Table Of Content

ON SOME QUOTIENTS MODULO NONREDUCTIVE GROUPS 0 1 MARIOMAICAN 0 2 n a J Contents 4 1. Introduction 1 1 2. Quotients Modulo Nonreductive Groups 3 ] 3. A Criterion for Morphisms of Type (2,1) 7 G 4. Morphisms of the Form mO(−d )⊕2O(−d )−→nO 11 1 2 A 5. Morphisms of the Form O(−d−1)⊕3O(−d)−→nO 14 . 6. Morphisms of the Form mO(−d−1)⊕3O(−1)−→nO 17 h 7. Morphisms of the Form O(−d−2)⊕3O(−d)−→nO 20 t a 8. Morphisms of the Form mO(−d )⊕O(−d )⊕O(−d )−→nO 24 1 2 3 m References 29 [ 3 v 9 1. Introduction 0 Quotients modulo nonreductive groups arise in the study of moduli spaces of 0 2 sheaves. Letkbeanalgebraicallyclosedfieldofcharacteristiczero. Wefixcoherent 1 algebraic sheaves E and F on the projective space Pr = Pr(k), which are direct 6 sums of simple sheaves. The linear algebraic group G = Aut(E)×Aut(F) acts 0 by conjugation on the finite dimensional vector space W = Hom(E,F). To every / h character of G Drézet and Trautmann associate subsets Wss ⊂ W of semistable at points andWs ⊂W of stable points. One expects there to be categoricalquotients m Wss//G and Ws/G. In [4] the author has found many examples of locally closed subsets inside moduli spaces of semistable sheaves (in the sense of Gieseker) on : v P2, with linear Hilbert polynomial, that are isomorphic to quotients of the form i X Wss//G. r This paper is not concerned with the study of moduli spaces but only with the a questionofexistence ofquotientsmodulo G. IfE hasmorethanonekindofsimple sheaf in its decomposition, then Aut(E) is nonreductive, so this situation falls out- side Mumford’s Geometric Invariant Theory. Drézet and Trautmann address this difficulty in [3] by embedding the action of G on W into the action of a reductive group G onto a larger space W. Their main result states that if certain compatibility conditions relating the semistable points of W and W are satisfied, then e f Wss//Gand Ws/G exist andare projective,respectivelyquasiprojectivevarieties. f Moreover, they find sufficient conditions, expressed in terms of linear algebra constants, under which the compatibility conditions are fulfilled. This allows Drézet and Trautmann to establish the existence of quotients for certain classes of morphisms, notably for morphisms of the form m O(−2)⊕m O(−1)−→nO, 1 2 1 2 MARIOMAICAN cf. 6.4 in [2]. The purpose of this paper is to give more examples to the Drézet- Trautmanntheory. We use their embedding into the actionofGbut wedo notuse their linearalgebraconstants. We areconcernedonly withthe geometricquotients e Ws/Gandwedonotdiscussproperlysemistablemorphisms,i.e. morphismswhich aresemistable but not stable. Applying 6.6.1 from[3] we establish the existence of geometric quotients in the following situations: • mO(−d )⊕2O(−d ) −→ nO on Pr such that 0 < λ < 1/(2a+m) and 1 2 1 either the conditions r−1+d −d 1 1 r+d −1 1 2 2 m< , λ ≤ 1− , 1 (cid:18) r−1 (cid:19) a+m−1(cid:18) n(cid:18) r (cid:19)(cid:19) or the conditions r+d −d 1 2 r+d −1 1 2 2 m< , λ ≤ − 1 (cid:18) r (cid:19) 3m 3mn(cid:18) r (cid:19) are satisfied. Here m and n are not both even; • O(−d−1)⊕3O(−d)−→nO on P2 such that 1 0<λ < , 1 10 1 3 2 3 − + (d2+d)≤λ ≤ − (d2+d), 1 2 4n 5 10n 3 3 −2+ (d2+2d)≤λ ≤1− (d2+3d); 1 n 4n • mO(−d−1)⊕3O(−1)−→nO on P2 with m<a and 1 n−3 n−6 0<λ < , λ (4m−3a+3b)≤ , λ ≤ . 1 1 1 3a+m n mn Here a=(d+1)(d+2)/2 and b=d(d+1)/2 and 3∤gcd(m,n); • O(−d−2)⊕3O(−d)−→nO on P2 such that 6 1 <λ < , 2 19 3 1 1 d+1 λ ≤ − , 2 2 2n(cid:18) 2 (cid:19) 1 d+2 1 d+1 λ ≤1− − , 2 n(cid:18) 2 (cid:19) n(cid:18) 2 (cid:19) 1 d+2 λ ≤ ; 2 n(cid:18) 2 (cid:19) • mO(−d )⊕O(−d )⊕O(−d )−→nO on Pr satisfying 3 2 1 d −d +r−1 1 2 m< , (cid:18) r−1 (cid:19) 1−mλ −a λ +a a λ 1 31 1 32 21 1 a λ <λ < , 21 1 2 1+a 32 λ <1−mλ , 2 1 1 1 d +r 3 λ ≤ − . 1 m+a mn+a n(cid:18) r (cid:19) 21 21 ON SOME QUOTIENTS MODULO NONREDUCTIVE GROUPS 3 Theprecisemeaningofλ andλ isrevealedinsection2: theyencodethecharacter 1 2 ofGrelativetowhichthesetsofstablepointshavebeendefined. Besidetheabove, we have at (3.3) a more general criterion giving sufficient conditions, expressed in terms of linear algebra constants, under which morphisms of type (2,1) m O(−d )⊕m O(−d )−→nO 1 1 2 2 satisfy the compatibility conditions from [3] that lead to existence of geometric quotients. The paper is organized as follows: in section 2 we supply background material about quotients modulo nonreductive groups. As this is not part of mainstram GeometricInvariantTheory,we felt it necessaryto reproducethe mainresults and definitionsfrom[3]. Section3containsourcriterionformorphismsoftype(2,1). We apply this criterion in section 7. The remaining sections are indirect applications of (3.3), by which we mean applications of the method of proof, rather than the statementof (3.3). All polarizationswe discuss are assumedto be nonsingularin a sense that is made precise in our very brief discussion of the Geometric Invariant Theory fan, cf. the beginning of section 3. 2. Quotients Modulo Nonreductive Groups In this section we reproduce some notations and definitions and we quote the main result from [3]. We also quote two particular cases of King’s Criterion of Semistability, as formulated in [3], which we will use in the subsequent sections. Fix a vectorspaceV overk ofdimensionn+1. DrézetandTrautmannconsider coherent algebraic sheaves E and F on the projective space Pn = P(V) having decompositions E = M ⊗E , F = N ⊗F . i i l l 1M≤i≤r 1M≤l≤s HereM , N arevectorspacesoverk ofdimensionsm ,n . In[3]itisassumedthat i l i l E andF aresimplesheaves,butforthepurposesofthispaperwewillassumethat i l they are line bundles: E =O(e ), e <...<e , F =O(f ), f <...<f . i i 1 r l l 1 s The linear algebraic group Aut(E)×Aut(F) acts by conjugation on the finite dimensional vector space W =Hom(E,F). The subgroup of homotheties, which we identify with k∗, acts trivially so, without losing any information, we can instead consider the action of the quotient G=Aut(E)×Aut(F)/k∗. Ifr>1,orifs>1,thegroupGisnonreductive,however,itcontainsthereductive subgroup G =GL(M )×...×GL(M )×GL(N )×...×GL(N )/k∗. red 1 r 1 s We represent elements of G by pairs (g,h), with red g =(g ,...,g ), h=(h ,...,h ), g ∈GL(M ), h ∈GL(N ). 1 r 1 s i i j j The characters of G are of the form red χ(g,h)= det(g )−λi · det(h )µl i l 1≤Yi≤r 1≤Yl≤s 4 MARIOMAICAN for integers λ ,...,λ ,µ ,...,µ . As χ must be trivial on the subgroup of homo- 1 r 1 s theties, we require the condition m λ =d= n µ . i i l l 1≤Xi≤r 1≤Xl≤s Clearly χ extends to a character of G, which we also denote by χ. Drézet and Trautmann call a polarization the tuple Λ=(−λ /d,...,−λ /d,µ /d,...,µ /d). 1 r 1 s Theyconsidera semistability notionfor G depending onΛ. We quotebelow the red definition. We point out that, relative to a certain linearization, this is the usual notion of semistability from Geometric Invariant Theory. This is made precise at lemma 3.4.1 in [3]. (2.1) Definition: Let Λ be a fixed polarization. A point ϕ∈W is called (i) semistable with respectto G and Λ if there areaninteger m≥1 andan red algebraic function f on W satisfying f(g.w) =χm(g)f(w) for all g ∈ G red and w ∈W, such that f(ϕ)6=0; (ii) stable withrespecttoG andΛ iftheisotropygroupofϕinG isfinite red red and there is f as above, but with the additional property that the action of G on the set {w∈W, f(w)6=0} has closed orbits. red This definition is consistent because proportionaltuples of integers give rise to the same sets of semistable (stable) points. Let T be the maximal torus of G . Note red that T is also a maximal torus in G. A point ϕ ∈ W is semistable (stable) with respect to G and Λ if and only if every point in its G -orbit is semistable red red (stable) with respectto T andthe restrictionofχ to T. Taking this equivalence as definition for semistability (stability) with respect to G we arrive at the following concept introduced by Drézet and Trautmann: (2.2) Definition: A point ϕ ∈ W is called semistable (stable) with respect to G andΛifeverypointinitsG-orbitissemistable(stable)withrespecttoG andΛ. red We denote by Wss(G,Λ) and Ws(G,Λ) the sets of semistable, respectively stable points in W. For checking semistability in concrete situations we need a criterion derived by A. King in [1] from Mumford’s Numerical Criterion. We use its formulation from [3]. Below we quote only a particular case. Let us represent a point ϕ ∈ W by a matrix (ϕ ) with ϕ ∈Hom(M ⊗H∗,N ), H =Hom(E ,F ). li 1≤l≤s,1≤i≤r li i li l li i l A family of subspaces M′ ⊂ M , N′ ⊂ N will be called admissible if not all i i l l subspaces are zero and we do not have M′ =M , N′ =N for all i, l. i i l l (2.3) Proposition: A morphism ϕ ∈ W is semistable (stable) with respect to G and Λ if and only if for each admissible family of subspaces M′ ⊂ M , N′ ⊂ N , i i l l which satisfies ϕ (M′⊗H∗)⊂N′ for all i, l, li i li l ON SOME QUOTIENTS MODULO NONREDUCTIVE GROUPS 5 we have s r µ dim(N′) ≥ (>) λ dim(M′). l l i i Xl=1 Xi=1 Drézet and Trautmann embed the action of G on W into the action of a reductive group G on a finite dimensional vector space W. They introduce associated polarizations e f Λ=(α ,...,α ,β ,...,β ) 1 r 1 s and define sets of semistableeand stable points Wss(G,Λ), respectively Ws(G,Λ), as at (2.1). In general they determine the following relationships: f e e f e e (2.4) Proposition: Let ζ : W −→ W denote the embedding map. Then we have the inclusions f ζ−1(Wss(G,Λ))⊂Wss(G,Λ) and ζ−1(Ws(G,Λ))⊂Ws(G,Λ). f e e f e e They also find sufficient conditions under which the reverse inclusions hold but, in this paper, we will not use them. The main result from [3] states that, if the sets of semistable (stable) points in W and W are compatible, then there are good or geometric quotients modulo G. We quote below the part that will be used in the f sequel: (2.5) Proposition: If ζ−1(Ws(G,Λ)) = Ws(G,Λ), then there exists a geometric quotient Ws(G,Λ)/G which is a smooth, quasiprojective variety. f e e Combining (2.4) and (2.5) one arrives at the following: (2.6) Proposition: If ζ(Ws(G,Λ)) ⊂ Ws(G,Λ), then there exists a geometric quotient Ws(G,Λ)/G which is a smooth, quasiprojective variety. f e e The goal of this paper is to give classes of examples of geometric quotients that result as applications of (2.6). We will avoid any discussion of properly semistable morphisms,i.e. morphismswhicharesemistablebutnotstable. DrézetandTraut- mann’stheoryworksundercertainapriorirestrictionsonthepolarizationΛ. First, it is noticed that the set of semistable (stable) points in W is nonempty only if λ ≥ (>)0 and µ ≥ (>)0 for all i, l. i l Other conditions can be found at 5.4.1 in [3]: Ws(G,Λ) is not empty only if (2.7) α2 >0 ffore (er,s)=(2,1), α >0, λ p <1 for (r,s)=(3,1). 3 1 1 We will describe the embedding only in the case (r,s)=(2,1). Let us write E =O(−d ), E =O(−d ), F =O. 1 1 2 2 1 The polarization Λ is a triple 1 Λ=(−λ ,−λ ,µ ), with µ = and m λ +m λ =1. 1 2 1 1 1 1 2 2 n 6 MARIOMAICAN Recall that H =Hom(E ,F )=Sd1V∗, H =Hom(E ,F )=Sd2V∗. 11 1 1 12 2 1 Consider also the spaces A =Hom(E ,E )=Sd1−d2V∗, 21 1 2 P =Hom(E ,E)=M ⊕ M ⊗A , 1 1 1 2 21 P =Hom(E ,E)=M , 2 2 2 W =Hom(P ⊗A ,P ) ⊕ Hom(P ,H ⊗N ). 2 21 1 1 11 1 Smallcaselettersfa=a21, p1 =m1+m2a21, p2 =m2 denotethe dimensions ofthe corresponding spaces. The group G acting by conjugation on W is G=GL(P1)e×GL(P2)×GL(N1)/k∗, f and the associated polaerization is 1 Λ=(−α ,−α ,β ) with α =λ , α =λ −aλ , β =µ = . 1 2 1 1 1 2 2 1 1 1 n 1 We reperesent morphisms in W as matrices ϕ=(ϕ′,ϕ′′), ϕ′ =(ϕ ) , ϕ ∈H , ij 1≤i≤n1,1≤j≤m1 ij 11 ϕ′′ =(ϕ ) , ϕ ∈H . ij 1≤i≤n1,m1<j≤m1+m2 ij 12 We write ϕ for the jth column of ϕ. We consider the row vector j X = X ··· X 1 a (cid:2) (cid:3) with entries forming a basis of A . The p ×p -matrix with entries in A 21 1 2 21 0 0 ··· 0  XT 0 ··· 0  0 XT ··· 0 ξ =    .. .. .. ..   . . . .     0 0 ··· XT    represents an element in Hom(P ⊗A ,P ). The n ×p -matrix with entries in 2 21 1 1 1 H 11 γ(ϕ)= ϕ′ ϕ X ··· ϕ X m1+1 m1+m2 represents an element in Ho(cid:2)m(P ,H ⊗N ). We put ζ(ϕ) =(cid:3)(ξ,γ(ϕ)). It is clear 1 11 1 that γ(h ϕ) = h γ(ϕ) for all h ∈ GL(N ). For this reason, when it comes to 1 1 1 1 semistability considerations, we can, and we will assume that h is the identity 1 automorphism. WefinishthissectionwiththeparticularcaseofKing’sCriterionofSemistability that applies to W: f (2.8) Proposition: A point (x,γ) of W is semistable (stable) with respect to G and Λ if and only if for each admissible family of subspaces P′ ⊂ P , P′ ⊂ P , f 1 1 2 2e N′ ⊂N satisfying 1 e 1 x(P′ ⊗A )⊂P′, γ(P′⊗H∗ )⊂N′, 2 21 1 1 11 1 we have β dim(N′) ≥ (>)α dim(P′)+α dim(P′). 1 1 1 1 2 2 ON SOME QUOTIENTS MODULO NONREDUCTIVE GROUPS 7 3. A Criterion for Morphisms of Type (2,1) We fix integers d >d >0, we fix a vector space V of dimension r+1 and we 1 2 consider morphisms ϕ=(ϕ′,ϕ′′):m O(−d )⊕m O(−d )−→nO on Pr =P(V). 1 1 2 2 The polarization Λ is uniquely determined by λ ∈ [0,1/m ]. The theory of the 1 1 GIT-fan, as developed in [5] and other works, informs us that there are finitely many values 0 = s < s < ... < s = 1/m such that when λ varies in an 0 1 q 1 1 interval (s ,s ) the set of semistable morphisms does not change and each open κ κ+1 interval(s ,s )ismaximalwiththisproperty. Theintervals(s ,s )arecalled κ κ+1 κ κ+1 chambers. The pointss arecalledsingular values forλ . Eitherλ orλ uniquely κ 1 1 2 determine Λ, so we can talk of singular polarizations Λ, or singular values for λ . 2 AccordingtoKing’sCriterionofSemistability(2.3),wheneverthesetofproperly semistable morphisms with respect to Λ is nonempty, there is an equality s r µ dim(N′)= λ dim(M′). l l i i Xl=1 Xi=1 Those polarizations for which there is an equality as above, for some choice of subspaces N′ and M′, will be called irregular, or we may say that λ or λ is l i 1 2 irregular. There are situations in which all polarizations are irregular, but those situations will not be addressedin this paper. The other possibility for morphisms of type (2,1), which we assume henceforth, is a finite set of irregular polarizations. From King’s Criterion of Semistability we see that Ws(G,Λ) depends only on the set of tuples of integers (a ,...,a ,b ,...,b ), 0≤a ≤m , 0≤b ≤n , for which 1 r 1 s i i l l there is an inequality s r µ b > λ a . l l i i Xl=1 Xi=1 Letnow Λ0 be a fixedregularpolarization. If Λ is sufficiently close to Λ0, meaning that λ is sufficiently close to λ0, then Λ is also regular and the sets of tuples of 1 1 integerswiththeabovepropertyforΛandΛ0 arethesame. Thus,thesetsofstable morphisms, which for regular polarizations coincide with the sets of semistable morphisms, are the same. In other words, any open interval bounded by two consecutive irregular values for λ is contained in a chamber. Thus, all singular 1 polarizations are irregular. The author does not know if the converse statement is also true, but he will give below an example in which the irregular polarizations are singular. These considerations also show that for Λ in a chamber we have Wss(G,Λ) = Ws(G,Λ) because, if Λ happens to be irregular, we can perturb it slightly to a regular polarization in the same chamber. Given integers 0 ≤ κ ≤ m and 0 ≤ κ ≤ m we denote by l the smallest 1 1 2 2 κ1κ2 integer satisfying l κ1κ2 >κ λ +κ λ 1 1 2 2 n and we consider morphisms of the form ϕ =(ϕ′ ,ϕ′′ ), where κ1κ2 κ1κ2 κ1κ2 ⋆ 0 ⋆ 0 ϕ′ = lκ1κ2,m1−κ1 , ϕ′′ = lκ1κ2,m2−κ2 . κ1κ2 (cid:20) ⋆ ⋆ (cid:21) κ1κ2 (cid:20) ⋆ ⋆ (cid:21) 8 MARIOMAICAN By 0 we denote the identically zero l×k-matrix. According to King’s Criterion lk (2.3), the morphism ϕ is semistable if and only if it is not equivalent to ϕ for κ1κ2 any choice of κ and κ . 1 2 (3.1) Example: In the simplest case m =m =1 the irregular polarizations are 1 2 of the form Λ=(κ/n, 1−κ/n, 1/n), 0≤κ≤n. The set of semistable morphisms may be empty for some polarizations, for instance, if n > κ+dim(Sd2V∗) and κ/n < λ < (κ + 1)/n. We see this from the semistability conditions: ϕ is in 1 Wss(G,Λ) if and only if ϕ is not equivalent to a morphism of the form ⋆ 0 0 ⋆ κ+1,1 or n−κ,1 . (cid:20) ⋆ ⋆ (cid:21) (cid:20) ⋆ ⋆ (cid:21) The following two conditions are sufficientto guaranteethe existence of semistable morphisms corresponding to all polarizations: r+d −1 r−1+d 2 1 n≤ and n≤ . (cid:18) r (cid:19) (cid:18) r−1 (cid:19) To see this we choose a nonzero linear form ψ ∈ V∗ and a linear complement U ⊂V∗ of the subspace generated by ψ. We choose linearly independent elements ϕ ,...,ϕ ∈Sd1U and ψ ,...,ψ ∈Sd2−1V∗. 11 n1 12 n2 We put ϕ = ψ ψ for 1 ≤ i ≤ n. We claim that ϕ is not equivalent to a i2 i2 matrixhavingazeroentry. Indeed,the entriesfromthesecondcolumnarelinearly independent, and no linear combination of entries from the second column can divide a linear combination of entries from the first column, because the former is divisible by ψ, whereas the latter is not. Under the above conditions on n we see that the singular polarizations are pre- cisely the irregular ones. Indeed, for λ = κ/n there exist properly semistable 1 morphisms: just choose ϕ ,...,ϕ linearly independent in Sd1U, put ϕ =0 for 11 κ1 i1 κ+1≤i≤n and ϕ =ψ ψ as above. i2 i2 We now turn to the embedding into the action of the reductive group. In order to apply the theory from section 2 we need to assume a priori that α >0, that is 2 λ >aλ . According to (2.8), a point (ξ,γ)∈W is semistable if and only if 2 1 l (ξ,γ)≁(ξ ,γ ) with f>kλ +iα . ki lk 1 2 n Here ξ , γ are matrices of the form ki lk ⋆ 0 ⋆ 0 ξ = k,m2−i , γ = l,p1−k . ki (cid:20) ⋆ ⋆ (cid:21) lk (cid:20) ⋆ ⋆ (cid:21) Because of the special form of ξ we must have k ≤ m +ai. We conclude that, in 1 order to ensure the semistability of (ξ,γ), it is enough to require the condition l γ ≁γ with >kλ +iα and k ≤m +ai. lk 1 2 1 n Actually, it is enough to require l (3.2) γ ≁γ with >kλ +iα , m +(i−1)a<k ≤m +ia, 0≤i≤m . lk 1 2 1 1 2 n Before we state the main result of this section we introduce some linear algebra constants very similar in definition to the constants c and d from [3]. Given l i ON SOME QUOTIENTS MODULO NONREDUCTIVE GROUPS 9 integers 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ m a we denote by k(i,j) the maximal 2 2 dimension of a vector space U ⊂ M ⊗H which is not contained in M′ ⊗H 2 12 2 12 for any subspace M′ ⊂ M of dimension i and for which there is a subspace in 2 2 M∗⊗A ofdimensionatleastj orthogonaltoU underthecanonicalbilinearmap 2 21 (M ⊗H )×(M∗⊗A )−→H . 2 12 2 21 11 Givenaninteger2≤i≤m andavectorspaceM ofdimensioniweletk(i)bethe 2 maximaldimensionofasubspaceU ⊂M⊗H whichisnotcontainedinM′⊗H 12 12 for any proper subspace M′ ⊂ M, and for which there is a nonzero subspace in M∗⊗A , orthogonal under the canonical bilinear map 21 (M ⊗H )×(M∗⊗A )−→H . 12 21 11 (3.3) Claim: Let W be the space of morphisms ϕ:m O(−d )⊕m O(−d )−→nO on Pr =P(V). 1 1 2 2 Assume that m <a=dim(Sd1−d2V∗). Let Λ be a nonsingular polarization satis- 1 fying the following conditions: λ >aλ , 2 1 inλ ≥nm λ +k(i,m a−ia−m ) for 1≤i≤m −1, 2 1 1 2 1 2 inλ ≥nm λ +k(i) for 2≤i≤m , 2 1 1 2 nλ +inλ ≥k(i,m a−ia−a+1) for 1≤i≤m −1. 1 2 2 2 Then Wss(G,Λ) admits a geometric quotient modulo G, which is a quasiprojective variety. Proof: LetϕbeinWss(G,Λ). Accordingto(2.6),weneedtoshowthat(ξ,γ(ϕ))is stable. PerturbingslightlyΛwecanensurethatΛisanonsingularpolarizationand, at the same time, that Wss(G,Λ) has not changed. Thus Wss(G,Λ) = Ws(G,Λ) e and it is enough to show that γ(ϕ) satisfies condition (3.2). f e e f e e Wewillarguebycontradiction. Assumethatγ(ϕ)isequivalenttosomeγ with lk l >kλ +iα , m +(i−1)a<k ≤m +ia, 0≤i≤m . 1 2 1 1 2 n Let ψ, ψ′, ψ′′ denote the truncated matrices consisting of the first l rows of ϕ, ϕ′, ϕ′′. Inviewofthe commentsbefore(2.8),wemayassumethatthe vectorsubspace in M ⊗H ⊕ M ⊗H spanned by the rows of ψ is orthogonal to a subspace, 1 11 2 12 denotedKer(ψ),insideM∗ ⊕M∗⊗A ,ofdimensionatleastp −k. Orthogonality 1 2 21 1 here is understood under the canonical bilinear map (M ⊗H ⊕ M ⊗H )×(M∗ ⊕ M∗⊗A )−→H . 1 11 2 12 1 2 21 11 By analogy, considering the pairings (M ⊗H )×M∗ −→H and (M ⊗H )×(M∗⊗A )−→H , 1 11 1 11 2 12 2 21 11 we define the orthogonal subspaces Ker(ψ′) and Ker(ψ′′). The dimensions of the corresponding spaces are denoted by ker(ψ), ker(ψ′), ker(ψ′′). Case i=0, 1≤k ≤m . As ker(ψ′′)≥m a−k >(m −1)a we see that Ker(ψ′′) 1 2 2 intersects nontrivially every copy of A inside M∗ ⊗ A ≃ km2 ⊗ A . Thus 21 2 21 21 ψ′′ = 0. Replacing possibly ϕ with an equivalent morphism we may assume that 10 MARIOMAICAN ker(ϕ′) ≥ m −k. We obtain that ϕ is equivalent to ϕ , which contradicts the 1 k0 semistability of ϕ. Case i=1, m <k <a. Again ψ′′ =0. As l/n>m λ +α >m λ we arrive at 1 1 1 2 1 1 the contradiction ϕ∼ϕ . m1,0 Case i = 1, a ≤ k ≤ m +a. If ψ′′ ∼ ψ′′, then ker(ψ′′) = (m −1)a hence, after 1 1 2 possibly replacing ϕ with an equivalent morphism, we may assume that ker(ψ′)≥m +m a−k−(m −1)a=m +a−k. 1 2 2 1 Moreover,froml/n>kλ +α =(k−a)λ +λ wegetl≥l . Thusϕ∼ϕ , 1 2 1 2 k−a,1 k−a,1 contradiction. If ψ′′ ≁ψ′′, then the rows of ψ′′ span a space of dimension at most 1 k(1,m a−k)≤k(1,m a−a−m ). 2 2 1 From the hypotheses of the claim we have l>nkλ +nα ≥naλ +nα =nλ ≥nm λ +k(1,m a−k) 1 2 1 2 2 1 1 2 forcing l ≥l +k(1,m a−k). We conclude that ϕ∼ϕ , contradiction. m1,0 2 m1,0 Case i ≥ 2, m +(i−1)a < k < ia. If ψ′′ ∼ ψ′′ , then, taking into account the 1 i−1 inequalities l >m λ +(i−1)aλ +iα >m λ +(i−1)λ , 1 1 1 2 1 1 2 n we obtain the contradiction ϕ ∼ϕ . Assume now that ψ′′ ≁ ψ′′ . The rows m1,i−1 i−1 of ψ′′ must span a vector space of dimension at most k(i−1,m a−k). From the 2 hypotheses of the claim we have the inequalities l>nkλ +niα ≥n(m +(i−1)a+1)λ +niα 1 2 1 1 2 >nm λ +nλ +n(i−1)λ 1 1 1 2 ≥nm λ +k(i−1,m a−ia+1) 1 1 2 ≥nm λ +k(i−1,m a−k). 1 1 2 From the above we get l ≥ l +k(i−1,m a−k), leading to the contradiction m1,0 2 ϕ∼ϕ . m1,0 Case i ≥ 2, ia ≤ k ≤ m +ia. As above, if ψ′′ ∼ ψ′′ we get a contradiction. 1 i−1 Assume now that ψ′′ ∼ψ′′ and ψ′′ ≁ψ′′ . If ker(ψ′′)=(m −i)a, then, replacing i i−1 2 possibly ϕ with an equivalent morphism, we may assume that ker(ψ′)≥m +m a−k−(m −i)a=m +ia−k. 1 2 2 1 Moreover,fromtheinequalitiesl/n>kλ +iα =(k−ia)λ +iλ wegetl≥l . 1 2 1 2 k−ia,i These lead to the contradiction ϕ∼ϕ . k−ia,i If ker(ψ′′) > (m −i)a, then the rows of ψ′′ span a space of dimension at most 2 k(i). From the hypotheses of the claim we have the inequalities l>nkλ +niα ≥niaλ +niα =niλ ≥nm λ +k(i). 1 2 1 2 2 1 1 Thus l≥l +k(i) leading to the contradiction ϕ∼ϕ . m1,0 m1,0