d, a0 and C. Let F(d, a0 , C) be the family of classes of coherent sheaves E on the fibers of X over S such that E is in F(d, a0 , C) if and only if it is of pure dimension d, 0 < a0 (E) ≤ a0 and µmax (E) ≤ C. There is an integer C 0 which is determined only by n, a0 , d and C and has the following property: (7.11.1) If a coherent sheaf E on Xs is in F(d, a0 , C) and if PSn−d−1 = Y ⊂ PnS is a linear subspace with Ys ∩ Supp(E) = ∅, then for the projection π : Pnk(s) \ Ys → Pdk(s) and for F = π∗ (E), we have µmax (F ) ≤ C 0 Proof. As in the argument in the above it is enough to prove our assertion for a fixed Y and E with Ys ∩ Supp(E) = ∅. Let Z be the closed subscheme of Pn = Pnk(s) defined by the ideal of annihilators of E. We denote the projection of Z to Pd = Pdk(s) by πZ . We claim (7.11.2) Z or the structure sheaf OZ is of pure dimension d. In fact, OZ is embedded in Hom(E, E) as the multiplication by constants and hence it is enough to show that Hom(E, E) is of pure dimension d. For suitable integers a and N , we have a surjection OPn (−a)N → E. Then Hom(E, E) can be regarded as a subsheaf of E(a)N . Since this is of pure dimension d, so is Hom(E, E).

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77

Pick a point z of Z such that dim OZ,z ≥ 1. Then the pure dimensionality of Z implies that depth OZ,z ≥ 1 and hence Z is Cohen-Macaulay in codimension 1. Since πZ is finite and Pd is smooth, this shows (7.11.3) There is an open set U of Pd such that Z is flat over U and Pd \ U is at least of codimension 2 in Pd . Let F1 be the first filter of the Harder-Narasimhan filtration of F . Since µ0 (F1 ) = µS (F1 ) = µmax (F ), we have to bound µ0 (F1 ) from above. By (7.11.2) the OPd algebra A = π∗ (OZ ) is torsion free. F is an A-module and we have a bijective correspondence between the set of coherent A-submodules of F and that of coherent subsheaves of E. The image F10 of the canonical homomorphism F1 ⊗OPd A → F is an A-submodule of F and hence it is the direct image of a coherent subsheaf E1 of E. By our hypothesis we have µ0 (F10 ) = µS (E1 ) ≤ C. By Lemma 7.12 below there is a positive integer m determined by n, d and a0 (E) such that A(m)|U is generated by its global sections, that is, there is a surjection ⊕α OU −→ A(m)|U .

Tensoring F1 to this and composing the above canonical homomorphism, we have a surjection of (F1 |U )⊕α to F10 (m)|U . Since F1 is µ-semi-stable, so is (F1 )⊕α and hence for every non-trivial quotient coherent sheaf Q, we have µ0 (Q) ≥ µ0 (F1 ). This and the fact that U contains all points of codimension 1 give rise to µ0 (F1 ) ≤ µ0 (F10 (m)) = µ0 (F10 ) + m ≤ C + m. Thus our remaining task is to prove the following. Lemma 7.12. There is a positive integer m determined by a0 (E), n and d such that A(m)|U is generated by its global sections. Proof. We shall first show a basic result: (7.12.1) If E is a coherent sheaf on a projective space PnK over a field K, then for the subscheme Z defined by the ideal of annihilators of E, we have a0 (OZ ) ≤ a0 (E)2 . In fact, we may assume that K is algebraically closed. Set r = a0 (E). As in the proof of (7.11.2) Hom(E, E) naturally contains OZ . Thus it is enough to show that a0 (Hom(E, E)) ≤ r2 . Our proof is by induction on d = dim Supp(E). First assume that d = 0. In this case E is generated by its global sections and r = dim H 0 (Pn , E). This means that there exists a surjection ⊕r OP n −→ E.

From this we obtain an injection Hom(E, E) → E r which leads us to a0 (Hom(E, E)) ≤ ra0 (E) = r2 .

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Assume next that d > 0. Take a hyperplane H which passes through none of the points in Ass(E). Then the multiplication by a linear form defining H provides us with the following exact sequence 0 −→ E(−1) −→ E −→ G −→ 0. We see that dim Supp(G) = d − 1 and a0 (G) = a0 (E). This sequence gives rise to another exact sequence 0 −−−−→ Hom(E, E(−1)) −−−−→ Hom(E, E) −−−−→ Hom(E, G).



o

o Hom(E, E)(−1)

Hom(G, G)

Since a0 (Hom(E, E)) = a0 (Hom(E, E)/Hom(E, E)(−1)), we obtain a0 (Hom(E, E)) ≤ a0 (Hom(G, G)). On the other hand, by our induction hypothesis a0 (Hom(G, G)) ≤ a0 (G)2 . Thus we come to the required inequality a0 (Hom(E, E)) ≤ a0 (G)2 = r2 . Let us go back to the proof of Lemma 7.12. By blowing-up PnS along Y , we obtain ⊕n−d a Pn−d -bundle p : P = P(N ) → Pd , where N is the vector bundle OPd (1)⊕OP . d a a For a positive integer a we denote the Hilbert scheme HilbP/Pd by H . We set H=

a a≤a0

Ha

(E)2

and Z˜ to be the universal subscheme of P ×Pd H. There is an open subscheme V of H which parametrizes the subschemes that do not meet the exceptional divisor D of the blowing-up. Z˜V ⊂ P ×Pd H −−−−→ P     pH  pV y py y V ⊂ H −−−−→ Pd Since the tautological line bundle OP (1) is isomorphic to OP (D) ⊗ p∗ (OPd (1)), we have OZ˜V (m) ∼ = OZ˜V ⊗ p∗V (OH (m)), where OH (1) is the pull-back of OPd (1) to H. By the projection formula we get an isomorphism (pV )∗ (OZ˜V (m)) ∼ = (pV )∗ (OZ˜V )(m). Since pV is finite and flat, this direct image is compatible with base changes. There is an integer m determined by a0 (E), n and d such that R1 (pV )∗ (IZ˜V (m)) = 0, where IZ˜V is the ideal of Z˜V . Then, for this m and the pull-back NV of N to V , we have a surjection S m (NV ) −→ (pV )∗ (OZ˜V (m)) ∼ = (pV )∗ (OZ˜V )(m). S m (NV ) is generated by its global sections because so is N . Hence (pV )∗ (OZ˜V )(m) is a quotient of a trivial vector bundle. On the other hand, since Z can be regarded as a subscheme of P, (7.11.2), (7.11.3) and the universality of the Hilbert

7

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79

scheme provides us with a morphism f of U to V such that ZU = Z˜ ×H U as subschemes of PU . Thus we have an isomorphism of A|U (m) = (pU )∗ (OZU )(m) to f ∗ (pV )∗ (OZ˜V )(m) and hence A|U (m) is generated by its global sections. Corollary 7.13. 5 If f : X → S is a geometrically integral, projective morphism of noetherian schemes and if r is a positive integer less than or equal to 0 00 3, then TX/S (n, r; a1 , . . . , an ; (α)), TX/S (n, r; a1 , a2 ; (α)) and TX/S (n, r; H(m); (α)) are bounded (without assuming f : X → S to be smooth) for all n, a0 , . . . , a1 , H(m) and (α), where the notation is the same as in section 3. Proof. By Theorem 7.9 our assertion is true for n = 2. Since Theorem 1.3 and Theorem 1.4 hold under the assumption that X is geometrically integral, the argument of Proposition 4.10 is valid in the present case, too. Thus the remaining problem is only Lemma 3.5 that we used to establish our induction steps. To prove Lemma 3.5 in this general situation we can embed Xs into a projective space and use a resolution by locally free sheaves on the projective space instead of those on Xs (see Lemma 7.14 below). To prove the openness of Simpson’s stability we shall show the openness of the pure d-dimensionality. There are two ways of the proof, using Proposition I 3.12 and the Quot-scheme or using a characterization of pure d-dimensionality in term of a resolution by locally free sheaves. We employ the second method because it is useful to study the pure dimensionality of the restriction of the sheaf to a general hyperplane section. Let E be a coherent sheaf on a projective scheme X over a field k. Since X is projective, we can embed X into a projective space Pn over k. As an OPn -module, E has a resolution by locally free coherent sheaves fn

fn−1

f2

f1

f0

0 −→ Fn −−−−→ Fn−1 −−−−→ · · · −−−−→ F1 −−−−→ F0 −−−−→ E −−−−→ 0. As remarked right after Definition 7.1, E is of pure dimension d if and only if each irreducible component of Supp(E) has dimension d and E satisfies the condition (S1 ). Taking this into account, the following lemma is a special case of Lemma 3.5. Lemma 7.14. Let Bi be the set of pinching points of Ki = ker(fi ). Then the following (1) and (2) are equivalent to each other (1) E is of pure dimension d. (2) Each irreducible component of Supp(E) has dimension d and dim Bi ≤ n − i − 3 for i ≥ n − d − 1. We can now prove the openness of pure d-dimensionality. Proposition 7.15. Let g : X → S be a projective morphism of locally noetherian schemes and E an S-flat coherent sheaf on X. There is an open set U of S such that a point s of S is in U if and only if E(s) is of pure dimension d. 5 See

Appendix A.

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Proof. Since our problem is local with respect to S, we may assume that S is noetherian, X is embedded in PnS and hence X itself is PnS . If m is a sufficiently large integer, then E(s)(m) is generated by global sections and H i (Xs , E(s)(m)) = 0 for all s ∈ S and i > 0. Then g∗ (E(m)) is locally free and the natural map g ∗ g∗ (E(m)) → E(m) is surjective. Thus there is a surjection f0 of a locally free sheaf F0 = g ∗ g∗ (E(m))(−m) to E. Applying the same procedure to ker(f0 ) we have an exact sequence F1 −→ F0 −→ E −→ 0. Repeating this we get an exact sequence fn

fn−1

f2

f1

f0

0 −→ Fn −−−−→ Fn−1 −−−−→ · · · −−−−→ F1 −−−−→ F0 −−−−→ E −−−−→ 0. where F0 , . . . , Fn−1 are locally free and Fn = ker(fn−1 ). Since E is flat over S and X = PnS is also flat over S, for every point s ∈ S, the restriction of the above sequence to the fiber Xs is exact. Hence Fn (s) is locally free on Xs . This and the flatness of Fn over S imply that Fn is locally free. For Ki = ker(fi ) and s ∈ S, Ki (s)x is a free OXs ,x -module if and only if (Ki )x is a free OX,x -module because Ki is flat over S. Thus if Bi is the set of pinching points of Ki , then (Bi )s is the set of pinching points. Since g is proper, S−1 = {s ∈ S | dim Supp(E)s > d} and Sj = {s ∈ S | dim(Bj )s > n − j − 3} are closed in S. By Lemma 7.14 the open set    n−1 [ [ U = S \ S−1  Sj  j=n−d−1

meets our requirement. After replacing torsion free by of pure dimension d, PE by PES and µ0 by µS , we can follow the same argument in the proof of Theorem I.4.2 to prove the openness of Simpson’s stability and semi-stability. Theorem 7.16. Let g : X → S and E be the same as in Proposition 7.15 and OX (1) a g-ample invertible sheaf. (1) For every sequence (α) = (α1 , . . . , αr−1 ), there is an open set U(α) of S such that for every algebraically closed field K, U(α) (K) is exactly the set {s ∈ S(K) | E(s) is of type (α) in the sense of Definition 7.6}. (2) There is an open set U s of S such that for every algebraically closed field K, U (K) is exactly the set {s ∈ S(K) | E(s) is stable in the sense of Simpson}. s

(3) There is an open set U ss of S such that for every algebraically closed field K, U (K) is exactly the set {s ∈ S(K) | E(s) is semi-stable in the sense of Simpson}. ss

Chapter 3

Construction of Moduli Spaces We are now going to construct the moduli spaces of stable sheaves. If we have a bounded family of stable sheaves, it can be parameterized by an open subscheme of a Quot-scheme on which a reductive group scheme acts. Two points of the subscheme correspond to the same sheaf if and only if they are in the same orbit of the group scheme action. Thus main part of the construction is to prove the existence of a quotient scheme of the scheme by the group scheme. Once we can overcome this difficulty, the universality of the moduli space is easily derived from that of the Quot-scheme and the quotient. In the first section we shall recall a proof of the representability of the Quotfunctors. We shall recall the results in Geometric Invariant Theory that are relevant to our aim. The most of results are only stated without proof and the author refers the readers to the famous “Geometric Invariant Theory” by D. Mumford and Seshadri’s work [Se]. After the study of rather technical notion of e-stability, the proof of a fundamental lemma to connect semi-stable sheaves with semi-stable points in Geometric Invariant Theory and the preparation of the general setting, we are going to construct moduli space of stable-sheaves. To compactify the moduli spaces we have to add points corresponding to the S-equivalence classes of semistable sheaves. The case over a filed of characteristic zero is treated in the sixth section along a beautiful idea of Simpson. To handle semi-stable sheaves in the general cases, we need a detailed analysis of closed orbits of a Grassmann type scheme. The seventh section is devoted to this analysis. A proof of the existence of moduli spaces in general cases will be given in the last section. To construct the moduli spaces of semi-stable sheaves on a singular variety, Simpson’s idea in [S] seems to be inevitable and it is essential throughout this chapter.

1 Construction of Quot-schemes In this section we prove that the functor QuotP F/X/S introduced at the beginning of Chapter I is represented. Before we go to our proof of the representability of QuotP F/X/S , let us show the “flattening stratification” by D. Mumford. Theorem 1.1. Let f : X → S be a projective morphism of noetherian schemes and OX (1) an f -ample line bundle. For a coherent sheaf F on X and a numerical polynomial P , there is a subscheme S P of S such that for all morphisms g : T → S of locally noetherian schemes, FT is flat over T and the Hilbert polynomial on each fiber of XT over T is the given P (m) if and only if g factors through S P . 81

82

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Proof. Step I First we shall treat the case that X = S and f is the identity morphism. Then F is a coherent sheaf on S, FT is g ∗ (F ) and g ∗ (F ) is flat and has the same Hilbert polynomial on each fiber if and only if it is locally free and of constant rank. P is, in this case, a non-negative integer e. Pick a point s of S such that dimk(s) F (s) = e. Choose members a1 , . . . , ae of the stalk Fs whose images to F (s) are a basis of the vector space. Extending these a1 , . . . , ae to sections of F over an open neighborhood U1 of s, we have a homomorphism ⊕e ϕ : OU −→ F |U1 . 1

Thanks to the choice of the sections and Nakayama’s lemma, ϕ is surjective at s and hence it is surjective on a smaller open neighborhood U2 . Applying the same procedure to ker(ϕ), we get an exact sequence ψ

⊕d ⊕e OU −−−−→ OU −−−−→ F |U3 −−−−→ 0 3 3 ⊕d ⊕e on an open neighborhood U3 of s in U2 . If we fix free bases of OU and OU , then 3 3 ψ is represented by an e × d matrix (aij ) whose entries are in Γ(U3 , OX ). It is clear now that g ∗ (F )|g−1 (U3 ) is locally free and of rank e if and only if g ∗ ψ = 0 or equivalently all the g ∗ (aij )’s are 0 in Γ(g −1 (U3 ), OT ). Thus the closed subscheme C of U3 defined by the ideal generated by aij ’s meets our requirement for F |U3 . By the universal property of C, we can glue them together to make up a closed subscheme S P of the union of the neighborhoods U3 ’s. Obviously this S P is what we wanted.

Step II Let S be a graded OS -algebra such that S is generated by S1 , S1 is of finite type and X = Proj(S). For the graded S-module M Γ∗ (F ) = f∗ (F (n)), n∈Z

we know that the natural homomorphism β : Γ^ ∗ (F ) → F is isomorphic. If g : T → S is a locally noetherian S-scheme, then FT is isomorphic to Γ^ ∗ (FT ). On the other hand, we see that ∗ ^ ∼ ^ ∼ g ∗ (Γ ∗ (F )) = (1X × g) (Γ∗ (F )) = FT

by [EGA, II Proposition 3.5.3]. This means that the natural homomorphism of g ∗ (Γ∗ (F )) to Γ∗ (FT ) is a TN-isomorphism [EGA, II Proposition 3.4.3]. Thus there is an integer mT such that for m ≥ mT , the natural homomorphism of g ∗ (f∗ (F (m))) to (fT )∗ (FT (m)) is isomorphic. Step III For an integer m, set Em = f∗ (F (m)). Using the generically flatness theorem [EGA, IV Theorem 11.1.1], we can decompose S into the disjoint union of a finite number of subschemes Si such that FSi is flat over Si . If we apply Step II

1

CONSTRUCTION OF QUOT-SCHEMES

83

to the case of T = Si , then we can find integers mSi such that for m ≥ mSi , we have Em |Si ∼ = (fSi )∗ (FSi (m)). By base change theorem, we have an integer m0Si such that for m ≥ m0Si and for all s ∈ Si , the natural map (fSi )∗ (FSi (m))(s) → H 0 (Xs , F (m)(s)) is bijective and H i (Xs , F (m)(s)) = 0 for i > 0. Thus setting m0 = maxi {mSi , m0Si } , we have (1.1.1) for all integers m ≥ m0 and all points s ∈ S, Em (s) is isomorphic to H 0 (Xs , F (m)(s)) and H i (Xs , F (m)(s)) = 0 for i > 0. Step IV Let us fix an integer m0 as in (1.1.1) and take an S-scheme g : T → S. Assume first that FT is flat over T and has the Hilbert polynomial P on each fiber of XT over T . Then, for m ≥ m0 and t ∈ T , the natural homomorphisms θ : g ∗ (Em ) → (fT )∗ (FT (m)) and τ : (fT )∗ (FT (m))(t) → H 0 (Xt , FT (m)(t)) induce the bijective map Em (s) ⊗ k(t) → H 0 (Xs , F (m)(s)) ⊗ k(t) ∼ = H 0 (Xt , FT (m)(t)) by (1.1.1), where s = g(t). Thus τ is surjective. Thus, by base change theorem, (fT )∗ (FT (m)) is locally free and for all t, τ is isomorphism. This and Nakayama’s lemma imply that θ is surjective. Since (fT )∗ (FT (m)) is locally free and θ(t) is isomorphic, we have ker(θ)(t) = 0 and hence θ is injective. Therefore, θ is isomorphic and g ∗ (Em ) is locally free. Conversely, if g ∗ (Em ) is locally free of rank P (m) for all sufficiently large integers m, then so is (fT )∗ (FT (m)) by Step II. By [EGA, III Proposition 7.9.14] we see then FT is flat over T and has the Hilbert polynomial P on each fiber of XT over T . Therefore, we see that FT is flat over T if and only if for every m ≥ m0 , g ∗ (Em ) is locally free. e Step V Let Sm be the universal subscheme of Step I where Em is locally free and of rank e. Consider ∞ \ P (m) Y = Sm m=m0

Since {F (s) | s ∈ S} is a bounded family, the set of Hilbert polynomials of the family is finite. Let n be the maximum of the degrees of the Hilbert polynomials. Then each Hilbert polynomial is determined by its values at mutually distinct n values of m’s. From this and (1.1.1) we deduce that a point s of S is in the set Y0 =

m\ 0 +n

P (m) Sm

m=m0

if and only if the Hilbert polynomial of F (s) is P . Thus set-theoretically Y must be equal to Y 0 which is locally closed. Thus Y is the limit of a descending chain of subschemes of S whose supports are the fixed Supp Y 0 . Since S is noetherian, Y is actually a subscheme of S. By Step IV it is now obvious that Y is the subscheme S P of S. Now let us go back to the Quot-scheme. Since the family of classes of F (s) on the fibers Xs of X over S is bounded, the number of the Hilbert polynomials of the members of the family is finite and there is a sequence (b) = (b0 , . . . , bn ) such that the family is a (b)-family. Fix a numerical polynomial P and consider the family

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F P of the classes of coherent quotient sheaves of F (s) with Hilbert polynomial P . Since the kernel of the surjection of F (s) to a member of F P is a (b)-sheaf and the number of the Hilbert polynomials is finite, the family of the classes of the kernels is bounded by Theorem I.3.11 and hence so is F P . There is an integer m0 such that for every integer m ≥ m0 , every geometric point s of S and for every exact sequence 0 −→ H −→ F (s) −→ E −→ 0 with the class of E in F P , we have the following (1.2.1) for every i > 0, H i (Xs , H(m)) = 0 and H(m) is generated by global sections, (1.2.2) for every i > 0, H i (Xs , F (s)(m)) = 0 and F (s)(m) is generated by global sections, (1.2.3) for every i > 0, H i (Xs , E(m)) = 0 and E(m) is generated by global sections. Our first result is the following. Proposition 1.3. If F is flat over S, then QuotP F/X/S is representable by a pair (Q, G) of a projective S-scheme Q and a quotient coherent sheaf G of FQ . For every m ≥ m0 with m0 defined in the above, Q is a closed subscheme of GrassP (m) (f∗ (F (m))) and det((fQ )∗ (G(m))) is an S-very ample invertible sheaf on Q. Proof. By replacing S by the union of some of connected components of S, we may assume that the Hilbert polynomials χ(F (s)(m)) are independent of the choice of geometric points s of S. Let T be a locally noetherian S-scheme and E a quotient sheaf of FT which is a member of QuotP F/X/S (T ). We have an exact sequence 0 −→ H(m) −→ FT (m) −→ E(m) −→ 0. Since FT and E are flat over T , so is H and then by the choice of m we obtain another exact sequence of locally free sheaves of constant ranks 0 −→ (fT )∗ (H(m)) −→ (fT )∗ (FT (m)) −→ (fT )∗ (E(m)) −→ 0. Moreover, this formation is compatible with base changes and (fT )∗ (FT (m)) ∼ = f∗ (F (m)) ⊗OS OT . Since (fT )∗ (E(m)) is locally free of rank P (m), the above exact sequence gives rise to a T -valued points Φ(T )(E) of A = GrassP (m) (f∗ (F (m))). The compatibility with base changes means that this map Φ(T ) is functorial with respect to T , that is, we have a functor Φ : QuotP F/X/S → hA . Conversely, assume that we have a T -valued point α f∗ (F (m)) ⊗OS OT ∼ = (fT )∗ (FT (m)) −→ N −→ 0

1

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85

of A. Setting N 0 = ker(α), we have a natural homomorphism β : fT∗ (N 0 ) → fT∗ (fT )∗ (FT (m)) → FT (m). Then coker(β)(−m) is a quotient coherent sheaf of FT . The functor AQS F/X/S is defined by assigning the set of isomorphism classes of all quotient coherent sheaves of FT to a locally noetherian S-scheme T . Then the above observation shows that each T -valued point of A provides us with a member of AQS F/X/S (T ) and this procedure is functorial. We denote this functor by Ψ : hA → AQS F/X/S . Let us look at Ψ(T )Φ(T )(E). Since for all points t of T , H(m)(t) is generated by global sections and (fT )∗ (H(m))(t) is isomorphic to H 0 (Xt , H(m)(t)), the image of fT∗ (fT )∗ (H(m)) → fT∗ (fT )∗ (FT (m)) → F (m) is exactly H(m). Thus Ψ(T )Φ(T )(E) recovers the original E. QuotP F/X/S ,−→ AQS F/X/S  Φ

Ψ

? hA ˜ on A defines a member The universal quotient bundle f∗ (F (m)) ⊗OS OA → N 0 FA → E of AQS F/X/S (A). Applying Theorem 1.1 to the triple fA : XA → A, 0 on E 0 and P , we have a subscheme Q of A and a Q-flat coherent sheaf G = EQ XQ . We shall show that the pair (Q, G) meet our requirement. We have proved in the above that for a member FT → E of QuotP F/X/S (T ), we have a morphism of g : T → A such that (1 × g)∗ (FA ) → (1 × g)∗ (E 0 ) is isomorphic to the given FT → E. Then the universality of Q implies that g is actually a morphism of T to Q and E ∼ = (1 × g)∗ (G). This means that Φ(T ) factors through hQ (T ). Since Φ(T ) is injective, we get an injective morphism of functors Θ : QuotP F/X/S −→ hQ Conversely, if g : T → Q is an S-morphism of locally noetherian schemes, then we have two exact sequences: ˜ ) −−−−→ 0 0 −−−−→ K −−−−→ f∗ (F (m)) ⊗OS OT −−−−→ g ∗ (N ko (fT )∗ (FT (m)) and F (m)T −→ (1 × g)∗ (G(m)) −→ 0.

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The construction of the second sequence gives rise to the following exact commutative diagram: 0 −−−−→ fT∗ (K) −−−−→ fT∗ (fT )∗ (FT (m)) −−−−→   y F (m)T

˜) fT∗ g ∗ (N   y

−−−−→ 0

−−−−→ (1 × g)∗ (G(m)) −−−−→ 0

The direct image of the above diagram by fT provides us with another commutative diagram: (fT )∗ (FT (m)) −−−−→   y

˜) g ∗ (N  u y

−−−−→ 0

(fT )∗ (FT (m)) −−−−→ (fT )∗ (1 × g)∗ (G(m)) −−−−→ 0 Pick a point t of T . By (1.2.2) the map (fT )∗ (FT (m))(t) → H 0 (Xt , FT (m)(t)) is bijective. Since (1 × g)∗ (G)(t) is a member of F P , the map H 0 (Xt , FT (m)(t)) → H 0 (Xt , (1 × g)∗ (G(m))(t)) is surjective by (1.2.1). Thus the commutativity of the ˜ ) to H 0 (Xt , (1 × g)∗ (G(m))(t)) is surjecfirst diagram implies that the map of g ∗ (N tive and hence we have the surjectivity of u by (1.2.3) and Nakayama’s lemma. Then ˜ ) and (fT )∗ (1 × g)∗ (G(m)) are locally free u must be isomorphic because both g ∗ (N and has the same rank P (m). This means that g is in the image of Θ(T ), and hence the surjectivity of Θ(T ). We have thus proved that QuotP F/X/S is representable by the pair (Q, G). Moreover, the Pl¨ ucker embedding of A is defined by the line bundle VP (m) ˜ N . On the other hand, applying the above argument to g = idQ , we see that ˜ |Q is isomorphic to fQ (G(m)). Therefore, det(fQ (G(m))) is very ample over S. N Since Q is quasi-projective, the projectivity of Q follows from the following lemma and then we complete our proof of Proposition 1.3. Lemma 1.4. Let f : X → S be a morphism of noetherian schemes, F a coherent sheaf on X and T = Spec(R) an S-scheme with R a discrete valuation ring. Assume that we have a quotient coherent sheaf FK → E on XK = X ×S Spec(K), where K ˜ of FT such is the field of quotient of R. Then there is a quotient coherent sheaf E ˜ ˜ that E is flat over T and FK → EK is the given quotient sheaf. Proof. Let E 0 be ker(FK → E) and i : XK → XT be the natural immersion. Since everything is noetherian, i is quasi-compact and since Spec(K) is separated over T , so is i. Thus i∗ (E 0 ) is a quasi-coherent subsheaf of the quasi-coherent sheaf i∗ i∗ (FT ). Let α : FT → i∗ i∗ (FT ) be the natural homomorphism. Then E˜0 = α−1 (i∗ (E 0 )) is a quasi-coherent subsheaf of the coherent sheaf FT and hence E˜0 is coherent. For 00 the coherent sheaf E 00 = FT /E˜0 , EK is isomorphic to E as quotient sheaves of FK . ˜ = E 00 /H is flat over T and E ˜K is Thus if H is the T -torsion part of E 00 . then E the given quotient sheaf of FK .

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87

Based on the above proposition, we are going to show the representability of the Quot-functor by a projective scheme. In contrast to general cases, QuotP F/X/S is representable by a closed subscheme of a single Grassmann scheme in the case of the above proposition. Theorem 1.5. If f : X → S is a projective morphism of noetherian schemes and F is a coherent sheaf on X, then QuotP F/X/S is representable by a pair (Q, G) of a projective S-scheme Q and a quotient coherent sheaf G of FQ . There is an integer m0 such that for every m ≥ m0 , det((fQ )∗ (G(m))) is an S-very ample invertible sheaf on Q. Proof. Let S = ∪Si be a finite open covering of S. Assume that the theorem is true for each pair of fSi : XSi → Si and FSi . Then the functor on Si is representable by a projective Si -scheme Qi and the universal quotient sheaf Gi . It is easy to see that the functor on Sij = Si ∩ Sj is representable by the pair ((Qi )Sij , (Gi )Sij ) and also by the pair ((Qj )Sij , (Gj )Sij ). Thus we can glue (Qi , Gi ) together to get an S-scheme Q and a quotient coherent sheaf G of FQ . Our functor QuotP F/X/S is obviously representable by (Q, G). Moreover, there is an integer mi such that for all m ≥ mi , det((fQi )∗ (Gi (m))) is very ample over Si . Then, setting m0 = maxi {mi }, we see that for all m ≥ m0 , det((fQ )∗ (G(m))) is an S-very ample invertible sheaf on Q. Thus we may assume that S is an affine scheme and that X is a closed subscheme of PnS and hence X itself is PnS . F is a quotient sheaf of a locally free sheaf U = OX (−a)⊕b and then QuotP F/X/S is a subfunctor of QuotP . We know by Proposition 1.3 that the latter is repreU/X/S sentable by the pair (Q, G) of a projective S-scheme Q and a quotient coherent sheaf G of UQ and that det((fQ )∗ (G(m))) is S-very ample for sufficiently large integer m. Let V be the kernel of the surjection U → F . If m is sufficiently large integer, then we have (1.5.1) for every q ∈ Q, VQ (m)(q) is generated by global sections and the natural map (fQ )∗ (VQ (m))(q) → H 0 (Xq , VQ (m)(q)) is surjective, (1.5.2) for every q ∈ Q, G(m)(q) is generated by global sections and for i > 0, H i (Xq , G(m)(q)) = 0. For every S-morphism g : T → Q of locally noetherian schemes, the homomorphism γ : VQ (m) → UQ (m) → G(m) gives rise to the following commutative diagram: w

fT∗ g ∗ (fQ )∗ (VQ (m)) −−−T−→ fT∗ g ∗ (fQ )∗ (G(m))     y yo fT∗ (fT )∗ (VT (m))   y VT (m)

v

−−−T−→ fT∗ (fT )∗ (GT (m))   y γT

−−−−→

GT (m)

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If γT = 0, then vT must be 0 and hence so is wT . Conversely, if wT = 0, then we see γT = 0 because the left vertical map fT∗ g ∗ (fQ )∗ (VQ (m)) → fT∗ (fT )∗ (VT (m)) → VT (m) is surjective by virtue of (1.5.1). Pick a small affine open set D = Spec(C) of Q. Then wD is determined by a homomorphism δ of a C-module M to a free C-module C ⊕d , where d = P (m). Let {u1 , . . . , ur } be a set of generators of M and e1 , . . . , ed a free basis of C ⊕d . Writing δ(uj ) =

d X

aij ei

i=1

with aij ∈ Γ(D, OQ ), we define BD to be the closed subscheme of D defined by the ideal generated by {aij }. For an S-morphism g : T → Q of locally noetherian schemes, wT = 0 on g −1 (D) if and only if g ∗ (aij ) = 0 for all i, j. Thus, covering Q by such small affine open sets D, we can glue BD together to obtain a closed subscheme Q0 of Q. Now wT = 0 if and only if g factors through Q0 . Since GT is a quotient sheaf of FT if and only if γT = 0, QuotP F/X/S is representable by the pair (Q0 , GQ0 ). Moreover, det((fQ0 )∗ (GQ0 (m))) is S-very ample for sufficiently large m because (fQ0 )∗ (GQ0 (m)) ∼ = (fQ )∗ (GQ (m))|Q0 by (1.5.2). By the above theorem the representability of QuotP F/X/S has been established and so has been QuotF/X/S . Definition 1.6. The projective S-scheme which represents QuotP F/X/S is denoted by QuotP F/X/S and is called the Quot-scheme of OX -module F over S with Hilbert polynomial P . Moreover, the universal family of quotient sheaves on X×S QuotP F/X/S is called the universal quotient sheaf. The functor QuotF/X/S is representable by ` the locally projective scheme P QuotP F/X/S and is simply denoted by QuotF/X/S . Remark 1.7. The proof of the above theorem shows that if F is a quotient coherent sheaf of an S-flat coherent OX -module F 0 , then QuotP F/X/S is a closed subscheme P P of QuotF 0 /X/S . Since QuotF 0 /X/S is a closed subscheme of a Grassmann scheme by Proposition 1.3, we see that QuotP F/X/S is embedded in the same Grassmann scheme. In particular if we assume (1.7.1) X is embedded in P(U ) over S with U a locally free coherent sheaf on S and OX (1) is the restriction of OP(U ) (1) to X then for sufficiently large integer n, F is a quotient coherent sheaf of OP(U ) (−n)⊕N with N = P (n) and hence there is an integer m0 depending on n such that for ⊕N every m ≥ m0 , QuotP )) by the F/X/S is embedded in GrassP (m) (π∗ (OP(U ) (m − n) Pl¨ ucker invertible sheaf det((fQ )∗ (G(m))), where π is the projection of P(U ) to S.

2 Geometric invariant theory In this section we shall go over the basic ideas and results of Geometric Invariant Theory. For the precise and detailed accounts of the contents of this section we refer the reader to Bible [GIT]. We shall start with the notion of a G-linearization.

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Definition 2.1. Let σ : G ×S X → X be an action of an S-group scheme G on an S-scheme X and p2 : G ×S X → X the second projection. For an OX -module F , a G-linearization of F is an isomorphism ∼ p∗ (F ) ϕ : σ ∗ (F ) −→ 2 which makes the following diagram commutative: (σ(1G × σ))∗ (F )

(1G × σ)∗ (ϕ) -

(p2 (1G × σ))∗ (F )

(σp23 )∗ (F )

(σ(µ × 1X ))∗ (F )

(µ × 1X )∗ (ϕ)

(p23 )∗ (ϕ)-

(p2 p23 )∗ (F )

- (p2 (µ × 1X ))∗ (F )

where µ : G ×S G → G is the group law and where p23 : G ×S G ×S X → G ×S X is the projection to the second and third factors. A G-linearization ϕ on F gives rise to an homomorphism σ∗

H 0 (ϕ)

H 0 (X, F ) −−−−→ H 0 (G ×S X, σ ∗ (F )) −−−−→ H 0 (G ×S X, p∗2 (F )) If S = Spec(k) with k a field and F is quasi-coherent, then H 0 (G ×S X, p∗2 (F )) is isomorphic to H 0 (G, OG ) ⊗k H 0 (X, F ) by K¨ unneth formula. Thus we have a homomorphism in this case ϕˆ : H 0 (X, F ) −→ H 0 (G, OG ) ⊗k H 0 (X, F ). The commutativity of the diagram in the above definition implies that ϕˆ is a dual action of G on the vector space H 0 (X, F ). A section v ∈ H 0 (X, F ) is said to be an invariant section if ϕ(v) ˆ = 1 ⊗ v. An affine group scheme G over a noetherian scheme S is called reductive if G is smooth over S and every geometric fiber of G over S is reductive or its unipotent radical is trivial. Definition 2.2. Assume that S = Spec(k) with k a field and σ : G ×S X → X is an action of a reductive algebraic group G on an algebraic scheme X over S. Let L be an invertible sheaf on X with a G-linearization ϕ : σ ∗ (L) → p∗2 (L). (1) A geometric point x of X is said to be semi-stable if there is a section v of H (X, L⊗n ) for some n such that v is invariant with respect to the linearization ϕn induced by ϕ, Xv = {z ∈ X | v(z) 6= 0} is affine and that v(x) 6= 0. 0

(2) A geometric point x of X is said to be stable if there is a section v of H 0 (X, L⊗n ) for some n such that v is invariant, Xv is affine, the action of G on Xv

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is closed and that v(x) 6= 0, where an action of G on an algebraic scheme Y over k is closed if for every K-valued geometric point y of Y , the GK -orbit of y in YK is closed. (3) A geometric point x of X is said to be properly stable if it is stable and the stabilizer group of x is finite. Let us interpret the above definition in the case where L is ample. The Glinearization of L induces a dual action of G on the vector space Vm = H 0 (X, L⊗m ). For a sufficiently large integer m, L⊗m is very ample and hence we have a quotient graded k-algebra B of the symmetric algebra S(V ) such that X = Proj(B) ⊂ ˆ be Proj(S(V )) and the dual action of G on B is induced from that on S(V ). Let X the affine cone Spec(B) embedded in the affine space Spec(S(V )). By the geometric reductivity of G (complete reducibility of linear representations of G in characteristic 0 and by Haboush’s solution of Mumford conjecture [Hab] in characteristic positive), we have the following [GIT, Proposition 2.2 and Appendix to Chapter 1, C]. Proposition 2.3. Let x be a K-valued geometric point of X, x∗ a K-valued geoˆ \ {0} over x and o(x∗ ) the GK -orbit of x∗ . metric point of X ˆ K does not contain (1) x is semi-stable if and only if the closure of o(x∗ ) in X the origin 0. ˆ K and its dimension is (2) x is properly stable if and only if o(x∗ ) is closed in X equal to dim G. Based on this we get a key result by Mumford which supplies us with an efficient criterion for semi-stability and stability of a geometric point. Since we are considering the stability and semi-stability of a geometric point, we assume for a while that the base field k is algebraically closed. Definition 2.4. A 1-parameter subgroup of an affine algebraic group G over k is a non-trivial homomorphism λ : Gm → G. Let σ : G ×k X → X be an action of G on a projective scheme X over k. Pick ∼ G × {x} −→ σ X. Identifying a closed point x. Then we have a morphism ψx : G → 1 Gm with A \ {0}, a 1-parameter subgroup λ of G gives rise to a morphism ψx λ : A1 \ {0} → X. Since X is proper over k, ψx λ extends to a morphism f of A1 → X and then f (0) must be a fixed point of the action of Gm via λ. Assume further that an ample G-linearized invertible sheaf L on X is given. The linearization induces that of Gm on L and hence an action of Gm on the bundle space L = Spec(S(L)) which is compatible with the action on X. Since f (0) is a fixed point of the Gm action, Gm acts on the fiber of L over f (0) that is a 1-dimensional vector space over k. The last action is given by (α, t) 7−→ αr t with α ∈ Gm (k), t ∈ L(f (0))(k) and with r an integer. A key definition by Mumford is the following.

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91

Definition 2.5. Under the notation in the above, we set µL (x, λ) = −r. Assume that we are given a G-action on the projective space Pnk over an algebraically closed field k and a G-linearization of OPn (1). For a non-trivial 1parameter subgroup λ of G, we can choose a system of homogeneous coordinates so that the action of λ is given as follows (x0 , x1 , . . . , xn ) 7−→ (tr0 x0 , tr1 x1 , . . . , trn xn ). Under this situation we have the following result on µOPn (1) (x, λ). Proposition 2.6. For a closed point x = (x0 , x1 , . . . , xn ) of Pnk , we have µOPn (1) (x, λ) = max{−ri | i such that xi 6= 0} A criterion of stability and semi-stability is stated as follows. Theorem 2.7. ([GIT, Theorem 2.1]) Let σ : G ×k X → X be an action of an affine reductive group G on a projective scheme X over an algebraically closed field k and L an ample invertible sheaf on X with a G-linearization. Then, for a closed point x of X, we have the following. (1) x is semi-stable if and only if for all 1-parameter subgroup λ of G, we have that µL (x, λ) ≥ 0. (2) x is properly stable if and only if for all 1-parameter subgroup λ of G, we have that µL (x, λ) > 0. To globalize the above ideas let us go back to a reductive group scheme G over a noetherian scheme S. Take a locally free coherent sheaf V of constant rank on S and a dual action θ : V → OS [G] ⊗OS V . θ provides us with an action of G on P(V ) and a G-linearization of OP(V ) (1). Let X be a G-stable closed subscheme of P(V ) and L be the restriction of OP(V ) (1) to X. Conversely, if f : Y → S is a flat projective morphism, G acts on Y and if M is an ample invertible sheaf on Y with a G-linearization such that the Hilbert polynomial on each fiber over S is constant, then for sufficiently large integer m, M ⊗m is very ample, VM = f∗ (M ⊗m ) is locally free coherent sheaf of constant rank and the G-linearization of M induces a dual action of G on VM . Y is a G-stable closed subscheme of P(VM ) with respect to the action of G on P(VM ) induced by the G-linearization and M ⊗m is isomorphic to OP(VM ) (1)|Y as G-linearized sheaves. Thus the first situation is the same as fixing a G-stable closed subscheme X in a projective, flat S-scheme Y with a G-action and G-linearized ample invertible sheaf M on Y . Proposition 2.8. ([Se, Proposition 7 and Remark 9]) Assume that a reductive S-group scheme G acts on P(U ) with U a locally free coherent sheaf on S such that OP(U ) (1) carries a G-linearization. If X is a G-stable closed subscheme of P(U ) and if L is an invertible sheaf on X which is isomorphic to OP(U ) (1)|X , then

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there is a G-stable open subscheme X ss (L) (or, X0s (L), resp.) such that for every K-valued geometric point t of S, X ss (L)t (K) (X0s (L)t (K), resp.) is exactly the set of semi-stable (properly stable, resp.) closed points of Xt with respect to L(t). The following is one of main results on the quotient of a projective scheme by a reductive group scheme. Theorem 2.9. ([Se, Theorem 4]) Under the same situation as in Proposition 2.8, there is a morphism ϕ : X ss (L) → Y of S-schemes with the following properties: (1) ϕ is a G-invariant affine morphism with respect to the trivial action of G on Y , we have OY = ϕ∗ (OX ss (L) )G . (2) ϕ is surjective, more precisely, for every K-valued geometric point t of S, ϕ induces the following identification: Yt (K) = X ss (L)t (K) modulo the equivalence relation x1 ∼ x2 if and only if o(x1 ) ∩ o(x2 ) 6= ∅. (3) For every G-stable closed subscheme Z of X ss (L), ϕ(Z) is closed in Y and if Z1 and Z2 are two G-stable closed subschemes of X ss (L) such that Z1 ∩ Z2 = ∅ then ϕ(Z1 ) ∩ ϕ(Z2 ) = ∅. (4) There is an open subscheme Y s of Y such that Xs (L) = ϕ−1 (Y s ). ∼ L⊗m |X ss (L) with m (5) There is an invertible sheaf L0 on Y such that ϕ∗ (L0 ) = a positive integer. If S is of finite type over a universally Japanese ring, then Y is projective over S and L0 is very ample. The pair (Y, ϕ) is unique up to a unique isomorphism of S-schemes. To see this we introduce various notions of quotient. Definition 2.10. Let σ : G ×S X → X be an action of reductive S-group scheme G on an S-scheme X and ϕ : X → Y an S-morphism. (1) (Y, ϕ) is called a geometric quotient if the following conditions are satisfied: (i)

the morphism ϕ is G-invariant, that is, the following diagram is commutative σ G ×S X −−−−→ X    ϕ p2 y y ϕ

(ii)

X −−−−→ Y every geometric fiber of ϕ is precisely the orbit of a geometric point of X,

(iii)

a subset U of Y is open if and only if ϕ−1 (U ) is open in X,

(v)

OY = ϕ∗ (OX )G .

(2) (Y, ϕ) is called a good quotient if it has the property (1), (2) and (3) of Theorem 2.9 for X ss (L) = X

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(3) (Y, ϕ) is called a categorical quotient if the condition (i) of (1) is satisfied and if for every pair (Z, ψ) of an S-scheme Z and an S-morphism ψ : X → Z with ψσ = ψp2 , there is a unique S-morphism θ : Y → Z such that ψ = θϕ. A geometric quotient is obviously a good quotient. It is clear by its definition that a categorical quotient is unique up to a unique isomorphism. Then the uniqueness of a good quotient follows from the next result. Proposition 2.11. A good quotient is a categorical quotient. Proof. Let (Y, ϕ) be a good quotient of an action σ; G ×S X → X and ψ : X → Z a G-invariant morphism. Take an affine open covering {Vi } of Z. Then Xi = X \ ψ −1 (Vi ) is a G-invariant closed set of X and hence Ui = Y \ ϕ(Xi ) is an open subset of Y . Pick a K-valued geometric point x of ψ −1 (Vi ) and let t be the image of x in S. Since the Gt -orbit o(x) of x is in the Gt -stable closed set ψt−1 (ψt (x)) in Xt , its closure o(x) is a subset of ψt−1 (ψt (x)) ⊂ ψt−1 ((Vi )t ). This and the property (2) of a good quotient imply that ϕ−1 (Ui ) = ψ −1 (Vi ). Since ϕ is an affine morphism, for an affine open subscheme U of Ui , ϕ−1 (U ) is a G-stable affine open subscheme of X such that ψ −1 (Vi ) contains ϕ−1 (U ). Writing ϕ−1 (U ) = Spec(C) and Vi = Spec(D), we have a homomorphism τ : D → C which is determined by ψ. By the property (1) of a good quotient we see that U = Spec(C G ). Thus ψ factors uniquely on ϕ−1 (U ) through ϕ: ϕ−1 (U ) ϕ

ψ @ ? R @ - Vi U @

Since making the ring of invariants is compatible with a flat base change, the above shows that ψ factors uniquely on the whole X through ϕ. Let S be a noetherian scheme, V a free OS -module of rank n and W a locally free OS -module of rank m. We have a canonical dual action of G = GL(V ) ∼ = GL(n, S) on V and hence on V ⊗ W = V ⊗OS W by letting G act trivially on W . Fix a basis e1 , . . . , en a free basis of V . Then, for a suitable affine system of coordinates {xij } of G over S, we have an isomorphism α : VG −→ VG by sending ej to

Pn

i=1

xij ei and hence another isomorphism β : (V ⊗ W )G −→ (V ⊗ W )G .

Let Z be the Grassmann scheme Grassr (V ⊗ W ) of r-dimensional quotient spaces of V ⊗ W and θ : (V ⊗ W )Z −→ U

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the universal quotient bundle on Z. Composing two homomorphisms p∗ (β)

p∗1 ((V ⊗ W )G ) −−1−−→ p∗1 ((V ⊗ W )G )

o p∗ (θ)

p∗2 ((V ⊗ W )Z ) −−2−−→ p∗2 (U ) we obtain the quotient locally free sheaf p∗2 (U ) of p∗1 ((V ⊗ W )G ) of rank r, where pi is the ith projection of G ×S Z. By the universality of the Grassmann scheme there exists a morphism σ : G × Z −→ Z which makes the following diagram commutative: σ ∗ (θ) ∼ p∗1 ((V ⊗ W )G ) −−−−→ σ ∗ ((V ⊗ W )Z ) −−−−→ σ ∗ (U )       ϕyo p∗ ψ yo 1 (β)y p∗ (θ) ∼ p∗1 ((V ⊗ W )G ) −−−−→ p∗2 ((V ⊗ W )Z ) −−2−−→ p∗2 (U )

Now it is not hard to see that σ is an action of G on Z and ψ (or, ϕ) defines a G-linearization of (V ⊗ W )Z (U , resp.). We have thus a G-morphism r r r r ^ ^ ^ ^ π : Z = P( U ) ,−→ P( (V ⊗ W )Z ) = P( (V ⊗ W )) ×S Z −→ P( (V ⊗ W )).

As is well-known, π is a closed immersion and called the Pl¨ ucker embedding. The Vr dual action of G on V ⊗ W induces that of (V ⊗ W ) and hence an action Vr on P( (V ⊗ W )). This is exactly the action we got in the above and Z is GVr stable under this action. Moreover, the G-linearization of U comes from that on Vr V OP( r (V ⊗W )) (1). Since the center GmS acts trivially on P( (V ⊗W )), the projecVr tive linear group scheme G = G/GmS acts on P( (V ⊗ W )) and OP(Vr (V ⊗W )) (a) Vr carries a G-linearization, where a is the rank of (V ⊗ W ). We shall show here one of key computations to construct the moduli spaces of stable sheaves. Proposition 2.12. Let k be an algebraically closed field and let S be Spec(k). An r-dimensional quotient k-vector space L of V ⊗k W is properly stable (or, semistable) with respect to the above action of G on the Grassmann variety Z and the Vr G-linearization of ( U )⊗a if and only if for every proper vector subspace V 0 of V , we have dim V 0 dim L, dim V (≥ resp.)

dim L0 >

where L0 is the image of V 0 ⊗k W to L.

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Proof. There is an isogeny of SL(V ) to G = P GL(V ). By the definition of the proper stability and the semi-stability, we may replace G by SL(V ) and then the Vr GL(V )-linearization of U induces its SL(V )-linearization. Thus we have only to check the proper stability and the semi-stability of L with respect to the Pl¨ ucker embedding of Z. Pick a 1-parameter subgroup λ of SL(V ) and fix a basis e1 , . . . , en so that the dual action of λ on V is defined by ei 7−→ tri ⊗ ei where r1 , . . . rn satisfy the condition (2.12.1)

n X

ri = 0, r1 ≤ r2 ≤ · · · ≤ rn and not all of ri ’s are 0.

i=1

Take a basis f1 , . . . , fm of W and set hβ = ei ⊗ fj with β = m(i − 1) + j. Then h1 , . . . , hmn is a basis of V ⊗k W . If we choose the homogeneous system of coordinate of P(V ⊗k W ) corresponding to this basis, the action of the 1-parameter subgroup λ is given by (z1 , . . . , zmn ) 7−→ (tr1 z1 , . . . , tr1 zm , tr2 zm+1 , . . . , tr2 z2m , . . . , trn zmn ). We set s1 ≤ s2 ≤ · · · ≤ smn in the following way: sβ = ri for β = (i − 1)m + j with 0 ≤ j < m. Let Uβ be the vector subspace of V ⊗k W generated by {h1 , . . . , hβ } and α : V ⊗k W → L the given surjection. For each integer j with 1 ≤ j ≤ r, there is a unique integer µj such that dim α(Uµj ) = j and dim α(Uµj −1 ) = j − 1. Then we have 0 < µ1 < µ2 < · · · < µr ≤ mn ¯ 1 = α(hµ ), . . . , h ¯ r = α(hµ )} forms a basis of L. Using these bases we can and {h 1 r represent the map α by the following (r × mn)-matrix:          

0 .. . .. . .. . 0

···

···

0 a1µ1 .. . 0 .. .. . . .. .. . . 0 0

···

a1µ2 −1

a1µ2

···

a1µ3 −1

···

a1µr

···

a1mn



···

0 .. . .. . 0

a2µ2

···

a2µ3 −1

···

···

0 .. . 0

···

0 .. . 0

···

a2µr .. . .. . arµr

a2mn .. . .. . armn

        

···

···

···

···

where aiµi = 1. Let pi1 ...ir be the Pl¨ ucker coordinate corresponding to hi1 ∧ · · · ∧ hir with i1 < · · · < ir . If ij < µj for a j, then {α(hi1 ), . . . α(hij )} is a subset of a vector

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subspace of dimension j − 1 and hence pi1 ...ir = 0 at the point L. On the other hand, we know that pµ1 ...µr (L) 6= 0. Using Proposition 2.6, we see therefore that r

µ(L, λ) = µ∧ U (L, λ) r X =− sµj j=1

=−

mn X

sβ (dim α(Uβ ) − dim α(Uβ−1 ))

β=1

= −rsmn +

mn−1 X

(sβ+1 − sβ ) dim α(Uβ )

β=1

= −rrn +

n−1 X

(rj+1 − rj ) dim α(Ujm )

j=1

Lemma 2.13. Let r1 , . . . , rn be a sequence of integers which satisfies the condition (2.12.1). Then there are positive rational numbers δ1 , . . . , δn−1 such that rj =

n−j X

−iδi +

i=1

n−1 X

(n − `)δ`

`=n−j+1

Proof. We have to solve the following system of linear equations   (n − 1)x1 + (n − 2)x2 + (n − 3)x3 + · · · + 2xn−2 + xn−1      −x1 + (n − 2)x2 + (n − 3)x3 + · · · + 2xn−2 + xn−1      ..   .  −x1 + · · · + (j − n)xn−j + (j − 1)xn−j+1 + · · · + xn−1      ..    .     −x − 2x − 3x − · · · − (n − 2)x − (n − 1)x 1

2

3

n−2

n−1

= rn = rn−1 .. . = rj .. . = r1

Summing up from the first to the (n − 1)-th equations, we obtain the last equation. Therefore, by Cramer’s formula, our assertion reduces to the fact that the determinant of the following matrix is equal to (rn−i − rn−i−1 )nn−2 :   n − 1 n − 2 n − 3 ··· n − i rn n − i − 2 ··· 2 1  −1 n − 2 n − 3 · · · n − i rn−1 n − i − 2 · · · 2 1     −1 −2 n − 3 · · · n − i rn−2 n − i − 2 · · · 2 1     .. .. .. .. .. .. .. ..    . . . . . . . .      −1 −2 −3 · · · n − i rn−i+1 n − i − 2 · · · 2 1     −1 −2 −3 ··· −i rn−i n − i − 2 ··· 2 1     .. .. .. .. .. .. .. ..   . . . . . . . .  −1

−2

−3

···

−i

r2

−i − 2

···

−n + 2

1

3

S-EQUIVALENCE AND E-SEMI-STABILITY

97

Let us go back to the proof of Proposition 1.11. Since the right hand side of the equation n−1 X µ(L, λ) = −rrn + (rj+1 − rj ) dim α(Ujm ) j=1

is a linear function of r1 , . . . , rn , the above lemma shows us that µ(L, λ) is positive if and only if so is for the extreme cases r1 = · · · = rp = p − n,

rp+1 = · · · = rn = p with 1 ≤ p ≤ n − 1.

Thus µ(L, λ) > 0 for all r1 , . . . , rn if and only if −rp + n dim α(Upm ) > 0 for 1 ≤ p ≤ n − 1. If Vi is the vector subspace of V spanned by e1 , . . . ei , then Upm = Vp ⊗k W and hence the above inequality can be rewritten in the form dim α(Vp ⊗k W ) >

dim Vp dim L. dim V

Since every p-dimensional subspace of V is Vp for a suitable 1-parameter subgroup and a suitable basis of V , the above inequality proves our assertion for the proper stability. For semi-stability we have only to replace positive by non-negative and > by ≥ in the final part of our argument.

3 S-equivalence and e-semi-stability Assume that E, E 0 and E 00 are coherent sheaves of pure dimension d on a projective scheme X over an algebraically closed field k such that they fit in an exact sequence 0 −→ E 0 −→ E −→ E 00 −→ 0. Such extensions are parameterized by the k-vector space Ext1OX (E 00 , E 0 ). If the above extension is non-trivial, then it corresponds to a non-zero element ξ. On the line T in Ext1OX (E 00 , E 0 ) joining ξ and the origin 0 we can construct a flat family of extensions ˜ −→ ET00 −→ 0 0 −→ ET0 −→ E ˜ such that for every k-rational point t of T , E(t) is the extension corresponding to 1 00 0 the t ∈ ExtOX (E , E ). Since the multiple αξ by a non-zero element α defines the ˜ ˜ same extension as E, E(t) is isomorphic to E for all t ∈ T (k) \ {0} and E(0) is the 0 00 0 00 direct sum of E and E . Assume now that both E and E are semi-stable and PES0 (m) = PES00 (m). If we could construct a space M parameterizing semi-stable ˜ gives sheaves and having the property like (I.1.1) for semi-stable sheaves, then E ˜ stated in the above means rise to a morphism u of T to M . The property of E that T \ {0} is sent to a single point. Since T is connected, g(T ) must be one point and hence the space M cannot separate extensions of the same sheaves. This phenomenon naturally leads us to an idea due to Seshadri.

98

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Proposition 3.1. Let (X, OX (1)) be a pair of a projective scheme over an algebraically closed field k and an ample invertible sheaf OX (1) on X and let E be a coherent sheaf on X of pure dimension d. Assume that E is semi-stable. (1) There exists a filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα−1 ⊂ Eα = E by coherent OX -modules of pure dimension d such that (a) Ei /Ei−1 is stable (1 ≤ i ≤ α) and (b) PESi (m) = PES (m) (1 ≤ i ≤ α). 0 (2) If 0 = E00 ⊂ E10 ⊂ · · · ⊂ Eβ−1 ⊂ Eβ0 = E is another filtration with the properties (a) and (b), then α = β and there is a permutation σ of {1, . . . , n} such 0 0 that Ei /Ei−1 is isomorphic to Eσ(i) /Eσ(i)−1 (1 ≤ i ≤ α).

Proof. (1) Let us prove our assertion by induction on a0 (E). Assume that (1) is true for semi-stable sheave F with a0 (F ) < a0 (E). If E is stable, then there is nothing to prove. Suppose that E is not semi-stable. Then the set A = {F | F is a coherent subsheaf of E with PFS (m) = PES (m)} is not empty and for every member F of A, E/F is of pure dimension d because of the semi-stability of E. Take a member E1 of A such that a0 (E1 ) is the smallest in the set {a0 (F ) | F ∈ A}. Then obviously E1 is stable and E/E1 is semi-stable. By our induction hypothesis there exists a filtration 0 = E1 /E1 ⊂ E2 /E1 ⊂ · · · ⊂ Eα−1 /E1 ⊂ Eα /E1 = E/E1 such S that (Ei /E1 )/(Ei−1 /E1 ) ∼ (m). = Ei /Ei−1 is stable and that PESi /E1 (m) = PE/E 1 S S S S S S Since PE1 (m) = PE (m), we see that PE (m) = PE/E1 (m) = PEi /E1 (m) = PEi (m). Thus the filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα−1 ⊂ Eα = E has the properties (a) and (b). (2) Our proof is by induction on α. If α = 1, then E is stable and hence our assertion is obvious. Assume that α > 1. Let γ be the smallest integer such 0 that Eγ0 contains E1 . The natural homomorphism ϕ : E1 → Eγ0 /Eγ−1 is not zero. 0 0 By Proposition II.7.3 we see that ϕ is injective. Since Eγ /Eγ−1 is stable, the equality PES1 (m) = PES (m) = PESγ /Eγ−1 (m) implies that ϕ is surjective. Thus E1 ¯ = E/E1 . Set is isomorphic to Eγ /Eγ−1 . Let us consider the semi-stable sheaf E ¯i = Ei+1 /E1 and E   Ei0 /Ei0 ∩ E1 0 ≤ i ≤ γ − 1 0 ¯ Ei =  0 Ei+1 /E1 γ ≤i≤β−1 ¯0 ⊂ E ¯1 ⊂ · · · ⊂ E ¯α−1 = E ¯ is a filtration of E ¯ with the property It is clear that 0 = E 0 0 ¯ ¯γ0 /E ¯0 (a) and (b). On the other hand, Ei is isomorphic to Ei for 0 ≤ i ≤ γ −1, E γ−1 = 0 0 0 0 0 0 0 0 0 ¯ /E ¯ Eγ+1 /(E1 + Eγ−1 ) = Eγ+1 /Eγ and E = E /E because E ∩ E 1 γ−1 = 0 j j−1 j+1 j 0 0 0 0 0 ¯ ¯ ¯ ¯ and E1 + Eγ−1 = Eγ . Thus the filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eβ−1 = E enjoys the property (a) and (b). Our induction hypothesis implies that α = β and there ¯i /E ¯i−1 ∼ ¯ 0 /E ¯0 is a permutation τ of {1, . . . , α − 1} such that E =E τ (i) τ (i)−1 . We define a permutation σ of {1, . . . , α} as follows:  γ if i = 1    σ(i) = τ (i − 1) if 1 ≤ τ (i − 1) ≤ γ − 1    τ (i − 1) + 1 if γ ≤ τ (i − 1) ≤ α − 1

3

S-EQUIVALENCE AND E-SEMI-STABILITY

99

The σ is a permutation which we want. The above proposition justifies the following definition. Definition 3.2. Let E and E 0 be semi-stable sheaves of pure dimension d on a projective scheme over an algebraically closed field. (1) A filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα−1 ⊂ Eα = E with the properties (a) and (b) of Proposition 3.1 is called Seshadri filtration. (2) For a Seshadri filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα−1 ⊂ Eα = E, we define α M Ei /Ei−1 . Each Ei /Ei−1 is called a component of gr(E). gr(E) to be i=1

(3) E and E 0 are said to be S-equivalent if gr(E) ∼ = gr(E 0 ) and then we denote 0 E ∼S E We don’t know the boundedness of semi-stable sheaves in the most general situation.1 For this reason we have to introduce a rather technical notion in the theory of stable sheaves. Let (X, OX (1)) be a pair of a projective scheme over an algebraically closed field k and a very ample invertible sheaf OX (1) on X. Pick a coherent sheaf E of pure dimension d. Take a general member H ∈ |OX (1)|. Then each irreducible component of Supp(E|H ) has dimension d−1 and E|H satisfies (S1 ) by Lemma II.3.5. So E|H is of pure dimension d − 1. Repeating this procedure, we can find general hyperplanes H1 , . . . Hd−1 such that for the scheme theoretic intersection Y = X∩H1 ∩· · ·∩Hd−1 , E|Y is a coherent OY -module of pure dimension 1. By Proposition I.3.12 the set {µS (F ) | F is a coherent subsheaf of E|Y of pure dimension 1 } is bounded above. Definition 3.3. Let (X, OX (1)) be a pair of a projective scheme over an algebraically closed field k and a very ample invertible sheaf OX (1) on X and let E be a coherent sheaf on X of pure dimension d. Fix a positive integers e. (1) E is said to be e-stable (or, e-semi-stable) if it is stable (semi-stable, resp.) and if for general hyperplane sections H1 , . . . , Hd−1 in |OX (1)|, for the subscheme Y = H1 ∩ · · · ∩ Hd−1 of X and for every coherent subsheaf F of E|Y with 0 < a0 (F ) < a0 (E), we have µS (F ) ≤ µS (E) + e. (2) E is said to be strictly e-semi-stable if E is e-semi-stable and if every quotient coherent sheaf of E with PES0 (m) = PES (m) is e-semi-stable. Note here that the condition PES0 (m) = PES (m) and the semi-stability of E imply that E 0 is of pure dimension d. What we have seen right before this definition is that every semi-stable sheaf is e-semi-stable for some e. As for the strict semi-stability we have a similar result. 1 See

Appendix A.

100

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CONSTRUCTION OF MODULI SPACES

Lemma 3.4. Let (X, OX (1)) be as in Definition 3.3 and let 0 −→ E 0 −→ E −→ E 00 −→ 0 be an exact sequence of coherent OX -module of pure dimension d. Assume that PES0 (m) = PES (m) = PES00 (m). (1) If E 0 and E 00 are e-semi-stable, then so is E. (2) If E is e-semi-stable, then E 0 is e-semi-stable and E 00 is a0 (E)e-semi-stable. Hence E is strictly a0 (E)e-semi-stable. Proof. For the semi-stability in the above assertions the results are a part of Proposition II.7.3. If we choose general members H1 , . . . , Hd−1 of |OX (1)|, then for the scheme theoretic intersection Y = H1 ∩ · · · ∩ Hd−1 , all the E 0 |Y , E|Y and E 00 |Y are of pure dimension 1, the sequence 0 −→ E 0 |Y −→ E|Y −→ E 00 |Y −→ 0 is exact and the condition in Definition 3.3 holds good for E or E 0 , E 0 according as E is e-semi-stable or E 0 and E 00 are e-semi-stable. (1) Let F be a coherent subsheaf of E|Y with 0 < a = a0 (F ) < a0 (E). Set F = F ∩ E 0 |Y and F 00 to be the image of F to E 00 |Y . Then we have 0

a0 (F ) = a0 (F 0 ) + a0 (F 00 ) a1 (F ) = a1 (F 0 ) + a1 (F 00 ) a1 (F 0 ) ≤ a0 (F 0 )µS (E 0 ) + a0 (F 0 )e a1 (F 00 ) ≤ a0 (F 00 )µS (E 00 ) + a0 (F 00 )e Since µS (E 0 ) = µS (E) = µS (E 00 ), the above relations give rise to µS (F ) ≤ µS (E) + e. (2) If F 0 is a coherent subsheaf of E 0 |Y with 0 < a0 (F 0 ) < a0 (E 0 ), then it is a subsheaf of E|Y . By e-semi-stability of E we have µS (F 0 ) ≤ µS (E) + e = µS (E 0 ) + e as required. Next let F 00 be a coherent subsheaf of E 00 with 0 < a0 (F 00 ) < a0 (E 00 ). If F is the inverse image of F 00 to E|Y . Then we get a1 (F ) = a1 (F 00 ) + a1 (E 0 ) a1 (F ) ≤ (a0 (E 0 ) + a0 (F 00 ))(µS (E) + e). Thus we obtain a1 (F 00 ) ≤ a0 (F 00 )(µS (E 00 ) + e) + a0 (E 0 )(µS (E 0 ) + e) − a1 (E 0 ) = a0 (F 00 )(µS (E 00 ) + e) + a0 (E 0 )e   a0 (F 00 ) + a0 (E 0 ) = a0 (F 00 ) µS (E 00 ) + e a0 (F 00 ) ≤ a0 (F 00 )(µS (E 00 ) + a0 (E)e).

3

S-EQUIVALENCE AND E-SEMI-STABILITY

101

The strict e-semi-stability behaves well with respect to the S-equivalence. Lemma 3.5. Let (X, OX (1)), E 0 , E and E 00 be as in Lemma 3.4. (1) If E is strictly e-semi-stable, then each component of gr(E) is e-stable. (2) E is strictly e-semi-stable if and only if so are both E 0 and E 00 . Proof. (1) Our proof is by induction on the number α of the components of gr(E). If α = 1, then we have nothing to prove. Assume that α > 1 and take a Seshadri filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα−1 ⊂ Eα = E. By Lemma 3.4 we know that ¯ = E/E1 is strictly e-semi-stable by the definition of strict e-semiE1 is e-stable. E ¯0 ⊂ E ¯1 = E2 /E1 ⊂ · · · ⊂ E ¯α−2 = Eα−1 /E1 ⊂ E ¯α−1 = E ¯ is a stability and 0 = E ¯ ¯ Seshadri filtration of E. Thus gr(E) = E1 ⊕gr(E) and our induction hypothesis tells ¯ is e-stable. We see therefore that each component us that each component of gr(E) of gr(E) is e-stable. (2) Note first that if E is an extension of a semi-stable sheaf E 00 by a semi-stable sheaf E 0 and if PES0 (m) = PES00 (m), then E is semi-stable, PES (m) = PES0 (m) = PES00 (m) and gr(E) = gr(E 0 ) ⊕ gr(E 00 ). If both E 0 and E 00 are strictly e-semistable, then this remark and our assertion (1) show that each component of gr(E) is e-semi-stable. Let F be a coherent quotient sheaf of E with PFS (m) = PES (m). Applying the above remark to E, F and ker(E → F ), we see that gr(F ) is a direct summand of gr(E) and hence each component of gr(F ) is e-stable. Using Lemma 3.4, (1) repeatedly, we see that F is e-semi-stable. Thus E is strictly e-semi-stable. Conversely, assume that E is strictly e-semi-stable. By the very definition of the strict e-semi-stability E 00 is obviously a strict e-semi-stable sheaf. Since gr(E 0 ) is a direct summand of gr(E), each component of gr(E 0 ) is e-stable. Then by using the former half of this proof repeatedly, we see that E 0 is strictly e-semi-stable. Corollary 3.6. E is strictly e-semi-stable if and only if so is gr(E). Now let us prove the openness of the strict e-semi-stability. Proposition 3.7. Let f : X → S be a projective morphism of locally noetherian schemes, OX (1) an f -very ample invertible sheaf and F an S-flat coherent OX module. Assume that the pair (X, OX (1)) satisfies the condition (1.7.1). Then there is an open set U of S such that for all algebraically closed field k, U (k) is exactly the set of k-valued points s of S with F (s) strictly e-semi-stable. Proof. Since the property that a coherent sheaf is of pure dimension d and semistable is open by Theorem II.7.16, we may assume that for every geometric point s of S, F (s) is a semi-stable sheaf of pure dimension d. By our assumption and the base change theorem V = f∗ (OX (1)) is a locally free coherent sheaf on S and X is a closed S-subscheme of P(V ). We may assume that V is of constant rank r. Let B be the Grassmann Grassr−d+1 (V ) of quotient bundles of V of rank r − d + 1 and VB → Q the universal quotient bundle. We shall consider the pull back F˜ of F to the scheme theoretic intersection Z of XB and P(Q) in P(VB ) = P(V ) ×S B. The

102

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set B 0 of points t of B such that dim Supp(F˜ (t)) = 1 and F˜ is flat at t is an open set of B. If I is the defining ideal of Z in XB , then we have an exact sequence u I ⊗ FB −−−−→ FB −→ F˜ −→ 0

and we see by Lemma I.3.10 that a point b of B is in B 0 if and only if u(b) is injective and dim Supp(F˜ (b)) = 1. For an over field K of k(b), u(b) is injective if and only if so is u(b) ⊗k(b) K. These mean that a geometric point t of B is a geometric point of B 0 if and only if u(t) is injective and dim Supp(F˜ (t)) = 1. Pick a geometric point s of S. We can choose members σ1 , . . . , σd−1 of V (s) = H 0 (Xs , OX (1)(s)) so general that σ1 is not zero at any point of Ass(F (s)) and the hypersurface σj+1 = 0 does not meet Ass(F (s)/(σ1 , . . . , σj )F (s)(−1)). Then σ1 , . . . , σd−1 are linearly independent in V (s) and hence gives rise to a geometric point t of B over s. By the choice of σ1 , . . . , σd−1 t is a geometric point of B 0 . Thus general points of Bs is in B 0 . Since F˜ is flat over B 0 , Theorem II.7.12 and Lemma I.2.16 imply that there is an open set B0 of B 0 such that a geometric point t of B 0 is in B0 if and only if F˜ (t) is of pure dimension 1 and has the property in Definition 3.3, (1). Since B is flat over S, the image U0 of B0 to S is an open set and by the construction of U0 a k-valued geometric point s of S is in U0 (k) if and only if F (s) is e-semi-stable. Replacing S by U0 , we may assume that for every geometric point s of S, F (s) is e-semi-stable and may assume also that the Hilbert polynomials of F (s) are independent of s ∈ U0 . Take an integer i with 1 ≤ i < a0 (F (s)) = a0 . SettingPi = i PFS(s) (m) and F0 = FU0 , let us consider the Quot-scheme Qi = QuotP F/X/S and the universal quotient sheaf Fi on X ×S Qi . Since Fi is flat over S, the above proof shows that there is a closed set Zi of Qi such that a k-valued geometric point z of Qi is in Zi (k) if and only if Fi (z) is not e-semi-stable. The image Ti of Zi in S is closed because Qi is proper over S. Now it is clear that U =S\

a[ 0 −1

Ti

i=1

is the open set we want. We have also boundedness of e-semi-stable sheaves. Proposition 3.8. Let f : X → S be a projective morphism of noetherian schemes, OX (1) an f -very ample invertible sheaf on X, H(x) a numerical polynomial of degree d and let e a positive integer. Then the family of the classes of e-semi-stable sheaves with Hilbert polynomial H(m) on the fibers of X over S is bounded. Proof. Let F be the family for which we have to prove the boundedness. There are integers a0 , . . . , ad such that H(m) =

d X i=0

ad−i

  m+i . i

4

GENERAL SETTING AND A FUNDAMENTAL LEMMA

103

Let E be an e-semi-stable sheaf E on a geometric fiber Xs of X over S such that χ(E(m)) = H(m). For general members D1 , . . . , Dd−1 of |OX (1)(s)|, for Y = D1 ∩ · · · ∩ Dd−1 , EY is of pure dimension 1 and for every coherent subsheaf F of E|Y with 0 < a0 (F ) < a0 , we have the inequality µS (F ) ≤ µS (E) + e, This means that E|Y has an fixed type determined by e and a0 . Since the Hilbert polynomial of E|Y is a0 (m + 1) + a1 , Theorem II.7.9 implies that the family {E|Y | E ∈ F} is bounded. Then there is an integer m0 such that for all integers m ≤ m0 , H 0 (Y, E(m)|Y ) = 0 because E|Y satisfies the condition (S1 ). Set Yi to be the scheme theoretic intersection D1 ∩ · · · ∩ Di . We have an exact sequence 0 −→ E(m − 1)|Yi −→ E(m)|Yi −→ E(m)|Yi+1 −→ 0. If we assume that for all m ≤ m0 , H 0 (Y, E(m)|Yi+1 ) = 0, then the above exact sequence and the fact that E|Yi satisfies the condition (S1 ) prove that for all m ≤ m0 , H 0 (Y, E(m)|Yi ) = 0(cf. Lemma II 3.5 (3)). Thus, by descending induction on i, we see that for every i with 0 ≤ i ≤ d − 1 and every m ≤ m0 , H 0 (Y, E(m)|Yi ) = 0. This implies that F(m0 ) = {E(m0 )) | E ∈ F } is a (b)-family with (b) = (0, . . . , 0). By Theorem I.3.11 F(m0 ) is bounded and hence so is F.

4 General setting and a fundamental lemma Let f : X → S be a morphism of projective schemes, OX (1) an f -very ample invertible sheaf on X. For positive integers a0 , e and d, we set F e (d, a0 ) to be the family of the classes of coherent sheaves E on the fibers of X over S with the following properties: (4.1.1) E is of pure dimension d. (4.1.2) 0 < a0 (E) ≤ a0 . (4.1.3) If E is on a geometric fiber Xs , D1 , . . . , Dd−1 are general members of |OX (1)(s)| and if Y is the scheme theoretic intersection D1 ∩ · · · ∩ Dd−1 , then E|Y is of pure dimension 1 and for every coherent subsheaf F of E|Y with 0 < a0 (F ) < a0 (E), we have µS (F ) ≤ µS (E) + e. The following is due to Simpson. Lemma 4.2. There is an integer B such that for every member E of F e (d, a0 ) and for every integer m ≥ −B − µS (E), we have dim H 0 (Xs , E(m)) ≤

a0 (E) (m + B + µS (E))d , d!

where E is on a geometric fiber Xs . Moreover, if m ≤ −B − µS (E), then we have that H 0 (Xs , E(m)) = 0.

104

Chapter 3

CONSTRUCTION OF MODULI SPACES

Proof. Note first that for every integer t and an E ∈ F e (d, a0 ) on a fiber Xs of X over S, a0 (E(t)) = a0 (E) and a1 (E(t)) = a1 (E)+ta0 (E) and that for every coherent subsheaf F of E|Y , a0 (F (t)) = a0 (F ) and a1 (F (t)) = a1 (F ) + ta0 (F ). Hence E(t) has the property (4.1.3). This means that for every integer t, E(t) is also a member of F e (d, a0 ). Thus if we take the integer t0 such that 0 ≤ a1 (E) + t0 a0 (E) < a0 (E), then 0 ≤ a1 (E(t0 )) < a0 (E(t0 )), that is, 0 ≤ µS (E(t0 )) < 1 and E(t0 ) is in F e (d, a0 ). Assume that our assertions are true for every member F of F e (d, a0 ) with 0 ≤ µS (F ) < 1. Then, for the above E, we have dim H 0 (Xs , E(m)) = dim H 0 (Xs , E(t0 )(m − t0 )) ≤

a0 (E) (m − t0 + B + µS (E(t0 )))d d!

for all m with m − t0 > −B − µS (E(t0 )). Since t0 = µS (E(t0 )) − µS (E), the above is equivalent to the condition that for every m > −B − µS (E), dim H 0 (Xs , E(m)) ≤

a0 (E) (m + B + µS (E))d . d!

Moreover, if m − t0 ≤ −B − µS (E(t0 )) or equivalently m ≤ −B − µS (E), then H 0 (Xs , E(m)) = H 0 (Xs , E(t0 )(m − t0 )) = 0. Therefore, we have to prove our assertion for the subfamily F e (d, a0 )1 of F e (d, a0 ) with an additional condition 0 ≤ µS (E) < 1. Take an E in F e (d, a0 )1 and Y for E in (4.1.3). Since χ(E|Y (m)) = a0 (E)(m + 1) + a1 (E) and since the range of a0 (E) and hence that of a1 (E) are between 0 and a0 , the set {χ(E|Y (m)) | E ∈ F e (d, a0 )1 } is a finite set. On the other hand, the property (4.1.3) implies that the type of E|Y is bounded by rational numbers determined by a0 and e. Then, by Theorem II.7.9 {E|Y | E ∈ F e (d, a0 )1 } is a bounded family. Using this and the fact that E|Y satisfies the condition (S1 ), we can find integers A and C such that for all E|Y and all integers m with m ≤ −C (or, m ≥ A), H 0 (Y, E|Y (m)) = 0 (H 1 (Y, E|Y (m)) = 0, resp.). For 1 ≤ i ≤ A + C − 1, we have dim H 0 (Y, E|Y (−C + i)) ≤ a0 (E)i. Let B1 be the integer max{C, 2a0 }. If −B1 ≤ m ≤ −C, then dim H 0 (Y, E|Y (m)) = 0 < a0 (E)(m + B1 ). For every integer m with −C + 1 ≤ m ≤ A − 1, we have that dim H 0 (Y, E|Y (m)) ≤ a0 (E)(m + C) ≤ a0 (E)(m + B1 ) ≤ a0 (E)(m + B1 + µS (E)). Finally, if m ≥ A, then dim H 0 (Y, E|Y (m)) = a0 (E)(m + 1) + a1 (E)   a0 (E) + a1 (E) = a0 (E) m + a0 (E)   2a0 ≤ a0 (E) m + a0 (E) ≤ a0 (E)(m + B1 ) ≤ a0 (E)(m + B1 + µS (E)).

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Since B1 ≥ C, we have that for every m ≤ −B1 − µS (E), dim H 0 (Y, E|Y (m)) = 0. Taking D1 , . . . , Dd−1 for E in (4.1.1), we put Yi to be the scheme theoretic intersection D1 ∩ · · · ∩ Dd−i . We may assume that E|Yi is of pure dimension i, Now suppose that there is an integer Bi such that for every E ∈ F e (d, a0 )1 and every m > Bi + µS (E), dim H 0 (Yi , E|Yi (m)) ≤

a0 (E) (m + Bi + µS (E))i i!

and that for every m ≤ −Bi − µS (E), dim H 0 (Yi , E|Yi (m)) = 0. The multiplication by the equation of Dd−i provides us with an exact sequence 0 −→ E|Yi+1 (m − 1) −→ E|Yi+1 (m) −→ E|Yi (m) −→ 0. If m ≤ −Bi −µS (E), then the map H 0 (Yi+1 , E|Yi+1 (m−1)) → H 0 (Yi+1 , E|Yi+1 (m)) is surjective. Since H 0 (Yi+1 , E|Yi+1 (m)) = 0 for sufficiently small m, this proves that for all m ≤ −Bi − µS (E), H 0 (Yi+1 , E|Yi+1 (m)) = 0. Moreover, the above sequence implies inequality dim H 0 (Yi+1 , E|Yi+1 (n)) − dim H 0 (Yi+1 , E|Yi+1 (n − 1)) ≤ dim H 0 (Yi , E|Yi (n)). If n > −Bi − µS (E), then the right hand side of the above inequality is less than a0 (E) (n + Bi + µS (E))i . Thus summing up these inequalities from or equal to i! n = −Bi to n = m and using the fact that H 0 (Yi+1 , E|Yi+1 (−Bi − 1)) = 0, we obtain m X a0 (E) dim H (Yi+1 , E|Yi+1 (m)) ≤ (n + Bi + µS (E))i i! 0

n=−Bi m X

a0 (E) (n + Bi + 1)i i! n=−Bi Z a0 (E) m+1 ≤ (x + Bi + 1)i dx i! −Bi −1 a0 (E) = (m + Bi + 2)i+1 . (i + 1)! ≤

Setting Bi+1 = Bi +2, we have that for every E ∈ F e (d, a0 )1 and every m > −Bi+1 , dim H 0 (Yi+1 , E|Yi+1 (m)) ≤

a0 (E) (m + Bi+1 + µS (E))i+1 (i + 1)!

and that for every m ≤ −Bi+1 , dim H 0 (Yi+1 , E|Yi+1 (m)) = 0. By induction on i we complete our proof. Let f : X → S and OX (1) be the same as in Lemma 4.2. For positive integers a0 , d and integers µ, b with b ≥ µ, we set G b (d, a0 , µ) to be the family of the classes of coherent sheaves E on the fibers of X over S with the properties (4.1.1), (4.1.2) and the properties

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(4.3.1) µS (E) ≤ µ. (4.3.2) If E is on a geometric fiber Xs , D1 , . . . , Dd−1 are general members of |OX (1)(s)| and if Y is the scheme theoretic intersection D1 ∩ · · · ∩ Dd−1 , then E|Y is of pure dimension 1 and for every coherent subsheaf F of E|Y with 0 < a0 (F ) < a0 (E), we have µS (F ) ≤ b. Lemma 4.4. There is an integer B, a non-negative integer C and a polynomial Φ(m) of degree ≤ d − 2 such that if E is on a geometric fiber Xs and a member of G b (d, a0 , µ), then (4.4.1) for every integer m with m ≥ −B − µ, we have a0 (E) dim H (Xs , E(m)) ≤ d! 0



µ + (a0 (E) − 1)b m+B+ a0 (E)

d + Φ(m),

(4.4.2) for every integer m with −B − b ≤ m ≤ −B − µ, we have dim H 0 (Xs , E(m)) ≤ C, (4.4.3) for every integer m with m ≤ −B − b, we have dim H 0 (Xs , E(m)) = 0. (4.4.4) for all real numbers x ≥ −B − µ, dΦ(x) ≥ 0 and Φ(−B − µ) ≥ 0. dx Moreover, B does not depend on µ.

Proof. Our proof is by induction on d. Pick a coherent sheaf E on a geometric fiber Xs which is a member of G b (d, a0 , µ). Assume first that d = 1. Let 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fα1 ⊂ Fα = E be the Harder-Narasimhan filtration of E and set Qi = Fi /Fi−1 . Since each Qi is semi-stable, Lemma 4.2 implies that there is an integer B1 which is independent of µ such that for every m > −B1 − µS (Qi ), dim H 0 (Xs , Qi (m)) ≤ a0 (Qi )(m + B10 + µS (Qi )) and for m ≤ −B1 − µS (Qi ), H 0 (Xs , Qi (m)) = 0. By the properties (4.3.1), (4.3.2) and by the definition of Harder-Narasimhan filtration, we see that µS (Qα ) ≤ µS (E) ≤ µ and µS (Qα ) < · · · < µS (Q2 ) < µS (Q1 ) = µS (F1 ) ≤ b. Thus the above

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107

implies dim H 0 (Xs , E(m)) ≤

α X

dim H 0 (Xs , Qi (m))

i=1

≤

α X

a0 (Qi ) max{m + B1 + µS (Qi ), 0}

i=1

≤

α−1 X

a0 (Qi ) max{m + B1 + b, 0}+

i=1

(a0 (Qα ) − 1) max{m + B1 + b, 0} + max{m + B1 + µ, 0} = (a0 (E) − 1) max{m + B1 + b, 0} + max{m + B1 + µ, 0}. If m ≥ −B1 − µ ≥ −B1 − b, then the above means   µ + (a0 (E) − 1)b . dim H 0 (Xs , E(m)) ≤ a0 (E) m + B1 + a0 (E) When m is between −B1 − b and −B1 − µ, we see dim H 0 (Xs , E(m)) ≤ (a0 (E) − 1)(b − µ) ≤ (a0 − 1)(b − µ) = C1 . Finally if m ≤ −B1 − b, then dim H 0 (Xs , E(m)) = 0. This completes the proof for d = 1. Assume next that d > 1 and our assertion is true for d−1. Let D = D1 , . . . , Dd−1 be general members of |OX (1)(s)| which have the property (4.3.2) for E. We may assume that E|D is of pure dimension d−1. Then E|D is a member of G b (d−1, a0 , µ). By our induction hypothesis we have Bd−1 , Cd−1 and Φd−1 (m) which satisfy the conditions (4.4.1), (4.4.2), (4.4.3) and (4.4.4) for E = E|D , B = Bd−1 , C = Cd−1 and Φ(m) = Φd−1 (m). We may also assume that Bd−1 is independent of µ. The exact sequence 0 −→ E(m − 1) −→ E(m) −→ E|D −→ 0. supplies us with the inequality dim H 0 (Xs , E(m)) − dim H 0 (Xs , E(m − 1)) ≤ dim H 0 (Xs , E|D (m)). Since H 0 (Xs , E|D (m)) = 0 for all m ≤ −Bd−1 − b and since H 0 (Xs , E(m)) = 0 for all sufficiently small m, we see that for all m ≤ −Bd−1 −b, H 0 (Xs , E(m)) = 0. Then, using this fact and (4.4.2) for E|D , we have that if −Bd−1 − b ≤ m ≤ −Bd−1 − µ, then dim H 0 (Xs , E(m)) ≤ Cd−1 (b − µ). The above inequalities between dimensions of cohomologies and (4.4.2) for E|D show that for all m ≥ −Bd−1 − µ, dim H 0 (Xs , E(m)) − dim H 0 (Xs , E(m − 1))  d−1 µ + (a0 (E) − 1)b a0 (E) m + Bd−1 + + Φd−1 (m). ≤ (d − 1)! a0 (E)

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Summing up this from −Bd−1 − b + 1 to m and using (4.4.2) for E|D again, we see the left hand side is dim H 0 (Xs , E(m)) and the right side is bounded from above by R = Cd−1 (b − µ)+ !  d−1 Z m+1 a0 (E) µ + (a0 (E) − 1)b x + Bd−1 + + Φd−1 (x) dx (d − 1)! a0 (E) −Bd−1 −µ because Φd−1 (x) is an increasing function on the interval of integration by (4.4.4) for Φd−1 (x). Since R=

 d µ + (a0 (E) − 1)b m + 1 + Bd−1 + + a0 (E)  d Z m+1 a0 (E) (a0 (E) − 1)(b − µ) + Φd−1 (x)dx, Cd−1 (b − µ) − d! a0 (E) −Bd−1 −µ

a0 (E) d!

we can find Φ(x) such that Φ(−Bd−1 − 1 − µ) ≥ 0 and dΦ(x)/dx = Φd−1 (x), by adding a suitable constant to Z

m+1

Φd−1 (x)dx. −Bd−1 −µ

By (4.4.4) for Φd−1 (x), Φd−1 (x + 1) must be non-negative if x ≥ −Bd−1 − 1 − µ. Therefore, B = Bd−1 + 1, C = Cd−1 (b − µ) and Φ(x) are integers and a polynomial we want. Thanks to the above lemma we can prove easily a fundamental lemma which connects semi-stable sheaves to semi-stable points of a Quot-scheme. Lemma 4.5. Let a0 and d be positive integers, µ and b be integers with b ≥ µ and let P (m) be a polynomial of degree d with rational coefficients whose leading term is md /d!. Then there are integers M and L such that for all members E of G b (d, a0 , M ) and for all integers m ≥ L we have dim H 0 (Xs , E(m)) < a0 (E)P (m), where E is on a fiber Xs of X over S. Proof. Let B be the integer given in Lemma 4.4 for the integers d, a0 and e. Let us write down the given polynomial P (m) in the form P (m) =

md md−1 + a1 + (terms of degree ≤ n − 2). d! (d − 1)!

Take an integer M such that M ≤ b and B+

M + (a − 1) < a1 for all integers a with 0 < a ≤ a0 . a

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109

Then there is an integer L ≥ −B − M such that for all integers m ≥ L and a with 0 < a ≤ a0 ,  d 1 M + (a − 1) Φ(m) m+B+ + < P (m). d! a a Pick a member E of G b (d, a0 , M ). Then by Lemma 4.4 we know that for every integer m > −B − M ,  d a0 (E) M + (a0 (E) − 1) 0 dim H (Xs , E(m)) ≤ + Φ(m), m+B+ d! a0 (E) Combining these, we see that if m is an integer such that m ≥ L ≥ −B − M , then  d a0 (E) M + (a0 (E) − 1) 0 dim H (Xs , E(m)) ≤ m+B+ + Φ(m) d! a0 (E) < a0 (E)P (m) as required. In this chapter, we shall fix, from now on, the following situation: Let S be a scheme of finite type over a universally Japanese ring Ξ, f : X → S a projective morphism and let OX (1) be an f -very ample invertible sheaf such (4.6) that X is embedded in P(U ) with U a locally free coherent sheaf on S, OX (1) is the restriction of OP(U ) (1) to X and that for all points s ∈ S and all integers i > 0, H i (Xs , OX (1)(s)) = 0. Since a coherent sheaf is stable (or, semi-stable) with respect to an ample invertible sheaf if and only if so is with respect to a positive power of the invertible sheaf, we will exploit the very ampleness and the vanishing of the cohomologies only to construct the moduli spaces and for the existence of the moduli spaces the ampleness of OX (1) is enough. Definition 4.7. Let (Sch/S) be the category of locally noetherian S-schemes, H(x) a numerical polynomial of degree d and let e be a positive integer. For an object T of (Sch/S) we define four sets as follows:   E is a T -flat coherent sheaf on X ×S T with . (1) ΣH (T ) = E ∼ X/S the property (4.7.1) where ∼ is the equivalence relation defined by (4.7.5).   E is a T -flat coherent sheaf on X ×S T with . H ¯ (2) ΣX/S (T ) = E ∼ the property (4.7.2) where ∼ is the equivalence relation defined by (4.7.6). The equivalence class of E is denoted by [E].   E is a T -flat coherent sheaf on X ×S T with . (3) ΣH,e (T ) = E ∼ X/S the property (4.7.3) where ∼ is the equivalence relation defined by (4.7.5).

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¯ H,e (T ) = (4) Σ X/S



CONSTRUCTION OF MODULI SPACES

 E is a T -flat coherent sheaf on X ×S T with . ∼ E the property (4.7.4)

where ∼ is the equivalence relation defined by (4.7.6). (4.7.1) For every geometric point t of T , E(t) is stable and its Hilbert polynomial is H(m). (4.7.2) For every geometric point t of T , E(t) is semi-stable and its Hilbert polynomial is H(m). (4.7.3) For every geometric point t of T , E(t) is e-stable and its Hilbert polynomial is H(m). (4.7.4) For every geometric point t of T , E(t) is strictly e-semi-stable and its Hilbert polynomial is H(m). ∼ (4.7.5) E ∼ E 0 if and only if there is an invertible sheaf L on T such that E = E 0 ⊗OT L. (4.7.6) E ∼ E 0 if and only if for some invertible sheaf L on T , (1) E ∼ = E 0 ⊗O T L or (2) there exist filtrations 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα = E and 0 = E00 ⊂ E10 ⊂ · · · ⊂ Eα0 = E 0 by coherent subsheaves such that ⊕α i=1 Ei /Ei−1 α 0 0 ∼ is flat over T , ⊕α E /E (⊕ E /E ) ⊗ L and that for every = i−1 OT i=1 i i=1 i i−1 geometric point t of T , 0 = E0 (t) ⊂ E1 (t) ⊂ · · · ⊂ Eα (t) = E(t) is a Seshadri filtration of E(t). If g : T 0 → T is a morphism in (Sch/S), then the above properties and the equivalence relations are preserved by making the pull back (1 ×S g)∗ (E). Thus all the H,e ¯H ¯ H,e ΣH X/S , ΣX/S , ΣX/S and ΣX/S define contravariant functors of (Sch/S) to (Sets). For convenience sake let us introduce a family of semi-stable sheaves. Definition 4.8. Let the pair (X, OX (1)) be the same as in (4.6), H(x) a numer¯ X/S (H), ical polynomial of degree d and let e be a positive integer. SX/S (H) (S ¯ X/S (e, H)) denotes the family of classes of coherent sheaves on the SX/S (e, H) or S fibers of X over S such that a coherent sheaf E on a geometric fiber is in SX/S (H) ¯ X/S (H), SX/S (e, H) or S ¯ X/S (e, H), resp.) if and only if it is stable (semi-stable, (S e-stable or strictly e-semi-stable, resp.) and the Hilbert polynomial of E is H. By the argument right before Definition 3.3 and Lemma 3.4 we get directly the following. Lemma 4.9. For every numerical polynomial H, SX/S (H) = ¯ X/S (H) = S S ¯ S e X/S (e, H).

S

e

SX/S (e, H) and

When s runs over all the geometric points of S, the set of equivalence classes of ¯ X/S (e, H). ¯ H,e (Spec(k(s))) covers the S E with [E] ∈ Σ X/S Proposition 4.10. Let H(x) be a numerical polynomial of degree d. There exists ¯ X/S (e, H), the following holds: an integer m0 such that for every E ∈ S

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111

(4.10.1) For every integer m ≥ m0 and every coherent subsheaf F of E with 0 6= F , dim H 0 (Xs , F (m))/a0 (F ) ≤ dim H 0 (Xs , E(m))/a0 (E). Moreover, the equality holds if and only if PFS (n) = PES (n) as polynomials in n. Proof. We can write H(m) in the form d X

  m+i ad−i (H) . i i=0

¯ X/S (e, H) by ProposiPut P (m) = H(m)/a0 (H). Because of the boundedness of S ¯ X/S (e, H), every integer tion 3.8, there is an integer m1 such that for every E ∈ S i m ≥ m1 and every integer i > 0, we have H (Xs , E(m)) = 0 and hence dim H 0 (Xs , E(m))/a0 (E) = PES (m) = H(m)/a0 (H) = P (m). Let F be the family of the classes of coherent sheaves on the fibers of X over S such ¯ X/S (e, H). that F on Xs is in F if and only if F is a non-zero subsheaf of an E ∈ S Put µ = a1 (H)/a0 (H) and b = µ + e. Since every member F of F is a subsheaf of a e-semi-stable sheaf E on a geometric fiber Xs with µS (E) = µ and since if D1 , . . . , Dd−1 is sufficiently general members of |OX (1)(s)|, the restriction of F to Y = D1 ∩ · · · ∩ Dd−1 is a subsheaf of E|Y , we see that F has the properties (4.3.1) and (4.3.2) for the above µ and b. F obviously enjoys the properties (4.1.1) and (4.1.2). Note here that if a0 (F ) = a0 (E), then a1 (F ) must be less than or equal to a1 (E) because for sufficiently large integers m, χ(F (m)) = dim H 0 (Xs , F (m)) ≤ dim H 0 (Xs , E(m)) = χ(E(m)). What we have seen in the above is F is a subfamily of G b (d, a0 , µ). Applying Lemma 4.5 to the case where a0 = a0 (H), d = d, b = b and P (m) = H(m)/a0 (H), we have integers M and L such that for every member F of FM = {F ∈ F | µS (F ) ≤ M } and for every integer m ≥ L, we have the inequality dim H 0 (Xs , F (m))/a0 (F ) < P (m). On the other hand, the subfamily F 0 = {F ∈ F | µS (F ) > M and F is a subsheaf ¯ X/S (e, H) such that E/F is of pure dimension d} is bounded by Proposiof E ∈ S tion I.3.12 and the boundedness of e-semi-stable sheaves. This means that the set {χ(F (m)) | F ∈ F 0 } is finite and for sufficiently large integers m and F ∈ F 0 , the higher cohomologies of F (m) vanish. Hence there is an integer m2 such that for every integer m ≥ m2 and for every F ∈ F 0 , F (m) is generated by global sections and dim H 0 (Xs , F (m))/a0 (F ) = χ(F (m))/a0 (F ) = PFS (m) ≤ PES (m) = P (m),

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CONSTRUCTION OF MODULI SPACES

¯ X/S (e, H) and the equality holds if and only where F is a subsheaf of a E ∈ S S S if PF (m) = PE (m) as polynomials in m. If F is not in FM and a subsheaf of ¯ X/S (e, H) , then there is a subsheaf F 0 of E such that F ⊂ F 0 , a0 (F 0 ) = a0 (F ) E∈S and F 0 is in F 0 . Thus for m ≥ m2 , we have dim H 0 (Xs , F (m))/a0 (F ) ≤ dim H 0 (Xs , F 0 (m))/a0 (F 0 ) and the equality holds if and only if F = F 0 because F 0 (m) is generated by global sections. Combining all the above together, we see m0 = max{m1 , L, m2 } is a desired integer. We shall fix a numerical polynomial H(m) of degree d and write down it in the form   d X m+i ad−i (H) H(m) = i i=0 with ai (H) integers. For an integer r with 1 ≤ r ≤ a0 (H), we denote rH(m)/a0 (H) by H (r) (m). By the boundedness of SX/S (e, H (r) ) and Proposition 4.10, there is an integer m(r, e) such that for all integers m ≥ m(r, e) and all E ∈ SX/S (e, H (r) ) on a geometric fiber Xs , we have (4.11.1) E(m) is generated by global sections and for all j > 0, H j (Xs , E(m)) = 0, (4.11.2) for all coherent subsheaves F with 0 6= F 6= E, dim H 0 (Xs , F (m))/a0 (F ) < dim H 0 (Xs , E(m))/r. (If SX/S (e, H (r) ) is empty, then we understand m(r, e) = 0.) Lemma 4.12. If m ≥

max

¯ X/S (e, H (r) ) on geo{m(r, e)}, then for all E ∈ S

1≤r≤a0 (H)

metric fibers Xs , we have (4.12.1) E(m) is generated by global sections and H j (Xs , E(m)) = 0 for all j > 0. (4.12.2) for all non-zero coherent subsheaves F of E, dim H 0 (Xs , F (m))/a0 (F ) ≤ dim H 0 (Xs , E(m))/r. and moreover the equality holds if and only if PFS (x) = PES (x) =

H(x) a0 (H)

as polynomials in x. Proof. Take a Seshadri filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ Eα−1 ⊂ Eα = E of E. It is easy to prove (4.12.1) by induction on α if we use (4.11.1). We shall prove (4.12.2) by induction on α. Let i be the smallest integer such that F ⊂ Ei . If i < α, then

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113

F is a subsheaf of Eα−1 . Since Eα−1 is strictly e-semi-stable by Lemma 3.5, our induction hypothesis implies that dim H 0 (Xs , F (m)) dim H 0 (Xs , Eα−1 (m)) ≤ a0 (F ) a0 (Eα−1 ) and the equality holds if and only if PFS (x) = PESα−1 (x) = PES (x). On the other hand, by (4.12.1) for E and Eα−1 we know that dim H 0 (Xs , E(m)) dim H 0 (Xs , Eα−1 (m)) = PESα−1 (m) = PES (m) = . a0 (Eα−1 ) r We may assume therefore that i = α. Set F 0 = F ∩ Eα−1 and F 00 = F/F 0 . Then F 00 is a non-zero subsheaf of E 00 = E/Eα−1 . If F 0 = 0, then dim H 0 (Xs , F 00 (m)) dim H 0 (Xs , F (m)) = a0 (F ) a0 (F 00 ) 0 dim H (Xs , E 00 (m)) dim H 0 (Xs , E(m)) ≤ = a0 (E 00 ) r because of (4.11.1) and (4.11.2) for E 0 and (4.12.1) for E. Moreover, the equality holds if and only if F 00 = E 00 , that is, PES (x) = PES00 (x) = PFS00 (x) = PFS (x). Assume that F 0 6= 0. Then, by our induction hypothesis we see dim H 0 (Xs , F (m)) ≤ dim H 0 (Xs , F 0 (m)) + dim H 0 (Xs , F 00 (m)) ≤ a0 (F 0 )

dim H 0 (Xs , E 00 (m)) dim H 0 (Xs , Eα−1 (m)) + a0 (F 00 ) a0 (Eα−1 ) a0 (E 00 )

= a0 (F 0 )PESα−1 (m) + a0 (F 00 )PES00 (m) = a0 (F )

dim H 0 (Xs , E(m)) . r

If the equality holds, then PFS0 (x) = PESα−1 (x) = PES (x) and PFS00 (x) = PES00 (x) = PES (x) and hence PFS (x) = PES (x). Conversely, if PFS (x) = PES (x), then the semistability of Eα−1 and E 00 implies that PFS0 (x) = PFS00 (x) = PES (x). This shows that F 0 is a strictly e-semi-stable sheaf with PFS0 (x) = PES (x). Then by (4.12.1) for F 0 we have H 1 (Xs , F 0 (m)) = 0 and then the equality holds in the above. Fix an integer m with m≥

max

{ m(r, e) }.

1≤r≤a0 (H)

Let H (r) [m] = H (r) [m](x) be the polynomial H (r) (x + m). H (r) [m] is the Hilbert ¯ X/S (e, H (r) )(m) = {E(m) | E ∈ S ¯ X/S (e, H (r) )}. We polynomial of members of S are going to consider only r’s for which H (r) (x) are numerical polynomials. Set Nr = H (r) [m](0) = H (r) (m). Then dim H 0 (Xs , E(m)) = Nr if E on a geometric ¯ X/S (e, H (r) ). fiber Xs is in S

114

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We fix a free Ξ-module Vr of rank Nr . If Y (or, A) is a Ξ-scheme (Ξ-algebra, resp.), then Vr (Y ) (Vr (A), resp.) denotes Vr ⊗Ξ OY (Vr ⊗Ξ A, resp.). Let us consider the Quot-scheme H (r) [m]

Qr = QuotVr (X)/X/S and the universal quotient sheaf ϕr : Vr (X ×S Qr ) → F˜r . Let Gr be the S-group scheme GL(Vr (S)). Using affine system of coordinate of Gr over s with respect to a basis z1 , . . . , zNr of Vr , we define an isomorphism θ of Vr (Gr ) to itself by sending zi PNr zj ⊗ xji . If p1 : Gr ×S X ×S Qr → Gr and p2 : Gr ×S X ×S Qr → X ×S Qr to j=1 are the projections, then we can compose two homomorphisms p∗ (θ)

1 Vr (Gr ×S X ×S Qr ) ∼ = Vr (Gr ×S X ×S Qr ), = p∗1 (Vr (Gr )) −−−−→ p∗1 (Vr (Gr )) ∼

p∗ (ϕr )

2 Vr (Gr ×S X ×S Qr ) ∼ = p∗2 (Vr (X ×S Qr )) −−−−→ p∗2 (F˜r )

to get a surjection ψ of Vr (Gr ×S X ×S Qr ) to p∗2 (F˜r ). By the universality of (Qr , ϕr , F˜r ) we have a morphism σ : Gr × Qr → Qr and an isomorphism ρ : ∗ ˜ (Fr ) → p∗2 (F˜r ) such that the diagram σX σ ∗ (ϕr )

∗ ∗ ˜ (Vr (X ×S Qr )) −−X−−−→ σX (Fr ) σX 



ρyo

ψ Vr (Gr ×S X ×S Qr ) −−−−→ p∗2 (F˜r )

is commutative. It is not hard to see that σ is an action of Gr on Qr and ρ is a Gr linearization on F˜r with respect to the trivial action of Gr on X. Moreover, since the action of the center of Gr induces the multiplication by units on F˜r , it is canceled by the multiplication by the inverse of the units, which means that the center acts ¯ r = P GL(Vr (S)) on Qr trivially on Qr . Consequently we have a natural action of G ˜ and a Gr -linearization on Fr . By Proposition 3.7, for each integer e0 with 0 ≤ e0 ≤ e, there exists an open 0 0 subscheme Rre,e of Qr such that a geometric point y of Qr is in Rre,e if and only if the following two conditions are satisfied: (4.13.1) Γ(ϕr ⊗ k(y)) : Vr (k(y)) → H 0 (Xy , F˜r (y)) is bijective. (4.13.2) F˜r (y) is strictly e0 -semi-stable. 0

¯ r -stable open set. For every geometric point s of It is clear that Rre,e is an G ¯ X/S (e0 , H (r) )(m), there is a surjective S and every coherent sheaf E on Xs in S homomorphism α : Vr (Xs ) → E such that Γ(α) : Vr (k(s)) → H 0 (Xs , E) is bijective by (4.12.1). By the universality of (Qr , ϕr , F˜r ), α corresponds to a geometric point 0 0 y of Qi . Since y is a geometric point of Rre,e , we obtain a surjective map ξre,e (s) for every geometric point s of S:

4

GENERAL SETTING AND A FUNDAMENTAL LEMMA

0

(4.13.3)

0

(r)

115

0

¯ H ,e (m)(Spec(k(s))) ξre,e (s) : Rre,e (k(s)) → Σ X/S (r)

0

¯ H ,e (Spec(k(s)))}. = {[E(m)] | [E] ∈ Σ X/S

0

On the other hand, if for two K-valued geometric points y1 , y2 of Rre,e over the same K-valued point s of S, F˜r (y1 ) is isomorphic to F˜r (y2 ), then (4.13.1) provides us with an isomorphism ∼ ∼ ∼ Vr (K) −−−−→ H 0 (Xs , F˜r (y1 )) −−−−→ H 0 (Xs , F˜r (y2 )) ←−−−− Vr (K) which gives rise to a K-valued geometric point g of Gr . By the definition of the action of Gr on Qr we see that σ(K)(g, y2 ) = y1 . Conversely, if two K-valued 0 geometric points y1 , y2 of Rre,e are in the same Gr (K)-orbit, then F˜r (y1 ) ∼ = F˜r (y2 ). Therefore, we get (4.13.4) for every K-valued geometric point s of S, the set of isomorphism classes of strictly e0 -semi-stable sheaves on Xs with Hilbert polynomial H (r) is in 0 bijective correspondence with the set of (Gr )s (K)-orbits of (Rre,e )s (K). Since Vr (X) is a quotient sheaf of Vr (P(U )), Remark 1.7 shows that for sufficiently large integer n, Qr is embedded in Zr (n) = GrassH (r) [m](n) (h∗ (Vr (P(U ))(n))), where h : P(U ) → S is the structure morphism. Fix a sufficiently large n and set W to be the locally free coherent sheaf h∗ (OP(U ) (n)) on S. Then h∗ (Vr (P(U ))(n)) is isomorphic to Vr ⊗ W = Vr ⊗Ξ W . To construct the moduli spaces of semi-stable sheaves we may assume that W is of constant rank, for example, by replacing S by a connected component of S. On one hand, the Gr -linearization on F˜r defines that on (fQr )∗ (F˜r (n)). On the other hand, the action of Gr on Vr (S) induces that on h∗ (Vr (P(U ))(n)) ∼ = Vr ⊗ W , which is the same action defined right before Proposition 2.12. We know moreover that det((fQr )∗ (F˜r (n))) is the restriction of the line bundle OZr (n) (1) on Zr (n) defining the Pl¨ ucker embedding to Qr (cf. Remark 1.7). There is a positive integer a such that Lr = (det((fQr )∗ (F˜r (n))))⊗a ¯ r -linearization (see Proposition 2.12). Combining all together, we see carries a G that ¯ r -equivariant and the Gr -linearized (4.13.5) the embedding of Qr into Zr (n) is G invertible sheaf Lr is the restriction of the Gr -linearized OZr (n) (a) to Qr . Notation 4.14. In the above notation, if r = a0 (H), then we omit r in the notation. Hence in the case of r = a0 (H) we use the notation H[m], N , V , Q, G, ¯ Re,e0 , Z(n) and L instead of H (r) [m], Nr , Vr , Qr , Gr , ϕr , F˜r , G ¯ r , Rre,e0 , ϕ, F˜ , G, Zr (n) and Lr , respectively.

116

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CONSTRUCTION OF MODULI SPACES

5 Moduli spaces of stable sheaves In this section we shall construct the coarse moduli scheme of the functor ΣH X/S under the situation of (4.6). Let us start with the following lemma. 0

Lemma 5.1. There is an integer n0 such that for all n ≥ n0 , Rre,e is embedded in Zr (n)ss (OZr (n) (a)) (for the notation see Proposition 2.8). A geometric point y of 0 Rre,e is in Zr (n)s0 (OZr (n) (a)) if and only if F˜r (y) is stable. Proof. Our proof is independent of r and hence we shall prove for r = a0 (H). First there is a positive integer n1 such that for all n ≥ n1 , Qr is embedded in Zr (n). 0 Pick a geometric point y of Re,e over a geometric point s of S and set E = F˜ (y). Take a vector subspace V 0 of V (k(y)) and let E 0 = E 0 (V 0 ) be the coherent subsheaf of E generated by Γ(ϕ)(V 0 ). Then, by(4.13.1) V 0 is a subspace of H 0 (Xs , E 0 ). The 0 family F = {E 0 (V 0 ) | V 0 ⊂ V (k(y)), y runs geometric points of Re,e } is bounded. Thus, by (4.12.2) there is a positive rational number  < 1 such that if E 0 ∈ F and PES0 (x) 6= H[m](x)/a0 (H) as polynomials in x, then we have dim H 0 (Xs , E 0 )
(1 − ). H[m](n) a0 (H) Now if n ≥ n0 = max{n1 , n2 , n3 }, then for W = f∗ (OX (n)), the image U 0 of V 0 ⊗ W (s) to U = H 0 (Xy , E(n)) is exactly H 0 (Xy , E 0 (n)) and its dimension is χ(E 0 (n)). If PES0 (x) 6= H[m](x)/a0 (H), then our computation proceeds as follows: dim V 0 ≤ dim H 0 (Xs , E 0 ) a0 (E 0 ) H(m)(1 − ) a0 (E) a0 (E 0 ) = dim H 0 (Xs , E)(1 − ) a0 (E) χ(E 0 (n)) < dim H 0 (Xs , E) H[m](n) dim U 0 = dim V (k(y)). dim U
1 − H(m) ≥ H(m) ≥ 0. a0 (H) a0 (H) Since dim V 0 and a0 (F ) are non-negative bounded integers, there is a positive number ε such that a0 (F ) H(m) > ε. dim V 0 − a0 (H) Set δ to be a positive number such that for all F δH(m)

a0 (F ) ≤ ε. a0 (H)

Since {χ(F (m)) | F ∈ B} is a finite set, there is an integer n0 such that for all integers n ≥ n0 and F ∈ B, we have a0 (H)PFS (n) ≤1+δ H[m](n) χ(F (n)) H(m) < dim V 0 H[m](n)

or

according to a0 (F ) 6= 0 or a0 (F ) = 0. Multiplying the first inequality by a0 (F )H(m) , a0 (H) we obtain

χ(F (n)) a0 (F ) H(m) ≤ H(m) + ε < dim V 0 . H[m](n) a0 (H)

Thus we get the following inequality for all F ∈ B and n ≥ n0 : χ(F (n))
βi0 (F2 ) and βi0 (F1 ) = γi0 (F1 ), then HomOXk (F1 , F2 ) = 0. Proof. If we take a coherent subsheaf F 0 of F , then we have aj (F 0 ) = aj (F 0 ∩ F1 ) + aj (F 0 /F 0 ∩ F1 ) = a0 (F 0 ∩ F1 )bj (F 0 ∩ F1 ) + a0 (F 0 /F 0 ∩ F1 )bj (F 0 /F 0 ∩ F1 ). We first prove (2). Set βi (Fj ) = (bj1 , . . . , bji , aj0 ). Since βi0 (F1 ) < βi0 (F2 ) by assumption, there is an ` ≤ i such that b1j = b2j for 1 ≤ j < ` and b1` < b2` . Suppose that F 0 ∩ F1 6= 0. There exists a p ≥ 1 such that bj (F 0 ∩ F1 ) = b1j and bj (F 0 /F 0 ∩ F1 ) = b2j for j < p, and bp (F 0 ∩ F1 ) < b1p or bp (F 0 /F 0 ∩ F1 ) < b2p If p < `, then ap (F 0 ) < a0 (F 0 ∩ F1 )b1p + a0 (F 0 /F 0 ∩ F1 )b2p = a0 (F 0 )b2p . This contradicts the choice of F 0 . If p ≥ `, then a` (F 0 ) ≤ a0 (F 0 ∩ F1 )b1` + a0 (F 0 /F 0 ∩ F1 )b2` < a0 (F 0 ∩ F1 )b2` + a0 (F 0 /F 0 ∩ F1 )b2` = a0 (F 0 )b2j . This also contradicts the choice of F 0 . Hence F 0 ∩ F1 = 0, which proves (2). By similar computations we have easily (1). To prove (3) we shall assume βi0 (F1 ) ≤ βi0 (F2 ) contrary to our conclusion. There exists a coherent subsheaf F 0 of F2 with γi0 (F 0 ) = βi0 (F2 ). If F 00 is the inverse image of F 0 to F , then the computation in the above shows that γi0 (F 00 ) ≥ γi0 (F1 ) because γi0 (F 0 ) = βi0 (F2 ) ≥ βi0 (F1 ) = γi0 (F1 ). Since γi0 (F1 ) = βi0 (F ), the only possibility is γi0 (F 00 ) = γi0 (F1 ), which implies however γi (F 00 ) > γi (F1 ). This violates our assumption. (4) Let τ be a homomorphism of F1 to F2 . The inequality γi0 (ker(τ )) ≤ βi0 (F1 ) implies γi0 (im(τ )) ≥ βi0 (F1 ) > βi0 (F2 ). This is not the case by definition. ˜ of G(E) ¯ and assume that E (0) = E ˜k is not semi-stable. Let ` Pick a member E 0 (0) 0 (0) ¯ be the integer min{i | βi (E ⊗ k) > γi (E )}, where k¯ is an algebraic closure of ¯ and k. We have a coherent subsheaf F of E (0) ⊗ k¯ such that γ` (F ) = β` (E (0) ⊗ k) ¯ E (0) ⊗ k/F is of pure dimension d. By using (3) and (4) of Lemma 8.3 and applying the same method as in the proof of Proposition I.2.11, we can find a coherent ¯ This means subsheaf F (0) of E (0) such that F (0) ⊗ k¯ = F as subsheaves of E (0) ⊗ k.

140

Chapter 3

CONSTRUCTION OF MODULI SPACES

¯ = β` (E (0) ). Moreover, we see that a coherent subsheaf F of E (0) that β` (E (0) ⊗ k) such that γ` (F ) = β` (E (0) ) and H (0) = E (0) /F is of pure dimension d is unique. Roughly speaking, F (0) is the first filter of the Harder-Narasimhan filtration of E (0) defined by using the first ` + 1 coefficients of the Hilbert polynomials. We call this F (0) the β` -subsheaf of E (0) . Define θ(0) to be the surjection ˜ → E (0) → H (0) . E Let t be a uniformanizing parameter of A. Then t = 0 defines the effective Cartier ˜ means that Ass(E) ˜ ∩ Xk = ∅. By Lemma divisor Xk on XT and the T -flatness of E (1) (0) ˜ ˜ (1) contains 8.1 E = ker(θ ) is a coherent sheaf of pure dimension d+1 on XT , E (1) (1) (0) ˜ (1) ∼ ¯ ˜ , there is an exact sequence tE , EK = E and for E = E k 2 0 −→ H (0) ⊗OXk IXk /IX −→ E (1) −→ F (0) −→ 0. k 2 2 ∼ Moreover, the image of tE (0) in E (1) is H (0) ⊗OXk IXk /IX . Since IXk /IX = OXk , k k the above exact sequence becomes

0 −→ H (0) −→ E (1) −→ F (0) −→ 0 ˜ (1) is a member of G(E). ¯ and then we see that E (1) is of pure dimension d. Thus E 0 (1) 0 (0) The above exact sequence and Lemma 8.3, (1) show that β` (E ) ≤ β` (F ) = γ`0 (F (0) ) = β`0 (E (0) ). If β`0 (E (1) ) = β`0 (E (0) ) and F 0 is a coherent subsheaf of E (1) with γ`0 (F 0 ) = β`0 (E (1) ), then F 0 can be regarded as a subsheaf of F (0) by Lemma 8.3, (2). Combining these, we get that β` (E (1) ) ≤ β` (E (0) ). Assume that β` (E (1) ) = β` (E (0) ) and let F (1) be the β` -subsheaf of E (1) . By Lemma 8.3, (2) and the above exact sequence, F (1) ∩ H (0) = 0 in E (1) and hence we have the following exact commutative diagram: 0H

HH j

*0  

F (1) ,−−−→ F (0) *   HH j (1) H *E H  j (1) H (0) ,−−−→ H  H * HH  j H  0 0

˜ (1) to get a member E ˜ (2) of Apply the elementary transformation along H (1) to E (2) (2) (1) (2) (2) ¯ and assume that β` (E ) = β` (E ) for E = E ˜ . Defining F G(E) and H (2) k as F (1) and H (1) , we obtain injections F (2) ,→ F (1) ,→ F (0) and H (0) ,→ H (1) ,→ H (2) . Performing the elementary transformations along the β` -subsheaf repeatedly, ˜ E ˜ (1) , . . . , E ˜ (i) of G(E) ¯ and assume that β` (E (i) ) = β` (E (0) ). we get members E,

8

LANGTON’S THEOREM

141

˜ (i+1) of G(E) ¯ by the elementary transformation Then we have another member E along the β` -subsheaf and the sequences of injections F (i+1) ,→ F (i) ,→ · · · ,→ F (1) ,→ F (0) , H (0) ,→ H (1) ,→ · · · ,→ H (i) ,→ H (i+1) . ˜ 0 of G(E) ¯ with β` (E ˜ 0 ) < β` (E ¯ (0) ) after finite steps If we cannot reach a member E k of elementary transformations, the above sequences of injections extend to infinite sequences. Since γ` (F (i) ) = β` (E (i) ) = β` (E (0) ) = γ` (F (0) ) and γ` (E (i) ) = ¯ = γ` (E (0) ), we have γ` (H (i) ) = γ` (H (0) ), which means that H (i) is isomorγ` (E) phic to H (0) in codimension ` in Supp(E (0) ) = Supp(E (i) ). Let Zi be the closed S set Supp(H (i) /H (0) ) and set Z = i≥1 Zi . Then Z is closed under specializations. 0 Applying [EGA, IV Proposition 5.11.1] to H = HX (H (0) ) 5 , we see that H is k /Z coherent. By [EGA, IV Proposition 5.10.2] there is a natural injection of H (i) to 0 H because H (i) is of pure dimension and H = HX (H (i) ). Thus we come to an k /Z infinite sequence of coherent sheaves H (0) ,→ H (1) ,→ · · · ,→ H (i) ,→ · · · ,→ H. The coherency of H implies that this sequence is stable after sufficiently large i, that is, there is an i such that for all j ≥ i, H (j) ∼ = H (i) . Since the Hilbert polynomial (j) of E is always H[m], the Hilbert polynomial of F (j) is independent of j ≥ i. ˜ (i) This implies that the injection F (i) ,→ F (j) must be isomorphic. Starting from E ˜ we may assume that for all i ≥ 0, the injections F (i+1) → F (i) and instead of E, H (i) → H (i+1) are isomorphism. Let Aˆ be the completion of A. Since Aˆ is faithfully flat over A, the above formation is kept after base change of A by Aˆ and hence we may assume that A is a complete local ring. The i-th infinitesimal neighborhood XT ⊗A A/ti+1 A is (j) ˜ (j) ⊗O OX . We also denote E ˜ ⊗O OX by denoted by Xi and Ei denotes E i i XT XT (i) (i) ˜ ⊂E ˜ induces a homomorphism ρi : E → Ei . We denote Ei . The inclusion E i

the image of ρi by Fi . The image of Fi in E0 = E (0) is F (0) for all i because F (i) is naturally isomorphic to F (0) . This and Nakayama’s lemma imply that the natural homomorphism ρji : Fi ⊗ OXj → Fj is surjective for all j ≤ i. By ignoring the multiplication by a global generator of conormal bundle of the divisor Xk , E (i) ˜ (i−1) /tE ˜ (i) . H (i−2) as a subsheaf of E (i−1) contains H (i−1) which is the subsheaf tE (i−1) is sent isomorphically to H . Applying this consideration repeatedly, we see ˜ generates H (i−1) in E (i) , in other word, that ti E ˜ (i) /(ti E ˜ + tE ˜ (i) ) ∼ E = F (i) ∼ = F (0) = F0 ˜ (i) /ti E, ˜ we get by natural homomorphisms. Since Fi ∼ =E ˜ (i) /(ti E ˜ + tE ˜ (i) ) ∼ Fi ⊗ OX0 ∼ =E = F0 5 See

0 page 74 for the notation HX

k /Z

142

Chapter 3

CONSTRUCTION OF MODULI SPACES

as we have seen in the above. Hence Fi ∩ tEi ⊂ tFi . Assume that for a j ≤ i, the inclusion Fi ∩ tj−1 Ei ⊂ tj−1 Fi holds. Pick a ∈ Fi ∩ tj Ei . By our induction hypothesis a ∈ tj−1 Fi and hence a = tj−1 b = tj c with b ∈ Fi and c ∈ Ei . Thus ˜ i E. ˜ By the T -flatness of E ˜ we see that b−tc is an element tj−1 (b−tc) = 0 in Ei = E/t i−j+1 i−j+1 of t Ei . If we write b − tc = t g for some g ∈ Ei , then b = t(c + ti−j g) is an element of Fi ∩ tEi ⊂ tFi . Thus there is an h in Fi such that b = th. Therefore, a = tj−1 b = tj h is an element of tj Fi . By induction on j we obtain Fi ∩ tj Ei ⊂ tj Fi , that is, ρji is injective. What we have seen in the above is that for j ≤ i and for the closed immersion ∗ νij : Xj → Xi , we get νij (Fi ) = Fj . Hence Fˆ = lim Fi is a coherent subsheaf ← i ˆ ˆ of E = lim Ei on the formal completion XT of XT along the closed fiber Xk . By ← i

Grothendieck’s existence theorem [EGA, III, Th´eor`eme 5.1.4] there exists a coherent ˜ such that the completion Fˆ 0 is Fˆ . Especially we have F 0 = F0 . subsheaf F 0 of E k 0 0 ˜k ) which ). The former is β`0 (E Since F is flat over T , γ`0 (Fk0 ) must be equal to γ`0 (FK 0 ) ≤ γ`0 (E˜K ) = γ`0 (E˜k ) because is greater than γ`0 (E˜k ). On the other hand, γ`0 (FK ˜K = E ¯ is semi-stable. This is a contradiction. Therefore, there is an i such E ¯ that β` (E (i) ) < β` (E (0) ). Since there are a finite number of values between γ` (E) (0) and β` (E ), after finite steps of elementary transformations along β` -subsheaf we ˜ of G(E) ¯ such that β 0 (G ˜ k ) = γ 0 (G ˜ k ). If G ˜ k is not semiarrive at a member G ` ` 0 0 ˜ 0 ˜ stable, then we take ` = min{i | βi (Gk ) > γi (Gk )} which is greater than `, and we apply elementary transformations along β`0 -subsheaf. Repeating these procedure, ˜ of G(E) ¯ such that H ˜ k is semi-stable. We completed we finally come to a member H the proof of the following theorem. Theorem 8.4. Under the situation (4.6) let A be a discrete valuation ring with quotient field K and residue field k. Assume that a morphism T = Spec(A) → S ¯ is a semi-stable sheaf on the generic fiber XK of XT , then there is a is given. If E ¯ and Ek is semi-stable. ˜ on XT such that E ˜K ∼ T -flat coherent sheaf E =E

9 Moduli spaces of semi-stable sheaves – general case We shall fix a pair (X, OX (1)) of a projective S-scheme f : X → S and an S-ample invertible sheaf OX (1) on X which satisfies the condition (4.6) and shall maintain the notation of sections 4, 5 and 6. Fix an integer m in Lemma 4.12 and consider the Quot-scheme H (r) [m]

Qr = QuotVr (X)/X/S , where r is a positive integer such that r ≤ a0 (H) and H (r) (x) = rH(x)/a0 (H) is a numerical polynomial and where Vr is a free Ξ-module of rank H (r) [m](0). Let Qr (d) be the scheme theoretic closure in Qr of the open set of parameterizing sheaves of pure dimension d as in the beginning of section 6 and let ϕr : Vr (X ×S Qr (d)) → F˜r 0 be the universal quotient sheaf on X ×S Qr (d). Then there is open set Rre,e defined by the properties (4.13.1) and (4.13.2). For sufficiently large integer n, we have an

9

MODULI SPACES OF SEMI-STABLE SHEAVES – GENERAL CASE

143

0

embedding of Qr (d) into Zr (n) so that Rre,e is in the open set of semi-stable points 0 Zr (n)ss (L) of Zr (n) and a geometric point y of Rre,e is in Zr (n)s0 (Lr ) if and only if F˜r (y) is stable (see Lemma 5.1). Lemma 9.1. There is an integer n0 such that for every integer n ≥ n0 , we have the following: 0

(9.1.1) For every positive integer r ≤ a0 (H), Rre,e is embedded in Zr (n)ss (Lr ). 0

0

(9.1.2) Let y and y 0 be k-valued geometric points of Rre,e and Rre,e over a geometric 0 point s of S, where r0 and r are positive integers such that r0 < r ≤ a0 (H). y 0 is a subpoint of y as points of Grassmann varieties if and only if F˜r (y) contains a coherent subsheaf F 0 which is isomorphic to F˜r0 (y 0 ). (9.1.3) For F 0 in (9.1.2), set F 00 = F˜r (y)/F 0 . We have a surjection H 0 (Xs , F 00 )⊗k 0

W (s) → H 0 (Xs , F 00 (n)) and it defines a point y 00 of Rre,e ⊂ Zr00 (n)ss (Lr00 ), 00 00 0 00 where r = r − r . Moreover, y is an extension of y by y 0 . Proof. We shall take n0 to be the maximum among n0 ’s in the proof of Lemma 5.1 for r ≤ a0 (H). (9.1.1) is Lemma 5.1. Assume that F = F˜r (y) contains a subsheaf F 0 which is isomorphic to F˜r0 (y). Set U 0 = H 0 (Xs , F˜r0 (y 0 )(n)) and U = H 0 (Xs , F˜r (y)(n)). The injections α0 : Vr0 (k) ∼ = H 0 (Xs , F˜r0 (y 0 )) ∼ = H 0 (Xs , F 0 ) ,→ H 0 (Xs , F˜r (y)) ∼ = Vr (k) 0 0 0 0 0 0 0 ∼ ˜ ˜ 0 β : U = H (Xs , Fr (y )(n)) = H (Xs , F (n)) ,→ H (Xs , Fr (y)(n)) = U provide us with an exact commutative diagram α0 ⊗1

(9.1.4)

0 −−−−→ Vr0 (k) ⊗k W (s) −−−−→ Vr (k) ⊗k W (s)    ψ ψ0 y y 0 −−−−→

U0

β0

−−−−→

U

By Lemma 5.1 ψ 0 and ψ represent y 0 and y, respectively. Thus y 0 is a subpoint of y. Conversely, suppose that the above exact commutative diagram is given and ψ 0 and ψ represent y 0 and y, respectively. Let K 0 and K be the kernel of ψ 0 and ψ, respectively and consider the following commutative diagram of coherent sheaves on Xs u0 ⊗1

1⊗w

u⊗1

1⊗w

K 0 ⊗k OXs (−n) −−−−→ Vr0 (k) ⊗k W (s) ⊗k OXs (−n) −−−−→ Vr0 (k) ⊗k OXs       0 α0 ⊗1⊗1y y yα ⊗1 K ⊗k OXs (−n) −−−−→ Vr (k) ⊗k W (s) ⊗k OXs (−n) −−−−→ Vr (k) ⊗k OXs where u0 and u are natural injections and where w : W (s) ⊗k OXs (−n) → OXs is defined by the map of global sections W (s) ⊗k OXs → OXs (n). The construction of the Quot-scheme in Proposition 1.3 tells us that F˜r (y) = coker((1 ⊗ w)(u ⊗ 1))

144

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CONSTRUCTION OF MODULI SPACES

and F˜r0 (y 0 ) = coker((1 ⊗ w)(u0 ⊗ 1)). So there is a morphism f : F˜r0 (y 0 ) → F˜r (y) such that the diagram Vr0 (k)   y

α0

−−−−→

Vr (k)   y

0

H (f ) H 0 (Xs , F˜r0 (y 0 )) −−−−→ H 0 (Xs , F˜r (y))

commutes. Put F 0 = im(f ). Since F˜r (y) and F˜r0 (y 0 ) are e0 -semi-stable with PFsr (y) (x) = PFs 0 (y0 ) (x), F 0 is e0 -semi-stable with PFsr (y) (x) = PFs 0 (x). Since H 0 (f ) r is injective, f is injective by the choice of m. This completes the proof of (9.1.2). Set F 00 = F˜r (y)/F 0 . Since P S (x) = P S0 (x) as polynomials in x and F is strictly F

F

e0 -semi-stable, we have PFS00 (x) = PFS (x) and then by Lemma 3.5 F 00 is strictly e0 semi-stable. The above proof shows that if we identify Vr0 (k) with H 0 (Xs , F˜r0 (y 0 )) and Vr (k) with H 0 (Xs , F˜r (y)), then α0 and β 0 are the maps induced by the injection F˜r0 (y 0 ) ∼ = F 0 ⊂ F˜r (y). By (4.12.1) for F˜r0 (y 0 ) we have an exact commutative diagram extending (9.1.4) α0 ⊗1

α00 ⊗1

0 −→ Vr0 (k) ⊗k W (s) −−−−→ Vr (k) ⊗k W (s) −−−−→ V 00 ⊗k W (s) −→ 0     ψ  ψ0 y ψ 00 y y 0 −→

U0

β0

−−−−→

β 00

−−−−→

U

U 00

−→ 0

where V 00 = H 0 (Xs , F 00 ), U 00 = H 0 (Xs , F 00 (n)), ψ 00 is the multiplication map and both α00 and β 00 are defined by the quotient homomorphism F˜r (y) → F 00 . Since ψ 00 is surjective, it gives rise to a k-rational point y 00 of Zr00 (n) and it comes from a point Vr00 (k) ⊗ OXs → F 00 of Qr00 . The strict e0 -semi-stability of F 00 means that y 00 0 is a geometric point of Rre,e 00 . Corollary 9.2. Let E1 , . . . , E` be e0 -stable sheaves on a geometric fiber Xt such ¯ t -stable that PESi (x) = H(x)/a0 (H) and a0 (E1 )+· · ·+a0 (E` ) = a0 (H). There is a G e,e0 closed set B(E1 , . . . , E` ) of (R )t with the following properties: (9.2.1) B(E1 , . . . , E` ) is closed in Z(n)ss (L)t . (9.2.2) For every algebraically closed field K containing k = k(t), 0

B(E1 , . . . , E` )(K) = {y ∈ (Re,e )s (K) | gr(F˜ (y)) ∼ = ⊕`i=1 Ei ⊗k K} (9.2.3) The orbit o(⊕`i=1 Ei ) of a point corresponding to ⊕`i=1 Ei is a unique closed orbit of B(E1 , . . . , E` )(k). Proof. Let us fix an integer n such that n ≥ n0 for the n0 in Lemma 9.1 and 0 embed each Rre,e into Zr (n). Set ri0 = a0 (Ei ). Then, by Lemma 5.1 Ei gives rise to a k-valued geometric point zi of Zri0 (n)s0 (Lri0 )t . Pick a permutation τ of {1, 2, . . . , `}.

9

MODULI SPACES OF SEMI-STABLE SHEAVES – GENERAL CASE

145

¯ t -stable closed set B(zτ (1) , . . . , zτ (`) ) in (Z(n)ss )t . If By Theorem 7.6 we have the G we put [ B(E1 , . . . , E` ) = B(zτ (1) , . . . , zτ (`) ), τ ∈S`

¯ t -stable closed set in (Z(n)ss )t and contains the unique then B(E1 , . . . , E` ) is G closed orbit o(z1 ⊕ · · · ⊕ z` ). Since 0

C = B(E1 , . . . , E` ) ∩ ((Z(n)ss \ Re,e )t ) ¯ t -stable closed set in (Z(n)ss )t , it must contain o(z1 ⊕· · ·⊕z` ) unless it is empty. is G On the other hand, by Corollary 3.6 the direct sum ⊕`i=1 Ei corresponds to a k-valued 0 geometric point y0 of (Re,e )t which is isomorphic to z1 ⊕ · · · ⊕ z` as points of the 0 Grassmann variety Z(n). Thus o(z1 ⊕ · · · ⊕ z` ) is in (Re,e )t and hence C is empty, 0 that is, B(E1 , . . . , E` ) ⊂ (Re,e )t . Pick a point x in B(zτ (1) , . . . , zτ (`) )(K). Setting r(τ )i = rτ0 (1) + · · · + rτ0 (i) , there is a sequence of K-valued points x1 , . . . , x` which has the property (7.6.1) for Z(Vi , ri ) = Zr(τ )i (n). If we take B(zτ (1) , . . . , zτ (i) ) as in 0

e,e Theorem 7.6, then we see as in the above that it is a subset of (Rr(τ )i )t . Thus each xi 0 e,e ˜ is a K-valued point of (Rr(τ )i )t . Assume that Fi−1 = Fr(τ )i−1 (xi−1 ) has a filtration ∼ Eτ (j) ⊗k K. Since xi is 0 = F 0 ⊂ F 0 ⊂ · · · ⊂ F 0 = Fi−1 such that F 0 /F 0 = 0

1

i−1

j

j−1

0

0

e,e e,e an extension of zτ (i) by xi−1 and since xi−1 and xi are in (Rr(τ )i−1 )t and (Rr(τ )i )t , respectively, Fi = F˜r(τ ) (xi ) contains a coherent subsheaf F 0 which is isomorphic i

i−1

0 0 to Fi−1 by Lemma 9.1. By the construction of Fi−1 , the point z 00 defined by Fi /Fi−1 as in (9.1.3) is isomorphic to zτ (i) over K as points of the Grassmann variety. This 0 0 ∼ means that Fi /Fi−1 as subsheaves of Fi , we = Eτ (i) ⊗k K. Regarding F10 , . . . Fi−1 0 0 0 0 ∼ have a filtration 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fi = Fi such that Fj0 /Fj−1 = Eτ (j) ⊗k K. Therefore, by induction on i, we get a filtration 0 = F00 ⊂ F10 ⊂ · · · ⊂ F`0 = F˜ (x) 0 ∼ such that for 1 ≤ j ≤ `, Fj0 /Fj−1 = Eτ (j) ⊗k K. Hence gr(F˜ (x)) ∼ = ⊕`i=1 Ei ⊗k K. 0 Conversely, it is not hard to see that if x is a K-valued geometric point of Re,e such that gr(F˜ (x)) ∼ = ⊕`i=1 Ei ⊗k K, then x is in B(E1 , . . . , E` ). This completes the proof that B(E1 , . . . , E` ) meats our requirement.

By Theorem 2.9 we have a good quotient π : Z(n)ss (L) → Y . By the properties 0 0 ¯ e,e0 = (2) and (3) of Theorem 2.9 and the above corollary, π −1 π(Re,e ) = Re,e and M 0 0 ¯ e,e of Y is, therefore, a good π(Re,e ) is an open set of Y . The open subscheme M e,e0 quotient of R . By a similar argument to the proof of Theorem 6.10 we have the following. 0

¯ e,e0 is a coarse moduli scheme of Σ ¯ H,e Proposition 9.3. M X/S 0 ¯ ¯ e,e0 is an open subscheme of M ¯ e,e . Since Re,e is a G-stable open set of Re,e , M 0 0 0 e ,e e,e ¯ ¯ M and M are coarse moduli schemes of the same functors and hence they are ¯ e0 ,e0 can be regarded isomorphic to each other by a unique isomorphism. Thus M ¯ e,e . Taking the inductive limit of {M ¯ e,e }, we naturally as an open subscheme of M e,e e,e ¯ ¯ get an S-scheme MX/S (H). Since each M contains M of section 5 as such an

146

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CONSTRUCTION OF MODULI SPACES

0 0 ¯ e0 ,e0 , the moduli scheme MX/S (H) of stable open subscheme that M e ,e = M e,e ∩ M ¯ X/S (H). Moreover, M ¯ X/S (H) is locally of finite sheaves is an open subscheme of M e,e ¯ type over S because each M is quasi-projective.

Theorem 9.4. Assume that f : X → S is a flat, projective morphism of schemes of finite type over a universally Japanese ring Ξ and OX (1) is an f -ample invertible ¯ X/S (H) is a coarse moduli scheme of the functor Σ ¯ H and sheaf on X. Then, M X/S it contains the moduli scheme MX/S (H) of stable sheaves as an open subscheme. ¯ X/S (H) is locally of finite type and separated over S. Moreover, M ¯ X/S (H) is M projective over S if and only if SX/S (H) is bounded. ¯ X/S (H) Proof. The proof of the former half is as in the proof of Theorem 5.3. If M is of finite type over S, then SX/S (H) is equal to SX/S (H, e) for some e and hence SX/S (H) is bounded by Proposition 3.8. Conversely, if SX/S (H) is bounded, then ¯ X/S (H) = M ¯ e,e for a sufficiently large e, which means that M ¯ X/S (H) is of finite M ¯ type over S. To show that MX/S (H) is proper over S, we use the valuative criterion of properness. Let A be a discrete valuation ring with quotient field K. Assume ¯ X/S (H) that we are given morphisms α : T = Spec A → S, β : Tη = Spec K → M ¯ such that p ◦ β = α ◦ ι, where ι : Tη → T is the inclusion map and p : MX/S (H) → S is the projection. There are a finite field extension K 0 of K and a morphism β 0 : Tη0 = Spec K 0 → Re,e such that π ◦ β 0 = β ◦ 0 , where 0 : Tη0 → Tη is the ¯ X/S (H) is the quotient morphism. morphism induced by K 0 ⊃ K and π : Re,e → M 0 The morphism β corresponds to a surjection V ⊗ OXK 0 → E with E semistable. Let A0 be a discrete valuation ring with quotient field K 0 which dominates A. Put T 0 = Spec A0 , and denote by  and ι0 the natural morphisms T 0 → T and Tη0 → T 0 respectively. By Langton’s theorem (Theorem 8.4), E extends to a T 0 -flat coherent ˜ such that the restriction of E ˜ to the closed fiber is semistable. Then OXA0 -module E 0 0 ˜ determines a morphism γ : T → M ¯ X/S (H). By construction, β ◦ 0 = γ 0 ◦ ι0 . E ¯ X/S (H) such that γ ◦ ι = β. Then there is a morphism γ : T → M Applying Theorem II.7.9 and Corollary II.7.13, we get the following. Corollary 9.5. is satisfied:

6

¯ X/S (H) is projective over S if one of the following conditions M

(1) Ξ is a field of characteristic zero. (2) H is a polynomial of degree not greater than 2. (3) f : X → S is geometrically integral, deg H = dim X/S and the rank of sheaves is not greater than 3.

6 See

Appendix A

Appendix A

On Langer’s work After the manuscript of this book was written, Langer made big progress on the boundedness of semi-stable sheaves and the dimension estimate of global sections of torsion free sheaves. The purpose of this appendix is to explain briefly his results and make it clear that some results in the preceding chapters have been greatly improved.

1 Boundedness of semi-stable sheaves Let X be a smooth projective variety of dimension n ≥ 2 over an algebraically closed field k and let OX (1) be a very ample line bundle on X. Set d = O(1)nX . We fix a nef line bundle A such that TX (A) is globally generated and set ( 0 if char k = 0 γr = r−1 n−1 if char k = p. p−1 AOX (1) For a torsion free sheaf E on X, ∆(E) denotes 2r(E)c2 (E) − (r(E) − 1)c1 (E)2 . If 0 = E0 ⊂ E1 ⊂ · · · ⊂ El = E is the Harder-Narasimhan filtration of E, then we set µmax (E) = µ(E1 ) and µmin (E) = µ(El /El−1 ). Langer proved the following restriction theorem ([L1, Corollary 3.11 and Corollary 2.5]). Theorem 1.1. Let E be a rank r torsion free sheaf on X and set µ = µ(E), µmax = µmax (E) and µmin = µmin (E). Then for general member Y of |OX (1)|, we have r 2 (µmax (E|Y ) − µmin (E|Y )) ≤d∆(E)OX (1)n−2 2 + 2r2 (γr + µmax − µ) (γr + µ − µmin ) . Now let us consider the relative situation to state the boundedness result. Let X → S be a smooth geometrically integral projective morphism of noetherian schemes and let OX (1) be an f -very ample line bundle. Assume that the dimension of the fibers of f are constant n. Fix integers r, a1 , . . . , an and a sequence (α) = (α1 , . . . , αr−1 ). Then for a coherent sheaf E on a geometric fiber Xs which belongs 0 to TX/S (n, r; a1 , . . . , an ; (α)) or TX/S (n, r; a1 , a2 ; (α)), the degree of discriminant n−2 ∆(E)OXs (1) is bounded above. By Theorem 1.1 and Proposition II.3.4, we obtain the boundedness of torsion free semi-stable sheaves. Theorem 1.2. Ln,r (Λ), L0n,r (Λ) and L00n,r (Λ) are true. 147

148

Appendix A

ON LANGER’S WORK

By this theorem, using the argument of the proof of Theorem II.7.9, we have the following complete generalization of Theorem II.7.9 ([L1, Theorem 4.4]). Theorem 1.3. Let the pair (X, OX (1)) be the same as in Definition II.7.7. Then 0 both SX/S (d; r, a1 , . . . , ad ; (α)) and SX/S (d; r, a1 , a2 ; (α)) are bounded. This in turn implies a complete generalization of Corollary III 9.5. ¯ X/S (H) is always projective over S. Theorem 1.4. M

2 Dimension estimate of global sections of torsion free sheaves In Section III 6 the construction of moduli spaces of semi-stable sheaves in characteristic 0 is explained. The key proposition is Proposition III 6.7, in which it is proved that the locus of the quot-scheme parametrizing semi-stable sheaves is equal to the locus of the quot-scheme consisting of semi-stable points with respect to the group action. In the proof of the proposition, it is essential that the constant B in Corollary III 6.4 does not depend on a2 (E). The dimension estimate in the corollary is proved by restricting a sheaf to a hypersurface and using the GrauertM¨ ulich-Spindler restriction theorem, available only in characteristic 0, which bounds the type of the restriction of a semi-stable sheaf E by a constant not depending on a2 (E). In positive characteristic, one must take a2 (E) into account as in Theorem 1.1 to bound the type of the restriction of a semi-stable sheaf E (see also [L2, Example 3.1]). As in Remark III 6.12, therefore, it is reasonable to believe that the constant B in Corollary III 6.4 must depend on a2 (E) in positive characteristic. Here comes a big surprise ! Langer proved that even in positive characteristic, B does not depend on a2 (E) ([L2, Corollary 3.4]): Theorem 2.1. Let X be a projective scheme over an algebraically closed field of arbitrary characteristic and let OX (1) be a very ample line bundle on X. For any coherent sheaf E of pure dimension d on X, we have

≤

dim H0 (X, E(m)) (
S 2 +f (a0 (E))+d) a0 (E) m+µmax (E)+a0 (E) d 0

if m + µSmax (E) + a0 (E)2 ≥ 0 if m + µSmax (E) + a0 (E)2 < 0.

Here µSmax (E) denotes the µS of the first filter of the Harder-Narasimhan filtration Pt of E and for a positive integer t, we set f (t) = −1 + i=1 1i . If you use Theorem 2.1 in stead of Corollary III 6.4, the argument of Section III 6 works regardless of the characteristic.

Appendix B

Some properties of the moduli The moduli space of semistable sheaves is constructed as a quotient space of an open subscheme of a quote-scheme. The aim of this appendix is to show that the quotient map is a PGL-bundle in the case of the moduli of stable sheaves. We first recall the setting from Chapter III. Let f : X → S be as in (4.6) of Chapter III. For a numerical polynomial H(x) of degree d, let ΣH X/S be the moduli functor of stable sheaves on X/S with its Hilbert polynomial H(x) defined in Definition III.4.7. By Langer’s theorem of boundedness of semistable sheaves with fixed Hilbert polynomial, we can fix an integer e such that every semistable sheaf with its Hilbert polynomial H(x) is strictly e-semistable. Fix a large integer m and put N := H(m). (Precisely, choose m as in Lemma III.4.12.) Fix a free Ξ-module V of rank N . Denote by H[m] = H[m](x) the polynomial H(x + m). H[m] Put Q := QuotV (X)/X/S , where V (X) = V ⊗Ξ OX , and let ϕ : V (X ×S Q) → F˜ be ¯ the universal quotient sheaf. Let G be the S-group scheme GL(V (S)), and let G ¯ on Q and a G-linearlization on F˜ . be P GL(V (S)). We have a natural action of G Let Rs be the open subscheme of Q such that a geometric point y of Q is in Rs if and only if the following two conditions are satisfied: • Γ(ϕ ⊗ k(y)) : V (k(y)) → H0 (Xy , F˜ (y)) is bijective, • F˜ (y) is stable. s ¯ Then the coarse moduli scheme MX/S of ΣH X/S is a geometric quotient of R by G (cf. Proposition III.5.2).

Proposition 1.1. The quotient map π : Rs → MX/S is a principal fiber bundle ¯ (see [GIT] Definition 0.10). with group G For the proof of the above proposition, we need two lemmas. Lemma 1.2. Let A be an artinian local ring with maximal ideal m and residue field k and let E be an A-flat coherent sheaf on XA = X ×S Spec A. Assume that the natural injection k → HomOXk (Ek , Ek ) is an isomorphism, where Ek = E ⊗A k. Then the natural homomorphism A → HomOXA (E, E) is an isomorphism. Proof. We shall prove this by induction on l(A) = length(A). If l(A) = 1, then A = k, and hence there is nothing to prove. Assume that our assertion is true if l(A) < l. If l(A) > 1, then there exists a principal ideal A such that A ' k ¯ = l(A) − 1, our assertion says that as A-modules. Since for A¯ = A/A, l(A) ¯ ¯ ¯ ¯ ¯ HomOXA¯ (E, E) = A, where E = E ⊗A A. Pick an element φ of HomOXA (E, E). If ¯ E) ¯ induced by φ, then φ¯ is the multiplication of an φ¯ is the member of HomOXA¯ (E, 149

150

Appendix B

SOME PROPERTIES OF THE MODULI

¯ Lift the a element a ¯ of A. ¯ to an element a of A and set ψ = φ − a · idE . Then ψ(E) is contained in E = E ⊗A A. If x is contained in mE, then ψ(x) = 0 because m = 0. Thus ψ induces a homomorphism ψ¯ : Ek → E ' Ek . By assumption on Ek , we can find a ¯b in k such that ψ¯ = ¯b · idE . Lift ¯b to a b in A. The definition of ψ¯ shows that ψ = (b)idE . Thus we obtain that φ = (a + b)idE . Pick a non-zero element c in A. The image of c·idE is cE. Since E is flat over A, cE = cA⊗A E 6= 0. Therefore A → HomOXA (E, E) is an isomorphism. ¯ on Rs is free, that is, Ψ = (¯ ¯ ×S R s → Lemma 1.3. The action σ ¯ of G σ , p2 ) : G s s R ×S R is a closed immersion. Proof. In the first place, we shall show that Ψ is proper. Let A be a discrete valuation ring over S with maximal ideal (α) and residue field k. Let K be the fractional field of A. Let (x1 , x2 ) be an A-valued point of Rs ×S Rs and let (¯ gK , x2 ) ¯ ×S Rs such that Ψ(K)(¯ be a K-valued point of G gK , x2 ) = (x1 , x2 ). We want to ¯ The A-valued point xi corresponds to a show that g¯K is an A-valued point of G. φi surjection V ⊗Ξ OXA −→ Ei on XA = X ×S Spec A such that Ei is A-flat, Ei ⊗A k is stable and Γ(φi ) : V ⊗Ξ A → H0 (XA , Ei ) is bijective. Let gK be a K-valued point ¯ is g¯K . σ of G whose image by the natural map G → G ¯ (¯ gK , x2 ) = x1 means that there is an isomorphism fK from E1 ⊗A K to E2 ⊗A K such that the diagram φ1 ⊗K

V ⊗Ξ OXK −−−−→ E1 ⊗A K    f gK y yK φ2 ⊗K

V ⊗Ξ OXK −−−−→ E2 ⊗A K commutes. For some integer a, αa fK extends to a morphism f : E1 → E2 of OXA modules such that f ⊗A k : E1 ⊗ k → E2 ⊗ k is non-zero. Since E1 ⊗ k and E2 ⊗ k are stable sheaves with the same Hilbert polynomial, f ⊗ k is an isomorphism. This implies that f is an isomorphism. Then we have a commutative diagram Γ(φ1 )

V ⊗Ξ A −−−−→ H0 (XA , E1 )    Γ(f ) gK y y Γ(φ2 )

V ⊗Ξ A −−−−→ H0 (XA , E2 ), where all arrows are isomorphisms. This means that gK is an A-valued point of G. ¯ This completes the proof of properness of Ψ. So g¯K is an A-valued point of G. For every algebraically closed field L, Ψ(L) is an injective map between the sets of L-valued points since the stabilizer of an L-valued point of Rs is trivial. This and the properness of Ψ implies that Ψ is an geometrically injective finite morphism. To conclude that Ψ is a closed immersion, it suffices to prove that for any artinian local ring A over S, the map ¯ Ψ(A) : G(A) ×S(A) Rs (A) → Rs (A) ×S(A) Rs (A)

151

is injective. Suppose that Ψ(A)(g1 , x) = Ψ(A)(g2 , x) for some A-valued points ¯ ×S Rs . Then Ψ(A)(e, x) = Ψ(A)(g −1 g2 , x). Thus we have (g1 , x) and (g2 , x) of G 1 only to show that if x = σ ¯ (A)(g, x), then g = e. To give a point x in Rs (A) is φ

just to give a surjection V ⊗Ξ OXA − → E on XA = X ×S Spec A such that E is A-flat, E ⊗A k is stable and Γ(φ) : V ⊗Ξ A → H0 (XA , E) is bijective. Let h be an ¯ is g. x = σ A-valued point of G whose image by the natural morphism G → G ¯ (g, x) means that there is an isomorphism f of E which makes the following diagram commutative φ V ⊗Ξ OXA −−−−→ E    f hy y φ

V ⊗Ξ OXA −−−−→ E. Since HomOXk (E ⊗A k, E ⊗A k) = k, f is the multiplication map of a unit a of A by Lemma 1.2. Then h is the multiplication of a because Γ(φ) is bijective. We see, therefore, that g = e. ¯ is a smooth group scheme over S Proof of Proposition 1.1. If one notes that G and that π : Rs → MX/S is a uniform geometric quotient, this is a consequence of the proposition below, which is a straightforward generalization of Proposition 0.9. of [GIT]. Proposition 1.4. Let X, Y be schemes, separated and of finite type over a noetherian scheme S. Let G be a group scheme, smooth and of finite type over S. Let σ : G ×S X → X be a free action of G on X over S. Let φ : X → Y be a uniform geometric quotient of X by G. Then X is a principal fiber bundle over Y with group G. Sketch of Proof. We need only to modify the proof of Proposition 0.9 of [GIT]. By assumption, the morphism Ψ = (σ, p2 ) : G ×S X → X ×S X induces a ¯ : G ×S X → X ×Y X. We need to show that φ is flat bijective closed immersion Ψ ¯ and Ψ is an isomorphism. For this, it suffices to show that for every point x ∈ X, closed in the fiber over S, the morphism φ] : OY,φ(x) → OX,x

(B.1)

is flat and that for every point z ∈ G ×S X, closed in the fiber over S, the morphism ¯ ] : OX× X,z → OG× X,z Ψ Y S

(B.2)

is an isomorphism. For this, we may assume that S = Spec R with R a noetherian local ring. Let G0 , X0 and Y0 be closed fibers of G, X and Y over S respectively. It suffices to show that for every closed point x ∈ X0 , the morphism (B.1) is flat and that for every closed point z ∈ G0 × X0 , the morphism (B.2) is an isomorphism. By [EGA, III, Proposition 10.3.1], there is a local homomorphism from R to a noetherian local ring R0 , flat over R, such that the residue field of R0 is algebraically

152

Appendix B

SOME PROPERTIES OF THE MODULI

closed. So replacing R if necessary, we may assume that the residue field k of R is algebraically closed. Let y ∈ Y0 be a closed point. By Replacing Y with Spec OY,y , [ and further replacing Spec OY,y with Spec O Y,y , we may assume that Y = Spec A, where A is a complete local ring with residue field isomorphic to k. Let x be a closed point of X0 . Let Ix ⊂ OX0 ,x be the defining ideal of the orbit G0 x. Since σ is free, the morphism ψx : G0 → X0 given by g 7→ gx defines an isomorphism OX0 ,x /Ix ' OG0 ,e . Let g = dim G0 and let f1 , . . . , fg ∈ mG0 ,e be a basis of this maximal ideal. Let x1 , . . . , xg ∈ mX,x be elements whose image in OX0 ,x /Ix are f1 , . . . , fg respectively. There is a unique closed subscheme H of X, finite over Y , such that H ∩ φ−1 (y) = {x} and the defining ideal of H at x is (x1 , . . . , xg ) ⊂ OX,x . Now the rest of the proof goes as the proof of Proposition 0.9 of [GIT]. We have corollaries of Proposition 1.1. Corollary 1.5. If S 0 is a noetherian S-scheme, then for X 0 = X ×S S 0 , we have MX 0 /S 0 = MX/S ×S S 0 Corollary 1.6. MX/S is smooth over S if and only if so is Rs .

Bibliography [B]

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153

Glossary of Notations Quot Quot r(E) ai (E) µ0 (E) PE (m) d(F, OX (1)) µ(E) 0 00 TX/S , TX/S , TX/S (Sk ) Ln,r (Λ), L0n,r (Λ), L00n,r (Λ) a0 (F ) µS (F ) PFS (m) 0 00 SX/S , SX/S , SX/S AQS X ss (L), X0s (L) ΣH X/S etc. SX/S (H) etc. H (r) (m) H (r) [m](x) Vr (Y ) (Vr (A), resp.)

Quot functor, 1 Quot scheme, 1 rank of E, 5 certain coefficients of the Hilbert polynomial χ(E(m)), 7 a1 (E)/r(E), 7 χ(E(m))/r(E), 7 degree of c1 (F ) with respect to OX (1), 7 d(E, OX (1))/r(E), 7 certain families of sheaves, 38 the serre condition, 38 statements of boundedness, 38 top coefficient of the Hilbert polynomial χ(F (m)), 72 a1 (F )/a0 (F ), 72 χ(F (m))/a0 (F ), 72 certain families of sheaves, 74 functor of all quotient sheaves, 85 locus of semistable (properly stable) points, 92 moduli functors of sheaves, 109 certain families of sheaves, 110 rH(m)/a0 (H), 112 polynomial H (r) (x + m), 113 Vr ⊗Ξ OY (Vr ⊗Ξ A, resp.), 114

Qr ¯r G 0 Rre,e Zr (n) 0 M e,e MX/S (H) Z(V, r) Y (V, U ) γi (F ),γi (F ) βi (F ), βi0 (F )

QuotVr (X)/X/S , 114 P GL(Vr (S)), 114 locus of e0 -semistable sheaves, 114 GrassH (r) [m](n) (h∗ (Vr (P(U ))(n))), 115 0 a coarse moduli scheme of ΣH,e X/S , 117 a coarse moduli scheme of ΣH X/S 119 Grassmann variety Grassr (V ⊗k W ), 129 an open subscheme of V(Homk (V ⊗k W, U )∨ ), 129 page 138 page 138

H (r) [m]

154