Available at: http://www.ictp.trieste.it/~pub-off IC/99/145

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

RATIONALITY OF MODULI OF VECTOR BUNDLESON CURVES

Aidan Schofield1

Department of Mathematics, University of Bristol,University Walk, Bristol BS8 1TW, United Kingdom

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

Alastair King2

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

October 1999

1E-mail: Aidan.Schofield@bristol.ac.uk2E-mail: kinga@ictp.trieste.it

1 Introduction

Let C be a smooth projective curve of genus g over an algebraically closed field k. Let Mr,dbe the moduli space of stable vector bundles of rank r and degree d over C. This is a smoothquasi-projective variety of dimension r2(g — 1) + 1, which is projective when r and d are coprime.Up to isomorphism, it depends only on the congruence class of d mod r. The rank 1 case M1,dis isomorphic to the Jacobian J(C) and every moduli space comes equipped with a determinantmap det: Mr,d —>• 1 , d whose fibre over L is Mr,L, the moduli space of bundles with fixeddeterminant L.

The goal of this paper is to describe these moduli spaces in the birational category, that is,to describe their function fields. We shall prove the following result.

Theorem 1.1. The moduli space Mr,d is birational to Mh,0 x A^r ~h ^g~l\ where h = hcf (r, d).

In other words, there is a dominant rational map /x: Mr,d —* SD?/i,o whose generic fibre isrational. We shall observe that this map restricts to a map between fixed determinant modulispaces (not necessarily with the same determinant) and so, in the case when r and d are coprime,we obtain the following long believed corollary, which has been proved in special cases ([?],[?],[?]).

Theorem 1.2. If L is a line bundle of degree d coprime to r, then Mr,L is a rational variety.

To ease the discussion, we use the following terminology to describe the relationship betweenMr,d and Mh,0. An irreducible algebraic variety X is birationally linear over another irreduciblealgebraic variety Y if there exists a dominant rational map φ: X —-> Y whose generic fibre isrational, that is, the function field k(X) is purely transcendental over the function field k(Y).Such a map φ will also be called birationally linear.

What we shall actually prove is a stronger statement that the map /x is birationally linearand preserves a suitable Brauer class. More precisely, for each type (r, d) with hcf (r, d) = h,the moduli space Mr,d carries a Brauer class ψr,d for its function field, represented by a centralsimple algebra of dimension h2, and the map /JL has the property that pi*(iph,o) = ψr,d. Thisstrengthening of the statement is the key to the proof, because it enables an induction on therank r.

In section ??, we construct an open dense subvariety of Mr,d as a quotient space of a suitablevariety Xr,d by a generically free action of PGLh where h = hcf(r, d). This arises becausewe are able to show that a general vector bundle E of type (r, d) arises as a quotient of aparticular bundle Fh in a unique way so that we can take Xr,d to be a suitable open subvarietyof Q U O T ( F H , r, d) on which PGLh acts in the natural way. We also arrange in this sectionthat the kernel of the surjection from Fh to E should have smaller rank (at least in the casewhere h / r) and this induces a rational map from M r , d to Mr1,d1 for some type (r1, d1) wherer1 < r. After this, in section ??, we show how this description of Mr,d as a quotient space fora generically free action of PGLh allows us to associate a Brauer class to its function field andwe use this Brauer class to describe birationally the rational map from Mr,d to M r 1,d1 in termsof "twisted Grassmannian varieties". In section ??, we use parabolic moduli spaces which giveus other "twisted Grassmannian varieties" which we may choose to be twisted "in the sameway" as our map between moduli spaces above. In the final section, we put these various resultstogether to construct a birationally linear rational map from Mr,d to Mh,0.

2 The first step

The purpose of this section is to show that the general bundle E of rank r and degree d maybe constructed as a quotient of Fh, where F is a fixed bundle of an appropriate type andh = hcf(r,d).

This will enable us to define the Brauer class on Mr,d that will be the focus of most of the

paper. Furthermore, we will see that the kernel of the quotient map q : Fh —>• E is also general

so that we may define a dominant rational map from Mr,d to Mr1,d1, where r1 < r when r does

not divide d. This will be the basis of the inductive construction of the birationally linear map

to Mh,0.

We will say that a vector bundle E of rank r and degree d has 'type' (r, d). For E of type

a = (rE, dE) and F of type β = (RF, DF), we write

χ(F,E) = hom(F, E) - ext(F, E)

= rFdE - rEdF - rErF(g - 1) = χ(β, α),

where hom(F, E) = dim Hom(F, E) and ext(F, E) = dim Ext (F, E). The middle equality is theRiemann-Roch Theorem.

We start with a lemma about the nature of generic maps between generic vector bundles.The proof is closely based on the proof given by Russo & Teixidor ([?] Theorem 1.2) that thetensor product of generic bundles is not special; a result originally due to Hirschowitz.

Lemma 2.1. Let E,F be generic vector bundles of fixed types. Suppose that there exists anon-zero map φ: F —> E and take φ to be a generic such map. Then ext(F,E) = 0 and φ hasmaximal rank. If rE / TF, then cokerφ is torsion-free; in particular, if rE < rF, then φ issurjective and if rE > rF then φ is injective.

Proof. Let [φ] denote the homothety class of φ in P(Hom(F,E)). Then the triple (F,E,

depends on

p0 = 1 - χ(F, F) + 1 - χ(E, E) + hom(F, E) - 1

parameters (cf. [?] Section 4). Let I = imφ, K = kerφ, Q = cokerφ and T be the torsionsubsheaf of Q. Further, let Q' = Q/T and /' be the inverse image of T in E. Thus we havethree short exact sequences

7^0 (2.1)

Q -> 0 (2.2)

Q ->• 0 (2.3)

in which all terms except Q are vector bundles. The triple (F, E, [φ]) is determined by the firstand last sequences (up to homothety) and a map t: I —> I' whose cokernel is T. The triple(I, /', [t]) depends on 1 — χ(I , I) + rIdT parameters and so the whole configuration depends onat most

0 -

0 -

0 -

4 K -

• + 7 -

> I' -

-4 F

> E

* E

, K) + l - χ(I, I) + rIdT + 1 - X(Q', Q')

+ ext(I, K)-l+ ext(Q', /') - 1

parameters. Now, E and F are stable, so hom(I, K) = hom(Q/,//) = 0 and hence ext(I,K) =

-X(I,K) and ext(Q',/') = -xiQ'J')- Furthermore xiQ',Q') = χ(Q,Q) and

X(Q', I') = rQ(dI + dT) - rI(dQ - dT) - rQrI(g - 1) = χ(Q, I) + rEdT

Hence, using the bilinearity of χ in short exact sequences, we get

p1 = 1 - χ(F, K) - χ(Q, E) - χ(I, I) - rQdT

But now p0 < p1 and so

hom(F,E) < X(F,I)+x(I,E)-x(I,I)-rQdT

= X(F,E)-x(K,Q)-rQdT

3

and hence

ext(F,E)<-x(K,Q)-rQdr (2.4)

A simple dimension count (cf. [?] Lemma 2.1) shows that, for general E and F to appear insequences (??) and (??), it is necessary that χ(K, I) > 0 and x(/',Q') > 0. In other words,

rKdI - rIdK > rKrI(g -

rIdQ - rQdI > rIrQ(g - 1) +

and hence

χ(K, Q) = (rK(ridQ - rQdI) + rQ(rKdI - rIdK) - rKrIrQ(g -

> rKrQ(g - 1) + dT(rKrE/rI)

Thus, subsituting this into (??), we finally deduce that

ext(F, E) < -rKrQ(g - 1) - dT(rQ + rKrE/rI)

This is only possible if (i) ext(F, E) = 0, (ii) RK = 0 or RQ = 0 and (iii) unless RQ = RK = 0, we

also have dT = 0. But (ii) means that rI has maximal rank and then (iii) means that cokerφ is

torsion-free, unless RE = RF .

We shall also use the following lemma which may be thought of as a generalisation of theresult that any (bounded) family of bundles on a curve may be extended to an irreducible family(cf. [?] Proposition 2.6).

Lemma 2.2. Let {Gx : x G X} be an irreducible family of vector bundles over C and let{Ey : y G Y} be any family of vector bundles over C of fixed type. Then there exists anirreducible family of extensions of vector bundles,

such that every vector bundle Q'z is isomorphic to some Gx and every extension 0 —G x —F T —>Ey —> 0 is isomorphic to one in this family.

Proof. After twisting by a suitable line bundle of positive degree, we may assume that Ext 1 (O, Gx)=0, for all x G X, and that every Ey is generated by global sections. Suppose that each Ey is of type(n,d). Extending the usual dimension counting argument in the Grassmannian G R ( N , H°(Ey)),we may choose n sections of Ey so that the induced map ρ : On —E y is an isomorphism of thefibres at the general point of C and drops rank by only 1 at other points. Thus the cokernel ofp is the structure sheaf Tξ of a subscheme ξ of degree d in C, that is, Ey is an extension of Tξ ontop of On.

The parameter space of such subschemes ξ is the d-fold symmetric product C^d\ which isan irreducible algebraic variety and which carries a universal family T. Since Tξ is torsion,Hom(Tξ, Gx © On) = 0 for all ξ G C(d) and all x G X. Hence there is a vector bundle λ : Z -•X x C(d) whose fibre above the point (x,ξ) is Ext(Tξ,Gx © On) and this carries a tautologicalfamily of extensions

{0 - + π1λ(z) ®On^T'z^ %2X{Z) ^0:zeZ}.

Letting Q'z = Gπ1λ(z) and S'z = T'ZIQ'Z, we may replace Z by the non-empty open set on whichT'z and S'z are vector bundles and obtain the required irreducible familty of extensions of vectorbundles. To see that every possible extension of Ey on top of Gx occurs note that every suchextension has a 3 step filtration with Tξ on top of On on top of Gx. But, since Ext 1(O, Gx) = 0,the extension at the bottom of this filtration splits and so it is simply an extension of Tξ on top

of Gx © On. •

Using these lemmas, we have the following result.

Proposition 2.3. For any type α = (r,d), let h = hcf(r,d). Then there is a unique type(3 = (s, e) satisfying

(i) χ(β,α)=h,

(ii) r/h < s < 2r/h, if h < r, or s = 2, if h = r.

Then, there exists a vector bundle F of type β such that for a general E of type α,

(iii) hom(F, E) = h and ext(F, E) = 0,

(iv) the natural map ΕF(E) : Hom(F, E) ®fc F —> E is surjective,

(v) the bundle E1 = kerΕF(E) is general and has ext(E1,F) = 0.

Proof. To solve (i) we simply need to solve sd — tr = h and set e = t — (g — 1)s. Given onesolution (s,t), the complete set of solutions is {(s,t) + k(r/h,d/h) : k G Z} which containsprecisely one solution in the range (ii). Part (iii) is provided by Lemma ??.

For the main part of the proof, the first step is to construct a short exact sequence

O- i-^-^-^O (2.5)

with E of type α, F of type β and Ext(E1, F) = 0 = Ext(F, E).First suppose that h = 1. Then Lemma ?? implies that for generic F and E we have

Ext(F, E) = 0 and, since r < s, the generic map is surjective. Let F —>• E' be a particularchoice of such generic bundles and map and let E[ be the kernel. At this stage, we haveExt(F', E') = 0, but may not have Ext(E[, F') = 0. On the other hand,

X(l3 -a, 13)= x(/3, a) + x(/3 - a,/3 - a) - X(a, a) > x(/3, a) = 1

since s — r < r. Hence, Lemma ?? also implies that, for generic E1 and F of types β — α and[3 respectively, Ext(E1,F) = 0 and, since s — r < s, the generic map is an injection of vectorbundles. Let E'[ —> F" be a particular choice of such generic bundles and map and let E" be itscokernel. This time, we have Ext(E'{,F") = 0, but may not have Ext(F",E") = 0.

But now we may include E[ and E" in an irreducible family {E1,x

: x G X} by [?] Proposition2.6. Then, by Lemma ??, there is an irreducible family of extensions

{0 -> z -^ T'z ->• S'z ->• 0 : z G Z}

which includes both 0 -• E[ -^ F' -^ E' -• 0 and 0 -^ E'[ -^ F" -^ E" -• 0. Hence we maychoose for (??) a general extension in this family and both Ext groups will vanish as required.

For an arbitrary value of h, we may obtain a sequence of the form (??) by taking the directsum of h copies of one for α = α/h.

For the second step, suppose that we have a sequence of the form (??). By [?] Proposition2.6, we may include E1 in an irreducible family {E1,x : x G X}, whose generic member is generaland for which every member satisfies ext(E1,x, F) = 0. There is then a vector bundle λ : Y —>• Xwhose fibre at x is Hom(E1,x,F

h) and over Y there is a tautological map fy : E1,λ(y) ~^ Fh.Replacing Y by the non-empty open set on which fy is injective, we have Ey = coker fy of typeα. We may further replace Y by the non-empty open set on which ext(F,Ey) = 0.

Now observe that the homomorphism Fh —>• y must be isomorphic to εF(Ey) : Hom(F, £y)®kF —>• p, because a linear dependence between the h components of the homomorphism from Fh

to Ey would imply that F is a summand of the kernel, which would contradict ext(E1,x, F) = 0.

5

If we consider just the family {Ey : y e Y}, then, as before, we may include this in an irre-ducible family {Ez : z G Z}, whose generic member is general and such that ext(F,Ez) = 0 andhom(F, Ez) = h for every z <E Z. Hence the kernel £'lz of the homomorphism ΕF(EZ)

: Hom(F, £z)®kF —E z is general and satisfies ext(£[ Z,F) = 0 , because this was already true over Y. Thus wehave all the properties we require. •

Proposition ?? shows that we have a dominant rational map

where E1 = kerΕF(E) has type (r1,d1) = h(s,e) — (r,d). One may immediately check thefollowing.

Lemma 2.4. The type (r1,d1) of E1 satisfies

(i) if h < r, then r1 < r,

(ii) h1 = hcf(r1,d1) is divisible by h,

(iii) det(Si) ^ det(F)hdet(E)~l.

The proof of Proposition ?? shows that the fibre of λF above a closed point [E1] is bira-tionally the Grassmannian of h-dimensional subspaces of Hom(E1, F). However, this bundle ofGrassmannians may be 'twisted', that is, it may not be locally trivial in the Zariski topology. Infact, it will fail to be locally trivial whenever h / 1 and will not be birationally linear wheneverhi / h, but we will be able to measure how twisted it is using a Brauer class on Mr1,d1 andthen compare λF to another Grassmannian bundle with the same twisting, but smaller fibres,to construct inductively our birationally linear map.

In fact, Proposition ?? also provides us with the way of constructing this Brauer class, be-cause it yields a description of Mr,d as a quotient of an open set in the quot scheme QUOT(F H , r, d)by PGLh. We describe this in detail in the next section.

3 Brauer classes and free PGL actions

In this section, we collect a number of results about free actions of the projective general lineargroup PGL, which allow us to define and compare the Brauer classes we are interested in.

Recall that the Brauer group of a field k may be described as consisting of classes representedby central simple algebras A over the field and that [A1] = [A2] in the Brauer group if and onlyif A1 and A2 are Morita equivalent or equivalently A1 <g> A2 is isomorphic to Mn(k) for a suitableinteger n where A° is the opposite algebra to A. This is equivalent to saying that there is anA1, A2 bimodule of dimension n where n2 = dimA1dimA2. The product in the Brauer groupis induced by the tensor product of algebras.

The reader may wish to consult [?] for further discussion of the Brauer group and centralsimple algebras.

Definition 3.1. Let X be an affine algebraic variety on which the algebraic group PGLn

acts freely. Over the quotient variety X/PGLn there is a bundle of central simple algebrasMn(k) xPGLn X of dimension n2. At the generic point, this is a central simple algebra over thefunction field k(X/PGLn) and hence defines a class in the Brauer group of k(X/PGLn). Weshall denote this class by Br(X/PGLn).

6

It is important to note that Br1 (X/PGLn) depends on the action of PGLn on X and notjust on the quotient space Y = X/PGLn. Note also that the bundle of central simple algebrasB = Mn(k) xPGLn X over Y is essentially equivalent to the PGLn action on X, because X canbe recovered as the Y-scheme that represents the functor of isomorphisms between B and thetrivial bundle of central simple algebras Mn(k) x Y over Y. The PGLn action is recovered viaits action on Mn(k). Moreover, we have the following.

Lemma 3.2. Let PGLn act freely on affine algebraic varieties X1 and X2. Let

4>:Xl/PGLn^X2/PGLn

be a dominant rational map. Then there is a PGLn-equivariant dominant rational map Φ: X —>•X2 making the following diagram commute

X, - * - > X 2

X2/PGLn

if and only ifBr(X1/PGLn) = (j)-l^8x{X2/PGLn).

Proof. After restricting to suitable open subvarieties and taking the pullback along φ we mayassume that φ is the identity map. We have two distinct PGLn bundles. These have the sameBrauer class if and only if over a suitable open subvariety of X/PGLn the associated bundlesof central simple algebras are isomorphic or equivalently the two PGLn bundles are isomorphicover this open subvariety. •

We can now define the Brauer classes on (the function fields of) our moduli spaces thatwe will use in the rest of the paper. For each type (r,d), fix one vector bundle F, which isgeneral in the sense of Proposition ?? and recall that h = hcf(r,d). Let Xr,d be the opensubset of QUOT(F H , r, d), which parametrizes (up to scaling) quotients q: Fh —>• E of type (r, d)which are stable bundles and for which the induced map kh —> Hom(F, E) is an isomorphism.The obvious action of GLh = Aut(Fh) induces a free action of PGLh on Xr,d and the mapXr,d —>• r,d, which forgets the quotient map q, identifies Xr,d/PGLh with an open dense subsetof Mr,d and, in particular, identifies their function fields. Since Mr,d is a projective variety wemay replace Xr,d by an open dense affine PGLh-equivariant subset of itself by taking the inverseimage of some open dense affine subset of Mr,d contained in the image of Xr,d.

Definition 3.3. For every type (r,d), the Brauer class ψr,d on Mr,d is the class correspondingto Br (Xr,d/PGLh) after we identify k(Xr,d

/PGLh) with k(Mr,d) as described above.

There are more general Brauer classes that arise naturally on X/PGLn, which we nowdescribe and relate to Br(X/PGLn). Let P be a vector bundle over the algebraic variety X onwhich GLn acts lifting the action of PGLn on X such that k* acts with weight w on the fibresof P. We will call such a bundle P a vector bundle of weight w on X; the GLn action on Plifting the PGLn action on X will be implicit. If P is a vector bundle of weight 0, then PGLn

acts on P and P/PGLn is a vector bundle over X/PGLn. If P is a vector bundle of weight wthen P <g) P is a vector bundle of weight 0 and P <g> P/PGLn is a bundle of central simplealgebras over X/PGLn. The bundle of central simple algebras associated to the PGLn actionof X is the special case where P is taken to be the bundle of weight 1 over X given by kn x X,with GLn acting diagonally, since we may identify Mn(k) with (kn) <g)kn. We define the Brauerclass defined by P to be the Brauer class of the central simple algebra over k(X/PGLn) definedby the generic fibre of the bundle of central simple algebras P <g> P/PGLn.

Lemma 3.4. Let P be a vector bundle of weight w over an algebraic variety X on which PGLn

acts freely. Then the Brauer class defined by P is wBr (X/PGLn).

Proof. Let P and Q be vector bundles of weight w. Then P <g> Q is a vector bundle of weight0 and P <g) Q/PGLn is a vector bundle over X/PGLn. It has a structure of a bimodule withPv ®P/PGLn acting on the left and Qv ®Q/PGLn acting on the right. Over the generic pointof X/PGLn it defines a Morita equivalence between (the generic fibres of) P <g> P/PGLn andQ <S> Q/PGLn. Hence the Brauer classes defined by P and Q are equal; in other words theBrauer class depends only on the weight.

Now, if w > 0, then Qw = (kn)®w x X with the diagonal action of GLn is a vector bundleof weight w and Q^ <g> Qw/PGLn is the wth tensor power of Q\ <g> Q1/PGLn. Since the classdefined by Q1 is Br(X/PGL n), the class defined by Qw is wBr(X/PGL n). On the otherhand, Q-w = {{kn)y)®w x X with the diagonal action of GLn is a vector bundle of weight—w. In particular, Q ^ <g> Q_i/PGLn is the sheaf of algebras opposite to Q 1 <8> Q1/PGLn andtherefore the class defined by Q-i is — Br (X/PGLn) and, as above, the class defined by Q-wis —wBr (X/PGLn). Finally, OX is a vector bundle of weight 0 and defines the class 0. •

Thus, if P is a vector bundle of weight 1 and rank r, then the Brauer class Br (X/PGLn)is represented by a central simple algebra of dimension r 2, namely P <g> P/PGLn. It willbe important to observe that, birationally, the converse is true. More precisely, we have thefollowing.

Lemma 3.5. Let PGLn act freely on an algebraic variety X and suppose that the Brauer classwBr (X/PGLn) is represented by a central simple algebra S of dimension s2 over k(X/PGLn).Then there exists a PGLn-equivariant open subset Y of X and a vector bundle Q of weight wover Y whose rank is s.

Proof. Let P be a vector bundle of weight w and rank p. It is enough to deal with the casewhere S is a division algebra since the remaining cases are all matrices over this and hence thevalues for s that arise are all multiples of this. In particular, therefore, we may assume that sdivides p. If s = p, there is nothing to prove so we may assume that s < p. Thus at the genericpoint of P <g> P/PGLn, there is an idempotent of rank s. This idempotent is defined over someopen subset of X which is PGLn-equivariant since the idempotent is PGLn-invariant and givesa decomposition P = P1 © P2 as a direct sum of vector bundles which are GLn-equivariant oneof which has rank s. These bundles have weight w since they are subbundles of P which hasweight w. •

We now come to the main object of this section, to describe the relationship between theBrauer classes considered above and twisted Grassmannian bundles such as λF: Mr,d —M r1,d1.We start in the general context of Grassmannian bundles associated to a vector bundle P ofweight w, although in the end we will only need to consider weights ± 1 . Let j < rk(P) bea positive integer. Then PGLn acts freely on the bundle of Grassmannians G R ( J , P) over Xand φ: Gr(j,P)/PGLn —> X/PGLn is a Grassmannian bundle over X/PGLn that is usuallynot trivial in the Zariski topology. Since the map from G R ( J , P) to X is PGLn-equivariant theBrauer class B r ( G r ( j , P)/PGLn) is just the pullback of the Brauer class Br (X/PGLn). Wecan also realise the algebraic variety Gr(j,P)/PGLn as a quotient variety for a free action ofthe algebraic group PGLj on the partial frame bundle of j linearly independent sections of thevector bundle P and we will need to know how to relate the two Brauer classes we obtain inthis way.

We must take care to differentiate two distinct ways of constructing the partial frame bun-dle. Let S be the universal sub-bundle on GR(J,P). Let FR(J,P) be the 'covariant' partial

8

frame bundle, whose fibre at x consists of isomorphisms k —>• Sx and let FR V ( J , P) be the 'con-travariant' partial frame bundle, whose fibre at x consists of isomorphisms (kj) —>• Sx. ThenGLj acts freely on both FR(J,P) and FR V ( J , P) and the quotient variety is GR(J,P) in bothcases. The difference is that the pullback of S to FR(J, P) is the trivial bundle with fibre kon which GLj acts with weight 1, while the pullback of S to FR V ( J , P) is the trivial bundlewith fibre (k^)w on which GLj acts with weight —1. The obvious isomorphism between the twoframe bundles is compatible with the transpose inverse automorphism of GLj, but not with theidentity automorphism.

The action of GLn lifts from GR(J,P) to FR(J,P) and FR V ( J , P), SO both carry an actionof GLj x GLn. The kernel of each action is isomorphic to k*, but in the covariant case it is{(twI,tI) : t e k*}, while in the contravariant case it is {{twl,rll) : t e k*}. (Recall thatw is the weight of the action of GLn on P.) Hence, both Fr(j,P)/GLn and FRV(j,P)/GLn

carry free actions of PGLj which determine Brauer classes on the quotient, which is equal toGr(j,P)/PGLn in both cases.

We summarise the maps considered above in the following commutative diagram for thecase of the covariant partial frame bundle. Note that the groups that appear as labels belowthe arrows indicate that the maps are quotient maps by a (generically) free action of the group.

Fr(j,P)

GR(J,P)

X

PGLn

X/PGLn (3.1)

The diagram for the contravariant frame bundle is of the identical form, but the need to dis-tinguish the two cases is made clear by the following result, which describes the relationshipbetween the Brauer classes determined by all the PGL actions in the diagram.

Lemma 3.6. Let P be a bundle of weight w on X. Then, in the notation described above,

Fr(j,P)/GLn

Gr(j,P)/PGLn

GR(J, P)/PGLn) =

J, P)/GLn)/PGLj) =

((FRV(J, P)/GLn)/PGLj) = -

(X/PGLn))

J, P)/PGLn)

R(J, P)/PGLn)

Proof. The first equality follows immediately from the fact that the lower diamond in (??) isan equivariant pullback. The action of GLn on P over X lifts naturally to an action of GLn

on the universal subbundle S over GR(J,P). Hence S has weight w and so the Brauer classon Gr(j,P)/PGLn represented by S eg) S/PGLn is wBr (GR(j,P)/PGLn) by Lemma ??.As already observed, the pullback S' of S to FR(J, P) is trivial with fibre kj and so thequotient by GLn also gives a trivial bundle S" with fibre kj on Fr(j,P)/GLn. But then<Bx((FR(j,P)/GLn)/PGLj) is equal to the Brauer class represented by (S")v eg) (S")/PGLj,which is equal to the Brauer class represented by S'v <g> S/PGLn, completing the proof in thecovariant case. The proof in the contravariant case is identical except that now S' and S"are trivial with fibre (kj)v so that 5St ( (FR V ( J , P)/GLn)/PGLj) is the negative of the classrepresented by {S")y eg) {S")/PGLj. •j .

9

We may now describe the rational map λF: Mr,d —>• r1,d1 as a Grassmannian bundle of the

type described above and determine the behaviour of the Brauer classes under pullback. Recall

that there exists a vector bundle F1 and an open subset Xr1,d1 of Quot(F1h1,r1,d1) such that

the map to Mr1,d1 is birational to the PGLh1 quotient map.

Proposition 3.7. On an open subset of Xr1,d1, there exists a vector bundle P of weight —1 and

of rank lh1 for some integer l such that the rational map

is birational to the Grassmannian bundle

</> : Gr(h,P)/PGLh1 -+ Xr1,d1/PGLh1.

Furthermore,

Proof. The idea of the proof is that since F is general, the set of quotients p

such that E1 := kerp e Mr1,d1, Ext(F,E) = 0 and Ext(E1,F) = 0 is not empty. It is bijective

to the set of h-dimensional subspaces Vv C Hom(E1, F) such that p : E 1> F <g> V is injective,

E := cokerp e Mr,d, Ext(E1,F) = 0 and Ext(F, E) = 0. We fill in the details below.

Consider the open set in Xr1,d1 parametrizing those q1: F 1 h>• E1 for which Ext(E1, F) = 0.

Over this open set there is a vector bundle P whose fibre at [q1] is Hom(E1,F). Since PGLh

acts with weight 1 on E1, it acts with weight —1 on P. We claim that Gr(h,P)/PGLh1

is birational to Mr,d. To see this, consider the contravariant partial frame bundle FRV(h,P)

whose fibre over [q1] e Xr1,d1 is naturally identified with the set of maps p: E —>• Fh for which

the induced map (kh) —>• Hom( E 1,F) is injective. On an open subset in FRV(h,P), the map

p is injective as a map of bundles and its cokernel q : Fh —> E gives a point in Xr,d. Since

p, but not q1, is determined by q we see that FRv(h, P)/GLh1 is birational to Xr,d and so

(FRV(h,P)/GLh1)/PGLh is birational to Mr,d. Since G R ( H , P ) is FR(H,P)/GLh, we deduce

that GR(H, P)/PGLh1 is birational to Mr,d.

We can arrange all the rational maps we have considered above into the following diagram

of the form of (??).

Gr(h, P) Xr,d

mri dl (3.2)

Since P has weight — 1, the first and last formulae in Lemma ?? give

which completes the proof. •

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4 The Hecke correspondence

One of the main ideas of the paper is to compare the (birationally) twisted Grassmannian bundle

Xp F :Mr,d —>• r1,d1 to another Grassmannian bundle which is twisted by the same amount but

has smaller fibres. This second bundle is provided by the Hecke correspondence, which we

describe in this section. Within this section, we may let h and h1 be arbitrary integers with

h<h\. Only later, will we need to use the fact that h actually divides h1.

Let Ph1,0,h be the moduli space of parabolic bundles, which parametrizes pairs consisting of

a bundle (or locally free sheaf) E1 of type (h1,0) together with a locally free subsheaf E2 C E1

such that the quotient E1/E2 is isomorphic to (Ox) for a fixed point l e C . In order to specify

a projective moduli space exactly, we would need to specify parabolic weights to determine

notions of stability and semistability. However, we are only interested in this space up to

birational equivalence and it is known ([?] Section 4) that the birational type of the moduli

space does not depend on the choice of parabolic weights. Indeed, we may choose to let Ph1,0,h

denote the dense open set of quasi-parabolic bundles E2 C E 1 that are stable for all choices of

parabolic weights.

The type of E2 must be (h1, —h) and there are two dominant rational maps

02- l-h: [E2 C E1] .-> [E2

The key point is that, like λF, the maps θ1 and θ2 are (birational to) twisted Grassman-nian bundles whose twisting is measured by the Brauer classes ψh1,0 and iphlt-h respectively.Furthermore, as we shall show below, these two Brauer classes pull back to the same class on

To construct Ph1,0,h birationally from Mh1,0, let H1 be the vector bundle over Xh1,0 whose

fibre over the point q1 : F1 —> E1 is Hom(E1, Ox), where Ox is the structure sheaf of the point

x e C. Then H1 is a vector bundle of weight —1 and Ph1,0,h is birational to GR(H, H1)/PGLh1.To see this, consider the contravariant frame bundle Fnv(h, H1). A point in the fibre over [q1]may be identified with a map

p:Ei^(Ox)h (4.1)

such that the induced map (kh) —>• Hom(E, Ox) is injective. If we restrict to the open set onwhich p is also surjective so that it determines a quasi-parabolic structure, then the map toPh1,0,h which forgets p and q1 is precisely the quotient by GLh that gives Gr(h,H1), followedby the quotient by PGLh1.

Thus we have another diagram of the form of (??).

Gr(h,H1) FR V (h, H1) /GLh1

X h1,0

(4.2)

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Hence, by Lemma ??, we have

0) = <8t ( ( F R V ( / * , Hl)/GLhl)/PGLh) (4.3)

Now we construct Ph1,0,h birationally from SOt ,— - To preserve the generality of this section,let m = hcf(h1,h), but note that m = h in the case of real interest. Let H2 be the vector bundleover Xhlt-h whose fibre above a point [q2 : F2 m> E2] is Ext(O x,E2). Then H 2 has weight 1and Ph1,0,h is also birational to GR(H, H2)/PGLm. This follows, as above, by considering theopen set in F R ( H , H2) parametrizing extensions

0 -• E 2 _ • # ! - • ( O x ) h ^ 0. (4.4)

such that the induced map kh —>• Ext(Ox,E2) is injective. The moduli space Ph1,0,h arises(birationally) by taking the quotient by GLh and then PGLm.

Thus, again, we have a diagram of the form of (??).

Fr(h,H2)

Gr(h,H2) Fr(h,H2)/GLm

PGL^\ /PGLh

Xhlt-h Ph1,0,h

θ2

,^,-h ( 4. 5)

and, by Lemma ??, we have

hlt-h) = Br ((Fr(h, H2)/GLm)/PGLh) (4.6)

But now we simply need to observe that the data in (??) and in (??) have the same form anddiffer only in the imposition of different open conditions. Thus we may identify open subsetsof F R ( H , H2)/GLm and F R ( H , H1)/GLh1. One could in principle identify both of these as opensubsets of an appropriate fine moduli space for such data. Furthermore, this identification iscompatible with the PGLh actions and so we may combine (??) and (??) to obtain

(4.7)

5 Construction of birationally linear maps

We may now proceed with the proof of the main theorem of the paper on the existence of abirationally linear map from the moduli space Mr,d of vector bundles of rank r and degree d tothe moduli space Mh,0. The proof goes by induction on the stronger statement that there is sucha birationally linear rational map that preserves the Brauer classes defined in the Section ??.More precisely, we prove the following theorem.

Theorem 5.1. Let ψr,d be the Brauer class on Mr,d defined for every type (r, d) in Definition ??

and let h = hcf(r,d). Then there exists a birationally linear map pi: Mr,d --•» h , 0 such that

12

Proof. If r divides d, then h = r and Mr,d is isomorphic to Mh,0 by tensoring with a linebundle of degree d/r. This isomorphism may be taken to be /x and it preserves the Brauer class.Otherwise, we saw in Section ?? how to construct a map λ F : M r,d -—» M r 1,d 1 with r1 < r andwe proved in Proposition ?? that

(5.1)

We construct the map /x, by induction on the rank r, as the composite of the top row of thefollowing commutative diagram of dominant rational maps which combines λF and the Heckecorrespondence described in Section ??. The other elements of the diagram we will explain next.

9K V ^ ^

Mh1,0 (5.2)

The maps \x\ and /X2 are of the same sort as /x and may be assumed to exist by induction, since

both n and h1 = hcf(r1,d1) are less than r. Thus they are birationally linear and satisfy

J"IW>/JI,O) = ψr1,d1 ( 5 . 3 )

/4(V\o) = iPhu-h- (5.4)

because h = hcf(h1, —h) by Lemma ??.

The central square in the diagram is a pull back. In particular, 9\: ^3 —-> M r 1,d1 is the pull

back of θ1 along [i\ and hence, by (??), it is a Grassmannian bundle over M r 1,d1 whose twisting

is measured by ψr1,d1. Thus 9\ and λF are twisted Grassmannian bundles associated to vector

bundles of weight —1 over Xr1,d1 and of ranks h1 and lh1 respectively (see Proposition ??).

We will prove in Lemma ?? below that this implies that there is a birationally linear map

ρ: Mr,d ---> ^3 such that λF = 9\p and hence

(5.5)

The pullback /2i of \x\ along θ1 is birationally linear and satisfies

K (0*1(^,0)) = 91 (vl(tphl,o) = 91 (W1)(il) . (5.6)

Thus fiip: Mr,d —* Ph1,0,h is birationally linear and pulls back Ol(iphi,o) to ψr,d. But by (??)and (??), this means that /JL = /X26l2AtiP pulls back ψh,0 to ψr,d as required and to complete theproof we need to show that θ2 is birationally linear. This follows from Lemma ?? below, because,as we saw in Section ??, θ2 is a twisted Grassmannian bundle of h-dimensional subspaces of avector bundle H2 of weight 1 over Xhlt_h and the Brauer class V'/n.-Ziis represented by a centralsimple algebra of dimension h2. Thus although θ2 is not locally trivial in the Zariski topologywhen h / 1, we can show that it is birationally linear since its generic fibre is birational to aGrassmannian over a division algebra; this is not the way it is expressed in Lemma ?? thoughthe translation to this is fairly simple. •

Thus (modulo two lemmas) we have proved Theorem ?? as we set out to do. To deduceTheorem ??, it is sufficient to observe that, by Lemma ??, the map λF restricts to a mapbetween moduli spaces of fixed determinant and that the Hecke correspondence restricts to acorrespondence between moduli spaces of fixed determinant. Therefore the map /x: Mr,d —*•

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restricts to a map between fixed determinant moduli spaces, although precisely how thedeterminants are related will depend on various choices made in the construction. In the caseh = 1, the fixed determinant moduli space is a point and we obtain Theorem ??.

We finish with the proofs of the two lemmas about birationally linear maps that we used inthe proof of Theorem ??. The first thing we need to understand is when a twisted Grassmannianbundle is birationally linear over its base. We shall provide a sufficient condition which is in factnecessary though we shall not prove that here since we do not need it.

Start by observing that, if P and Q are vector bundles of weight w over X, then M =P <S> Q/PGLn is a bundle of left modules for P <g> P/PGLn and that this correspondence isinvertible because Q = P (Sipv^p 7*M, where γ : X —>• X/PGLn is the quotient map.

Lemma 5.2. Let P be a vector bundle of weight w over X. Assume that the Brauer class associ-ated to P is represented by a central simple algebra of dimension j 2 . Then π : G R ( J , P)/PGLn —>•X/PGLn is a birationally linear map.

Proof. Let A be the central simple algebra given by the bundle of central simple algebras P <g>P/PGLn over the field k(X/PGLn). Then by assumption A has a left ideal of dimension j rk(P)which is of necessity a direct summand of A. Therefore, over some dense open subvariety ofX/PGLn, Pv(g>P/PGLn ^ L i © L 2 where L1 and L 2 are bundles of left ideals for Pv(g>P/PGLn

and rk(L1) = j r k ( P ) . We may as well assume that this happens over X/PGLn. We obtain acorresponding direct sum decomposition of P, P = P1 © P 2 where P1 and P 2 are GLn stablesubbundles of P and hence both of weight w. Also rk(P1) = j . Now consider the vectorbundle P1 <S> P2. Let λ: P —>• P 2 be the universal homomorphism of vector bundles defined onP1 eg) P2 and consider the map of vector bundles over P1 eg) P 2 , (Id, λ): P1 -> P1 © P2 = P.This representation of P1 as a subbundle of P defines a map from P 1 <g> P 2 to G R ( J , P) whichis PGLn-equivariant, injective and onto an open subvariety of G R ( J , P) . Hence P1 <S> P2/PGLn

which is a vector bundle over X/PGLn is an open subvariety of G R ( J , P) . •

It remains to show that two twisted Grassmannian bundles of equal dimensional subspaces

arising from vector bundles of the same weight have a birationally linear map between them.

Lemma 5.3. Let P and Q be vector bundles of weight w over X and suppose that j < rk(Q) <rk(P). Then there is a birationally linear rational map

ρ: Gr(j,P)/PGLn -> Gr(j,Q)/PGLn.

compatible with the bundle maps to X/PGLn.

Proof. Pv(S>Q/PGLn is a bundle of left modules for Pv(g>P/PGLn of rank equal to rk(P) rk(Q)

and since rk(Q) < rk(P) there is an open subvariety of X/PGLn on which

P eg) P/PGLn ^ P eg) Q/PGLn © L

for some vector bundle of left ideals L since this is true at the generic point of X/PGLn. Hencewe may assume that on X/PGLn, P ^ Q © Q' for GLn stable subbundles Q and Q'. Let S bethe universal subbundle on GR(J,Q). Then S and Q' are both vector bundles of weight w onG R ( J , Q). We consider the vector bundle Sy ®Q' over G R ( J , Q). Let λ: S —>• Q' be the universalhomomorphism of vector bundles defined on S>v <g> Q' and let ι: S —> Q be the universal inclusionof S in Q pulled back to S <g> Q'; now consider the map of vector bundles

(ι, λ ) : S -> Q © Q' ^ P

defined on 5 V <S>Q'. This gives a subbundle of P of rank j and hence defines a map from S'v <g> Q'to G R ( J , P) . This map is injective and onto an open subvariety of G R ( J , P) and it is alsoPGLn-equivariant. Hence S'v <g> Q'/PGLn is an open subvariety of Gr(j,P)/PGLn. However,S <S> Q'/PGLn is a vector bundle over G R ( J , Q)/PGLn which gives our lemma. •

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References

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[3] H. Lange, Zur Klassification von Regelmannigfaltigkeiten, Math. Annalen 262 (1983), 447-459.

[4] M.S. Narasimhan and S. Ramanan, Deformation of the moduli space of vector bundles overan algebraic curve, Annals of Math. 101 (1975), 391-417.

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