Jean-Marc Dr

G Ünther Trautmann

MODULI SPACES OF DECOMPOSABLE MORPHISMS OF SHEAVES AND QUOTIENTS BY NON-REDUCTIVE GROUPS

We extend the methods of geometric invariant theory to actions of nonreductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non-reductive. Given a linearization of the natural action of the group Aut(E) × Aut(F ) on Hom(E,F), a homomorphism is called stable if its orbit with respect to the unipotent radical is contained in the stable locus with respect to the natural reductive subgroup of the automorphism group. We encounter effective numerical conditions for a linearization such that the corresponding open set of semistable homomorphisms admits a good and projective quotient in the sense of geometric invariant theory, and that this quotient is in addition a geometric quotient on the set of stable homomorphisms.

Let X be a projective algebraic variety over the field of complex numbers. Given two coherent sheaves E, F, on X the algebraic group G = Aut(E) × Aut(F) acts naturally on the affine space W = Hom(E, F) by (g, h).w = h • w • g -1 . If two morphisms are in the same G-orbit then they have isomorphic cokernels and kernels. Therefore it is natural to ask for good quotients of such actions in the sense of geometric invariant theory.

Morphisms of type (r, s).

In general E and F will be decomposable such that G is not reductive. More specifically let E and F be direct sums

E = ⊕ 1≤i≤r M i ⊗ E i and F = ⊕ 1≤l≤s N l ⊗ F l ,
where M i and N l are finite dimensional vector spaces and E i , F l are simple sheaves, i.e. their only endomorphisms are the homotheties, and such that Hom(E i , E j ) = 0 for i > j and Hom(F l , F m ) = 0 for l > m. In this case we call homomorphisms E → F of type (r, s). Then the groups Aut(E) and Aut(F) can be viewed as groups of matrices of the following type. The group Aut(E), say, is the group of matrices

      g 1 0 • • • 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 • • • u r,r-1 g r      
where g i ∈ GL(M i ) and u ji ∈ Hom(M i , M j ⊗ Hom(E i , E j )).

In the literature on moduli of vector bundles and coherent sheaves many quotients of spaces P Hom(E, F) of type [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF][START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF] by the reductive group Aut(E) × Aut(F) have been investigated, see for example [START_REF] Drézet | Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur P 2 (C)[END_REF], [START_REF] Ellingsrud | On the variety of nets of quadrics defining twisted cubic curves[END_REF], [START_REF] Ellingsrud | On the Chow ring of a geometric quotient[END_REF], [START_REF] Karpov | Semi-stable sheaves on a two-dimensional quadric and Kronecker modules[END_REF], [START_REF] Miró-Roig | The moduli scheme M (0, 2, 4) over P 3[END_REF]. The moduli spaces described in this way are the simplest ones, and this allows to test in these cases some conjectures that are expected to be true on more general moduli spaces of sheaves (cf. [START_REF] Drézet | Cohomologie des variétés de modules de hauteur nulle[END_REF], [START_REF] Tjøtta | Rational curves on the space of determinantal nets of conics[END_REF]). We think that the moduli spaces of morphisms of type (r, s) will be as useful to treat other less simple moduli problems of sheaves. In fact, if one wants to use the spaces Hom(E, F) as parameter spaces for moduli spaces of sheaves, which are as close as possible to the moduli spaces, the higher types (r, s) are unavoidable.

The homomorphisms in a Beilinson complex of a bundle on projective n-space, for example, have in general arbitrary type (r, s) depending on the dimensions of the cohomology spaces of the bundle. In several papers, see [START_REF] Miró-Roig | Some moduli spaces for rank 2 stable reflexive sheaves on P 3[END_REF], [START_REF] Okonek | Moduli extremer reflexiver Garben auf P n[END_REF] for example, semi-stable sheaves or ideal sheaves of subschemes of projective spaces, are represented as quotients of injective morphisms of type (r, s), and one should expect that the moduli spaces of such sheaves are isomorphic to a good quotient of an open subset of the corresponding space of homomorphisms. In some cases of type [START_REF] Bialynicki-Birula | Open subsets of projective space with a good quotient by an action of a reductive group[END_REF][START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF] this has been verified for semi-stable sheaves on P 2 in [START_REF] Drézet | Variétés de modules extrémales de faisceaux semi-stables sur P 2 (C)[END_REF].

In case of type (r, s) there are good and projective quotients if one restricts the action to the reductive subgroup

G red = GL(M i ) × GL(N l ).
This has been shown recently by A. King in [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF]. The quotient problem for Hom(E, F) of type (r, s) with respect to the full group Aut(E) × Aut(F) is however the generic one and indispensable.

Unfortunately the by now standard geometric invariant theory (GIT) doesn't provide a direct answer for these quotient problems in case Aut(E)×Aut(F) is not reductive. There are several papers dealing with the action of an arbitrary algebraic group like [START_REF] Fauntleroy | Geometric invariant theory for general algebraic groups[END_REF], [START_REF] Fauntleroy | Invariant theory for linear algebraic groups II[END_REF], [START_REF] Dixmier | Quelques aspects de la théorie des invariants[END_REF], [START_REF] Dixmier | Sur le quotient d'une variété algébrique par un groupe algébrique[END_REF] and older ones, but their results are insufficient for the above problem. The conditions of [START_REF] Fauntleroy | Geometric invariant theory for general algebraic groups[END_REF] are close to what we need, but they don't allow a concrete description of the set of semi-stable points in our case and they don't guarantee good or projective quotients, see remark 4.1.2.

The main idea

Our procedure is very close to standard GIT and we finally reduce the problem of the quotient to the one of a reductive group action. We introduce polarizations Λ ∈ Q r+s of tuples of rational numbers for the action of G on the affine space in analogy to the ones of A. King in [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF], which are refinements of the polarizations by ample line bundles on the projective space PW , and then introduce open sets W s (G, Λ) ⊂ W ss (G, Λ) of stable and semi-stable points depending on Λ and study the quotient problem for these open subsets. There are chambers in Q r+s such that the polarizations in one chamber define the same open set, in accordance with the chamber structure in Neron-Severi spaces of polarizations in the reductive case, see f.e. [START_REF] Dolgachev | Variation of Geometric Invariant Theory Quotients[END_REF] I. Dolgachev -Y. Hu and [START_REF] Thaddeus | Geometric invariant theory and flips[END_REF] M. Thaddeus. However, in contrast to the reductive case, good quotients W ss (G, Λ)//G don't exist for all polarizations, see 4.2. As a main achievement we are providing numerical conditions on the polarizations, depending on the dimensions of the spaces M i and N l , under which such quotients exist. The main step for that is to embed the group actions G × W → W into an action G × W → W of a reductive group G and to compare the open sets W ss (G, Λ) and W ss (G, Λ), where Λ is a polarization for the G-action associated to Λ.

Construction of quotients by non reductive groups.

To be more precise, a polarization Λ is a tuple (λ 1 , . . . , λ r , µ 1 , . . . , µ s ) of positive rational numbers, called weights of the factors M i ⊗ E i and N l ⊗ F l respectively, which satisfy λ i m i = µ l n l = 1, where m i , n l denote the dimensions of the spaces of the same name. We use then the numerical criterion of A. King, [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF], as definition for semistability with respect to the reductive group G red . An element w ∈ W is (G red , Λ)-stable if for any proper choice of subspaces M i ⊂ M i , N l ⊂ N l such that w maps ⊕(M i ⊗ E i ) into ⊕(N l ⊗ F l ), we have λ i m i < µ l n l , or semi-stable if equality is allowed. Let W s (G red , Λ) ⊂ W ss (G red , Λ) denote the set of stable and semi-stable points so defined.

If H ⊂ G is the unipotent radical of G, which is generated by the homomorphisms E i → E j and F l → F m for i < j and l < m, we say that w is (G, Λ)-(semi-)stable if h.w is (G red , Λ)-(semi-)stable for any h ∈ H, see 4.1. We thus have open subsets W ss (G, Λ) ⊂ W ss (G red , Λ) and W s (G, Λ) ⊂ W ss (G red , Λ).

The main result of our paper is that there are sufficient numerical and effective bounds for the polarizations Λ such that W ss (G, Λ) admits a good and even projective quotient W ss (G, Λ)//G and that in addition W s (G, Λ) admits a geometric quotient, which is smooth and quasi-projective, see proposition 6.1.1 and the results 7.2.2, 7.5.3, and section 8.

The definitions of good and geometric quotients are recalled in 6.1. By using correspondences between spaces of morphisms, called mutations, it is possible to deduce from our results other polarizations such that there exists a good projective quotient (see [START_REF] Drézet | Quotients algébriques par des groupes non réductifs et variétés de modules de complexes[END_REF], [START_REF] Drézet | Espaces abstraits de morphismes et mutations[END_REF]).

All this is achieved by embedding the action G × W → W into an action G × W → W of a reductive group and then imposing conditions for the equality W ss (G, Λ) = W ∩ W ss (G, Λ), where Λ is the associated polarization. The quotient is then the quotient of the saturated subvariety GW ss (G, Λ) ⊂ W ss (G, Λ). The quotient will be projective if G.W G.W doesn't meet W ss (G, Λ). Also for this, numerical conditions can be found in section 8.

The idea of embedding the non-reductive action G×W → W into the action G×W → W is simply to replace the E i by E 1 using the evaluation maps Hom(E 1 , E i ) ⊗ E 1 → E i . It is explained in 5.1 and 5.1.1 that this is the outcome when we start to replace the sheaves E i step by step and similarly for the sheaves F l . Since we have to deal everywhere with the dimensions of the vector spaces Hom(E i , E j ) and Hom(F l , F m ) which form the components of the unipotent group H, we have translated the whole setup into an abstract multilinear setting and related actions by technical reasons. This gives more general results although we have only applications in the theory of sheaves. The reader should always keep in mind the motivation in 5.1.

The results obtained in the simplest case (morphisms of type [START_REF] Bialynicki-Birula | Open subsets of projective space with a good quotient by an action of a reductive group[END_REF][START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF] or [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF][START_REF] Bialynicki-Birula | Open subsets of projective space with a good quotient by an action of a reductive group[END_REF]) are stated in 1.5. They are characteristic for the general case in which only the conditions are more complicated.

Remark on finite generatedness

One would expect that the quotients of W could be obtained by first forming the quotient W/H with respect to the unipotent radical H and then in a second step a quotient of W/H by G/H ∼ = G red . However, the actions of unipotent groups behave generally very badly, [START_REF] Greuel | Geometric quotients of unipotent group actions[END_REF], and we are not able to prove that the algebra C[W/H] is finitely generated. This would be an essential step in a direct construction of the quotient. Of course, the main difficulty also in this paper arises from the presence of the group H. The counterexample of M.Nagata, [START_REF] Nagata | On the 14th problem of Hilbert[END_REF], also shows that the finite generatedness depends on the dimensions of the problem. So from a philosophical point of view we are determining bounds for the dimensions involved under which we can expect local affine G-invariant coordinate rings which are finitely generated, and thus to obtain good quasi-projective quotients, even so the bounds might not be the best. The simple examples 4.2, 4.3 show that a good quotient W ss (G, Λ)//G might not exist if the conditions are not fulfilled.

1.5. Morphisms of type [START_REF] Bialynicki-Birula | Open subsets of projective space with a good quotient by an action of a reductive group[END_REF][START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF] In this case the homomorphisms of sheaves are of the type

m 1 E 1 ⊕ m 2 E 2 -→ n 1 F 1 ,
where we use the notation mE for C m ⊗ E. For this type a polarization is given by a pair (λ 1 , λ 2 ) of positive rational numbers such that λ 1 m 1 + λ 2 m 2 = 1. It is determined by the rational number t = m 2 λ 2 which lies in [0, 1]. Writing W ss (t) for W ss and W s (t) for W s for the moment, our results depend on constants c(k) defined as follows : Let τ : Hom(E 1 , F 1 ) * ⊗ Hom(E 1 , E 2 ) -→ Hom(E 2 , F 1 ) * be the linear map induced by the composition map Hom(E 2 , F 1 ) ⊗ Hom(E 1 , E 2 ) → Hom(E 1 , F 1 ) , and

τ k = τ ⊗ I C k : Hom(E 1 , F 1 ) * ⊗ (Hom(E 1 , E 2 ) ⊗ C k ) -→ Hom(E 2 , F 1 ) * ⊗ C k .
Let K be the set of proper linear subspaces K ⊂ Hom(E 1 , E 2 ) ⊗ C k such that for every proper linear subspace

F ⊂ C k , K is not contained in Hom(E 1 , E 2 ) ⊗ F . Let c(k) = sup K∈K ( codim(τ k (Hom(E 1 , F 1 ) * ⊗ K) codim(K)
).

Theorem:

There exists a good projective quotient W ss (t)//G and a geometric quotient

W s (t)/G if t > m 2 dim(Hom(E 1 , E 2 )) dim(Hom(E 1 , E 2 )) + m 1 and t > dim(Hom(E 1 , E 2 )).c(m 2 ) m 2 n 1 .
In the case of morphisms m 1 O(-2)⊕m 2 O(-1) -→ n 1 O on projective spaces the constants have been computed in [START_REF] Drézet | Espaces abstraits de morphismes et mutations[END_REF] and we obtain the more explicit result :

1.5.2. Theorem: Let n ≥ 2 be an integer. There exists a good projective quotient W ss (t)//G and a geometric quotient W s (t)/G in the case of morphisms

m 1 O(-2) ⊕ m 2 O(-1) -→ n 1 O on the projective space P n if t > (n + 1)m 2 (n + 1)m 2 + m 1 , t > (n + 1)m 2 2 (m 2 -1) 2n 1 (m 2 (n + 1) -1) if 2 ≤ m 2 ≤ n + 1, t > (n + 1) 2 m 2 2(n + 2)n 1 if m 2 > n + 1.

Construction of fine moduli spaces of torsion free sheaves

In section 10 we construct smooth projective fine moduli spaces of torsion free coherent sheaves on P n using morphisms

( * ) O(-2) ⊗ C 2 -→ O(-1) ⊕ (O ⊗ C k ), (for (n + 1)(n + 2)/2 < k < (n + 1) 2 )
. More precisely we prove that for all polarizations, semi-stable morphisms are injective outside a closed subvariety of codimension ≥ 2, hence their cokernels are torsion free sheaves. A generic morphism is injective and its cokernel is locally free. In this case we can construct

q = (n + 1)(n + 2) 2 - n + k + 1 2 distinct smooth projective moduli spaces M 1 , • • • , M q of such morphisms, of dimension 2(n -1) + k((n + 1) 2 -k)
. Moreover, all the M i are birational to each other. For 1 ≤ i ≤ q, we construct a coherent sheaf E i on M i × P n , flat over M i , such that for every closed point z ∈ M i , E iz is isomorphic to the cokernel of the morphism ( * ) corresponding to z.

We prove that M i is a fine moduli space of torsion free sheaves with universal sheaf E i . In particular, this means that for every closed point z ∈ M i , the Kodaïra-Spencer map

T z M i -→ Ext 1 (E iz , E iz )
is bijective, and for any two distinct closed points z 1 , z 2 ∈ M i , the sheaves E iz 1 , E iz 2 are not isomorphic.

Open problems

Even in the simplest case of morphisms of type (2,1) we do not know what all the polarizations are for which a good quotient W ss //G exists. More generally it would be interesting to find all the saturated open subsets U of W such that a good quotient (quasiprojective or not) U//G exists, or all the open subsets U such that a geometric quotient U/G exists.

The corresponding problem for reductive groups has been studied in [START_REF] Mumford | Geometric invariant theory[END_REF], 1.12, 1.13, and in [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF], [START_REF] Bialynicki-Birula | Open subsets of projective space with a good quotient by an action of a reductive group[END_REF].

Organization of the paper

In section 2 we describe our problem in terms of multilinear algebra.

In section 3 we recall results of A. King, [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF]. The reductive group actions considered in this paper, the action of G red on W and that of G on W, are particular cases of [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF]. We also discuss the relation of Λ-(semi-)stability in W with that in the projective space PW . But we cannot work solely on the projective niveau, because the embedding W ⊂ W is not linear.

After defining G-(semi-)stability for the non-reductive group in section 4 we describe the embedding in section 5 and introduce the associated polarizations.

Section 6 contains the step of constructing the quotient W ss (G, Λ)//G using the GITquotient W ss (G, Λ)//G of A. King.

Sections 7 and 8 are the hard parts of the paper. Here the conditions of the weights which define good polarizations are derived. It seems that the constants appearing in these estimates had not been considered before.

In section 9 we are investigating a few examples in order to test the strength of the bounds. Here we restrict ourselves to small type (2, 1), (2, 2), [START_REF] Dixmier | Quelques aspects de la théorie des invariants[END_REF][START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF] in order to avoid long computations of the constants which give the bounds for the polarizations. What we discover in varying the polarizations are flips between the moduli spaces, as one has to expect from the general results on the variation of linearizations of group actions, cf. [START_REF] Reid | What is a flip[END_REF], [START_REF] Dolgachev | Variation of Geometric Invariant Theory Quotients[END_REF], [START_REF] Thaddeus | Geometric invariant theory and flips[END_REF]. In example 9.2 we have a very simple effect of a flip, but in example 9.5 the chambers of the polarizations look already very complicated.

In section 10 we define new fine moduli spaces of torsion free sheaves using our moduli spaces of morphisms.

Acknowledgement. The work on this paper was supported by DFG. The first author wishes to thank the University of Kaiserslautern, where the work was started, for its hospitality.

The moduli problem for decomposable homomorphisms

Let E = ⊕ E i ⊗ M i and F = ⊕ F l ⊗ N l be semi-simple sheaves as in the introduction.

In order to describe the action of G = Aut(E) × Aut(F) on W = Hom(E, F) in greater detail we use the abbreviations

H li = Hom(E i , F l ) A ji = Hom(E i , E j ) B ml = Hom(F l , F m ) ,
such that we are given the natural pairings

H lj ⊗ A ji → H li for i ≤ j A kj ⊗ A ji → A ki for i ≤ j ≤ k B ml ⊗ H li → H mi for l ≤ m B nm ⊗ B ml → B nl for l ≤ m ≤ n.
The group G consists now of pairs (g, h) of matrices

g =       g 1 0 • • • 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 • • • u r,r-1 g r       and h =       h 1 0 • • • 0 v 21 h 2 . . . . . . . . . . . . 0 v s1 • • • v s,s-1 h s       with diagonal elements g i ∈ GL(M i ), h l ∈ GL(N l ) and u ji ∈ Hom(M i , M j ⊗ A ji ), v ml ∈ Hom(N l , N m ⊗ B ml ). Similarly a homomorphism w ∈ Hom(E, F) is represented by a matrix w = (ϕ li ) of ho- momorphisms ϕ li ∈ Hom(M i , N l ⊗ H li ) = Hom(H * li ⊗ M i , N l ).
Using the natural pairings, the left action (g, h).w = hwg -1 of G on W is described by the matrix product

      h 1 0 • • • 0 v 21 h 2 . . . . . . . . . . . . 0 v s1 • • • v s,s-1 h s       •      ϕ 11 • • • ϕ 1r . . . . . . ϕ s1 • • • ϕ sr      •       g 1 0 • • • 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 • • • u r,r-1 g r       -1
, where the compositions v ml • ϕ li and ϕ lj • u ji are compositions as sheaf homomorphisms but can also be interpreted as compositions induced by the pairings of the vector spaces above. Thus the group G, the space W and the action are already determined by the vector spaces A ji , B ml , H li and the pairings between them. Therefore, in the following we define G, W and the actions G × W → W by abstractly given vector spaces and pairings. The resulting statements can then be applied to systems of sheaves by specifying the spaces as spaces of homomorphisms as above.

The abstract setting

Let r, s be positive integers and let for 1 ≤ i ≤ j ≤ r, 1 ≤ l ≤ m ≤ s finite dimensional vector spaces A ji , B ml , H li be given, where we assume that A ii = C and B ll = C. Moreover we suppose that we are given linear maps, called compositions,

H lj ⊗ A ji → H li for 1 ≤ i ≤ j ≤ r, 1 ≤ l ≤ s A kj ⊗ A ji → A ki for 1 ≤ i ≤ j ≤ k ≤ r B ml ⊗ H li → H mi for 1 ≤ i ≤ r, 1 ≤ l ≤ m ≤ s B nm ⊗ B ml → B nl for 1 ≤ l ≤ m ≤ n ≤ s.
We assume that all these maps and the induced maps

H * li ⊗ A ji → H * lj and H * mi ⊗ B ml → H * li
are surjective. This is the case when all the spaces are spaces of sheaf homomorphisms as above for which the sheaves E i and F l are line bundles on a projective space or each of them is a bundle Ω p (p).

We may and do assume that these pairings are the identities if i = j, l = m etc. . Finally, we suppose that these maps verify the natural associative properties of compositions. This means that the induced diagrams

A kj ⊗ A ji ⊗ A ih / / A ki ⊗ A ih A kj ⊗ A jh / / A kh B on ⊗ B nm ⊗ B ml / / B om ⊗ B nl B on ⊗ B nl / / B ol H lk ⊗ A kj ⊗ A ji / / H lj ⊗ A ji H lk ⊗ A ki / / H li B nm ⊗ B ml ⊗ H li / / B nl ⊗ H li B nm ⊗ H mi / / H ni B ml ⊗ H lj ⊗ A ji / / H mj ⊗ A ji B ml ⊗ H li / / H mi
are commutative for all possible combinations of indices.

In our setup we also let finite dimensional vector spaces M i for 1 ≤ i ≤ r and N l for 1 ≤ l ≤ s be given and we consider finally the vector space

W = ⊕ i,l Hom(M i , N l ⊗ H li ) = ⊕ i,l Hom(H * li ⊗ M i , N l )
where summation is over 1 ≤ i ≤ r and 1 ≤ l ≤ s. This is the space of homomorphisms in the abstract setting. The group G and its action on W are now also given in the abstract setting as follows.

The group G

We define G as a product G L × G R of two groups where the left group G L replaces Aut(E) and the right group G R replaces Aut(F) in our motivation. Let G L be the set of matrices

      g 1 0 • • • 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 • • • u r,r-1 g r       with g i ∈ GL(M i ) and u ji ∈ Hom(M i , M j ⊗ A ji ) = Hom(A * ji ⊗ M i , M j ).
The group law in G L is now defined as matrix multiplication where we define the compositions u kj * u ji naturally according to the given pairings as the composition

M i u ji -→ M j ⊗ A ji u kj ⊗id ----→ M k ⊗ A kj ⊗ A ji id⊗comp -----→ M k ⊗ A ki .
Explicitly, if g has the entries g i , u ji and g has the entries g i , u ji then the product g = g g in G L is defined as the matrix with the entries g i = g i • g i in the diagonal and

u ki = u ki • g i + i<j<k u kj * u ji + (g k ⊗ id) • u ki for 1 ≤ i < k ≤ r.
The verification that this defines a group structure on G L is now straightforward.

As a set G L is the product of all the GL(M i ) and all Hom(M i , M j ⊗ A ji ) for i < j and thus has the structure of an affine variety. Since multiplication is composed by a system of bilinear maps it is a morphism of affine varieties. Hence G L is naturally endowed with the structure of an algebraic group. The group G R is now defined in the same way by replacing the spaces M i and A ji by N l and B ml . Finally G = G L × G R is defined as an algebraic group.

The action of G and W

We will define a left action of G R and a right action of G L on W such that the action of G on W can be defined by (g, h).w = h.w.g -1 . Both actions are defined as matrix products as described above in the case of sheaf homomorphisms using the abstract compositions as in the definition of the group law.

If w has the entries ϕ li ∈ Hom(H * li ⊗ M i , N l ) and g ∈ G L has the entries g i and u ij then w.g is defined as the matrix product

     ϕ 11 • • • ϕ 1r . . . . . . ϕ s1 • • • ϕ sr      =      ϕ 11 • • • ϕ 1r . . . . . . ϕ s1 • • • ϕ sr            g 1 0 • • • 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 • • • u r,r-1 g r       with ϕ li = ϕ li • g i + i<j ϕ lj * u ji (if i = r the last sum is 0),
where ϕ lj * u ji is the composition

M i → M j ⊗ A ji → N l ⊗ H lj ⊗ A ji → N l ⊗ H li
or dually the composition

H * li ⊗ M i → H * lj ⊗ A * ji ⊗ M i → H * lj ⊗ M j → N l .
The left action of G R is defined in the same way. In the next two sections we give an analysis of stability and semi-stability for the action of G and its natural reductive subgroup G red . In the reductive case this is due to A. King.

Canonical subgroups of G

We let H L ⊂ G L and H R ⊂ G R be the maximal normal unipotent subgroups of G L and G R defined by the condition that all g i = id M i and all h l = id N l . Then H = H L × H R is a maximal normal unipotent subgroup of G. Similarly we consider the reductive subgroups G L,red and G R,red of G L and G R defined by the conditions u ji = 0 and v ml = 0 for all indices. Then G red = G L,red ×G R,red is a reductive subgroup of G and it is easy to see that G/H ∼ = G red . The restricted action of G red is much simpler and reduces to the natural actions of GL(M i ) on M i and GL(N l ) on N l 3. Actions of reductive groups

Results of A. King

Let Q be a finite set, Γ ⊂ Q × Q a subset such that the union of the images of the two projections of Γ is Q. For each α ∈ Q, let m α be a positive integer, M α a vector space of dimension m α and for each (α, β) ∈ Γ, let V αβ be a finite dimensional nonzero vector space. Let

W 0 = ⊕ (α,β)∈Γ Hom(M α ⊗ V αβ , M β ).
On W 0 we have the following action of the reductive group

G 0 = α∈Q GL(M α )
arising naturally in this situation. If (f βα ) ∈ W 0 and (g α ) ∈ G 0 , then

(g α ).(f βα ) = (g β • f βα • (g α ⊗ id) -1 ).
Let (e α ) α∈Q be a sequence of integers such that α∈Q e α m α = 0.

To this sequence is associated the character χ of G 0 defined by

χ(g) = α∈Q det(g α ) -eα .
This character is trivial on the canonical subgroup of G 0 isomorphic to C * (for every λ ∈ C * , the element (g α ) of G 0 corresponding to λ is such that g α = λ.id for each α). This subgroup acts trivially on W 0 . A point x ∈ W 0 is called χ-semi-stable if there exists an integer n ≥ 1 and a polynomial f ∈ C[W 0 ] which is χ n -invariant and such that f (x) = 0 (f is called χ n -invariant if for every w ∈ W 0 and g ∈ G 0 we have

f (gw) = χ n (g)f (w)). The point x is called χ-stable if moreover dim(G 0 x) = dim(G 0 /C * ) and if the action of G 0 on {w ∈ W 0 , f (w) = 0} is closed.
A. King proves in [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF] the following results :

(1) A point x = (f βα ) ∈ W 0 is χ-semi-stable (resp. χ-stable) if and only if for each family (M α ), α ∈ Q, of subspaces M α ⊂ M α which is neither the trivial family (0) nor the given family (M α ) and which satisfies

f βα (M α ⊗ V αβ ) ⊂ M β
for each (α, β) ∈ Γ, we have α∈Q e α dim(M α ) ≤ 0 (resp. < 0 ).

(2) Let W ss 0 (resp. W s 0 ) be the open subset of W 0 consisting of semi-stable (resp stable) points. Then there exists a good quotient π : W ss 0 -→ M by G 0 /C * which is a projective variety.

(3) The restriction of this quotient

W s 0 -→ M s = π(W s 0
) is a geometric quotient and M s is smooth.

Polarizations

The (semi-)stable points of W 0 remain the same if we replace (e α ) by (ce α ), c being a positive integer. So the notion of (semi-)stability is fully described by the reduced parameters ( eα t ), where t = α∈Q,eα>0

e α m α .

So we can define the polarization of the action of G 0 on W 0 by any sequence (c α ) α∈Q of nonzero rational numbers such that

α∈Q c α m α = 0 , α∈Q,cα>0 c α m α = 1.
By multiplying this sequence by the smallest common denominator of the c α we obtain a sequence (e α ) of integers and the corresponding character of G 0 . Therefore the loci of stable and semi-stable points of W 0 with respect to G 0 and a polarization Λ 0 = (c α ) are well defined and denoted by W s 0 (G 0 , Λ 0 ) and W ss 0 (G 0 , Λ 0 ).

Conditions imposed by the non-emptiness of the quotient

If W s 0 is not empty, the e α must satisfy some conditions. We will derive this only in the three situations which occur in this paper. Polarizations satisfying these necessary conditions will be called proper. The first is that of the action of G red in 2.4 and the second is that of G and W in section 5, and the third is the case in between occurring in 7.4.2.

First case

Let r, s be positive integers. We take

Q = {α 1 . . . , α r , β 1 , . . . , β s }, Γ = {α 1 . . . , α r } × {β 1 , . . . , β s }.
This is the case of morphisms of type (r, s). For 1 ≤ i ≤ r, let M α i = M α i if e α i > 0, and {0} otherwise, and for 1 ≤ l ≤ s, let M β l = M β l . Then if one e α i is not positive, we have α∈Q e α dim(M α ) ≥ 0 and (M α ) = (M α ), so in this case no point of W 0 is stable. So we obtain , if W s 0 is non-empty, the conditions e α i > 0 , for any i, and e β l < 0 , for any l.

A proper polarization is in this case a sequence (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) of rational numbers such that the λ i and the µ l are positive and satisfy

1≤i≤r λ i m α i = 1≤l≤s µ l m β l = 1.

Second case

This case appears when we use a bigger reductive group to define the quotient (this is the case of W later on). Let r, s be positive integers. Here we take

Q = {α 1 . . . , α r , β 1 , . . . , β s }, Γ = {(α i , α i-1 ), 2 ≤ i ≤ r, (α 1 , β s ), (β l , β l-1 ), 2 ≤ l ≤ s}.
Then the necessary conditions for W s 0 to be non-empty are: i≤j≤r e α j m α j > 0 for any i, and 1≤l≤m e β l m β l < 0 for any m.

To derive the first set of conditions we consider for any i the family (M γ ) for which

M α j = 0 if i ≤ j ≤ r and M γ = M γ for all other γ ∈ Q. Then f αβ (M α ⊗ V αβ ) ⊂ M β for any f ∈ W 0 and any (α, β) ∈ Γ. If f is stable we obtain - i≤j≤r e α j m α j = γ∈Q e γ dim(M γ ) < 0 Moreover, if the family (M γ ) is defined by M α j = 0 for 1 ≤ j ≤ r, M β l = 0 if m < l ≤ s and M γ = M γ else, we obtain directly 1≤l≤m e β l m β l = γ∈Q e γ dim(M γ ) < 0.
A proper polarization in this case is then a sequence (ρ 1 , . . . , ρ r , -σ 1 , . . . , -σ s ) of rational numbers satisfying

1≤i≤r ρ i m α i = 1≤l≤s σ l m β l = 1.
and i≤j≤r ρ j m α j > 0 for any i and 1≤l≤m σ l m β l > 0 for any m.

We could also drop the normalization condition.

Third case

This case is a combination of the first and second case. It appears in the proof of the equivalence of semi-stability in 7.3. Here Q is the same as in the previous cases and

Γ = {(α i , α i-1 ) , 2 ≤ i ≤ r , (α 1 , β l ) , 1 ≤ l ≤ s}.
Now the necessary conditions for W s 0 to be non-empty are: i≤j≤r e α j m α j > 0 for any i, and e β l < 0 for any l.

The first condition follows as in the second case when we consider the family (M γ ) with M α j = 0 for i ≤ j ≤ r and M γ = M γ for all other γ ∈ Q. The second condition follows when all M γ are zero except M β l = M β l for one l. Again a proper polarization in this case is a sequence (ρ 1 , . . . , ρ r , -µ 1 , . . . , -µ l ) with

1≤i≤r ρ i m α i = 1≤l≤s µ l m β l = 1
and i≤j≤r ρ j m α j > 0 for any i and µ l > 0 for any l.

3.4.

The action of G 0 on P(W 0 )

We suppose that we are in one of the first two preceding cases and that there exist stable points in W 0 . Let P be a nonzero homogeneous polynomial, χ n -invariant for some positive integer n. The χ n -invariance implies that P has degree n.t where in case 1 (action of G red on W )

t = 1≤i≤r e α i m α i ,
and in case 2 (action of G on W)

t = 1≤i≤r ie α i m α i - 1≤l≤s (s -l)e β l m β l .
To see this let λ ∈ C * and let g be given by g α i = λ -1 id and g β l = id in the first case and by g α i = λ -i id and g β l = λ l-s id in the second case. Then gx = λx and χ n (g) = λ nt in both cases, such that P (λx) = λ nt P (x). Now we will see that there exists a G 0 -line bundle L on P(W 0 ) such that the set W ss 0 of semi-stable points is exactly the set of points over P(W 0 ) ss (G 0 , L), which is the set of semi-stable points in the sense of Geometric Invariant Theory corresponding to

L = O P(W 0 ) (t),
cf. [START_REF] Mumford | Geometric invariant theory[END_REF], [START_REF] Newstead | Introduction to moduli problems and orbit spaces[END_REF], [START_REF] Popov | Algebraic Geometry IV: Linear algebraic groups, invariant theory[END_REF]. Here the action of G 0 on L is the natural action multiplied by χ. More precisely, the action of G 0 on W 0 induces an action of this group on S t W 0 and on S t W * 0 by:

(g.F )(w) = F (g -1 w)
for all g ∈ G 0 , w ∈ W 0 and F ∈ S t W * 0 , viewed as an homogeneous polynomial of degree t on W 0 . The line bundle space L of L is acted on by G 0 in the same way : if ξ ∈ L <w> then g.ξ ∈ L <gw> is the form on < gw > * ⊗t = L <gw> given by (g.ξ)(y) = ξ(g -1 y). We modify now the action of G 0 on L (resp. S t W * 0 ) by multiplying with χ(g) :

g * ξ = χ(g)g.ξ for ξ ∈ L <w> , or g * F = χ(g)g.F for F ∈ H 0 (P(W 0 ), L) = S t W * 0 .
Now P ∈ H 0 (P(W 0 ), L ⊗n ) is an invariant section if and only if P is a homogeneous polynomial of degree tn which satisfies

P (gw) = χ n (g)P (w).
From the definition of semi-stable points in W 0 and P(W 0 ) with respect to the modified G 0 -structure on L = O P(W 0 ) (t), we get immediately 3.4.1. Lemma: Assume that W s 0 (G 0 , Λ 0 ) = ∅ and let t be defined as above in the two cases of W 0 . Then the set W ss 0 (G 0 , Λ 0 ) is the cone of the set P(W 0 ) ss (G 0 , O P(W 0 ) (t)) as defined in G.I.T.

There are two definitions of stable points in P(W 0 ), the classical one, given in [START_REF] Mumford | Geometric invariant theory[END_REF], [START_REF] Newstead | Introduction to moduli problems and orbit spaces[END_REF], and a more recent one, given in [START_REF] Popov | Algebraic Geometry IV: Linear algebraic groups, invariant theory[END_REF]. If we take D. Mumford's definition, the cone of the set of stable points in P(W 0 ) does not coincide with W s 0 because every point of P(W 0 ) has a stabilizer of positive dimension. In fact there is a subgroup of G 0 /C * of positive dimension which acts trivially on P(W 0 ). In the first case for example such a group is given by g α i = λid and g β l = µid with λ, µ ∈ C * . If we want to keep the coincidence between the sets of stable points for one and the same group, we would have to consider the action of a smaller reductive group in order to eliminate additional stabilizers. We will do this in 3.5 only in the first case. If we take the definition of V.L. Popov and E.G. Vinberg, then we obtain that the set W s 0 (G 0 , Λ 0 ) is exactly the cone of the set

P(W 0 ) s (G 0 , O P(W 0 ) (t))

The group G

Let G and W be as in section 2 and let Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) be a proper polarization as in 3.3.1 for the action of G red on W . It is then convenient to use the subgroup G red of G red consisting of elements ((g i ), (h l )) satisfying

1≤i≤r det(g i ) a i1 = 1≤l≤s det(h l ) b sl = 1
, where a ji = dim(A ji ) and b ml = dim(B ml ).

We consider the action of G red on L induced by the modified χ-action of G red . Now the set W s (G red , Λ) of χ-stable points of W is exactly the cone over the locus P(W ) s (G red , L) of stable points of P(W ) in the sense of Geometric Invariant Theory.

Semi-stability in the non-reductive case

Let G and W be as in section 2. A character χ on G red as in King's setup can be extended to a character of G. Also the modified action of G red on L can be extended to an action of G. Let G be the subgroup of G defined by the same equations as for G red . It contains H and G red , and we have G /H G red .

In the case of the action of G red on W a proper polarization is given by a sequence λ 1 , . . . , λ r , µ 1 , . . . , µ s of positive rational numbers such that

1≤i≤r λ i m i = 1≤l≤s µ l n l = 1.
More precisely, the polarization is exactly the sequence (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ). The parameter λ i (resp. µ l ) will be called the weight of the vector space M i (resp. N l ). We see that the dimension of the set of possible proper polarizations is r + s -2. Let t denote the smallest common denominator of the numbers λ i and µ l and χ the character of G red defined by the sequence of integers (-tλ 1 , . . . , -tλ r , tµ 1 , . . . , tµ s ). Let

L = O P(W ) (t) with t = 1≤i≤r m i tλ i .
As we have seen, if we consider the modified action of G red on L, then the χ-semistable points of W are exactly those over the semi-stable points of P(W ) in the sense of Geometric Invariant Theory with respect to the action of G red /C * on L. The χ tn -invariant polynomials are the G red -invariant sections of L n .

We are now going to define a notion of (semi-)stability for the points of W with respect to the given action of the non-reductive group G. Let H ⊂ G be the above unipotent group, see also 2.4.

Definition:

A point w ∈ W is called G-semi-stable (resp. G-stable) with respect to the (proper) polarization Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) if every point of Hw is G red - semi-stable (resp.
G red -stable) with respect to this polarization.

We denote these sets by W ss (G, Λ) resp. W s (G, Λ).

For many of the quotient problems for the spaces of homomorphisms between ⊕m i E i and ⊕n j F j and their cokernel sheaves this is a fruitful notion. In 4.2 we investigate an example with an explicit description of the open sets W s (G, Λ) ⊂ W s (G red , Λ). This example also shows that the existence of a good quotient depends on the choice of the polarization.

4.1.1. Situation for type (2,1):

In the case of morphisms of type (2, 1) we have µ 1 = 1/n 1 and the polarization is completely described by the single parameter t = m 2 λ 2 . We must have 0 < t < 1.

A polarization such that there exists integers

m 1 , m 2 , n 1 , with 0 < n 1 < n 1 , 0 ≤ m i ≤ m i , such that m 1 n 1 -m 1 n 1 , m 2 n 1 -m 2 n 1
are not both 0, and that

λ 1 m 1 + λ 2 m 2 = n 1 n 1 is called singular.
There are only finitely many singular polarizations, corresponding to the values 0

< t 1 < t 2 < • • • < t p < 1 of t. Let t 0 = 0, t p+1 = 1. If Λ, Λ are polarizations corresponding to parameters t, t such that for some i ∈ {0, • • • , p} we have t i < t, t < t i+1 , then W ss (G, Λ) = W ss (G, Λ ) and W s (G, Λ) = W s (G, Λ ).
Hence there are exactly 2p + 1 notions of G-(semi-)stability in this case. Moreover, if m 1 , m 2 and n 1 are relatively prime, and Λ is a non singular polarization, we have

W ss (G, Λ) = W s (G, Λ).
In the general case of morphisms of type (r, s), it is not difficult to see that there are only finitely many notions of G-(semi-)stability.

Remark:

In [START_REF] Fauntleroy | Geometric invariant theory for general algebraic groups[END_REF] semi-stability is defined as follows : A point w ∈ W is semi-stable if there exists a positive integer k and a G -invariant section s of L k such that s(w) = 0 (there is also a condition on the action of H). It is clear that a semi-stable point in the sense of Fauntleroy is also G-semi-stable with respect to (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ). It is proved in [START_REF] Fauntleroy | Geometric invariant theory for general algebraic groups[END_REF] that there exists a categorical quotient of the open subset of semi-stable points in the sense of [START_REF] Fauntleroy | Geometric invariant theory for general algebraic groups[END_REF], but it is not clear that all G-semi-stable points are semi-stable. Moreover, in the general situation of [START_REF] Fauntleroy | Geometric invariant theory for general algebraic groups[END_REF] there is no way to impose conditions which would imply that the categorical quotient is a good quotient or even projective. Using definition 4.1 we are able to derive a criterion for the existence of a good and projective quotient of W under the action of G.

Existence and non-existence of good quotients, an example

We show here that we cannot expect that a good quotient W ss (Λ, G)//G will exist for any polarization Λ.

We consider morphisms 2O(-2) → O(-1) ⊕ O on P 2 . There are 3 notions of G-(semi-)stability in this case, two corresponding to non singular polarizations. For one of the non singular polarizations the quotient W s (Λ, G)/G exists and for the other we prove the inexistence of a good quotient W s (Λ, G)//G.

Let V be a complex vector space of dimension 3, and P 2 = PV . Let W = Hom(2O(-2), O(-1) ⊕ O) on P 2 . A polarization for the action of G on W is a triple (1/2, -µ 1 , -µ 2 ) of positive rational numbers such that µ 1 + µ 2 = 1. As in 4.1.1 a polarization depends only on µ 1 . There is only one singular polarization, corresponding to µ 1 = 1/2. Hence if we consider only non singular polarizations there are only two notions of G-(semi-)stability, the first one corresponding to polarizations such that µ 1 > 1/2 and the second to polarizations such that µ 1 < 1/2. In both cases semi-stable points are already stable. We are going to show that in the first case W s (G, Λ) has a geometric quotient which is projective and smooth and that in the second case W s (G, Λ) doesn't even admit a good quotient.

The elements x ∈ W and g ∈ G are written as matrices

x = z 1 z 2 q 1 q 2 and g = σ, α 0 z β where z 1 , z 2 ∈ V * , q 1 , q 2 ∈ S 2 V * , σ ∈ GL(2), α, β ∈ C * and z ∈ V * . 4.2.1. The case µ 1 > 1/2
In this case W s (G, Λ) has a geometric quotient which is the universal cubic Z ⊂ PV × PS 3 V * of the Hilbert scheme of plane cubic curves in P 2 = PV . The quotient map is given by x → (

z 1 ∧ z 2 , z 1 q 2 -z 2 q 1 )
Remark: If µ 1 > 3/4, then µ 1 > 3µ 2 and the conditions of 1.5.1 (in the dual case (1, 2))for a good and projective quotient to exist in this case are satisfied.

The proof is done in several steps.

(1) claim 1: Let x ∈ W be as above. Then

(i) x ∈ W s (G red , Λ) if and only if z 1 ∧ z 2 = 0 in Λ 2 V * and q 1 , q 2 are not both zero. (ii) x ∈ W s (G, Λ) if and only if z 1 ∧ z 2 = 0 and det(x) = z 1 q 2 -z 2 q 1 = 0 in S 3 V * .
Proof of claim 1. (i) follows easily from the criterion (1) in 3.1. As for (ii) let x ∈ W s (G red , Λ) with det(x) = 0. Then det(h.x) = det(x) = 0 for any h = ( 1 0 z 1 ) which implies that also h.x ∈ W s (G red , Λ). Let conversely x ∈ W s (G, Λ). Then det(x) = 0 because otherwise there is a linear form z ∈ V * with q 1 = zz 1 and q 2 = zz 2 and with h = ( 1 0 -z 1 ) the element h.x is the matrix ( z 1 z 2 0 0 ) which is not in W s (G red , Λ).

(2) By the result of A. King in 3.1, (3), there is a geometric quotient W s (G red , Λ)/G red which is smooth and projective.

claim 2: W s (G red , Λ)/G red ∼ = P(Q * ⊗ S 2 V * ). Here Q * = Ω 1 (1)
is the dual of the tautological quotient bundle over PV . (The dimension of this quotient variety is 13 while dim W = 18 and dim G red /C * = 5).

To verify claim 2 we consider the map

x = z 1 z 2 q 1 q 2 α → ( z 1 ∧ z 2 , z 1 ⊗ q 2 -z 2 ⊗ q 1 ) from W s (G red , Λ) to PV × P(V * ⊗ S 2 V * ) ⊂ P(V * ⊗ S 2 V * ⊗ O PV ) where we identify PΛ 2 V * with PV via z 1 ∧z 2 ↔ a , z 1 (a) = z 2 (a) = 0. Then each α(x) ∈ P(Q * a ⊗S 2 V * ) because Q * a ⊂ V * is the subspace of forms vanishing in a . It follows immediately that α is a morphism W s (G red , Λ) → P(Q * ⊗ S 2 V * )
which is surjective and G red -equivariant. It induces a morphism of the geometric quotient to P(Q * ⊗ S 2 V * ) which is even bijective. Since both, the quotient and the target are smooth, this is an isomorphism.

(3) Since Q * ⊂ V * ⊗ O PV we have an induced homomorphism Q * ⊗ S 2 V * → S 3 V * ⊗ O PV . It is the middle part of the canonical exact sequence 0 → Λ 2 Q * ⊗ V * → Q * ⊗ S 2 V * → S 3 V * ⊗ O PV ev -→ O PV (3) → 0
of vector bundles on PV . Let Z be the kernel of ev. From the left part of the sequence we obtain the affine bundle

P(Q * ⊗ S 2 V * ) P(Λ 2 Q * ⊗ V * ) β - → P(Z) ⊂ PV × PS 3 V * .
Here P(Z) = Z is nothing but the universal cubic and the fibres of β are isomorphic to V * .

claim 3: W s (G, Λ) ⊂ W s (G red , Λ) is the inverse image of P(Q * ⊗ S 2 V * ) P(Λ 2 Q * ⊗ V * ) under α and α |W s (G,Λ) is a geometric quotient with respect to G red . Proof of claim 3. z 1 ⊗ q 2 -z 2 ⊗ q 1 belongs to Λ 2 Q * a ⊗ V * if and only if z 1 q 2 -z 2 q 1 = 0, see (ii) of claim 1. (4) Let now π = β • α be the morphism W s (G, Λ) → Z, given by x → ( a , z 1 q 2 -z 2 q 1 ), where z 1 (a) = z 2 (a) = 0.
It is obviously G-equivariant and its fibres coincide with the G-orbits. Since α is a geometric quotient and β is an affine bundle, then π is also a geometric quotient.

Remark:

The variety Z is isomorphic to the moduli space M = M P 2 (3m + 1) of stable coherent sheaves on P 2 with Hilbert polynomial χF(m) = 3m + 1. This had been verified by J. Le Potier in [START_REF] Potier | Faisceaux semi-stables de dimension 1 sur le plan projectif[END_REF]. The space W s (G, Λ) is a natural parametrization of M because any F ∈ M can be presented in an extension sequence 0 → O C → F → C p → 0 where C is the cubic curve supporting F and p ∈ C, and then F has a resolution

0 → 2O(-2) x - → O(-1) ⊕ O → F → 0.
This resolution is the Beilinson resolution as can easily be verified. Moreover, x is (G, Λ)stable if and only if F is stable. (If p is a smooth point of C, then F is the line bundle O C (p) and if p is a singular point of C, then F is the unique Cohen-Macaulay module on C with the given polynomial). There is an obvious universal family F on W s (G, Λ)× H PV which defines a G-equivariant morphism W s (G, Λ) → M and then a bijective morphism Z → M , which by smoothness, is an isomorphism. One knows that M carries a universal family E. This family can be obtained as the non-trivial extension

0 → O Z× H Z → E → O ∆ → 0,
where H = PS 3 V * and Z × H Z ⊂ Z × PV, or can be obtained as the descent of the family F. More details can be found in [START_REF] Freiermuth | On the moduli space M p (P 3 ) of semi-stable sheaves on P 3 with Hilbert polynomial P (m) = 3m + 1[END_REF].

4.2.2. The case µ 1 < 1/2
We suppose now that the polarization Λ is such that µ 1 < 1/2. In this case an element x of W is G-stable if and only if z 1 , z 2 are not both zero, and if for every z ∈ V * , q 1 -zz 1 and q 2 -zz 2 are linearly independent.

Proposition:

For this polarization there does not exist a good quotient W s (G, Λ)//G.

Proof. Let z 1 be a non-zero element of V * , let q ∈ S 2 V * \z 1 V * , and let x ∈ W be the matrix

z 1 0 q z 2 1 .
Then x is stable.

Claim : The orbit Gx is closed and if y ∈ W s (G, Λ) is such that Gy meets Gx, then y ∈ Gx.
Before proving the claim, we will show that it implies proposition 4.2.3. The stabilizer of a generic point in W s (G, Λ) is isomorphic to C * : it consists of pairs of homotheties (λ, λ). It follows that if M = W s (G, Λ)//G exists, then all the fibers of the quotient morphism π : W s (G, Λ) → M are of dimension at least dim(G) -1. Now suppose that the claim is true. Then this implies that π -1 (π(x)) = Gx. But the stabilizer G x of x has dimension 2 : it consists of pairs

( α 0 β α , α 0 βz 1 α ) with α ∈ C * , β ∈ C, and hence has dimension 2. It follows that dim(π -1 (π(x))) < dim(G) -1, a contradiction. Proof of the claim. Let y ∈ W s (G, Λ) such that x ∈ Gy. Let y = z z 2 q 1 q 2 .
Then z 1 is contained in the vector space spanned by z and z 2 . Hence by replacing y with an element of Gy we can assume that z = z 1 and that z 2 = 0 if z 2 is a multiple of z 1 .

According to lemma 4.2.4 there exists a smooth irreducible curve C, x 0 ∈ C, and a morphism θ :

C\{x 0 } -→ G such that θ : C\{x 0 } / / W t / / θ(t)y
can be extended to θ : C → W , with θ(x 0 ) = x. We can write, for t ∈ C\{x 0 },

θ(t) = a(t)z 1 + b(t)z 2 c(t)z 1 + d(t)z 2 q 1 (t) q 2 (t) with (1) q 1 (t) = λ(t) a(t)q 1 + b(t)q 2 + u(t)z 1 , (2)
q 2 (t) = λ(t) c(t)q 1 + d(t)q 2 + u(t)z 2 ,
where λ, a, b, c, d are morphisms C\{x 0 } → C and u : C\{x 0 } → V * . The morphisms λ, a, b, c, d can be extended to morphisms C → P 1 = C ∪ {0, ∞}, denoted by λ, a, b, c, d respectively, and u extends to u : C → P(V * ⊕ C). Now we use the fact that θ is defined at x 0 . The first consequence is that a(x 0 ) = 1, c(x 0 ) = 0, and if z 2 = 0 then b and d also vanish at x 0 . The second is that the morphisms q 1 , q 2 : C\{x 0 } → S 2 V * can be extended to q 1 , q 2 : C → S 2 V * , and we have q 1 (x 0 ) = q, q 2 (x 0 ) = z 2 1 . We will now consider three cases :

λ(x 0 ) = 0, λ(x 0 ) = ∞, λ(x 0 ) ∈ C * .
Suppose that λ(x 0 ) = 0. If z 2 = 0, then (1) implies that q 1 (x 0 ) = q is a multiple of z 1 , but this is not true. If z 2 = 0 then (2) implies that q 2 is a multiple of z 2 1 and (1) implies then that q is also a multiple of z 1 , which is not true. Hence we cannot have λ(x 0 ) = 0.

Suppose that λ(x 0 ) = ∞. If z 2 = 0, then (1) implies that µ : C\{x 0 } / / S 2 V * t / / a(t)q 1 + b(t)q 2 + u(t)z 1 and η : C\{x 0 } / / S 2 V * t / / c(t)q 1 + d(t)q 2 + u(t)z 1
extend to morphisms C → S 2 V * which vanish at x 0 . It follows from the fact that µ(x 0 ) = 0 that u = u(x 0 ) ∈ V * , and that q 1 = -uz 1 . Since q 1 = 0 (by G-stability of y), we have u = 0. But since c(x 0 ) = d(x 0 ) = 0, this contradicts the fact that η(x 0 ) = 0.

If z 2 = 0 then we deduce from the fact that µ(x 0 ) = 0 that q 1 ∈ <q 2 , V * z 1 >, which contradicts the G-stability of y.

It follows that we have δ = λ(x 0 ) ∈ C * . If z 2 = 0, using the fact that a(x 0 ) = 1 and b(x 0 ) = c(x 0 ) = d(x 0 ) = 0 we see that u = u(x 0 ) ∈ V * and that z 2 1 = δuz 2 , which contradicts the fact that z 1 ∧ z 2 = 0.

Hence we have z 2 = 0. It follows from (2) that d(x 0 ) ∈ C * and that z 2 1 = δd(x 0 )q 2 . By (1) we see that

: C\{x 0 } / / S 2 V * t / / b(t)q 2 + u(t)z 1
extends to C and that

(x 0 ) = 1 δ q -q 1 .
We have, if t = x 0

(t) = z 1 ( b(t) δd(x 0 ) z 1 + u(t)).
It follows that (x 0 ) is a multiple of z 1 : (x 0 ) = z 1 v. We have then

q 1 = 1 δ q -z 1 v and y = z 1 0 q 1 q 2 = z 1 0 1 δ q -z 1 v 1 δd(x 0 ) z 2 1 ∈ Gx as claimed.
It remains to show that Gx is closed. This can be proved easily by computing the stabilizers of all the points in W s (G, Λ). We see then that G x has the maximal possible dimension, hence Gx is closed.

We now give a proof of the lemma used in the preceeding proposition :

4.2.4. Lemma : Let W be a finite dimensional vector space, G a linear algebraic group acting algebraically on W , y ∈ W and x ∈ Gy\Gy. Then there exists a smooth curve C, x 0 ∈ C and a morphism

θ : C\{x 0 } -→ G such that the morphism θ : C\{x 0 } / / W t / / θ(t)y extends to θ : C → W and that θ(x 0 ) = x.
Proof. Let n = dim(W ), d = dim(Gy). The generic (n -d + 1)-dimensional affine subspace F ⊂ W through x meets Gy on a curve, and meets Gy\Gy in a finite number of points. Hence we can find a curve X ⊂ Gy that meets Gy\Gy only at x. Taking the normalization of X and substracting a finite number of points or unnecessary components if needed, we obtain a morphism α : Z → Gy (where Z is a smooth curve) and a point z 0 ∈ Z such that α(z 0 ) = x and α(Z\{z 0 }) ⊂ Gy. Consider now the restriction of α

Z\{z 0 } -→ Gy G/G y .
There exists a smooth curve Z and an etale surjective morphism φ : Then we can take C = U ∪ {y 0 }, x 0 = y 0 , and for t ∈ C\{x 0 } = U , we have

Z → Z\{z 0 } such that the principal G y -bundle φ * α * G on Z is locally trivial. By considering completions Z , Z of Z , Z
θ(t) = ψ(γ -1 (t, e)),
where ψ is the canonical morphism Γ → G.

More general counterexamples of inexistence of geometric quotients

Let W be the space of homomorphisms

O(-2) ⊕ O(-1) → C 2n ⊗ O(1)
over P n and let the homomorphism φ 0 ∈ W be given by the matrix

          z 2 0 z 1 z 2 1 . . . . . . z 2 0 z n z 2 n z 0 z 2 1 0 . . . . . . z 0 z 2 n 0          
where the z ν are homogeneous coordinates. The stabilizer of φ 0 contains C * and the pairs

1 0 az 0 1 , I n -aI n 0 I n in Aut(O(-2) ⊕ O(-1)) × GL(C 2n ) and thus has dimension ≥ 2. If Λ = (λ 1 , λ 2 , -µ 1 ) is a polarization with 0 < λ 1 , 0 < λ 2 < 1 2
, then it is easy to see that φ 0 is Λ-stable in the sense of 4.1. For example (m 1 , m 2 , n ) = (0, 1, n) is the dimension vector of a φ 0 -invariant choice of subspaces with

λ 1 m 1 + λ 2 m 2 -µ 1 n = λ 2 -1/2 < 0.
There are however stable homomorphisms φ ∈ W with stabilizer C * . Therefore W s (G, Λ)/G can never admit the structure of a geometric quotient. We will see in 7.2.2 that a sufficient condition for that in the case of this W is

λ 2 > (n + 1)λ 1 or λ 2 > n+1 n+2 because λ 1 + λ 2 = 1.

Embedding into a reductive group action

We will construct an algebraic reductive group G, a finite dimensional vector space W on which G acts algebraically, and an injective morphism

ζ : W -→ W
compatible with a morphism of groups

θ : G -→ G.
The traces of G-orbits on ζ(W ) will be exactly the G-orbits. The space W is of the same type as those studied in 3.1. We will associate naturally to any polarization of the action of G on W a character χ of G/C * , i.e. a polarization of the action of G on W.

We will prove that in certain cases a point w of W is G-(semi-)stable with respect to the given polarization if and only if ζ(w) is χ-(semi-)stable with respect to the associated polarization. The existence of a good and projective quotient of the open set of G-semistable points will follow from this.

Motivation in terms of sheaves

The idea for the embedding of W into a space W with a reductive group action is to replace the sheaves

E i in E = ⊕(E i ⊗ M i ) by E 1 ⊗ Hom(E 1 , E i ) and dually the sheaves F l in F = ⊕(F l ⊗ N l ) by F s ⊗ Hom(F l , F s ) * and then to consider the induced composed homomorphisms γ(Φ) for Φ ∈ Hom(E, F) = W E 1 ⊗ Hom(E 1 , E) → E → F → F s ⊗ Hom(F, F s ) *
in the bigger space W of all homomorphisms between E 1 ⊗ Hom(E 1 , E) and F s ⊗ Hom(F, F s ) * . This space is naturally acted on by the reductive group

G = GL(Hom(E 1 , E)) × GL(Hom(F, F s ) * ).
However it is not suitable enough for our purpose by two reasons. It does not allow enough polarizations as in section 3 for direct sums in order to have consistency of (semi-)stability and, secondly the group actions G×W → W and G× W → W don't have consistent orbits. Both insufficiencies are however eliminated when we consider the following enlargement of W . We set

P i = Hom(E i , E) and Q l = Hom(F, F l ) * ,
and introduce the auxiliary spaces

W L = ⊕ 1<i≤r Hom(P i ⊗ Hom(E i-1 , E i ), P i-1 ), W R = ⊕ 1≤l<s Hom(Q l+1 ⊗ Hom(F l , F l+1 ), Q l ),
and define

W = W L ⊕ Hom(E 1 ⊗ P 1 , F s ⊗ Q s ) ⊕ W R .
There are distinguished elements

(ξ 2 , • • • , ξ r ) ∈ W L , (η 1 , • • • , η s-1 ) ∈ W R
whose components are the natural composition maps. The embedding of W into W will be defined as the affine map

W ζ -→ W, Φ → ((ξ 2 , • • • , ξ r ), γ(Φ), (η 1 , • • • , η s-1 )),
where γ(Φ) is the above composition for a given Φ ∈ W. The components of W L and W R will guarantee a compatible action of a reductive group and at the same time the possibility of choosing enough polarizations for this action.

5.1.1. Remark: One might hope to be able to do induction on r and/or s by simply replacing

M r-1 ⊗ E r-1 ⊕ M r ⊗ E r by (M r-1 ⊕ M r ⊗ Hom(E r-1 , E r )
) ⊗ E r-1 and keeping the other E i for i < r -1. But then we drop the information about the homomorphisms E i → E r . Therefore we are lead to replace all E i , i ≥ 2, by E 1 at a time, i.e. by

P 1 ⊗ E 1 = (M 1 ⊕ M 2 ⊗ A 21 ⊕ • • • ⊕ M r ⊗ A r1 ) ⊗ E 1 ,
where A ji = Hom(E i , E j ). Moreover, in order to keep the information of the homomorphisms E i → E j for 2 ≤ i ≤ j we consider also the spaces

P i = M i ⊕ M i+1 ⊗ A i+1,j ⊕ . . . ⊕ M r ⊗ A ri
together with the maps P i ⊗ A i,i-1 → P i-1 in the following. The reader may convince himself that only because of this the actions of the original group is compatible with the action of the bigger reductive group. It is a beautiful outcome that then we are able to compare the semi-stability with respect to related polarizations in section 7.

The abstract definition of W

The above motivating definition of the space W can immediately be turned into the following final definition using the spaces H li , A ji and B ml and the pairings between them. For any possible i and l we introduce the spaces

P i = ⊕ i≤j≤r M j ⊗ A ji and Q l = ⊕ 1≤m≤l N m ⊗ B * lm ,
and we denote by p i and q l their dimensions. For 1 < i and l < s we let

P i ⊗ A i,i-1 ξ i -→ P i-1 and Q l+1 ⊗ B l+1,l η l -→ Q l
be the canonical morphisms, defined as follows. On the component M j ⊗ A ji of P i , the map ξ i is the map

(M j ⊗ A ji ) ⊗ A i,i-1 -→ M j ⊗ A j,i-1
induced by the composition map of the spaces A. The map η l is defined in the same way. As in 5.1 we set

W L = ⊕ 1<i≤r Hom(P i ⊗ A i,i-1 , P i-1 ), W R = ⊕ 1≤l<s Hom(Q l+1 ⊗ B l+1,l , Q l ),
and

W = W L ⊕ Hom(P 1 , Q s ⊗ H s1 ) ⊕ W R .
In order to define the embedding ζ we define the operator γ as follows. Given w = (φ li ) ∈ W with φ li ∈ Hom(M i , N l ⊗ H li ), we let

γ(w) ∈ Hom(P 1 , Q s ⊗ H s1 ) = Hom(P 1 ⊗ H * s1 , Q s )
be the linear map defined by the matrix (γ li (w)), for which each γ li (w) is the composed linear map

M i ⊗ A i1 -→ N l ⊗ H li ⊗ A i1 -→ N l ⊗ H l1 -→ N l ⊗ B * sl ⊗ H s1 ,
where the first map is induced by φ li , the second by the composition H li ⊗ A i1 → H l1 and the third by the dual composition H l1 → B * sl ⊗ H s1 . The map ζ can now be defined by

W ζ -→ W, w → ((ξ 2 , • • • , ξ r ), γ(w), (η 1 , • • • , η s-1 )).

Lemma:

The linear map γ is injective and hence the morphism ζ is a closed embedding of affine schemes.

Proof. From the surjectivity assumptions in 2.1 we find that dually the composition

H li -→ H l1 ⊗ A * i1 -→ B * sl ⊗ H s1 ⊗ A * i1
is injective. Now it follows from the definition of γ li (w) that φ li can be recovered from γ li (w), by shifting A i1 to its dual.

The new group G

We consider now the natural action on W as described in 3.1 in the general situation, where the group is

G = G L × G R , with G L = 1≤i≤r GL(P i ), G R = 1≤l≤s GL(Q l ).
To be precise, this action is described in components by

g i-1 • x i-1,i • (g i ⊗ id) -1 , h s • ψ • (g 1 ⊗ id) -1 and h l • y l,l+1 • (h l+1 ⊗ id) -1 , with x i-1,i ∈ Hom(P i ⊗A i,i-1 , P i-1 ), ψ ∈ Hom(P 1 ⊗H * s1 , Q s ), y l,l+1 ∈ Hom(Q l+1 ⊗B l+1,l , Q l )
and with

g i ∈ GL(P i ), h l ∈ GL(Q l ).
The first and third expression describe the natural actions of G L on W L and of G R on W R .

There are also natural embeddings of

G L , G R , G into G L , G R , G respectively. For that it is enough to describe the embedding of G L in G L . Given an element g ∈ G L , g =       g 1 0 . . . 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 . . . u r,r-1 g r       with g i ∈ GL(M i ) and u ji ∈ Hom(M i , M j ⊗ A ji ) we define θ L,i (g) ∈ GL(P i ) as the matrix θ L,i (g) =       gi 0 . . . 0
     
with respect to the decomposition of P i with the following components: gj = g j ⊗ id on M j ⊗ A ji and for i ≤ j ≤ k the map ũkj is the composition

M j ⊗ A ji -→ M k ⊗ A kj ⊗ A ji -→ M k ⊗ A ki ,
where the second arrow is induced by the given pairing. In case j = i we have gi = g i and ũki = u ki . Now we define the map

G L θ L → G L by g → (θ L,1 g, • • • , θ L,r g).
It is then easy to verify that θ L is an injective group homomorphism and defines a closed embedding of algebraic groups. With this embedding we consider G L as a closed subgroup of G L . In the same way we obtain a closed embedding

θ R of G R ⊂ G R . Finally we obtain the closed embedding θ = (θ L , θ R ) of G ⊂ G. 5.3.1. Lemma: The subgroup G L ⊂ G L (respectively G R ⊂ G R ) is the stabilizer of the distinguished element (ξ 2 , . . . , ξ r ) ∈ W L (respectively (η 1 , . . . , η s-1 ) ∈ W R )
Proof. It is enough to prove the statement only for G L because of duality. The fact that G L stabilizes (ξ 2 , . . . , ξ r ) is an easy consequence of the properties of the composition maps.

The converse can be proved by induction on r. It is trivial for r = 1. Suppose that r ≥ 2 and that the statement is true for r -1. Let (γ 1 , . . . , γ r ) be an element of the stabilizer of (ξ 2 , . . . , ξ r ). When we replace the space W by W , corresponding to the spaces M 2 , . . . , M r and the same spaces N l and similarly W L by W L , then (γ 2 , . . . , γ r ) is an element of the stabilizer of (ξ 3 , . . . , ξ r ), so by the induction hypothesis it belongs to G L and there exists an element

g =       g 2 0 • • • 0 u 32 g 3 . . . . . . . . . . . . 0 u r2 • • • u r,r-1 g r       such that (γ 2 , . . . , γ r ) = θ L (g ). Let now γ 1 ∈ GL(P 1 ) have the components M i ⊗ A i1 y ji -→ M j ⊗ A j1 for all 1 ≤ i, j ≤ r.
The identity γ 1 • ξ 2 = ξ 2 • γ 2 then shows that y ji = 0 for j < i, y ii = g i for 2 ≤ i and y ji = u ji for 2 ≤ j < i. Now let g 1 = y 11 , u j1 = y j1 , for 2 ≤ j ≤ r, which are linear mappings M 1 -→ M j ⊗ A j1 . Then

g =       g 1 0 • • • 0 u 21 g 2 . . . . . . . . . . . . 0 u r1 • • • u r,r-1 g r      
is an element of G L and we have (γ 1 , . . . , γ r ) = θ L (g).

Remark: since the action of G L on W L is linear, it is clear that we have an isomorphism

G L /G L G L (ξ 2 , . . . , ξ r ), and similarly G R /G R G R (η 1 , . . . , η s-1 ).
We will use this fact in section 8.

Using the associativity of the composition maps it is again easy to verify that the actions of G on W and G on W are compatible, i.e. that the diagram

G × W ---→ W θ×ζ     ζ G × W ---→ W
is commutative, in which the horizontal maps are the actions. In addition we have the In the sequel we will use the following notation: the dimension of a vector space will be the small version of its name. So

m i = dim(M i ), n l = dim(N l ), p i = dim(P i ), q m = dim(Q m ) a ji = dim(A ji ), b ml = dim(B ml ) etc.
A proper polarization of the action of G on W is a tuple Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ), where λ i and µ l are positive rational numbers such that

1≤i≤r λ i m i = 1≤l≤s µ l n l = 1.
We define the new sequence of rational numbers α 1 , . . . , α r , β 1 , . . . , β s by the conditions

      λ 1 . . . . . . λ r       =       1 0 • • • 0 a 21 1 . . . . . . . . . . . . . . . 0 a r1 • • • a r,r-1 1             α 1 . . . . . . α r       ,       µ 1 . . . . . . µ s       =       1 b 2,1 • • • b s1 0 1 . . . . . . . . . . . . . . . b s,s-1 0 • • • 0 1             β 1 . . . . . . β s      
.

Then we have

1 = 1≤i≤r λ i m i = 1≤i≤r α i p i and 1 = 1≤l≤s µ l n l = 1≤l≤s β l q l .
In particular the tuple Λ = (α

1 , • • • , α r , -β 1 , • • • , -β s )
P i = ⊕ i≤j M j ⊗ A ji , and Q l = ⊕ l≤m N l ⊗ B *
ml respectively then we have

1≤i≤r λ i m i = 1≤i≤r α i p i , and 1≤l≤s
µ l n l = 1≤l≤s β l q l .
If the set of stable points in W with respect to the associated polarization is non-empty then by 3.3.2 the weights satisfy the conditions i≤j≤r α j p j > 0 for any i and 1≤l≤m β l q l > 0 for any m.

Equivalently the conditions may also be written as i≤j≤r α j p j > 0 for 2 ≤ i ≤ r and 1 -m≤l≤s

β l q l > 0 for 2 ≤ m ≤ s.
Substituting the weights of the original polarization on W , we can reformulate these conditions. In the cases treated in the examples they reduce to the following

5.4.1. Weight conditions.
Let W be of type (r, s) and let Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) be a proper polarization of W with positive λ i and µ l . If the set W s (G, Λ) of stable points of W with respect to the associated polarization Λ is non-empty, then in case of type (2, 1):

λ 2 -a 21 λ 1 > 0, type (3, 1): λ 3 -a 32 λ 2 + (a 32 a 21 -a 31 )λ 1 > 0, λ 1 (m 1 + a 21 m 2 + a 31 m 3 ) < 1, type (2, 2): λ 2 -a 21 λ 1 > 0, µ 1 -b 21 µ 2 > 0.

Comparison of invariant polynomials

In the following we assume that Λ = (α 1 , . . . , α r , -β 1 , . . . , -β s ) is the polarization on W associated to the polarization Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ). The semi-stable locus W ss (G, Λ) with respect to this polarization is more precisely defined by the character X associated to it as in 3.1. If q is lowest common denominator of α 1 , . . . , α r , β 1 , . . . , β s , we have

X (g) = ( 1≤i≤r det(g i ) -qα i )( 1≤l≤s det(h l ) qβ l )
for an element g ∈ G with components g i and h l . By the matrix relations between the polarizations q is also a common denominator of λ 1 , . . . , λ r , µ 1 , . . . , µ s , such that, if p denotes the lowest, we have q = pu for some u. The character χ with respect to the given polarization can be defined by

χ(g, h) = 1≤i≤r det(g i ) -pλ i 1≤l≤s det(h l ) pµ l ,
where the g i resp. h l are the diagonal components of g resp. h, see 2.2. Now the relations between the polarizations imply by a straightforward calculation that

X (θ(g, h)) = χ(g, h) u .
If F is a X m -invariant polynomial on W it follows that

F (ζ((g, h).w)) = F (θ(g, h).ζ(w)) = χ(g, h) um F (ζ(w)),
i.e. that F • ζ is a χ um -invariant polynomial on W . As a consequence we obtain the 5.5.1. Lemma:

ζ -1 (W ss (G, Λ)) ⊂ W ss (G, Λ),
i.e. if w ∈ W and ζ(w) is G-semi-stable in W with respect to the polarization Λ = (α 1 , . . . , α r , -β 1 , . . . , -β s ) then w is G-semi-stable in W with respect to the polarization Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) (in the sense of 4.1).

Proof. There exists a X m -invariant polynomial F on W such that F (ζ(w)) = 0. Then

F (ζ((g, h).w)) = F (ζ(w)) = 0
for any element (g, h) in the unipotent subgroup H ⊂ G. This means that w is G-semistable.

Remark:

When we consider the subgroup G ⊂ G defined by the condition det(g 1 ) = det(h s ) = 1, we have θ(G ) ⊂ G as follows from the definition of G in 3.5. With respect to these groups the semi-stable points are those over the semi-stable loci in P(W ) resp. P(W), with respect to the line bundles

L = O P(W ) (t) and L = O P(W) (t),
where t and t is defined as in 3.4 in the different cases endowed with the modified action defined by the characters. However, we cannot compare P(W ) and P(W) directly because the morphism ζ does not descend.

We need the analogous statement of Lemma 5.5.1 also in the case of stable points. For that is is more convenient to use the subspace criterion (1) of A. King in the case of G red and G. This gives also another proof in the semi-stable case.

Lemma:

With the same notation as in the previous Lemma

ζ -1 (W s (G, Λ)) ⊂ W s (G, Λ)
Proof. Let w = (φ li ) be a point of W with maps M i ⊗ H * li φ li -→ N l and suppose that w is not G-stable with respect to the polarization Λ. We can assume that it is not G red -stable, too. Then there are linear subspaces M i ⊂ M i and N l ⊂ N l for all i and l such that the family ((M i )), (N l )) is proper and such that

φ li (M i ⊗ H * li ) ⊂ N l and i λ i m i - l µ l n l ≥ 0.
With these subspaces we can introduce the subspaces P i ⊂ P i and Q l ⊂ Q l as

P i = ⊕ i≤j M j ⊗ A ji and Q l = ⊕ m≤l N m ⊗ B * lm .
They form a proper family of subspaces and satisfy

ξ i (P i ⊗ A i,i-1 ) ⊂ P i-1 , γ(w)(P 1 ⊗ H * s1 ) ⊂ Q s , η l (Q l+1 ⊗ B l+1,l ) ⊂ Q l
for the possible values of i and l. But by the definition of the spaces and because Λ is the associated polarization, the formulas of 5.4 imply the dimension formula

i α i p i - β l q l = i λ i m i - l µ l n l ≥ 0.
This states that also ζ(w) is not G-stable.

In section 7 we will derive sufficient conditions for the equality

ζ -1 (W s (G, Λ)) = W s (G, Λ) and ζ -1 (W ss (G, Λ)) = W ss (G, Λ).
In the following section we show how this equality implies the existence of a good and projective quotient W ss (G, Λ)//G using the result for W ss (G, Λ)//G from Geometric Invariant Theory.

Construction and properties of the quotient

We keep the notation of the previous sections and let Λ be the polarization on W associated to the polarization Λ on W . We do not require that they are proper here, but we will do that later for the examples. In addition we introduce the saturation

Z = Gζ(W ) ⊂ W
of the image of W with respect to the action of G.

6.1. Construction of the quotient 6.1.1. Proposition: Let W and W together with their G-and G-structure be as in section 2 and 5, let Λ be a polarization for (W, G) and Λ be the associated polarization for (W, G).

(

) If ζ -1 (W s (G, Λ)) = W s (G, Λ), then there exists a geometric quotient W s (G, Λ) - → M s of W s by G, which is a quasi-projective nonsingular variety. (2) If in addition ζ -1 (W ss (G, Λ)) = W ss (G, Λ) and ( Z Z) ∩ W ss (G, Λ) = ∅, then there exists a good quotient W ss (G, Λ) π - → M , such that M is a normal projective variety, M s is an open subset of M , and W s (G, Λ) → M s is the restriction of π. 1
We recall here the definition of a good and a geometric quotient of C.S. Seshadri, see [START_REF] Newstead | Introduction to moduli problems and orbit spaces[END_REF], [START_REF] Mumford | Geometric invariant theory[END_REF]. Let an algebraic group G act on an algebraic variety or algebraic scheme X. Then a pair (ϕ, Y ) of a variety and a morphism X ϕ -→ Y is called a good quotient if (i) ϕ is G-equivariant (for the trivial action of G on Y ), (ii) ϕ is affine and surjective, (iii

) If U is an open subset of Y then ϕ * is an isomorphism O Y (U ) ≈ O Y (ϕ -1 U ) G ,
where the latter denotes the ring of G-invariant functions, (iv) If F 1 , F 2 are disjoint closed and G-invariant subvarieties of X then ϕ(F 1 ), ϕ(F 2 ) are closed and disjoint.

If in addition the fibres of ϕ are the orbits of the action and all have the same dimension, the quotient (ϕ, Y ) is called a geometric quotient.

As usual we write X//G for a good quotient space and X/G for a geometric quotient space.

Proof. We will prove the second statement first, assuming that the conditions of ( 1) and ( 2) are satisfied. We use the abbreviations W ss = W ss (G, Λ), W ss = W ss (G, Λ) and similarly W s , W s for the subsets of the stable points. By the result of A. King, 3.1, there exists a good projective quotient of W ss by the reductive group G. So there exists also a good and projective quotient of the closed invariant subvariety Z ∩ W ss which we denote by Z ∩ W ss π 0 -→ M.

By assumption (2) Gζ(W ss ) = Z ∩ W ss = Z ∩ W ss . We let π be the composition

W ss ζ - → Gζ(W ss ) π 0 -→ M.
We know already that M is projective. We will then verify that (π, M ) is the good quotient of the proposition. We consider first the commutative diagram

G × W ss µ / / p Gζ(W ss ) π 0 W ss π / / M
in which p is the projection and µ is defined by (g, w) → gζ(w). There is an action of G on G × W ss by g.(g, w) = (gθ(g) -1 , g.w) and it follows that µ is G-equivariant.

Claim:

The morphism µ is a geometric quotient of G × W ss by G.

Proof of the claim:

We show first that the fibres of µ are the G-orbits. So let (g, w) , (g , w ) be two elements in G × W ss such that µ(g, w) = µ(g , w ). Then ζ(w) = g -1 g ζ(w ). By Lemma 5.3.1 g = g -1 g ∈ G and g.(g, w) = (g , w ). The claim will be proved if we show that µ has local sections. For this it suffices to use the remark following Lemma 5.3.1 and a local section of the quotient map G → G/G. Now we are going to verify the 4 properties of a good quotient for π. Clearly (i) is satisfied by the definition of π.

Proof of (ii). It is clear that π is surjective. The morphism π is affine because π = π 0 • ζ and π 0 and ζ are affine.

Proof of (iii). Let U ⊂ M be an open subset. Then

O(U ) ⊂ O(π -1 (U )) G since π is G-invariant. Conversely let f ∈ O(π -1 (U )) G . The f • p ∈ O(G × π -1 (U )) G ,
and since µ is a geometric quotient, f • p descends to an f ∈ O(µ(G × π -1 (U ))), which is G-invariant. Now again f descends because π 0 is a good quotient. This proves equality

O(U ) = O(π -1 (U )) G .
Proof of (iv). Let F 1 , F 2 be disjoint, closed, G-invariant subvarieties of W ss . Then p -1 (F 1 ), p -1 (F 2 ) are disjoint, closed and G-invariant subvarieties of G × W ss . Since µ is a good quotient, µ(p -1 (F 1 )), µ(p -1 (F 2 )) are disjoint, closed and G-invariant in Gζ(W ss ). Finally, since π 0 is a good quotient,

π 0 • µ(p -1 (F 1 )), π 0 • µ(p -1 (F 2 )) are disjoint and closed subvarieties of M . But π 0 • µ(p -1 (F i )) = π(F i ), which proves (iv).
The normality of M follows from the fact that Gζ(W ss ) is smooth and π 0 is a good quotient, [START_REF] Mumford | Geometric invariant theory[END_REF], with respect to the reductive group G. That π becomes a geometric quotient on the open set W s of stable points follows from the fact that the G-orbits in Gζ(W s ) = Z ∩ W s intersect W s in G-orbits. In particular the stabilizers of w in G and of ζ(w) in G are isomorphic, such that all orbits have the same dimension.

The proof of ( 1) is a modification of the above. In any case π 0 induces the geometric quotient

Z ∩ W s π 0 -→ M 0 with M 0 open in M . Now Gζ(W s ) = Z ∩ W s is a π 0 -saturated open subset of Z ∩ W s , such that we obtain a geometric quotient Gζ(W s ) π 0 -→ M s with M s ⊂ M 0 open
. By the same arguments as above applied to the diagram related to G × W s → Gζ(W s ) we conclude that W s π -→ M s is a geometric quotient.

Remarks: 1) The idea of this proof comes from [START_REF] Seshadri | Mumford's conjecture for GL(2) and applications[END_REF], and has already been used in [START_REF] Drézet | Fibrés stables et fibrés exceptionnels sur P 2 (C)[END_REF] and [START_REF] Drézet | Variétés de modules extrémales de faisceaux semi-stables sur P 2 (C)[END_REF].

2) If the second condition of ( 2) is not satisfied, we cannot even prove that W ss (G, Λ) admits a good quasi-projective quotient, because Z ∩ W ss might not be saturated. Of course the projectivity of the quotient depends on this condition.

S-equivalence

We suppose that the hypotheses of proposition 6.1.1 are satisfied, with polarization Λ for (W, G) and associated polarization Λ for (W, G).

It is easy to define the Jordan-Hölder filtration of G-semi-stable elements of W with respect to Λ (cf. [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF] for a more general situation). Using the preceding results we can also define a Jordan-Hölder filtration of a G-semi-stable element of W with respect to Λ. Let w = (φ li ) ∈ W ss (G, Λ). Then there exist a positive integer p, an element h ∈ H and filtrations

M 0 i = {0} ⊂ M 1 i ⊂ • • • ⊂ M p i = M i , N 0 l = {0} ⊂ N 1 l ⊂ • • • ⊂ N p l = N l , with i λ i dim(M j i ) = l µ l dim(N j l )
for each j, such that h.w = (φ li ) satisfies

φ li (H * li ⊗ M j i ) ⊂ N j l , and that if φ j li : H * li ⊗ (M j i /M j-1 i ) -→ N j l /N j-1 l
is the induced morphism, then (φ j li ) li is G-stable with respect to Λ for any j. This filtration and h need not be unique, but p is unique and the (φ j li ), too, up to the order and isomorphisms. Conversely, an element of W having such a filtration is G-semi-stable with respect to Λ. We say that two elements (φ li ) and (φ li ) of W ss (G, Λ) are S-equivalent if they have Jordan-Hölder decompositions (φ j li ), (φ li j ) respectively of the same length, and if there exists a permutation σ of {1, . . . , p} such that (φ li j ) is isomorphic to (φ σ(j) li ) for any j.

The following result is also easily deduced from 6.1.1.

6.2.1. Proposition: Let w, w ∈ W ss (G, Λ). Then π(w) = π(w ) if and only if w and w are S-equivalent.

It follows that the set of closed points of M is exactly the set of S-equivalence classes of elements of W ss .

Comparison of semi-stability

We are going to investigate conditions for the weights of the polarizations under which a (semi-)stable point w ∈ W is mapped to a (semi-)stable point ζ(w) ∈ W. For the estimates we need the following constants which depend on the dimensions m i and the composition maps H li ⊗ A i1 → H l1 .

Constants

Let K be the family of proper linear subspaces

K ⊂ ⊕ 2≤i M i ⊗ A i1 such that K is not contained in ⊕ 2≤i M i ⊗ A i1 for any family (M i ) = (M i ) of subspaces. For any l we let the map ⊕ 2≤i M i ⊗ A i1 ⊗ H * l1 δ l -→ ⊕ 2≤i M i ⊗ H * li
be induced by the maps A i1 ⊗ H * l1 → H * li associated to the composition maps, which are supposed to be surjective, see 2.1. We introduce the constant

c l (m 2 , . . . , m r ) = sup K∈K ρ l (K) with ρ l (K) = codim(δ l (K ⊗ H * l1 )) codim(K)
.

Similarly we define the constants d i (n 1 , . . . , n s-1 ) in the dual situation. Let Proof. It will be sufficient to assume that m i = mi for all i except one, m 2 < m2 say. Then let Mi be vector spaces of dimensions mi and suppose that

⊕ l<s N * l ⊗ H * li δ ∨ i ←-⊕ l<s N * l ⊗ B sl ⊗
d i (n) = d i (n 1 , . . . , n s-1 ) = sup L∈L codim(δ ∨ i (L ⊗ H * si )) codim(L) .
M2 = L 2 ⊕ M 2 and Mi = M i for i ≥ 3.
For any K ∈ K we consider the subspace

K = (L 2 ⊗ A 21 ) ⊕ K ⊂ ( M2 ⊗ A 21 ) ⊕ ( ⊕ 2<j M j ⊗ A j1 ).
Then codim( K) = codim(K) and also codim(

δ l ( K ⊗ H * l1 )) = codim(δ l (K ⊗ H * l1 )) because δ l is a direct sum of the surjective operator A j1 ⊗ H * l1 → H * l1 such that δ l (L 2 ⊗ A 21 ⊗ H * l1 ) equals L 2 ⊗ H * l2 and δ l ( K ⊗ H * l1 ) = (L 2 ⊗ H * l2 ) ⊕ δ l (K ⊗ H * l1
). Therefore ρ l (K) = ρ l ( K). Once we have shown that also K belongs to the analogous family K, the Lemma is proved.

To see this let M 2 ⊂ M2 and M i = M i ⊂ M i for i ≥ 3 be subspaces such that K ⊂ ⊕ 2≤i M i ⊗ A i,1 .
Then in particular

L 2 ⊗ A 21 ⊂ M 2 ⊗ A 21 and thus L 2 ⊂ M 2 . But then M 2 = L 2 ⊕ M 2 with M 2 = M 2 ∩ M 2 and it follows that K ⊂ ⊕ 2≤i M i ⊗ A i1 .
Since K ∈ K we obtain M i = M i for all i and then also M 2 = M2 .

Study of the converse I

Let Λ = (λ 1 , . . . λ r , -µ 1 , . . . , -µ s ) be a polarization on W and let Λ = (α 1 , . . . , α r , -β 1 , . . . , -β s ) be the associated polarization on W (the associated polarization has been defined in 5.4). We had shown in 5.5.1 and 5.5.3 that if w ∈ W and ζ(w) is (semi-)stable in W with respect to G and Λ, then so is w with respect to G and Λ. We are going to derive sufficient conditions for the converse, i.e. whether ζ(w) is (semi-)stable if w is (semi-)stable.

In the sequel we are going to use the following notation: Given a family M = (M i ) of subspaces M i ⊂ M i we set

P i (M ) = ⊕ i≤j M j ⊗ A ji
and call a subspace P i ⊂ P i saturated if there is such a family with P i = P i (M ). Note that in this case

i α i p i = i λ i m i .
Similarly we introduce the spaces Q l (N ) for a subfamily N = (N l ) of (N l ) and call them saturated.

Let w = (φ li ) be given and assume that ζ(w) is not semi-stable with respect to Λ. Then there exist linear subspaces P i ⊂ P i and Q l ⊂ Q l such that

ξ i (P i ⊗ A i,i-1 ) ⊂ P i-1 , γ(w)(P 1 ⊗ H * s1 ) ⊂ Q s , η l (Q l+1 ⊗ B l+1,l ) ⊂ Q l and such that i α i p i - l β l q l > 0,
where as before the small characters denote the dimension of the spaces. If there were subspaces M i ⊂ M i and N l ⊂ N l with P i = P i (M ) and

Q l = Q l (N ) as in 5.5.3, then γ(w)(P 1 ⊗ H * s1 ) ⊂ Q s would imply that ϕ li (M i ⊗ H * li )
⊂ N l and we would have

i λ i m i - l µ l n l = i α i p i - l β l q l > 0,
and w would not be semi-stable. In the following we are going to construct families M , N of subspaces M i ⊂ M i and N l ⊂ N l such that P i = P i (M ) and Q l = Q l (N ) are as close to P i , Q l as possible and such that there is a useful estimate for

i λ i m i - l µ l n l .
Step 1: We can assume that P i has a decomposition

P i = M i ⊕ X i in M i ⊕ ( ⊕ i<j M j ⊗ A ji )
and such that X r = 0. To derive this, we remark that for a subspace S of a direct sum E ⊕F of vector spaces there exists a linear map E u -→ F such that the isomorphism 1 0 u 1 of E ⊕ F transforms S into S ⊕ S , where S is the projection of S in E and S = S ∩ F . Using this and descending induction on i we can find an element h ∈ H L ⊂ G L , see 2.4, such that the truncations θ L,i (h) ∈ GL(P i ), see 5.3, map P i onto a direct sum M i ⊕ X i for any i. Since ξ i (P i ⊗ A i,i-1 ) ⊂ P i-1 we easily derive that

⊕ i<j M j ⊗ A ji ⊂ X i ⊂ ⊕ i<j M j ⊗ A ji
for all possible i. We put

ρ i = codim( ⊕ i<j M j ⊗ A ji , X i ) = codim(P i (M ), P i ).
Note that ρ r = 0.

Step 2: Let M 1 , . . . M r be subspaces of M 1 , . . . , M r respectively such that P i (M ) ⊃ P i is minimal over P i for any i. Then M i ⊂ M i since these spaces are the first components of P i ⊂ P i (M ) respectively and we have M 1 = M 1 . We let

σ i = i≤j (m j -m j )a ji = codim(P i (M ), P i (M )).
Step 3: We are going to define the subspaces N l ⊂ N l ⊂ N l as images.

Let P 1 ⊗ H * l1 γ l (w)
---→ N l be the map which is the sum of the composed maps

M i ⊗ A i1 ⊗ H * l1 → M i ⊗ H * li φ li -→ N l .
Then we define

N l = γ l (w)(P 1 ⊗ H * l1 ) = φ l1 (M 1 ⊗ H * l1 ) + γ l (w)(X 1 ⊗ H * l1 ) and N l = γ l (w)(P 1 (M ) ⊗ H * l1 ) = φ l1 (M 1 ⊗ H * l1 ) + 2≤j φ lj (M j ⊗ H * lj ).
It follows N l ⊂ N l for any l.

Step 4: If the weights β l are supposed to be positive, we may assume that

γ(w)(P 1 ⊗ H * s1 ) = Q s and η l (Q l+1 ⊗ B l+1,l ) = Q l for l < s.
Otherwise we could choose subspaces Q l ⊂ Q l by descending induction as images. Thenl β l q l ≥ -β l q l would improve the assumption on the choice of the spaces P i and Q l . Now it follows that for any l

Q l ⊂ Q l (N ) because P 1 ⊗ H * s1 is mapped to ⊕ l≤s N l ⊗ B *
sl and the maps η l are the identity on the spaces N m . Note that we even have

Q l ⊂ Q l (N ) since γ l | P 1 ⊗H * s1 factorises through ⊕ l≤s N L ⊗B * sl
as follows from the definition of N l .

Lemma:

Suppose that all β 1 , . . . , β s > 0, and let

∆ = i λ i m i - l µ l n l . Then ∆ > l β l q l - l µ l n l + i α i (σ i -ρ i ) - l µ l c l (m 2 , . . . , m r )(σ 1 -ρ 1 ). Proof. Let Y l = δ l (X 1 ⊗ H * l1 ) ⊂ Z l = ⊕ 2≤i M i ⊗ H * li .
Since X 1 is not contained in a direct sum with spaces smaller than

M i we get codim(Y l , Z l ) ≤ c l (m 2 , . . . , m r ) codim(X 1 , ⊕ 2≤j M j ⊗ A j1 ).
By Lemma 7.1.1 and above definitions we get

codim(Y l , Z l ) ≤ c l (m 2 , . . . , m r )( 1≤i m i a i1 -p 1 ) = c l (m 2 , . . . , m r )( 2≤i (m i -m i )a i1 -ρ 1 ).
The map

j φ lj sends (M 1 ⊗ H * l1 ) ⊕ Z l onto N l by definition of N l and also maps (M 1 ⊗ H * l1 ) ⊕ δ l (X 1 ⊗ H * l1 ) onto N l . Therefore, since M 1 = M 1 , we have a surjection Z l /Y l → N l /N l
and the dimension estimate

n l -n l ≤ c l (m 2 , . . . , m r )( 2≤i (m i -m i )a i1 -ρ 1 ).
Now we can derive the estimate of the Lemma. If there is no summation condition it is understood that the sum has to be taken over all indices of the given interval. We have

∆ = i λ i m i - l µ l n l = i λ i m i - l µ l n l + j λ j (m j -m j ) - l µ l (n l -n l ).
Substituting for λ j in the third sum and replacing the first by

i λ i m i = i α i dim( ⊕ i≤j M j ⊗ A ji ) = i α i (p i -ρ i )
and using the definition of σ i we get

∆ = i α i p i - l µ l n l + i α i (σ i -ρ i ) - l µ l (n l -n l ).
Now using the assumed estimate for the first sum and the derived estimate for n l -n l we get ∆ >

l β l q l - l µ l n l + i α i (σ i -ρ i ) - l µ l c l (m 2 , . . . , m r )(σ 1 -ρ 1 ).

Corollary:

Suppose that s = 1, let Λ = (λ 1 , . . . , λ r , -1 n 1 ) and let Λ be the associated polarization (α 1 , . . . , α r , -1 n 1 ). If all α i > 0 and if

λ 2 ≥ a 21 n 1 c 1 (m 2 , . . . , m r ) then ζ -1 W ss (G, Λ) = W ss (G, Λ) and ζ -1 W s (G, Λ) = W s (G, Λ).
Remarks: (1) Note that by the normalization of the polarizations we must have µ 1 n 1 = 1 such that 1/n 1 is the only possible value for µ 1 = β 1 .

(2) If all α i > 0, then the necessary conditions for W s (G, Λ) = ∅ and W s (G, Λ) = ∅ are both satisfied, see 5.4. The condition of the corollary is an extra condition.

Proof. Let us first assume that ζ(w) is not semi-stable and let the spaces P i and Q 1 be as at the beginning of 7.2. The only β 1 = 1/n 1 is positive. Let the other spaces be chosen as in 7.2. The difference β l q l -µ l n l reduces to q 1 /n 1 -n 1 /n 1 , and since

N 1 = γ(w)(P 1 ⊗ H * 11 ) = Q 1 , this difference is zero. Therefore ∆ > i α i (σ i -ρ i ) - 1 n 1 c 1 (m 2 , . . . , m r )(σ 1 -ρ 1 ).
Since all the α i are positive we have

i α i (σ i -ρ i ) ≥ α 1 (σ 1 -ρ 1 ) + α 2 (σ 2 -ρ 2 ).
Moreover, ξ 2 induces a surjection

P 2 (M ) ⊗ A 21 /P 2 ⊗ A 21 → P 1 (M )/P 1 because M 1 = M 1 . Therefore we obtain the dimensions estimate (σ 2 -ρ 2 )a 21 ≥ σ 1 -ρ 1 . It follows that ∆ > (- 1 n 1 c 1 (m 2 , . . . , m r ) + α 1 + α 2 a 21 )(σ 1 -ρ 1 ).
Since λ 2 = a 21 α 1 + α 2 ≥ a 21 n 1 c 1 (m 2 , . . . , m r ) the last expression is non-negative. This proves the case of semi-stability. For the case of stability we assume that w is stable and that ζ(w) is already semi-stable. If ζ(w) were not stable, we would find subspaces P i and N 1 as in 7.2 such that α i p i -µ 1 n 1 = 0 and such that at least one P i is different from P i . Now let the spaces M i and N I be constructed as above. Then we have

∆ ≥ i α i s i - c 1 n 1 s 1 ≥ 2<i α i s i + (λ 2 - c 1 n 1 a 21 ) s 1 a 21 ≥ 0,
where

s i = σ i -ρ i = dim(P i (M )/P i )
, and where we use that s 2 a 21 ≥ s 1 . If the family M is different from M , then 0 > ∆, and if it is equal, then ∆ = 0. In order to obtain a contradiction we have to show that M is different from M . Assume that it is not. Then s i = dim(P i /P i ) and we must have s i = 0 for i ≥ 3 and s 1 (λ 2 -c 1 n 1 a 21 ) = 0. If also s 1 = 0, then by the above estimate also s 2 = 0, contradicting the choice of the P i . Therefore s 1 = 0 and λ 2 = c 1 n 1 a 21 . But then ∆ = α 2 (s 2 -s 1 a 21 ) and we have s 2 a 21 = s 1 . From this it is easy to see that P i = P i ( M ) where Mi = M i for i = 2 and M2 = M 2 = M 2 . Then we have

i α i m i -µ 1 n 1 = i λ i p i -µ 1 n 1 = 0
which contradicts the stability of w.

Study of the converse II

We keep the notation of 7.2 and compare the (semi-)stability of points in W and W in two steps, each reducing to the case s = 1. We consider the intermediate space

V = W L ⊕ ⊕ 1≤l≤s Hom(P 1 ⊗ H * l1 , N l )
and the maps

W ζ 1 -→ V ζ 2 -→ W.
Here ζ 1 is defined by

w → (ξ 2 , . . . , ξ r , γ 1 (w), . . . , γ s (w)),
where γ l (w) is the map defined by w = (φ li ) as in 7.2. The map ζ 2 is defined by (x 2 , . . . , x r , γ 1 , . . . , γ s ) → (x 2 , . . . , x r , γ, η 1 , . . . , η s-1 ),

where now γ : P 1 ⊗ H * s1 → Q s is induced by the tuple (γ 1 , . . . , γ s ) as the sum of the compositions

P 1 ⊗ H * s1 → N l ⊗ H l1 ⊗ H * s1 → N l ⊗ B * sl
which are induced by the γ l and the pairings

B sl ⊗ H l1 → H s1 . It is obvious that ζ = ζ 2 • ζ 1 .
Note that both ζ 1 and ζ 2 are injective by the same reason as for ζ.

On V the group G L × G R acts naturally and we have the embedding

G = G L × G R θ L ×id → G L × G R , see 5.3
G L × G R → G L × G R = G
and ζ 2 is equivariant and satisfies the analogous statements for the orbits. Given the polarization Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) for (W, G) we consider the polarization Λ = (α 1 , . . . , α r , -µ 1 , . . . , -µ s ) for (V, G L × G R ) where the α i are defined as in 5.4. As in 5.5.1, 5.5.3 it is easy to show that

ζ -1 1 V ss (G L × G R , Λ) ⊂ W ss (G, Λ) and ζ -1 1 V s (G L × G R , Λ) ⊂ W s (G, Λ) and similarly that ζ -1 2 W ss (G, Λ) ⊂ V ss (G L × G R , Λ) and ζ -1 2 W s (G, Λ) ⊂ V s (G L × G R , Λ
). Note that as for W ss , W s , we have unipotent sub-orbits in V ss and V s , see 4.1. We are going to show that in all 4 cases equality holds under suitable conditions on the weights of the polarizations. Then the same is true for ζ.

imply the dimension estimates

p i -p i ≤ a i1 (p 1 -p 1 )
for i ≥ 2. Next we consider the homomorphism

Z 1 = ⊕ l<s (N l /N l ) * ⊗ H * l1 δ ∨ 1 ← -⊕ l<s (N l /N l ) * ⊗ B sl ⊗ H * s1 .
We have X s ⊂ l<s (N l /N l ) * ⊗ B sl and consider the subspace

Y 1 = δ ∨ 1 (X s ⊗ H * s1 ) ⊂ Z 1 . By the definition of the constant d 1 (n) = d 1 (n 1 , . . . , n s-1 ) we get dim(Z 1 /Y 1 ) ≤ d 1 (n) codim(X s ) = d 1 (n)(σ s -ρ s )
where

σ l = codim(Q * l (N/N ), Q * l (N/N )) and ρ l = codim(Q * l (N/N ), (Q l /Q l ) *
). Further we have a surjective map

Z 1 /Y 1 → (P 1 /P 1 ) * /(P 1 /P 1 ) * which is induced by the map Q * s ⊗H * s1 → P * 1 and the induced surjection Q * s (N/N )⊗H * s1 → (P/P 1 ) * , since N s = N s . So we get p 1 -p 1 ≤ d 1 (n)(σ s -ρ s ).
Now we can estimate the discriminant in 7.5.1. Lemma: Let all the α i be non-negative and let ∆ :

= i α i p i - l µ l n l . Then ∆ > l β l (σ l -ρ l ) - i α i a i1 d 1 (n)(σ s -ρ s ). Proof. Since i α i p i = l µ l n l we also have ∆ = l µ l (n l -n l ) - i α i (p i -p i ).
with the same steps as in the previous proofs we get

∆ = i α i p i - l β l q l + l β l (σ l -ρ l ) - i α i (p i -p i )
Inserting the assumption on the first difference and the estimate for p i -p i we get the result.

As in the previous cases we obtain the 7.5.2. Corollary: In the above notation let all α i > 0, and all β l > 0, and let

µ s-1 ≥ b s,s-1 d 1 (n) i α i a i1 . Then ζ -1 2 W ss (G, Λ) = V ss (G L × G R , Λ) and ζ -1 2 W s (G, Λ) = V s (G L × G R , Λ).
Proof. : In the notation of 7.5 there is a surjection

(Q s-1 (N )/Q s-1 ) * ⊗ B s,s-1 → (Q s (N )/Q s ) * because N s = N s . Therefore (σ s-1 -ρ s-1 )b s,s-1 ≥ σ s -ρ s .
If the condition of the Corollary is satisfied, then ∆ > 0 follows, where we use

µ s-1 = β s b s,s-1 + β s-1 .
Combining the results of 7.4.2 and 7.5.2 we get the 7.5.3. Proposition: Let Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) be a polarization for (W, G) and let Λ = (α 1 , . . . , α r , -β 1 , . . . , -β s ) be the associated polarization for (W, G). Suppose that all α i > 0, and all β l > 0 and that

λ 2 ≥ a 21 l µ l c l (m) and µ s-1 ≥ b s,s-1 d 1 (n) i α i a i1 . Then ζ -1 W ss (G, Λ) = W ss (G, Λ) and ζ -1 W s (G, Λ) = W s (G, Λ).

Projectivity conditions

The projectivity of the quotient in 6.1.1 depends on the second condition in (2), i.e. whether the boundary Z Z of the saturated set contains no semi-stable points of W.

Again this condition depends on the chosen polarization and conditions for the weights.

In order to derive these conditions in some cases we describe the boundary in terms independent of the group action.

Saturated boundary.

The elements of W are tuples w = (x 2 , . . . , x r , γ, y 1 , . . . , y s-1 ) of linear maps

P i ⊗ A i,i-1 x i -→ P i-1 , P 1 ⊗ H * s1 γ - → Q s , Q l+1 y l -→ B * l+1,l ⊗ Q l If w ∈ Z, there are an element w ∈ W and automorphisms ρ i ∈ Aut(P i ) , σ l ∈ Aut(Q l ) such that x i = ρ i-1 • ξ i • (ρ -1 i ⊗ id), γ = σ 1 • γ(w) • (ρ -1 1 ⊗ id), y l = (id ⊗ σ l ) • η l • σ -1 l+1 .
Here id stands for the different identities of the spaces A, B and H. We let x i respectively ξ i be the mapping

P i ⊗ A i,i-1 ⊗ . . . ⊗ A 21 → P i-1 ⊗ A i-1,i-2 ⊗ . . . ⊗ A 21
induced by x i respectively ξ i for i ≥ 3. From the relations between the x i and ξ i it follows easily that for each i ≥ 3 the composition x 2 • x 3 • . . . • x i has a factorization

P i ⊗ A i,i-1 ⊗ • • • ⊗ A 21 / / P 1 P i ⊗ A i1 x 1i 6 6
where the vertical map is the surjection induced by the pairings. This follows from the commutative diagrams induced by the automorphism ρ i and because ξ 2 • ξ 3 •• • •• ξ i admits such a factorization for each i ≥ 3. We put x 21 = x 2 . By the dual description for the maps y l we are given factorizations

B * sl ⊗ Q l Q s y ls 6 6 / / B * s,s-1 ⊗ . . . ⊗ B * sl ⊗ Q l of the maps y l • . . . • y s-2 • y s-1 for l ≤ s -2.
By similar arguments there are also factorizations

P i ⊗ A i1 ⊗ H * s1 x 1i ⊗id / / P 1 ⊗ H * s1 γ / / Q s P i ⊗ H * si γ si 3 3 (L i )
for all i ≥ 2 and dually factorizations

Q l ⊗ H l1 P 1 γ l1 3 3 γ / / Q s ⊗ H s1 y ls ⊗id / / B * sl ⊗ Q l ⊗ H s1 (R l )
for all l. Moreover, there are further factorizations of the induced composed maps

P i ⊗ H * si ⊗ B sl γ si ⊗id / / Q s ⊗ B sl y ls / / Q l P i ⊗ H * li Φ li 4 4
(L li ) and dually

Q l ⊗ H li P i Ψ li 4 4 x 1i / / P 1 ⊗ A * i1 γ l1 ⊗id / / Q l ⊗ H l1 ⊗ A * i1 (R li )
All these factorizations are based on mappings induced by the pairings. All factorization conditions are independent of the chosen automorphisms. One can rediscover the original components φ li of w from Φ li or Ψ li if x j = ξ j and y l = η l for all j and all l. In fact we have 8.1.1. Lemma: Let w = (x 2 , . . . , x r , γ, y 1 , . . . , y s-1 ) ∈ W. Then w ∈ Z if and only if

(1) rank x i = i≤j m j a j,i-1 for i ≥ 2 (1 * ) rank y l = k≤l b l+1,k n k for l ≤ s -1 (2) x 2 • x 3 • . . . • x i has a factorization P i ⊗ A i1 x i1 -→ P 1 for i ≥ 3 (2 * ) y l • . . . • y s-2 • y s-1 has a factorization Q s y ls -→ B * sl ⊗ Q l for l ≤ s -2 (3) γ • (x 1i ⊗ id) has factorizations (L i ) and (L li ) (3 * ) (y ls ⊗ id) • γ has factorizations (R l ) and (R li ).
Proof. If w ∈ Z, the three conditions are satisfied by the above, where rank x i is the dimension of the image of ξ i and rank y l is the rank of η l as the map Q l+1 → B * l+1,l ⊗ Q l . Let conversely w satisfy these conditions. We proceed by descending induction to find automorphisms ρ i by which the x i can be identified with the ξ i . Note that the factorization conditions are maintained under automorphisms. Since x r has maximal rank it is an injection M r ⊗ A r,r-1 → M r-1 ⊕ M r ⊗ A r,r-1 = P r-1 . Hence we can find an automorphism ρ r-1 of P r-1 such that ρ r-1 • x r becomes ξ r . Let us assume now that modulo some automorphisms ρ r-1 , . . . , ρ i we have x j = ξ j for j > i. We are going to find an automorphism ρ i-1 such that ρ i-1 • x i = ξ i . Because of the rank condition we can assume that ⊕ i≤j M j ⊗ A j,i-1 is the image of x i in P i-1 . Now using all the x i • ξ i+1 • . . . • ξ k we find that x i has a factorization through the standard map

P i ⊗ A i,i-1 → ⊕ i≤j M j ⊗ A j,i-1 xi -→ M i-1 ⊕ ⊕ i≤j M j ⊗ A j,i-1 .
induced by the pairings. Now the rank condition implies that xi induces an automorphism on ⊕ i≤j M j ⊗ A j,i-1 . This can be used to make xi the identity via an automorphism ρ i-1 . Now x i = ξ i . By the analogous dual procedure we can also find automorphism σ l ∈ Aut(Q l ) such that we can assume that y l = η l . Finally the factorizations (L li ) or (R li ) resulting from (3) and (3 * ) yield mappings Φ li or Ψ li from which we get φ li as composition

M i ⊗ H * li P i ⊗ H li Φ li -→ Q l N l .
It follows from the special type of the ξ i and η l that these are original components of an element w = (φ li ) inducing γ(w) = γ. We are going to derive effective sufficient conditions for the projectivity of the quotient in the cases (2, 1), (2, 2), (3, 1). Proof. We present only the case (ii), case(i) is an easier version of (ii). Let (x 2 , γ, y 1 ) ∈ Z Z and let us assume that rank x 2 is not maximal. Let K be the kernel of M 2 ⊗ A 21

x 2

-→ P 1 and let M 2 ⊂ M 2 be the smallest subspace such that K is contained in M 2 ⊗ A 21 . We put

P 2 = M 2 , P 1 = x 2 (M 2 ⊗ A 21 ) , Q 2 = γ(P 1 ⊗ H * 21 ) and Q 1 = y 1 (Q 2 ⊗ B 21 ) and consider ∆ = α 1 p 1 + α 2 p 2 -β 1 q 1 -β 2 q 2 .
By definition p 1 = dim(M 2 ⊗A 21 /K). Diagram (L 2 ) reduces in our case, with M 2 replaced by M 2 , to

M 2 ⊗ A 21 ⊗ H * 21 δ 2 x 2 ⊗id / / / / P 1 ⊗ H * 21 γ / / / / Q 2 M 2 ⊗ H * 22 γ 22 3 3 3 3 and γ 22 vanishes on δ 2 (K ⊗ H * 21 ) because K is the kernel of x 2 . Therefore q 2 ≤ dim(M 2 ⊗ H * 22 /δ 2 (K ⊗ H * 21 )) ≤ c 2 (m 2 )p 1 .
In order to estimate q 1 we consider diagram (L 21 ) enlarged by the commutative square of induced pairings

M 2 ⊗ A 21 ⊗ H * 21 ⊗ B 21 / / / / M 2 ⊗ H * 22 ⊗ B 21 γ 22 ⊗id / / / / Q 2 ⊗ B 21 y 1 / / / / Q 1 M 2 ⊗ A 21 ⊗ H * 11 δ 1 / / / / M 2 ⊗ H * 12 Φ 12 3 3 3 3
.

Again the map Φ 12 vanishes on δ 1 (K ⊗ H * 11 ) and we get

q 1 ≤ dim(M 2 ⊗ H * 12 /δ 1 (K ⊗ H * 11 )) ≤ c 1 (m 2 )p 1 .
Now we have the estimate

∆ ≥ α 2 p 2 + (α 1 -β 1 c 1 (m 2 ) -β 2 c 2 (m 2 ))p 1 .
Therefore the condition α 1 ≥ β 1 c 1 (m 2 ) + β 2 c 2 (m 2 ) would be sufficient, because α 2 p 2 > 0.

We modify the last estimate as follows. Since the weights in case (2, 2) are related by

λ 1 = α 1 λ 2 = a 21 α 1 + α 2 and µ 2 = β 2 µ 1 = β 1 + β 2 b 21
and since we have This shows that ∆ > 0 if x 2 is degenerate and the first condition of (ii) is satisfied. In case rank y 1 is not maximal the second condition follows by the dual procedure.

The case (3,1)

In order to derive a similar result in case (3, 1) we introduce the additional constant c 3 (m 3 ) analogous to c 3 (m 3 ) := c 1 (0, m 3 ) in 7.1. Let

M 3 ⊗ A 32 ⊗ H * 12 τ - → M 3 ⊗ H * 13
be the linear map induced by the pairing and let K be the family of all proper subspaces K ⊂ M 3 ⊗ A 32 which are not contained in M 3 ⊗ A 32 for any subspace M 3 ⊂ M 3 different from M 3 . We put

c 3 (m 3 ) = sup K∈K codim(τ (K ⊗ H * 12 )) codim(K)
For brevity we write c 3 = c 3 (m 3 ), c 3 = c 3 (m 3 ) = c 1 (0, m 3 ) and c 1 = c 1 (m 2 , m 3 ). 8.2.1. Proposition: Let (r, s) = (3, 1), let Λ = (λ 1 , λ 2 , λ 3 , -µ 1 ) be a polarization for (W, G) and Λ = (α 1 , α 2 , α 3 , -µ 1 ) be the associated polarization for (W, G), and assume that all α i > 0. (In this case

µ 1 = 1 n 1 .) If (1) α 2 c 3 + λ 1 c 3 ≥ µ 1 c 3 c 3 (2) λ 2 ≥ a 21 µ 1 c 1 (3) λ 3 ≥ a 31 µ 1 c 1 then Z Z contains no semi-stable point.
Moreover, condition (1) may be replaced by any of the conditions

(i) λ 3 ≥ µ 1 c 3 a 32 + a 31 λ 1 (ii) λ 3 ≥ µ 1 c 3 a 31 + a 32 α 2 (iii) λ 3 ≥ µ 1 c 3 a 32 a 21
Remark: Z Z contains no semi-stable point also in each of the following cases

(a) λ 1 ≥ µ 1 c 3 (b) α 2 ≥ µ 1 c 3 (c) α 3 ≥ µ 1 c 3 a 31 or α 3 ≥ µ 1 c 3 a 32 .
This can be seen by a direct estimate of the discriminant ∆ after substituting for q 1 in the following proof.

Proof. Let (x 2 , x 3 , γ) ∈ Z Z. We distinguish the following cases of degeneracy of x 2 and x 3 . case 1: x 3 is injective: Then by the proof of 8.1.1 we can assume that x 3 = ξ 3 is the canonical embedding and that x 13 and x 2 have a factorization x2 in the following diagram

M 3 ⊗ A 32 ⊗ A 21 / / ξ 3 / / (M 2 ⊕ M 3 ⊗ A 32 ) ⊗ A 21 x 2 / / P 1 M 3 ⊗ A 31 x 13 2 2 / / ξ 3 / / M 2 ⊗ A 21 ⊕ M 3 ⊗ A 31 x2 6 6 .
Here also ξ 3 is the canonical embedding. Moreover it is easy to verify that in this case also the composed map γ • (x 2 ⊗ id) admits a decomposition

(M 2 ⊗ A 21 ⊗ H * 11 ) ⊕ (M 3 ⊗ A 31 ⊗ H * 11 ) δ 1 -→ (M 2 ⊗ H * 12 ) ⊕ (M 3 ⊗ H * 13 ) γ - → Q 1 .
Here K = Ker(x 2 ) = 0 since x2 cannot be injective by the assumption on its rank. We choose subspaces M 2 , M 3 such that

K ⊂ M 2 ⊗ A 21 ⊕ M 3 ⊗ A 31
and such that these subspaces are minimal with this property. Now we consider the spaces

P 3 = M 3 , P 2 = M 2 ⊕ (M 3 ⊗ A 32 ), P 1 = x 2 (P 2 ⊗ A 21 ), Q 1 = γ(P 1 ⊗ H * 11 ) and their discriminant ∆ = α 1 p 1 + α 2 p 2 + α 3 p 3 -β 1 q 1 .
By the definition of the constant c 1 (m 2 , m 3 ) and the diagram

(M 2 ⊗ A 21 ⊕ M 3 ⊗ A 31 ) ⊗ H * 11 / / / / P 1 ⊗ H * 11 / / / / Q 1 M 2 ⊗ H * 12 ⊕ M 3 ⊗ H * 13 3 3 3 3
we obtain the estimate

q 1 ≤ c 1 (m 2 , m 3 )p 1 ≤ c 1 (m 2 , m 3 )p 1 ,
where by the definition of P 1 we have p 1 = m 2 a 2 + m 3 a 31 -k. Inserting this we obtain

∆ ≥ (µ 1 c 1 -λ 1 )k + (λ 2 -µ 1 c 1 a 21 )m 2 + (λ 3 -µ 1 c 1 a 31 )m 3 If µ 1 c 1 -λ 1 > 0, conditions (2)
and ( 3) imply that ∆ > 0. If, however,

λ 1 ≥ µ 1 c 1 we have the direct estimate ∆ ≥ (λ 1 -µ 1 c 1 )p 1 + α 2 p 2 + α 3 p 3 > 0.
This proves the proposition in the first case.

case 2: x 3 is not injective

Here we let K denote the kernel of x 3 and we choose a subspace M 3 ⊂ M 3 such that K ⊂ M 3 ⊗ A 32 and M 3 is minimal with this property. Then we consider the subspaces

P 3 = M 3 , P 2 = x 3 (M 3 ⊗ A 32 ), P 1 = x 2 (P 2 ⊗ A 21 ), Q 1 = γ(P 1 ⊗ H * 11 ). We have the exact sequences 0 → K → M 3 ⊗ A 32 x 2 -→ P 2 → 0 0 → L → M 3 ⊗ A 31 x 13 --→ P 1 → 0
where L denotes the kernel of x 13 . From the factorization properties restricted to the spaces P i and Q 1 we extract the following commutative diagram of surjections

M 3 ⊗ A 31 ⊗ H * 11 δ 1 / / P 1 ⊗ H * 11 γ M 3 ⊗ A 32 ⊗ A 21 ⊗ H * 11 4 4 * * M 3 ⊗ H * 13 γ 13 / / Q 1 M 3 ⊗ A 32 ⊗ H * 12 / / τ O O P 2 ⊗ H * 12 γ 12 O O .
From this we get again the estimates

q 1 ≤ c 3 (m 3 )p 1 ≤ c 3 (m 3 )p 1 and q 1 ≤ c 3 (m 3 )p 2 ≤ c 3 (m 3 )p 2 ,
where p 1 = m 3 a 31 -l and p 2 = m 3 a 32 -k. Let 0 < t < 1 be a real number. Then we have

q 1 ≤ tc 3 p 2 + (1 -t)c 3 p 1 .
Substituting this into the discriminant we get

∆ ≥ (λ 1 -(1 -t)µ 1 c 3 )p 1 + (α 2 -tµ 1 c 3 )p 2 + α 3 m 3 .
Now condition (1) enables us to find t with

1 - λ 1 µ 1 c 3 ≤ t ≤ α 2 µ 1 c 3 ,
such that the first two terms of the estimate are non-negative. Therefore ∆ > 0, and again (x 2 , x 3 , γ) is not semi-stable.

In order to show that (1) can be replaced by one of (i), (ii) or (iii) we substitute α i and p i and get after cancelation

∆ = -λ 1 l -α 2 k + λ 3 m 3 -µ 1 q 1 ≥ -λ 1 l -α 2 k + λ 3 m 3 -µ 1 c 3 (m 3 a 32 -k) = -λ 1 l + (µ 1 c 3 -α 2 )k + (λ 3 -µ 1 c 3 a 32 )m 3 .
If α 2 ≥ µ 1 c 3 , then by a direct estimate we get ∆ > 0. Therefore we may assume that µ 1 c 3 -α 2 > 0. Since in addition l ≤ m 3 a 31 , we get ∆ > (λ 3 -µ 1 c 3 a 32 -a 31 λ 1 )m 3 .

This shows that (1) can be replaced by (i). In the same way one shows that (1) can be replaced by (ii), using the other estimate of q 1 . That finally (1) can be replaced by (iii) can be shown by substituting first m 3 ≥ under this condition. Since 2λ 1 + λ 2 = 1 and 3µ 1 = 1, we can replace the polarization by the rational number t = λ 2 > 3 5 (cf. 9.3). The numerical condition for (semi-)stability then becomes

∆ = 1 -t 2 m 1 + tm 2 - 1 3 n < 0 (≤ 0),
where (m 1 , m 2 , n) is the dimension vector of a (φ 1 , φ 2 )-invariant sub-family of vector spaces, such that m 1 ≤ 2, m 2 ≤ 1, n ≤ 3. One can easily check that t = 2 3 is the only value for which ∆ might be zero, and this is the case for the values (0, 1, 2) and (2, 0, 1). And indeed, the homomorphisms φ given by matrices

  * * 0 * * z 2 * * z 3   and   0 0 z 1 0 0 z 2 * * z 3  
with generically chosen entries and linear forms z i are semi-stable and not stable for t = 2 3 .

The case t > 2 3

It is easy to show that in this case (φ 1 , φ 2 ) is t-stable if and only if

• φ 2 is nowhere zero • for any (ψ 1 , ψ 2 ) = h.(φ 1 , φ 2 ) with h ∈ H and any 1-dimensional subspace M 1 ⊂ C 2 we have ψ 1 (M 1 (⊗O(-2)) = 0.

The first condition says that coker(φ 2 ) is isomorphic to the universal quotient bundle Q on P 2 , and the second that φ 1 induces a 2-dimensional subspace of H 0 Q(2). It follows that the sets W s (t) of stable points are the same for t > 2 3 , which we denote by W s + . Moreover, from the above characterization of stable homomorphism we deduce that the geometric quotient M + = W s + /G is isomorphic to the Grassmannian

M + ∼ = Gr(2, H 0 Q(2))
which is smooth of dimension 26. There is an interesting subvariety Z ⊂ M + which consists of the images of the homomorphisms

  0 0 z 1 0 0 z 2 * * z 3   (1)
which belong to W s + . These are those (φ 1 , φ 2 ) for which the induced homomorphism 2O(-2) → Q is not injective. We will see next that Z is isomorphic to the non-stable locus of M 0 below and is smooth of dimension 10.

The case t = 2

We write W ss 0 for W ss ( 2 3 ). When considering the matrix representations we find that W s + ⊂ W ss 0 and that the remaining part W ss 0 W s + consists of those homomorphisms for which φ 2 is zero in exactly one point. Such homomorphisms are equivalent to matrices

  * * z * * w f g 0   (2)
where z, w are independent linear and f, g are independent quadratic forms. Note, however, that W s + intersects the non-stable locus of W ss 0 in matrices equivalent to those of type [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF]. But the orbit closures in W ss 0 of both types (1) and ( 2) of matrices contain the direct sums

  0 0 z 0 0 w f g 0   (3)
of independent linear and quadratic forms. From that it follows that the induced morphism M + → M 0 of the quotients is bijective and moreover an isomorphism by Zariski's main theorem, because both spaces are normal. The points of the non-stable locus M 0 M s 0 are represented by matrices of type [START_REF] Dixmier | Quelques aspects de la théorie des invariants[END_REF]. It is again routine to deduce from this observation that

M 0 M s 0 ∼ = P 2 × Gr(2, H 0 O(2)).
The subvariety Z ⊂ M + corresponds to this set under the isomorphism. We can also identify the set M s 0 of stable points with Gr(2, H 0 Q(2)) Z. 9.2.3. The case 3 5 < t < 2 3

Similarly to the case W s + we find that here W s -= W s (t) is independent of t and that W s -⊂ W ss 0 . The remaining part consists now of all homomorphisms which are equivalent to a matrix of type [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF]. Note that now homomorphisms of type (2) are contained in W s -. The induced morphism M -→ M 0 is again surjective but not injective over M 0 M s 0 . Let Y be the inverse image of M 0 M s 0 . Then Y consists of the points which are represented by matrices of type (2) which are not equivalent to matrices of type [START_REF] Dixmier | Quelques aspects de la théorie des invariants[END_REF]. It is easy to check that the restricted morphism

M -Y -→ ≈ M s 0
is bijective and therefore also an isomorphism by Zariski's main theorem. We are going to verify that Y is a divisor in M -. There is a morphism Y p -→ P 2 which assigns to the class of (φ 1 , φ 2 ) the point x at which φ 2 is degenerate. In this case coker(φ 2 ) ∼ = O ⊕ I x [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF] where I x is the ideal sheaf of x. For such (φ 1 , φ 2 ) we are given an exact diagram 0 2O(-2) Remarks: (1) One would like to interpret the matrices of type (2) as representing extensions of the sheaves coker(f, g) and I x (1) = coker z w . Indeed a matrix of type (2) defines such an extension, but this extension is isomorphic to the direct sum.

O O O 0 / / 2O(-2) ⊕ O(-1) O O (Φ 1 ,Φ 2 ) / / 3O / / F / / O O 0 0 / / O(-1) O O Φ 2 / / 3O / / O ⊕ I x (1) O O / / 0 0 O O 2O(-2) O O 0 O O such that (φ 1 , φ 2 ) corresponds to a 2-dimensional subspace Γ ⊂ H 0 (O(2) ⊕ I x ( 3
(2) The above correspondence between (φ 1 , φ 2 ) and Γ indicates that the quotient spaces considered here are spaces of coherent systems as in [START_REF] Potier | Systèmes cohérents et structures de niveau[END_REF].

The flip

The diagram M -→ M 0 ≈ ← M + can be interpreted as a flip. It is induced by the inclusions W s -⊂ W ss 0 ⊃ W s + . The orbits of stable points of type (2) in W s -and of type (1) in W s + don't intersect in W ss 0 but so do their closures in W ss 0 . Thus the fibres of M -→ M 0 and M 0 ← M + correspond to the two different types of semi-stable orbits in W ss 0 defining the same points in M 0 M s 0 .

General homomorphisms of type (2, 1)

In a more general situation of type (2, 1) we consider homomorphisms

m 1 O(-2) ⊕ m 2 O(-1) → n 1 O over P n .
A polarization in this case is determined by the rational number t = m 2 λ 2 with 0 < t < 1 and 1 -t = m 1 λ 1 , µ 1 = 1/n 1 . A Λ-(semi-)stable homomorphism is then called t-(semi-)stable. We write W ss (t) and W s (t) for W ss (G, Λ) and W s (G, Λ). In terms of t the conditions are

1 > t > (n + 1)m 2 (n + 1)m 2 + m 1 and t ≥ (n + 1)m 2 n 1 c 1 (m 2 ).
The constant c 1 (m 2 ) is given in proposition 9.1.1. Such polarizations exist if and only if

n 1 > (n + 1)m 2 c 1 (m 2 ).
In order to measure t-stability we introduce the numbers

r 1 = m 1 m 1 , r 2 = m 1 m 2 , s 2 = n 1 n 1 and call (r 1 , r 2 , s 1 ) φ-admissible if there are subspaces M 1 ⊂ M 1 , M 2 ⊂ M 2 , N 1 ⊂ N 1 of dimensions m 1 , m 2 , n 1 such that φ maps M 1 ⊗ O(-2) ⊕ M 2 ⊗ O(-1) into N 1 ⊗ O.
Then φ is t-(semi-)stable if and only if for any φ-admissible proper triple (r 1 , r 2 , s 1 ), i.e. a triple which is neither (0, 0, 0) or (1, 1, 1), we have

∆ t = (1 -t)r 1 + tr 2 -s 1 < 0 (≤ 0).
A polarization t is called critical if there are proper triples with ∆ t = 0. Thus the critical values of t are the rational numbers

s 1 -r 1 r 2 -r 1 ,
where we may assume s 1 = 0, 1 and thus r 1 = r 2 . We let t max be the maximal critical value if there are such with 0 < t < 1 and put t max = 0 otherwise. If t is not critical we have W s (t) = W ss (t). 9.3.1. Lemma: Suppose that m 2 and n 1 are relatively prime and that t max < t < 1.

Then φ = (φ 1 , φ 2 ) is t-stable if and only if

(1) φ 2 is stable with respect to the group GL(M 2 ) × GL(N 1 ).

(2) For any 1-dimensional subspace C j → M 1 , and any h

∈ Hom(M 1 ⊗ O(-2), M 2 ⊗ O(-1)) the map (ϕ 1 + h • ϕ 2 ) • j : O(-2) → N 1 ⊗ O is not zero.
Proof. By the characterization of stability in section 3 the homomorphism φ 2 is stable if and only if for any proper pair M 2 ⊂ M 2 , N 1 ⊂ N 1 of φ 2 -admissible subspaces r 2 < s 1 . Now let (φ 1 , φ 2 ) be stable. If φ 2 were not stable there would be a proper φ 2 -admissible pair (r 2 , s 1 ) with s 1 ≤ r 2 . But then s 1 < r 2 because m 2 , n 1 are supposed to be relatively prime. Then s 1 /r 2 < t because s 1 /r 2 is a critical value and thus ∆ t = r 2 t -s 1 > 0, contradicting the stability of (φ 1 , φ 2 ). The condition ( 2) is trivially satisfied if (φ 1 , φ 2 ) is t-stable, because otherwise (1, 0, 0) would be admissible with ∆ t = 1 -t > 0. We have to show now that conversely (1), (2) imply that (φ 1 , φ 2 ) is t-stable. For this let (r 1 , r 2 , s 1 ) be a proper (φ 1 , φ 2 )-admissible triple. If r 1 ≤ r 2 and r 2 = 0, there is nothing to prove. If r 2 > 0 then r 2 < s 1 by (1) and we have t(r 2 -r 1 ) < s 1 -r 1 and hence ∆ t < 0. If however r 2 < r 1 we have ∆ t < 0 in case r 1 ≤ s 1 . Since the case s 1 ≤ r 2 is only possible if s 1 = r 2 = 0 and then r 1 = 0 by (2), we can assume that r 2 < s 1 < r 1 . But then

r 1 -s 1 r 1 -r 2 < t
because the fraction is a critical value, and last inequality is the inequality ∆ t < 0.

Now we are able to describe the space M + = W s (t)/G for t max < t which is independent of t. According to the lemma W s (t) can only be non-empty if there are stable morphisms φ 2 . This is the case if and only if

1 σ(n) < n 1 m 2 < σ(n)
where σ(n) = 1 2 (n+1+ (n + 1) 2 -4), see [START_REF] Drézet | Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur P 2 (C)[END_REF]. We restrict ourselves now to the case where in addition to the previous conditions on n 1 , m 2 we have n 1 ≥ nm 2 and (n 1 , m 2 ) = 1. Then a stable φ 2 is injective and a subbundle (except at finite number of points in case n 1 = nm 2 , see [START_REF] Drézet | Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur P 2 (C)[END_REF], [START_REF] Drézet | Exceptional bundles and moduli spaces of stables sheaves on P n , In Vector Bundles in Algebraic Geometry[END_REF]). The quotient space of this space of stable homomorphisms by GL(M 2 ) × GL(N 1 ) is denoted by N = N (n + 1, m 2 , n 1 ). It is a smooth projective variety and there is a universal sheaf E on N × P n . For x ∈ N let E x denote the fibre sheaf representing x. Since it is the cokernel of the representing homomorphism φ 2 , we get

h 0 E x (2) = (n + 1) n 1 (n + 2) 2 -m 2 .
Therefore p * E(2) is locally free on N where p denotes the first projection of N × P n . Now M + can be non-empty only if

m 1 ≤ (n + 1) n 1 (n + 2) 2 -m 2 .
If conversely this is the case for any stable φ 2 and any subspace M 1 ⊂ H 0 E x (2) where x = [φ 2 ], there is a lifting φ

1 : M 1 ⊗ O(-2) → N 1 ⊗ O of M 1 ⊗ O(-2) → E x , and (φ 1 , φ 2 )
satisfies ( 1), (2) of the lemma. It follows now easily by considering corresponding families that

M + ∼ = Gr N (m 1 , p * E(2))
where Gr N denotes the relative Grassmannian. It is more difficult to characterize the other moduli spaces M (t) = W ss (t)/G for the intervals between the critical values or for the critical values and to interpret the flips between them.

Example of type (2, 2)

We consider now a simple example of type (2, 2) on P 3 of homomorphisms

O(-2) ⊕ O(-1) φ - → O ⊕ 3O(1).
Again the polarizations Λ = (λ 1 , λ 2 , -µ 1 , -µ 2 ) are supposed to be proper for W and W such that we have λ i > 0, µ l > 0 and

λ 2 > 4λ 1 and µ 1 > 4µ 2 .
All constants c l (m 2 ) and d i (n 1 ) are again zero, because m 2 = n 1 = 1. Then by the above conditions also the conditions for proposition 7.5.3 and proposition 8.1.3 are satisfied, such that there exists a good and projective quotient W ss (G, Λ)//G for any polarization satisfying the conditions. Since we have λ 1 + λ 2 = 1 and µ 1 + 3µ 2 = 1, the polarization Λ is determined already by λ 2 and µ 1 , for which the above conditions become

1 > λ 2 > 4 5 and 3 7 > 1 -µ 1 > 0. (1)
Next we derive the conditions for the occurrence of true semi-stable points. If (m 1 , m 2 , n 1 , n 2 ) is the dimension vector of a φ-invariant sub-family we have to consider the equation

∆ = (1 -λ 2 )m 1 + λ 2 m 2 -µ 1 n 1 - 1 3 (1 -µ 1 )n 2 = 0.
By inserting all possible dimension vectors we get the 6 conditions

1 -µ 1 = 3 k λ 2 , 1 -µ 1 = - 3 k λ 2 + 3 k (2)
for k = 1, 2, 3. If one of these is satisfied, there might be non-stable points in W ss (G, Λ).

In the following figure 1 the lines with the equations ( 2) are shown together with the rectangle (1) (lower right), for the points of which we get good and projective quotients. where the z i are homogeneous coordinates of P 3 , is easily verified to be G-stable for each polarization Λ in the rectangle [START_REF] Bialynicki-Birula | A recipe for finding open subsets of vector spaces with a good quotient[END_REF]. Therefore the moduli spaces are not empty. On each of the 3 lines in the rectangle (1) each point defines one and the same open set W ss (G, Λ) and hence one and the same moduli space with semi-stable and non-stable points. Similarly, on each of the 4 open triangles we have one and the same moduli space, which is a smooth projective geometric quotient. Each of the 7 spaces has dimension 77. The reader may also verify that the moduli space for an open triangle admits a morphism to the moduli space of each of its edges, thereby defining a chain of flips. Let W be the space of those homomorphisms. A proper polarization Λ = (λ 1 , λ 2 , -µ 1 , -µ 2 ) for W satisfies

m 1 λ 1 + 2λ 2 = 1 , 2µ 1 + n 2 µ 2 = 1
with λ 1 , λ 2 , µ 1 , µ 2 positive. We will also assume that α 2 > 0, β 1 > 0, i.e. λ 2 > 4λ 1 and µ 1 > 4µ 2 . These four conditions can be replaced by

The first condition of (3) follows already from the first of (2). After replacing λ 1 and µ 2 conditions (2) and (3) are equivalent to (5) has a solution µ 1 . For this we distinguish the cases n 2 < 8, n 2 = 8, 8 < n 2 . If n 2 < 8 the first inequality of (5) has a solution µ 1 < 1 2 if m 1 < 6. If n 2 = 8, then m 1 ≤ 6, which is case (i'). If n 2 > 8, the first inequality of [START_REF] Dolgachev | Variation of Geometric Invariant Theory Quotients[END_REF] They are all satisfied if we suppose (ii) of the claim.

In figure 2 the lines of the critical values of the polarizations i.e. of the pairs (λ 2 , µ 1 ) are shown together with the small region of those pairs which satisfy the sufficient conditions (4) for the existence of a good and projective quotient, based on the values m 1 = 3 and n 2 = 5.

Example of type (3, 1)

As an example of type (3, 1) we consider only the space of homomorphisms O(-4) ⊕ O(-2) ⊕ O(-1) → 5O over P 3 . We assume again that all λ i and all α i are positive. Then the conditions of 7.2.2 together with the normalization of the polarization are

λ 1 + λ 2 + λ 3 = 1 λ 2 > 10λ 1 λ 3 -4λ 2 + 20λ 1 > 0. µ 1 = 1 k λ 2 ≥ 4 5 c 1 (1, 1)
As additional condition for the projectivity of the quotient we use condition (a) of the remark following proposition 8.2.1. Since in this case both the constants c 3 (1) and c 3 (1) are zero, this condition is just λ 1 ≥ 0 and is already satisfied by our assumption.

For homomorphisms of the above type the condition λ 3 < 4 5 is necessary if W s (G, Λ) = ∅. For if φ = (φ 1 , φ 2 , φ 3 ) is an element of W then φ 3 has degree 1 and thus contains at most 4 independent components. Then m 1 = m 2 = 0 and m 3 = 1, n 1 = 4 is a choice of dimensions of φ-invariant subspaces and the discriminant becomes ∆ = λ 3 -1 5 .

By 9.1.2 the value of c 1 (1, 1) is 1 5 . Now it is easy to see that there exist polarizations Λ which satisfy the above inequalities. That W s (G, Λ) is then indeed non-empty follows from the existence of generic matrices as in 9.4. Moreover there are again regions of polarizations for which the sets W ss (G, Λ) are the same and which are responsible for flips.

Construction of fine moduli spaces of torsion free sheaves

Let n,k be integers such that n ≥ 2 and

(n + 1)(n + 2) 2 < k ≤ (n + 1) 2 .

Let V be a vector space of dimension n + 1, P n = P(V ). We will study in this chapter morphisms of sheaves on P n of type

Φ = (Φ 1 , Φ 2 ) : O(-2) ⊗ C 2 -→ O(-1) ⊕ (O ⊗ C k ).
Let

f 1 : C 2 -→ V *
the linear map induced by Φ 1 . For semistable morphisms (with respect to a given polarization) f 1 is non zero. So it is of rank 1 or 2. Morphisms Φ such that f 1 is of rank 2 are called generic, and those such that f 1 is of rank 1 are called special.

Generic morphisms

Suppose that Φ = (Φ 1 , Φ 2 ) is a generic morphism. Let P = Im(f 1 ) and P n-2 ⊂ P n be the linear subspace of zeroes of linear forms in P . Then (cf. [START_REF] Drézet | Variétés de modules alternatives[END_REF]).

For example moduli spaces of stables sheaves admitting a universal sheaf are fine moduli spaces of sheaves. The critical polarizations in our range are given by

λ 1 = 1 2p , t = λ 2 = 1 - k 2p , n + 1 + k 2 < p ≤ (n + 1)(n + 2) 2 . Let q = (n + 1)(n + 2) 2 - n + 1 + k 2 + 1
Now we prove that the Kodaïra-Spencer map of E at z ∈ M (t) is bijective. Let w ∈ π -1 (z).

Then we have a commutative diagram

T w W T π / / ωw T z M (t) ωz Ext 1 (F w , F w ) Ext 1 (E z , E z )
The tangent map T π is surjective because M (t) is a geometric quotient. So it suffices to prove that ω z is surjective and that dim(Ext 1 (E z , E z )) = dim(M (t)). Consider the exact sequence

0 -→ A 0w = O(-2) ⊗ C 2 -→ B 0w = O(-1) ⊕ (O ⊗ C k ) -→ F w -→ 0.
It is well known that (up to a sign) ω w is the composition Hom(A 0w , B 0w ) -→ Hom(A 0w , F w ) -→ Ext 1 (F w , F w ) of maps induced by the preceeding exact sequence. Now the result follows easily from the exact sequence 0 -→ End(F w ) -→ End(B 0w ) -→ Hom(A 0w , B 0w )/ End(A 0w ) -→ Ext 1 (F w , F w ) -→ 0.

We must now verify that if z 1 , z 2 ∈ M (t) are distinct closed points, then E z 1 and E z 2 are not isomorphic. This follows from the more general following result : if two injective morphisms of vector bundles on

P n O(-2) ⊗ C n 1 -→ (O(-1) ⊗ C m 2 ) ⊕ (O ⊗ C m 1 )
have isomorphic cokernels, then they are in the same orbit.

The property (iii) of the definition of a fine moduli space is easily verified.

It follows that the q moduli spaces of morphisms M 1 , • • • , M q , with their corresponding universal sheaves, are also fine moduli spaces of torsion free sheaves on P n . The moduli space M q is the same as the obvious one M (cf. 10.2), and the corresponding universal sheaf is the same (up to an element of P ic(M)) as E.

These examples are generalizations of the case of P 2 (with k = 7) that was treated in [START_REF] Drézet | Variétés de modules alternatives[END_REF]. But in this case our results are not needed, because we get only two fine moduli spaces : one is the obvious moduli space and the other is the corresponding moduli space of stable sheaves on P 2 .

On P n , n ≥ 3, our moduli spaces are new. We don't know if the corresponding moduli space of stable sheaves is among them.

10.4.5. Remark: it is not hard to prove that all the moduli spaces M 1 , • • • , M q are distinct.

2 .

 2 The moduli problem for decomposable homomorphisms 3. Actions of reductive groups 4. Semi-stability in the non-reductive case 5. Embedding into a reductive group action 6. Construction and properties of the quotient 7. Comparison of semi-stability 8. Projectivity conditions 9. Examples 10. Construction of fine moduli spaces of torsion free sheaves References 1. Introduction

  and an extension of φ to a morphism Z → Z we obtain a smooth curve Y , y 0 ∈ Y and a morphism β : Y → Z such that β(y 0 ) = z 0 and that the principal G ybundle Γ = β * α * G is defined on Y \{y 0 } and locally trivial. Let U ⊂ Y be a nonempty open subset such that we have a G y -isomorphism γ : Γ |U U × G y

5. 3 . 2 . 5 . 4 .

 3254 Corollary: Let w, w ∈ W . Then w and w are in the same G-orbit in W if and only if ζ(w) and ζ(w ) are in the same G-orbit in W. Proof. It follows from the compatibility of the actions that if g.w = w in W then also θ(g).ζ(w) = ζ(w ) in W by the last diagram. Conversely, if g ∈ G and g.ζ(w) = ζ(w ) then g stabilizes (ξ 2 , • • • , ξ r , η 1 , • • • , η s-1 ) by the definition of ζ in 5.2. By Lemma 5.3.1 g ∈ G. The associated polarization In 3.3.1 and 3.3.2 we had introduced polarizations for the different types of actions of G red on W and of G on W. In the following we will describe polarizations on W and W which are compatible with the morphism ζ : W -→ W. Their weight vectors are related by the following matrix equations and determine each other. The entries of the matrices are just the dimensions of the spaces A ji and B ml .

  H * si be induced by the maps B sl ⊗ H * si → H * li and let L be the family of proper subspaces L ⊂ ⊕ l<s N * l ⊗ B sl which are not contained in ⊕ l<s N l ⊗ B sl for any family (N l ) = (N * l ) of subspaces. Then we define

7. 1 . 1 .

 11 Lemma: If m i ≤ mi for all i ≥ 2, then c l (m 2 , . . . , m r ) ≤ c l ( m2 , . . . , mr ).

8. 1 . 2 .

 12 Corollary: With the same notation as in 8.1.1, if w ∈ Z Z, then (1) rank x i ≤ rank ξ i and rank y l ≤ rank η l with strict inequality for at least one i or l, and (2), (2 * ), (3), (3 * ) of 8.1.1 are satisfied. Proof. All conditions are closed and thus hold for points in Z. If w ∈ Z Z then by 8.1.1 equality in (1) cannot hold for all i and l.

8. 1 . 3 .

 13 Proposition: Let the polarizations Λ and Λ be as in proposition 7.5.3 and let Z = Gζ(W ). Then Z Z contains no semi-stable point in the following cases(i) (r, s) = (2, 1) and λ 2 ≥ c 1 (m 2 )a 21 µ 1 (ii) (r, s) = (2, 2) and λ 2 ≥ (µ 1 c 1 (m 2 )+µ 2 (c 2 (m 2 )-b 21 c 1 (m 2 ))a 21 , µ 1 ≥ (λ 1 (d 1 (n 1 ))-d 2 (n 1 )a21 )+λ 2 d 2 (n 1 ))b 21 .

λ 2 -( λ 2 a 21 -µ 1 c 1 (m 2 )

 22112 a 21 λ 1 > 0 and p 2 a 21 -p 1 > 0, we get the estimate ∆ > -µ 2 c 2 (m 2 ) + µ 2 c 1 (m 2 )b 21 )p 1 .

p 2 a 32 and canceling α 2 p 2 and then substituting p 2 ≥ p 1 a 8 . 3 .

 2183 21 to get a 32 a 21 ∆ ≥ λ 1 p 1 (a 32 a 21 -a 31 ) + (λ 3 -µ 1 c 3 a 32 a 21 )p 1 . Proof of theorems 1.5.1 and 1.5.2 Theorem 1.5.1 is an immediate consequence of proposition 6.1.1, corollary 7.2.2 and proposition 8.1.3. Theorem 1.5.2 follows immediately from theorem 1.5.1 and 9.1.

Figure 1

 1 Figure 1

9. 5 . 2 ) 1 ( 2 ) = d 2 ( 2

 521222 More general homomorphisms of type (2, More general homomorphisms for which we know the constants explicitly are homomorphisms of type m 1 O(-2) ⊕ 2O(-1) → 2O ⊕ n 2 O(1) over P 3 , say. By proposition 9.1 the constants are here c

7 4 n 2 λ 2 ≥

 72 (n 2 -8)µ 1 + 4

Figure 2

 2 Figure 2Here the horizontal axis represents m 2 λ 2 and the vertical axis represents n 1 µ 1 for m 1 = 3 and n 2 = 5.

4 .

 4 Φ 1 is isomorphic to the canonical morphism O(-2) ⊗ P -→ O(-1) hence we have ker(Φ 1 ) O(-3), and Im(Φ 1 ) I P n-2 (-1) (the ideal sheaf of P n-2 twisted by O(-1)). Let Φ : O(-3) -→ O ⊗ C k be the restriction of Φ 2 to ker(Φ 1 ). It vanishes on P n-2 and induces a linear mapf : C k * -→ H 0 (I P n-2 (3)).10.1.1. Lemma : If Φ is semi-stable (for some polarization) then f is injective.Proof. Let K 0 = ker(f ) ⊥ ⊂ C k . Then Im(Φ ) ⊂ O ⊗ K 0 . The morphism O(-2) ⊗ C 2 -→ O(-1) ⊕ (O ⊗ C k /K 0 )induced by Φ vanishes on O(-3) = ker(Φ 1 ). Hence it induces a morphism (ψ 1 , ψ 2 ) :I P n-2 (-1) -→ O(-1) ⊕ (O ⊗ C k /K 0 )where ψ 1 is the inclusion. Since Hom(I P n-2 (-1), O) = Hom(O(-1), O), we can (by replacing Φ by an element of its Hom(O(-1), O ⊗ C k )-orbit) suppose that ψ 2 = 0. It follows that Im(Φ) ⊂ O(-1) ⊕ (O ⊗ K 0 ), and since Φ is semi-stable, we have K 0 = C k , i.e. f is injective. Now it remains to prove that Φ 2 is generically injective on H, but this follows easily from the fact that Φ 2 is defined by an injection C 2 → H 0 (O H (2)) ⊗ C k . 10.Fine moduli spaces of torsion-free sheaves 10.4.1. Définition : Let S be a smooth variety, F a coherent sheaf on S × P n , flat on S. We say that S is a fine moduli space of sheaves with universal sheaf F if the following properties are verified : (i) For every closed point s ∈ S the Kodaïra-Spencer map ω s : T s S -→ Ext 1 (F s , F s ) is bijective. (ii) For every closed points s 1 , s 2 ∈ S with s 1 = s 2 , F s 1 and F s 2 are not isomorphic. (iii) For every flat family E of coherent sheaves on P n parametrized by an algebraic variety T , and for any closed points s ∈ S, t ∈ T such that F s E t , there exists an open neighbourhood U of t in T , and a morphism f : U → S such that f (t) = s and (f × I Pn ) * (F) E |U .

10. 4 . 2 .

 42 Application of theorem 1.5.2Polarizations for morphismsO(-2) ⊗ C 2 -→ O(-1) ⊕ (O ⊗ C k ) are defined by pairs (λ 1 , λ 2 ) of positive rational numbers such that λ 2 + λ 1 k = 1 (so here λ 2 is associated to O(-1) and λ 1 to O ⊗ C k ).By theorem 1.5.2, there exists a projective good quotient of the open subset W ss of semi-stable points as soon as t = λ 2 > n + 1 n + 1 + k .

  is a polarization on W such that α i is the weight of P i and -β l the weight of Q l . It is called the associated polarization on W. It is compatible with ζ in the following sense: If M i ⊂ M i , and N l ⊂ N l are linear subspaces, and if the subspaces of P i and Q l are defined by

  It follows that Y x is a variety of dimension 23. Using the techniques of this paper for this quotient, we can even prove that Y is smooth. Finally Y has dimension 25 and thus is a divisor in the irreducible and normal variety M -.

)). The condition of defining a element of Y is that Γ is neither contained in H 0 I x (3) nor in H 0 (O(2))s for any section s of O ⊕ I x (1). We let U x ⊂ Gr(2, H 0 (O(2) ⊕ I x (3)) denote the open subvariety of such Γ. By assigning to Γ the class of (φ 2 , φ 2 ) where φ 1 is defined by a lifting in the above diagram, we get a morphism U x → M -whose image is the fibre

Y x = p -1 (x). The morphism U x Y x

is nothing but the quotient of U x by the algebraic group Aut(O ⊕ I x (1)).

  There are polarizations Λ such that W ss (G, Λ) admits a good and projective quotient in the following cases(i) m 1 < 6 and n 2 < 8 (i') m 1 ≤ 6 and n 2 = 8 (ii) 8 ≤ m 1 + 3 ≤ n 2 and 8m 1 + 8 < 7n 2Proof. The conditions of 7.2.2 for the equivalence of (semi-)stability become

	4 8 + m 1	< λ 2 <	1 2	and	4 8 + n 2	< µ 1 <	1 2	(1)
	9.5.1. Claim: λ 2 ≥	4 7	(µ 1 + 4µ 2 ) and µ 1 ≥	16 7	(4λ 2 -15λ 1 )	(2)
	and the conditions of 8.1.3 for the projectivity of the quotient become
				λ 2 ≥	4 7	µ 1 and µ 1 ≥	4 7	λ 2 .

  Then[START_REF] Dolgachev | Variation of Geometric Invariant Theory Quotients[END_REF] has a solution µ 1 if and only if7n 2 -4m 1 -32 > 0 (7n 2 -4m 1 -32)(n 2 + 8) > 4(n 2 -8)(m 1 + 8) 7(7n 2 -4m 1 -32) > 16(n 2 -8)These inequalities reduce to 7n 2 > 4m 1 + 32 7n 2 > 8m 1 + 8 33n 2 > 28m 1 + 96

		reduces to
	7n 2 -4m 1 -32 (n 2 -8)(m 1 + 8)	≥ µ 1 >	4 n 2 + 8	.	(6)

Estimate for ζ 1

Let w = (φ li ) in W be given and assume that ζ 1 (w) is not semi-stable. Then there are linear subspaces P i ⊂ P i and N l ⊂ N l and a unipotent element h ∈ H R such that for (γ 1 , . . . , γ s ) = h . (γ 1 , . . . , γ s ) we have ξ i (P i ⊗ A i,i-1 ) ⊂ P i-1 and γ l (P l ⊗ H * l1 ) ⊂ N l for all i ≥ 2 and all l, and such that

We may assume that h = id because H R acts on W in the same way and we can replace w by h.w. Moreover, we may assume that all N l are equal to γ l (P 1 ⊗ H * l1 ) since all µ l > 0. Now we proceed as in 7.2 replacing the spaces Q l by N l . Therefore we find subspaces M i ⊂ M i ⊂ M i such that M 1 = M 1 and such that

and the family M is minimal with this property. We denote ρ i = codim(P i (M ), P i ) , σ i = codim(P i (M ), P i (M )) and let N l = γ l (P 1 (M ) ⊗ H * l1 ) ⊃ N l . As in 7.2.1 we consider the surjection

where Y l ⊂ Z l are the same, and we get the estimate n l -n l ≤ c l (m 2 , . . . , m r )(σ 1 -ρ 1 ) for any l. The estimation of the discriminant ∆ is now simpler than in 7.2.

Lemma: With the above notation

where c l (m) = c l (m 2 , . . . , m r ).

Proof. By replacing dimensions and inserting the estimate for n l -n l as in 7.2 we get

7.4.2. Corollary: Let Λ = (λ 1 , . . . , λ r , -µ 1 , . . . , -µ s ) be a polarization for W and let Λ = (α 1 , . . . , α r , -µ 1 , . . . , -µ s ) be the associated polarization for V as in 7.3. If all

Proof. The proof is the same as for 7.2.2, because the spaces P i and P i (M ) are defined in the same way and we thus get the estimate (σ 2 -ρ 2 )a 21 ≥ σ 1 -ρ 1 .

Estimate for ζ 2

The analogous estimate for ζ 2 follows by duality while we can assume that s = 1 or r = 1. The proof could be done by formally transform it into a dual situation which is similar to that of 7.4, but it is better to keep direct track of the weights. Let (x 2 , . . . , x r , γ 1 , . . . γ s ) be given in W L ⊕ V and assume that its image under ζ 2 is not semi-stable. Then there are subspaces

where γ is defined as in 7.3, and such that

We assume that all α i ≥ 0, and then we may assume that P i is maximal, i.e. the inverse image of P i-1 ⊗ A * i,i-1 under P i → P i-1 ⊗ A * i,i-1 for i ≥ 2, and similarly P 1 in P 1 under P 1 → Q s ⊗ H s1 . As in 7.4 we can find subspaces N l ⊂ N l such that

We choose subspaces N l ⊂ N l which are maximal such that

We have N s = N s . We let P 1 be the inverse image of Q s (N ) under P 1 → Q s ⊗ H s1 . Then P 1 ⊂ P 1 . Furthermore we let inductively P i ⊂ P i be the inverse images for i ≥ 2. Then we have injections

and induced by factorization the images

The induced injections P i /P i (P 1 /P 1 ) ⊗ A * i1 9. Examples

Constants

We give here some constants (cf. 7.1) used in the examples. The following result is proved in [START_REF] Drézet | Espaces abstraits de morphismes et mutations[END_REF], prop. 6.1.

9.1.1. Proposition : For homomorphisms of type

on a projective space of dimension n we have

9.1.2. Lemma: For homomorphisms of type

Proof. We put s(p) = dim(S p V ). The homomorphisms δ 1 of 7.1 reduces here to the canonical map

If K is a proper subspace of S d-2 V * ⊕ S d-1 V * which is not contained in one of the summands, it contains elements (f, g) with f = 0 or elements (f, g) with g = 0. But since

. But now we can find subspaces which realize this bound. For any z ∈ V * we let K be the space of all (f,

First example of type (2, 1)

We use the abbreviation mF for C m ⊗ F for a sheaf and a positive integer and consider here homomorphisms

----→ 3O over P 2 of type (2, 1). The polarization Λ = (λ 1 , λ 2 , -µ 1 ) is supposed to be proper for W and W, i.e. λ i > 0 and α i > 0 for all i. The only constant involved here is c 1 (m 2 ) = c(1) = 0. Therefore the conditions of 7.2.2 and 8.1.3 are automatically satisfied by α 2 = λ 2 -3λ 1 > 0. Hence all the quotients of W ss (G, Λ) will be good and projective Note that we have taken k ≤ (n + 1) 2 = h 0 (I P n-2 (3)), to allow the injectivity of f . Suppose that f is injective. Let K = Im(f ). then Φ is isomorphic to the canonical morphism

It is easy to see that P and K depend only on the G-orbit of Φ. Conversely, suppose P and K are given. We can define an element (Φ 1 , Φ 2 ) of W associated to P and K as follows : let (z 1 , z 2 ) be a basis of P . Let (z 1 q 1i + z 2 q 2i ) 1≤i≤k be a basis of K, with q 1i , q 2i ∈ S 2 V * . Using this basis we can identify K and K * with C k . We define

by

10.1.2. Lemma : Let K ⊂ H 0 (I P n-2 (3)) a linear subspace of dimension k. Then Φ K is injective outside of a closed subvariety of codimension 2.

Proof. Let x ∈ P n . Then Φ K is non injective at x if and only if all the elements of K (which are homogeneous polynomials of degree 3) vanish at x. Suppose that Φ K is non injective on an irreducible hypersurface S. Then all the polynomials in K vanish on S.

Let f be an irreducible equation of S. Then all the elements of K are multiple of f . It follows that f is of positive degree d ≤ 3, and

Suppose that there exists a polarization such that Φ is semi-stable. Then Φ is generically injective and coker(Φ) has no torsion. Moreover, if K is generic, Φ is injective.

Proof. Lemma 10.1.2 implies that Φ is injective outside a closed subvariety of codimension ≥ 2. It follows that Φ is generically injective and that coker(Φ) has no torsion. To prove that Φ is injective for a generic K, it suffices to find a K such that Φ is injective. Let (z 1 , z 2 ) be a basis of P . Let q 1 , • • • , q r , (resp. q 1 , • • • , q s ) be linearly independant elements of S 2 V * that have no common zeroes in P n , with r + s = k (this is possible since 2n + 2 ≤ k ≤ (n + 1) 2 ). Let

It is easy to see that for such a K, Φ is injective.

The obvious moduli space of morphisms and its universal sheaf

Let P ⊂ V * a plane, P n-2 ⊂ P n the subspace defined by P and K ⊂ H 0 (I P n-2 (3)) a linear subspace of dimension k. Let E(P, K) = coker(Φ), where Φ is a morphism associated to P and K. Since the G-orbit of Φ is determined by P and K, E(P, K) is well defined. We will give another construction of E(P, K).

Let F K = coker(Φ K ). It is a torsion free sheaf according to lemma 10.1.2.

10.2.1. Lemma : We have Ext 1 (O P n-2 (-1), F K ) C, and the non-trivial extension of

Proof. The exact sequence

where θ comes from the multiplication

The kernel of µ is canonically isomorphic to ∧ 2 P ⊥ ⊗ V * and it is easy to see that i K is contained in the image of µ ⊗ I K * . It follows that we have an exact sequence

The last assertion follows from the commutative diagram with exact rows and columns :

Let M be the projective variety of pairs (P, K), where P is a plane of V * and K ⊂ H 0 (I P n-2 (3)) is a vector subspace of dimension k (P n-2 beeing the codimension 2 linear subspace of P n defined by P ). We can view M as a moduli space for generic morphisms. We will give a construction of a universal sheaf E on M × P n , i.e E is flat on M and for every (P, K) ∈ M, E (P,K) is isomorphic to the cokernel of a generic morphism associated to (P, K). It is also possible to define a universal morphism whose cokernel is isomorphic to E, but we will see this more generally in 10.4.

Let Gr(2, V * ) be the grassmannian of planes in V * and q : M → Gr(2, V * ) be the obvious projection. Let U be the universal subsheaf of O × V * on Gr(2, V * ). Let p M : M × P n → M, p 2 : M × P n → P n be the projections. Then we have a canonical obvious morphism of vector bundles on

Let P be its cokernel. It is a flat family of sheaves on P n . For every (P, K) ∈ M we have P (P,K) = O P n-2 . Let K be the universal sheaf on M × P n , such that K (P,K) = K. Then we have a canonical obvious morphism of vector bundles on M × P n p * 2 (O(-3)) -→ K * . Let F be its cokernel. Then for every (P, K) ∈ M, F (P,K) is the sheaf that was noted F K before. By lemma 10.2.1, the sheaf Ext 1 p M (P ⊗ p * 2 (O(-1)), F) is a line bundle L on M. Then we have a universal extension

on M × P n . Then using lemma 10.2.1 it is easy to see that for every (P, K) ∈ M, E (P,K) is isomorphic to the cokernel of a generic morphism associated to (P, K).

Special morphisms

Let Φ = (Φ 1 , Φ 2 ) be a special morphism. Let f 1 : C 2 → V * the associated application of rank 1. Let H be the hyperplane of P n defined by Im(f 1 ). We have an exact sequence

be the linear map induced by Φ 2 .

10.3.1. Lemma : If Φ is semi-stable (for a given polarization) then Φ 2 is injective.

be a basis of V * , such that z 1 is an equation of H. By using the action of Hom(O(-1)

Now Φ 2 is not zero on C 1 : otherwise we would have q 1i ∈ z 1 V * , and Im(Φ

and this would contradict the semi-stability of Φ. Hence, by considering the action of GL [START_REF] Bialynicki-Birula | Open subsets of projective space with a good quotient by an action of a reductive group[END_REF], it suffices to prove that Φ 2 does not vanish on C 2 . Suppose it does. Then Φ 2 vanishes on O(-2)⊗C 2 , because q 2i ∈ S 2 < z 2 , • • • , z n+1 >, and again Im(Φ 2 ) ⊂ O ⊗ C k , with k ≤ dim(S 2 V * ) < k, which contradicts the semi-stability of Φ.

10.3.2. Lemma : Suppose that Φ is semi-stable with respect to some polarization. Then it is injective outside of a closed subvariety of codimension ≥ 2, and coker(Φ) has no torsion.

Proof. It suffices to prove the first statement. Let x ∈ P n and u ∈ C 2 such that Φ 1 (x 2 ⊗ u) = 0. Then we have either u ∈ C 1 or u ∈ C 1 and x ∈ H. Suppose that Φ is not injective at all points of an irreducible hypersurface D = H. Then the same is true for Φ |O(-2)⊗C 1 . Suppose that this morphism is defined by quadratic forms q 1 , • • • , q k . These forms vanish on D, hence they are all multiple of an equation of D. It follows as in the proof of 10.3.1 that Im(Φ 2 ) ⊂ O ⊗ C k , with k < k, which contradicts the semi-stability of Φ.

(where [x] denotes the integer part of x). Then we obtain exactly q moduli spaces of morphisms corresponding to non critical values : M 1 , • • • , M q , where for 1 ≤ i < q

, beeing a sufficiently small positive rational number. We have M q = M (cf. the end of 10.2).

Fine moduli spaces

Suppose that we choose a polarization such that t is not a critical value. In this case we have W ss = W s , and the stabilizer in G of the points of W s is the canonical subgroup isomorphic to C. Let M (t) = W s /G, and π : W s → M (t) be the quotient map. On W s × P n we have a universal morphism

is a flat family of torsion free sheaves on P n parametrized by W s (this is a consequence of lemmas 10.1.3 and 10.3.2). There is a canonical action of G on F such that C acts by multiplication.

Recall that a G-sheaf E on W s × P n descends to M (t) × P n if there exists a coherent sheaf E on M (t) × P n and a G-isomorphism (π × I Pn ) * (E ) E.

Theorem :

There exists a G-line bundle L on M (t) × P n such that F ⊗ L descends to M (t). Let E be the corresponding sheaf on M (t) × P n . Then M (t) is a fine moduli space of sheaves on P n with universal sheaf E.

Proof. On W s we have a canonical action of G on the bundles O W s ⊗ C 2 , L = O W s and O W s ⊗ C k . On these bundles C acts as ordinary multiplication by scalars. Let A 0 , B 0 be the G-bundles

) (where p W is the projection W s × P n → W s ). On these bundles C acts trivially. We can multiply the universal morphism with p * W (L -1 ) and we obtain a new universal morphism Ψ 0 : A 0 -→ B 0 . Now it is easy to see that the bundles A 0 , B 0 descend to M (t) × P n either directly from our construction of the quotient, or by using the more general results of [START_REF] Drézet | Quotients algébriques par des groupes non réductifs et variétés de modules de complexes[END_REF], 2.3. Let A = A 0 /G, B = B 0 /G. The G-morphism Ψ 0 also descends and we get a universal morphism of vector bundles on M (t) × P n Ψ : A -→ B.

We define now E = coker(Ψ), and it is clear that π * (E) F ⊗ L -1 .