A duality result for moduli spaces of semistable sheaves supported on projective curves

A Duality Result for Moduli Spaces of Semistable Sheaves Supported on Projective Curves

MARIOMAICAN(*)

ABSTRACT- We showthat the map sending a sheaf to its dual gives an isomorphism of the moduli space of semistable sheaves with fixed multiplicity and Euler char- acteristic and supported on projective curves to the moduli space of semistable sheaves of dimension one on the projective space with the same multiplicity but with opposite Euler characteristic.

Letkbe an algebraically closed field of characteristic zero. LetPⁿ be the projective space of dimension n over k. We recall that a coherent algebraic sheaf F on Pⁿ has support of dimension one if and only if its Hilbert polynomial P_F has degree one that is, if we can write PF(m)ˆrm‡x. Here rˆr(F) is a positive integer called the multi- plicityofF, whilexis the Euler characteristicx(F)ˆh⁰(F) h¹(F).

We fix integersr1 andxand we denote by M_Pⁿ(r;x) the moduli space of semistable sheaves on Pⁿ with Hilbert polynomial P(m)ˆrm‡x. In this paper we will prove that M_Pⁿ(r;x) and M_Pⁿ(r; x) are isomorphic. The isomorphism maps a point represented byF to the point represented by the dual sheafExt^{n 1}(F;v_Pⁿ). We will use the concept of semistability due to Gieseker and defined in terms of the lexicographic order on the coeffi- cients of the reduced Hilbert polynomial, cf. definition 1.2.4 in [12]. We recall that a sheaf F on Pⁿ with one-dimensional support is semistable (stable) if and only if it is pure, meaning that there are no subsheaves with support of dimension zero, and for every proper subsheafE F we have

x(E)

r(E) (5) x(F) r(F):

It was first noticed in [5] thatF ! Ext¹(F;v_P²) gives a birational map from M_P²(r;x) to M_P²(r; x) whenrandxare mutually prime.

(*) Indirizzo dell'A.: Str. Avrig 9-19, Ap. 26, Bucharest, Romania.

E-mail: m-maican@wiu.edu

We begin by recalling the Beilinson free monad. LetFbe a sheaf onPⁿ. AmonadforF is a sequence of sheaves

0 ! C ^p !. . . ! C⁰ !. . . ! C^q !0

which is exact, except atC⁰, where the cohomology isF. If eachCⁱis a direct sum of line bundles we talk of afree monad. In the sequelF will be a co- herent sheaf onPⁿ. According to [1], cf. also [4], there is a free monad forF

0 ! C ⁿ !. . . ! C⁰ !. . . ! Cⁿ !0

with

Cⁱˆ M

0pn

H^i‡p(F V^p(p)) O( p):

LEMMA1. Consider a free monad forF

0 ! A ⁿ !. . . ! A⁰ !. . . ! Aⁿ !0

withAⁱˆ L

0pnWip O( p), where Wip are vector spaces over k of di- mension wip. Then wip ˆh^i‡p(F V^p(p))for all i and p if and only if each map W_i;p O( p) !W_i‡1;p O( p)occuring in the monad is zero.

PROOF. The two spectral sequences⁰E and⁰⁰E converging to the hy- percohomology ofAV^j(j) are given by

0E^pq₂ ˆH^p(Pⁿ;H^q(AV^j(j)));

00E^pq₁ ˆH^q(Pⁿ;A^pV^j(j)):

By hypothesisH^q(AV^j(j)) isF V^j(j) forqˆ0 and is zero forq6ˆ0.

Thus

0E^pq₂ ˆ H^p(F V^j(j)); forqˆ0;

0 forq6ˆ0;

(

which shows that ⁰E degenerates at level two. The differentials in the second spectral sequence⁰⁰E^p;q₁ !⁰⁰E^p‡1;q₁ are induced on cohomology by the mapsA^pV^j(j) ! A^p‡1V^j(j). According to Bott's formulas on p. 8 in [15] we have h^q(V^j(j i))ˆ0 for q6ˆj, 0in or for qˆj, i6ˆj, 0in. Thus⁰⁰E^pq₁ ˆ0 ifq6ˆjand we have the identification

00E^pj₁ ˆW_pjH^j(Pⁿ;V^j)'W_pj:

The differentialsf_pj:⁰⁰E^p;j₁ !⁰⁰E^p‡1;j₁ are identified with the maps from the monadWp;j O( j) !Wp‡1;j O( j).

Assume nowthat all mapsfpjare zero. Then⁰⁰E degenerates at its first term and, comparing⁰E with⁰⁰E, we get

h^p(F V^j(j))ˆdim(⁰E^p0₂ )ˆdim(⁰E^p₁)ˆdim(⁰⁰E^p₁)ˆdim(⁰⁰E^{p j;}₁ ^j)ˆw_{p j;j}: Conversely, assume thatf_pjare not all zero. Then⁰⁰E degenerates at level two and we have

h^p(F V^j(j))ˆdim(⁰⁰E^p₁)ˆdim(⁰⁰E^{p j;}₂ ^j)5dim(⁰⁰E^{p j;}₁ ^j)ˆw_{p j;j} for at least one choice of indices p and j. This finishes the proof of the lemma.

COROLLARY 2. All the maps Hⁱ(F V^p(p)) O( p) !

!H^i‡1(F V^p(p)) O( p) occuring in the Beilinson free monad for F are zero.

In the sequel we will assume that the schematic support ofF has co- dimensioncinPⁿ. Thedual sheafF^DofFis defined byF^Dˆ Ext^c(F;v_Pⁿ).

We remark that the hypothesis on the dimension of the support ofF en- sures that the extension sheavesExtⁱ(F;v_Pⁿ) vanish for 0i5c, see (iii) 7.3 in [9].

LEMMA3. LetFbe a coherent sheaf onPⁿwith support of codimension c1. Let

0 ! C ^p !. . . ! C⁰ !. . . ! C^q !0

be a free monad forF. Assume thatExtⁱ(F;v_Pⁿ)ˆ0for i>c. We consider the dual bundlesCⁱ_Dˆ Hom(C ^{i c};v_Pⁿ). Then the dual sequence

0 ! C_D^{q c} !. . . ! C⁰_D !. . . ! C^{p c}_D !0

is a free monad forF^D.

PROOF. We compare the spectral sequences⁰E and⁰⁰E converging to the hyper-derived functorExt^p‡q(C;v_Pⁿ). They are given by

0E^pq₁ ˆ Ext^q(C ^p;v_Pⁿ);

00E^pq₂ ˆ Ext^p(H ^q(C);v_Pⁿ):

SinceCⁱare projective sheaves, we have⁰E^pq₁ ˆ0 forq6ˆ0. This shows that

0E degenerates at level two. The same is true of⁰⁰E because⁰⁰E^pq₂ ˆ0 for q6ˆ0. By virtue of the hypothesis and of the remark preceding the lemma,

we have

00E^p0₂ ˆ F^D forpˆc;

0 forp6ˆc:

In conclusion, the zero rowof⁰E1provides the desired monad forF^D. For any sheaf F onPⁿ there is a natural homomorphismF ! F^DD which is injective if and only if F is pure. We recall that a sheaf with support of dimensiondis calledpureif it does not have a nonzero subsheaf with support of dimension smaller thand. We say thatFisreflexiveif the map F ! F^DD is an isomorphism. According to 1.1.10 in [12] the hy- potheses of lemma 3 are satisfied for pure sheaves of dimension one and for reflexive sheaves of dimension two.

REMARK4. LetFbe a sheaf onPⁿwith support of dimension one. We assume thatFis pure and we notice that this is equivalent to saying thatF has no zero-dimensional torsion. According to lemma 3.1(i) and proposition 3.3(iv) from [7], this is further equivalent to saying that at every closed pointxin the support ofFwe have depth(Fx)1. From this we see thatF satisfies Serre's condition S_{2;n 1}:

depth(F_x)minf2;dim(O_Pⁿ_;x) n‡1gfor all x2Supp(F):

From 1.1.10 in [12] we conclude thatF is reflexive.

PROPOSITION 5. Let F be a coherent sheaf on Pⁿ with support of codimensionn dˆc1. Assume thatExtⁱ(F;v_Pⁿ)ˆ0fori > c. Then for alli,jwe have

hⁱ(F V^j(j))ˆh^{d i}(F^DV^{n j}(n j‡1)):

PROOF. We apply lemma 3 to the Beilinson free monad forF. We get a monad forF^Dwith terms

Cⁱ_Dˆ Hom(C ^{i c};v_Pⁿ)ˆ M

0pn

H ^{i c‡p}(F V^p(p)) O(p n 1):

We tensor the above monad withO(1) to get a monad forF^D(1). In viewof corollary 2, this monad satisfies the necessary and sufficient condition from lemma 1. We conclude that for alli,p

h^i‡p(F^DV^p(p‡1))ˆh^{d i p}(F V^{n p}(n p)):

This proves the proposition.

COROLLARY6. LetFbe as in the previous proposition. Then for all i, j we have

hⁱ(F)ˆh^{d i}(F^D):

REMARK 7. The above corollary also follows from Serre duality and the degeneration of the local-to-global spectral sequence E^pq₂ ˆH^p(Pⁿ;Ext^q(F;v_Pⁿ)), which converges to Ext^p‡q(F;v_Pⁿ).

Next we would like to relate the Hilbert polynomialsPF andP_F^D ofF, respectively F^D. We recall thatPF has degree equal to the dimension of Supp(F). As the support ofF^D is included in the support ofF, we have deg(P_F^D)deg(P_F).

COROLLARY8. LetFbe as in the previous proposition. Then for all m we have

P_FD(m)ˆ( 1)^dP_F( m):

In particular, if F is pure of dimension one with Hilbert polynomial PF(m)ˆrm‡x, then the dual sheaf has Hilbert polynomial P_F^D(m)ˆrm x.

LEMMA 9. LetF be a coherent sheaf onPⁿ, n2, pure of dimension one. ThenF is semistable (stable) if and only ifF^Dis semistable (stable).

PROOF. According to 1.2.6 in [12], the sheafFis semistable (stable) if and only if for all pure one-dimensional quotientsGofF we have

x(G)

r(G) (>) x(F) r(F):

Take a pure one-dimensional destabilizing quotientG ˆ F=K. SinceKhas support of dimension one, we haveExt^{n 2}(K;v_Pⁿ)ˆ0, henceG^Dis a sub- sheaf ofF^D. From corollary 8 we get

x(G^D)

r(G^D)ˆ x(G)

r(G)> x(F)

r(F)ˆx(F^D) r(F^D):

Thus G^D is a destabilizing subsheaf of F^D. This proves sufficiency. Ne- cessity follows from the fact thatFis reflexive, cf. remark 4.

Two semistable sheavesFandGonPⁿwith Hilbert polynomialPgive the same point in M_Pⁿ(P) if and only if they areS-equivalent, meaning that

there are filtrations by subsheaves

0ˆ F₀ F₁. . . F_{k 1} F_kˆ F;

0ˆ G₀ G₁. . . G_{k 1} G_kˆ G;

such that all quotientsF_i=F_{i 1}andG_i=G_{i 1}are stable and there is a per- mutationsof the set of indicesf1;. . .;kgsuch that for alli

F_i=F_{i 1}' G_s(i)=G_{s(i) 1}:

For stable sheaves S-equivalence means isomorphism. In other words, if the above conditions are satisfied, thenFis stable if and only ifGis stable and thenkˆ1 andF ' G. AJordan-HoÈlder filtrationof a semistable sheaf is a filtration as above by subsheaves such that all quotients are stable.

As in the proof of 9.3 from [14], a Jordan-HoÈlder filtration

0ˆ F₀ F₁. . . F_{k 1} F_kˆ F

for a semistable sheaf of dimension one onPⁿ gives a Jordan-HoÈlder fil- tration

0ˆ(F=F_k)^D(F=F_{k 1})^D. . .(F=F₁)^D(F=F₀)^Dˆ F^D

for the dual sheaf with quotients (Fi=Fi 1)^D. The latter are stable by virtue of lemma 9. We arrive at the following lemma:

LEMMA 10. Let F andGbe semistable sheaves on Pⁿ with the same Hilbert polynomial P(m)ˆrm‡x. IfF andGare S-equivalent, then so are their duals.

LEMMA11. Let S be an algebraic scheme over k and letFbe an S-flat coherent sheaf on SPⁿ. Assume that for every s in S the restriction F_s ˆ F_jfsgPⁿhas support of codimension c. Then for all i5c we have

Extⁱ(F;v_SPⁿ_=S)ˆ0:

PROOF. Without loss of generality we may assume thatSis affine. As in the proof of (iii) 7.3 from [9], we reduce to showing the vanishing of Extⁱ(F;v_SPⁿ_=S(q)) for largeq. By 11.2(f) on p. 213 of [8] we have the duality

Extⁱ_OSPn(F;v_SPⁿ_=S(q))'HomOS(R^{n i}f(F( q));O_S)

where f :SPⁿ !S is the projection onto the first component. As n i exceeds the dimension of each restriction F( q)_s, we have

H^{n i}(F( q)_s)ˆ0 for allsinS. From exercise (iii) 11.8 in [9] we deduce that R^{n i}f(F( q))ˆ0, so the above extension group vanishes.

In the sequel we will need the following relative version of the Hilbert syzygy theorem. LetSbe an algebraic scheme overkandFa coherentS- flat sheaf onSPⁿ. Then there exists a locally free resolutionE ! F of length at mostn. Such a resolution can be constructed using the relative Beilinson spectral sequence with first term

E^pq₁ ˆR^qf(F(m)gV ^p( p))IO_Pⁿ(p);

which converges toF(m) in degree zero and to 0 in degree different from zero (see 4.1.11 in [15]). Heref :SPⁿ !Sandg:SPⁿ !Pⁿare the canonical projections. Ifmis large enough, then E^pq₁ ˆ0 forq6ˆ0 and the push-forward sheaves f(F(m)gV ^p( p)) are locally free. Thus the spectral sequence degenerates at level two and the zero row of E₁provides a locally free resolution ofF(m). We can coverSwith open affine subsets U, such that the restriction of each term of the resolution toUPⁿ is a direct sum of line bundles of the formgO_Pⁿ( l).

LEMMA12. Let S be an algebraic scheme over k and letF be an S-flat coherent sheaf on SPⁿ. Assume that for every s in S the restriction Fsˆ F_jfsgPⁿhas support of codimension c and thatExtⁱ(Fs;v_Pⁿ)ˆ0for i>c. Then

Extⁱ(F;v_SPⁿ_=S)ˆ0 for i6ˆc:

Moreover, the sheafF^Dˆ Ext^c(F;v_SPⁿ_=S)is flat over S and(Fs)^D'(F^D)_s for all s in S.

PROOF. Consider a finite locally free resolution E ! F. The exten- sion sheaf Ext^p(F;v_SPⁿ_=S) is the p-th cohomology of the complex Hom(E;v_SPⁿ_=S). We fix a pointsinSand denotei:fsg Pⁿ !SPⁿ the canonical inclusion. The functor Hom( ;v_SPⁿ_=S) sends projective sheaves to i-acyclic sheaves, hence there is the Grothendieck spectral sequence (see (viii) 9.3 in [10])

E^pq₂ ˆL^piR^qHom( ;v_SPⁿ_=S)(F)ˆ Tor^O_p^SPn(Ext_O^q_SPn(F;v_SPⁿ_=S);O_fsgPⁿ);

which converges to the hyper-derived functorsL^p‡qi(Hom(E;v_SPⁿ_=S)).

We recall that the above is the second term of the spectral sequence as- sociated to the bigraded complex obtained by taking a fourth-quadrant Cartan-Eilenberg resolution ofHom(E;v_SPⁿ_=S) and then applyingi. The

other spectral sequence associated to the bigraded complex is given by

0E^pq₁ ˆL^qi(Hom(E ^p;v_SPⁿ_=S)):

0E degenerates at level two because⁰E^pq₁ ˆ0 forq6ˆ0. Thus L^pi(Hom(E;v_SPⁿ_=S))'⁰E^p0₁ ˆ⁰E^p0₂

is the cohomology at positioniHom(E ^p;v_SPⁿ_=S) of the complex iHom(E;v_SPⁿ_=S)' Hom(iE;iv_SPⁿ_=S)' Hom(E_s;v_Pⁿ):

As every term of the resolutionE ! FisS-flat, restricting tofsg Pⁿwe obtain a locally free resolutionE_s ! F_s. Thus, for allj,

L^ji(Hom(E;v_SPⁿ_=S))' Ext^j(F_s;v_Pⁿ):

By hypothesis the above sheaves are zero for j6ˆc, forcing E^pq₁ ˆ0 for p‡q6ˆc. Assumec5n. From the vanishing of

E⁰ⁿ₁ ˆE⁰ⁿ₂ ˆ Extⁿ(F;v_SPⁿ_=S)_s

for an arbitrary points in S, we deduce that Extⁿ(F;v_SPⁿ_=S)ˆ0. Thus E^pn₂ ˆ0 for all p, forcing E^0;n₂ ¹ˆE^0;n₁ ¹. Ifc5n 1 we can repeat the argument and conclude thatExt^{n 1}(F;v_SPⁿ_=S)ˆ0. By induction we obtain that the sheavesExt^q(F;v_SPⁿ_=S) vanish forq>c. In viewof lemma 11 they also vanish forq5c. Thus E^pq₂ ˆ0 forq6ˆcand E degenerates at level two:

E^pc₂ ˆE^pc₁'L^p‡ci(Hom(E;v_SPⁿ_=S))' Ext^p‡c(Fs;v_Pⁿ):

Forpˆ0 we arrive at the isomorphism (F^D)_s '(Fs)^D. Forpˆ 1 we get Tor^O₁^SPn(F^D;O_fsgPⁿ)ˆ0:

From this we deduce thatF^DisS-flat. Indeed, localizing at a pointx lying overs, we obtain

Tor^O₁^x((F^D)_x;OxOsk(s))ˆ0:

Herek(s) denotes the residue field ofs. We nowuse the local criterion of flatness 4.1.3 in [13] to conclude that the stalk (F^D)_xis flat overO_s.

Before we proceed any further we need to review the construction of the moduli space M_Pⁿ(P) of semistable sheaves onPⁿ with fixed Hilbert polynomial P. There exists an integer m0 such that for every semi- stable sheafFonPⁿwith Hilbert polynomialPthe twisted sheafF(m) is generated by global sections and the higher cohomology groups Hⁱ(F(m)), i1, vanish. Thus h⁰(F(m))ˆP(m) and F occurs as a quotient

r:V O( m)!! F, whereV is a fixed vector space overkof dimension P(m). We consider the quotient scheme QˆQuot_Pⁿ(V O_Pⁿ( m);P) and the universal quotient sheaf Fe on QPⁿ. Let RQ be the open subset of equivalence classes of quotients [r:V O_Pⁿ( m)!! F]

for which F is semistable and the map on global sections H⁰(r(m)):V !H⁰(F(m)) is an isomorphism. The reductive group SL(V) acts onQvia its action on the first component ofV O_Pⁿ( m). ClearlyR is invariant under this action. Ifmis chosen large enough, then M_Pⁿ(P) is a good quotient ofRby SL(V). We denote byp:R !M_Pⁿ(P) be the good quotient map.

THEOREM 13. For any integers n2, r1 and x there is an iso- morphism

M_Pⁿ(r;x) !M_Pⁿ(r; x)

which maps the S-equivalence class of a sheafFto the S-equivalence class ofF^D.

PROOF. According to remark 4, corollary 8, lemma 9 and lemma 10 the above map, call ith, is well-defined and bijective. It remains to show thathis a morphism. By symmetry it will follow thath ¹is also a morphism.

We put P(m)ˆrm‡x, so P^D(m)ˆrm x. We adopt the notations preceeding the theorem. The sheaf Fe^D on RPⁿ defined by Fe^Dˆ

ˆ Ext^{n 1}(Fe;v_RPⁿ_=R) is coherent and, by the previous lemma, R-flat.

Moreover, for eachsinR, its restrictionFe^D_s tofsg Pⁿis isomorphic to (Fe_s)^D. According to corollary 8, the latter has Hilbert polynomialP^D. The moduli space property of M_Pⁿ(P^D) gives a morphismp^D:R !M_Pⁿ(P^D) which mapssto the S-equivalence class ofFe^D_s.

From lemma 10 we see that p^D is constant on the fibers of p. The good quotient property of p shows thatp^D factors through a morphism M_Pⁿ(P) !M_Pⁿ(P^D). This morphism is h, which finishes the proof of the theorem.

REMARK14. IfPhas degree at least 2, then the above proof yields a morphism

U !M_Pⁿ(P^D); [F] ![F^D];

defined on the open set U in M_Pⁿ(P) of isomorphism classes of stable sheaves F with F^D stable and satisfying Extⁱ(F;v_Pⁿ)ˆ0 for i>c. For sheaves supported on surfaces the last condition is equivalent to saying that Fis reflexive. We notice thatUis nonempty for allPof degree 2 for which

there is a smooth surfaceXin Pⁿ and a line bundleLonX with Hilbert polynomialP. This is so because any locally freeOX-module of rank one, withXa reduced subscheme ofPⁿ, is stable as a sheaf onPⁿ. Moreover,Lis reflexive as a sheaf on Pⁿ, which can be seen from the isomorphism (iL)^D'i(L^D), wherei:X !Pⁿis the embedding map.

In the remaining part of this paper we will apply the duality result to the following situation. Assume that M_Pⁿ(r;x) is the quotient of a certain parameter space by an algebraic group. This could be the case, for in- stance, if all sheaves giving a point in M_Pⁿ(r;x) are quasi-isomorphic to monads of a certain kind. Constructions of moduli spaces of sheaves as quotients of parameter spaces are ubiquitous in the literature, mostly in- volving actions of reductive groups, and, more recently, as in [2], [3] or [6], for actions of nonreductive groups. Under the above hypothesis on M_Pⁿ(r;x), we will show that M_Pⁿ(r; x) is the quotient of the ``dual para- meter space'' modulo the ``dual group.'' This will generalize our partial results from section 9 of [14], where we dealt with locally closed sub- varieties inside M_P²(r;x) for certain choices ofrandx.

We fix sheavesCⁱonPⁿ, piq, that are finite direct sums of line bundles. We fix a polynomialP(m)ˆrm‡x. We assume that each semi- stable sheaf onPⁿ with Hilbert polynomial P(m)ˆrm‡x occurs as the cohomology of a monad

0 ! C ^{p W}!^p . . . ^W!¹ C^{0 W}!⁰ . . . ^Wq!¹ C^q !0:

For a monad of this form we writeWˆ(W _p;. . .;W_{q 1}) and we denote byF_W its cohomology. LetW^ssbe the set of all thoseWfor whichFWis semistable.

We assume thatW^ssis a constructible subset inside the finite dimensional vector space

W ˆHom(C ^p;C ^p‡1). . .Hom(C^{q 1};C^q):

We equipW^ss with the induced structure of a reduced variety. The alge- braic group

GˆAut(C ^p). . .Aut(C^q)

acts onW in an obvious manner: givengˆ(g p;. . .;gq) and a tuple Was above we put

gWˆ(g _p‡1W _pg ¹_p;. . .;g_qW_{q 1}g_q¹₁):

We remark that, if at least one Cⁱ has a direct summand of the form O_Pⁿ(a) O_Pⁿ(b),a5b, thenGis nonreductive, i.e. it has a nontrivial uni-

potent radical, cf. 19.5 in [11]. Indeed, the subgroup of Aut(O_Pⁿ(a) O_Pⁿ(b)) given by matrices of the form

1 0? 1

is a normal connected unipotent subgroup. This shows that the unipotent radical of Aut(O_Pⁿ(a) O_Pⁿ(b)) is nontrivial. A similar argument applies in general toG.

ClearlyFWandFgWare isomorphic, so one is semistable if and only if the other is. This shows thatW^ssisG-invariant.

Letp:W^ssPⁿ !Pⁿ be the projection onto the second component.

We puteCⁱˆpCⁱ. OnW^ssPⁿwe have the universal monad eC ^{p F}!^p . . . ^F!¹ eC^{0 F}!⁰ . . . ^F!^q1 eC^q

with F_ijfWgPⁿˆW_i. Let Fe denote the middle cohomology of this se- quence. Using arguments as in the proof of lemma 12 one can see that its restriction Fe_W to any fiber fWg Pⁿ is isomorphic to F_W. In parti- cular, allFe_Whave the same Hilbert polynomialP. AsW^ssis reduced we deduce thatFe isW^ss-flat. The moduli space property of M_Pⁿ(P) yields a morphism

p:W^ss !M_Pⁿ(P); p(W)ˆ[FW]:

ClearlypisG-equivariant. We will be interested in the case in whichpis a good or a geometric quotient.

We nowapply the duality result to the above set-up. Under the hy- pothesis that each sheaf giving a point in M_Pⁿ(P) is the cohomology of a monadC and by virtue of lemma 3, corollary 8 and lemma 9, we see that each semistable sheaf on Pⁿ with Hilbert polynomialP^D(m)ˆrm xis the cohomology of a monad

0 ! C_D^{q c WD}!^{q c} . . . ^W!^D1 C⁰_D ^W!^D0 . . .^W!

Dp c 1

C^{p c}_D !0 with

Cⁱ_Dˆ Hom(C ^{i c};v_Pⁿ); W^D_i ˆ Hom(W _{i c 1};v_Pⁿ):

We letW_D^ssbe the set of all those tuplesW^Dˆ(W^D_{q c};. . .;W^D_{p c 1}) for which F_WD is semistable. We viewW_D^ssas a subvariety inside the affine variety

W_DˆHom(C_D^{q c};C_D^{q c‡1}). . .Hom(C^{p c}_D ¹;C^{p c}_D )

equipped with the analogously defined action of the group of auto- morphisms

G_DˆAut(C_D^{q c}). . .Aut(C^{p c}_D ):

As before, there is aGD-equivariant map

p^D:W_D^ss !M_Pⁿ(P^D); p^D(W^D)ˆ[F_W^D]:

COROLLARY15. pis a good (geometric) quotient map if and only ifp^D is a good (geometric) quotient map.

PROOF. First we notice thatG'GDand that the mapW !W^Dacts by transposition, so it gives an equivariant isomorphism W^ss !W_D^ss. According to lemma 3 this isomorphism fits into a commutative diagram

By virtue of the previous theorem the bottom map [F] ![F^D] is an iso- morphism, which proves the statement.

We finish with an example. Fix a vector spaceV overkof dimension 4.

According to [6] a sheafFonP³ˆP(V) with Hilbert polynomial 3m‡1 is semistable if and only if it has a resolution

0 !2O( 3) !^c O( 1)3O( 2) !^W O O( 1) ! F !0 withWnot equivalent to a matrix of the form

? ? ? ?

0 0 ? ?

:

Moreover, M_P³(3;1) is a geometric quotient of the space of parameters (c;W) modulo the action of the group of automorphisms. The concrete description of the space of parametersW^ssis given at 5.2 in [6]. It consists of pairs (c;W) for which the sequence of global sections

0 !k^2H0(c(3))! S²V(k³V)^H0(W(3))! S³VS²V

is exact and either W₂₁6ˆ0 or the entries fW₂₂;W₂₃;W₂₄g are linearly in- dependent as elements ofV.

The above considerations tell us that each sheafF onP³with Hilbert polynomial 3m 1 is semistable if and only if there is (c;W) inW^ssand a resolution

0 ! O( 4) O( 3) ^W!^D O( 3)3O( 2) ^c!^D 2O( 1) ! F !0:

From the above corollary we deduce that M_P³(3; 1) is a geometric quo- tient of the space of parameters W_D^ss of transposed matrices (W^D;c^D) modulo the group of automorphisms.

Acknowledgements. The author thanks the referee for providing sev- eral corrections and improvements and for numerous helpful comments.

The referee suggested the proof of lemma 12, shortened our original proof of lemma 3 and pointed out that the two conditions from lemma 1 are equivalent (originally we stated only sufficiency).

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Manoscritto pervenuto in redazione il 23 gennaio 2009.