A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES INDRANIL BISWAS, AJNEET DHILLON, JACQUES HURTUBISE, AND RICHARD A. WENTWORTH Abstract. Let be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d p and d z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on of rank r and degree d z d p, and f : O r F is a meromorphic homomorphism which an isomorphism outside a finite subset of and has pole (respectively, zero) of total degree d p (respectively, d z ). Two such pairs (F 1, f 1 ) and (F 2, f 2 ) are called isomorphic if there is a holomorphic isomorphism of F 1 with F 2 over that takes f 1 to f 2. We construct a natural compactification of the moduli space equivalence classes pairs of the above type. The Poincaré polynomial of this compactification is computed. 1. Introduction Take a compact connected Riemann surface. Fix positive integers r and d. Consider pairs of the form (E, f), where E is a holomorphic vector bundle on of rank r and degree d, and f : O r E is an O linear homomorphism which is an isomorphism outside a finite subset of. This implies that the total degree of zeros of f is d. Two such pairs (E 1, f 1 ) and (E 2, f 2 ) are called equivalent if there is a holomorphic isomorphism φ : E 1 E 2 such that φ f 1 = f 2. Such pairs are examples of vortices [BDW], [Br], [BR], [Ba], [EINOS]. For any pair (E, f) of the above type, consider the dual homomorphism f : E (O r ) = O r. The quotient O r /image(f ) is an element of the Quot scheme Quot(r, d) that parametrizes all torsion quotients of O r of degree d. Conversely, given any torsion quotient O r of degree d, consider the homomorphism O r ψ T = (O r ) ψ kernel(ψ) 2000 Mathematics Subject Classification. 14H60, 14D21, 14D23. Key words and phrases. Generalized Quot scheme, meromorphic vortices, moduli space, Poincaré polynomial. 1

2 I. BISWAS, A. DHILLON, J. HURTUBISE, AND R. A. WENTWORTH induced by the inclusion kernel(ψ) O r. The pair (kernel(ψ), ψ ) is clearly of the above type. Therefore, the moduli space of equivalence classes of pairs (E, f) is identified with the Quot scheme Quot(r, d). Here we consider pairs of the form (E, f), where E is a holomorphic vector bundle on of rank r and degree d, and f : O r E is an O linear meromorphic homomorphism which is an isomorphism outside a finite subset of. We assume that the total degree of the poles of the meromorphic homomorphism is d p. This implies that the total degree of the zeros of the meromorphic homomorphism is d + d p. As before, two such pairs (E 1, f 1 ) and (E 2, f 2 ) will be called equivalent if there is a holomorphic isomorphism φ : E 1 E 2 such that φ f 1 = f 2. The equivalence classes of pairs can be considered as examples of meromorphic vortices. We construct a natural compactification of the moduli space of these meromorphic vortices. We compute the Poincaré polynomial of this compactification. 2. Notations and Conventions Let S be a scheme and Y S a smooth projective morphism. Given a coherent sheaf F on Y flat over S and a numerical polynomial r(t), we denote by Quot(F/S, r(t)) the Grothendieck Quot scheme over S parametrizing quotients of F with Hilbert polynomial r(t) [Gr]. There is a universal exact sequence on Quot(F/S, r(t)) S Y 0 K univ Quot(F/S,r(t)) π Y F Q univ Quot(F/S,r(t)) 0, where π Y : Quot(F/S, r(t)) S Y Y is the natural projection. Often we will just drop the subscripts and write K univ or Q univ instead. This construction is well behaved with respect to pull-backs, so let us record the following: Lemma 2.1. For any morphism g : T S, the base change holds. Quot(g F/T, r(t)) = Quot(F/S, r(t)) S T Proof. This follows by examining the corresponding representable functors. We will mostly be interested in the case where Y S is a smooth, connected and of relative dimension one, that is a relative curve, and F is locally free of rank r. Further, we will only consider torsion quotients of rank zero and degree d. This Quot scheme will be denoted by Quot(F/S, d). When r = 1 and S is a point, then the d-th symmetric power of Y. Quot(O, d) = Sym d (Y ),

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES 3 Given an positive integer d by a partition of length k > 0 of d we mean a sequence P = (p 1, p 2,, p k ) of non-negative integers with k i=1 p i = d. For such a partition define d(p) := k i=1 (i 1)p i. We will write Sym P (Y ) = Sym p 1 (Y ) Sym pr (Y ). 3. A relative Quot scheme Let be a compact connected Riemann surface. Let E and F be two holomorphic vector bundles on of common rank r. Take a dense open subset U, such that the complement S := \ U is a finite set, and take an isomorphism of coherent analytic sheaves f : E U F U over U. This homomorphism f will be called meromorphic if there is a positive integer n such that f extends to a homomorphism of coherent analytic sheaves f : E F O (ns) F over, where S is the reduced divisor defined by the finite subset S. Note that since the divisor S is effective, we have F F O (ns). Therefore, f is meromorphic if and only if the homomorphism f is algebraic with respect to the algebraic structures on E U and F U given by the algebraic structures on E and F respectively. Take a meromorphic homomorphism f as above. We note that the extension f is uniquely determined by f because f and f coincide over U. The inverse image E(f) := f 1 (F) E (recall that F F O (ns)) is clearly independent of the choice of n. We note that both E(f) and f(e(f)) are holomorphic vector bundles on because they are coherent analytic subsheaves of holomorphic vector bundles. Both of then are of rank r, and the restriction (3.1) f E(f) : E(f) f(e(f)) is an isomorphism of holomorphic vector bundles. Define (3.2) Q p (f) := E/E(f) and Q z (f) := F/( f(e(f))) (the subscripts p and z stand for pole and zero respectively). We note that both Q p (f) and Q z (f) are torsion coherent analytic sheaves on. In particular, their supports are finite subsets of. From (3.2) it follows that (3.3) degree(q p (f)) = degree(e) degree(e(f)) and degree(q z (f)) = degree(f) degree( f(e(f))). Fix positive integers r, d p and d z. Set the domain E to be the trivial vector bundle O r of rank r. Consider all triples of the form (F, U, f), where F is a holomorphic vector bundle on of rank r, U is the complement of a finite subset of, and f : O r U = O r U F U is a meromorphic homomorphism such that degree(q p (f)) = d p and degree(q z (f)) = d z.

4 I. BISWAS, A. DHILLON, J. HURTUBISE, AND R. A. WENTWORTH Since f E(f) in (3.1) is an isomorphism, from (3.3) we conclude that (3.4) degree(f) = d z d p + degree(o r ) = d z d p. Two such triples (F 1, U 1, f 1 ) and (F 2, U 2, f 2 ) will be called equivalent if there is a holomorphic isomorphism of vector bundles over such that β : F 1 F 2 β (f 1 U1 U 2 ) = f 2 U1 U 2. Therefore, the equivalence class of (F, U, f) depends only on (F, f) and it is independent of U. More precisely, (F, U, f) is equivalent to (F, W, f W ) for every W U such that the complement U \ W is a finite set. Let (3.5) Q 0 = Q 0 (r, d p, d z ) be the space of all equivalence classes of triples of the above form. We will embed Q 0 as a Zariski open subset of a smooth complex projective variety. Take any triple (F, U, f) as above that is represented by a point of Q 0. Consider the short exact sequence (3.6) 0 E(f) := kernel(q p ) E = O r q p Qp (f) 0, where q p denotes the projection to the quotient in (3.2). We also have E(f) = f(e(f)) F (recall that f E(f) in (3.1) is an isomorphism). Let (3.7) 0 F E(f) be the dual of the above inclusion of E(f) in F. From (3.6) we have degree(e(f) ) = degree(q p (f)) = d p. Therefore, from (3.4) it follows that degree(e(f) /F ) = degree(e(f) ) degree(f ) = d p + d z d p = d z as degree(f ) = degree(f). These imply that we can recover the equivalence class of (F, f) once we know the following two: the torsion quotient Q p (f) of O r of degree d p, and the torsion quotient E(f) /F of E(f) of degree d z. (It should be clarified that knowing the torsion quotient Q p (f) means knowing the sheaf Q p (f) along with the surjective homomorphism O r Q p(f); similarly knowing the torsion quotient E(f) /F means knowing the sheaf E(f) /F along with the surjective homomorphism from E(f) to it.) Indeed, once we know Q p (f), we know the kernel E(f) and hence know E(f) ; if we know the quotient E(f) /F, then we know the subsheaf F of E(f). The dual of this inclusion F E(f), namely the homomorphism E(f) F,

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES 5 gives the meromorphic homomorphism f. In other words, we have the diagram 0 0 E(f) O r Q p 0 f F Q z 0 Let Quot(r, d p ) be the Quot scheme parametrizing the torsion quotients of O r of degree d p. We have the tautological short exact sequence of coherent analytic sheaves on Q(r, d p ) (3.8) 0 K univ p O r Quniv 0, where p is the projection of Quot(r, d p ) to. We write K = K univ. Now consider the dual vector bundle K Quot(r, d p ) p Q Quot(r, dp ), where p Q is the natural projection. Using p Q, we will consider K as a family of vector bundles on parametrized by Quot(r, d p ). For any point y Quot(r, d p ), the vector bundle K {y} over will be denoted by K y. Let (3.9) ϕ : Quot(r, d p, d z ) := Quot(K / Quot(r, d p ), d z ) Q(r, d p ) be the relative Quot scheme over Quot(r, d p ), for the family K, parametrizing the torsion quotients of degree d z. Therefore, for any point y Quot(r, d p ), the fiber ϕ 1 (y) is the Quot scheme parametrizing the torsion quotients of degree d z of the vector bundle K y. Both Quot(r, d p ) and the fibers of ϕ are irreducible smooth projective varieties. The morphism ϕ is smooth. Therefore, the projective variety Quot(r, d p, d z ) is irreducible and smooth. Consider Q 0 defined in (3.5). We have a map η : Q 0 Quot(r, d p ) that sends any triple (F, U, f) Q 0 to the point representing the quotient Q p (f) in (3.6). Let (3.10) η : Q 0 Quot(K, d z ) = Quot(r, d p, d z ) be the map that sends any point α = (F, U, f) Q 0 to the point of ϕ 1 (η (α)) that represents the quotient E(f) /F in (3.7). This map η is injective because, as observed earlier, the equivalence class of the pair (F, f) can be recovered from the quotient Q p (f) of O r and the quotient E(f) /F of E(f). The image of η is clearly a Zariski open subset of Quot(r, d p, d z ).

6 I. BISWAS, A. DHILLON, J. HURTUBISE, AND R. A. WENTWORTH Let r K r p O r = p O be the r-th exterior power of the homomorphism in (3.8). Considering it as a family of subsheaves of O of degree d p parametrized by Quot(r, d p ), we have the corresponding classifying morphism Let δ 1 : Quot(r, d p ) Quot(1, d p ) = Sym dp (). (3.11) δ 1 := δ 1 ϕ : Quot(r, d p, d z ) =: Q Sym dp () be the composition, where ϕ is constructed in (3.9). Next, consider the tautological subsheaf S (Id ϕ) K on Q. Let r S r (Id ϕ) K be the r-th exterior power of the above inclusion. Let (3.12) δ 2 : Q Sym dz () be the morphism that sends any y Q to the scheme theoretic support of the quotient sheaf ( r (Id ϕ) K ϕ(y))/( r S {y} ). Now define the morphism (3.13) δ := (δ 1, δ 2 ) : Quot(r, d p, d q ) = Q Sym dp () Sym dz (), where δ 1 and δ 2 are constructed in (3.11) and (3.12) respectively. It can be shown that δ is surjective. In fact, in Section 4 we will construct, and use, a section of δ. Remark 3.1. Let M (r, d z d p ) denote the moduli stack of vector bundles on of rank r and degree d z d p. Since there is a universal bundle over Quot(r, d p, d q ), we get a morphism Quot(r, d p, d q ) M (r, d z d p ). 4. Fundamental group of Quot(r, d p, d z ) Proposition 4.1. The homomorphism between fundamental groups induced by the morphism δ in (3.13) is an isomorphism. Proof. We will first construct a section of δ. Let D(d p ) Sym dp () be the divisor consisting of all (x, {y 1,, y dp }) such that x {y 1,, y dp }. Then the subsheaf O Sym dp () ( D(d p)) O r 1 Sym dp () O r Sym dp () produces a classifying morphism (4.1) θ 1 : Sym dp () Quot(r, d p ). Let ξ 1 (respectively, ξ 2 ) denote the projection of Sym dp () Sym dz () to Sym dp () (respectively, Sym dz ()). Like before, D(d z ) Sym dp () be the divisor consisting of all (x, {y 1,, y dz }) such that x {y 1,, y dz }. The subsheaf ((Id ξ 1 ) (O Sym dp () (D(d p))) (Id ξ 2 ) (O Sym dz () ( D(d z))))

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES 7 (O r 1 Sym dp ()Sym dz () ) (Id ξ 1 ) (O Sym dp () ( D(d p)) O r 1 Sym dp () ) produces a classifying morphism (4.2) θ : Sym dp () Sym dz () Quot(r, d p, d q ). We note that ϕ θ = θ 1, where ϕ and θ 1 are the morphisms constructed in (3.9) and (4.1) respectively. It is straightforward to check that (4.3) δ θ = Id Sym dp () Sym dz (), where δ is constructed in (3.13). In view of this section θ, we conclude that the induced homomorphism between fundamental groups δ : π 1 (Quot(r, d p, d q )) π 1 (Sym dp () Sym dz ()) is surjective (the base points of fundamental groups are suppressed in the notation). Let (4.4) U Sym dp () Sym dz ()) be the Zariski open subset consisting of all (x, y) = ({x 1,, x dp }, {y 1,, y dz }) Sym dp () Sym dz ()) such that the d p + d z points {x 1,, x dp, y 1,, y dz } are all distinct, equivalently, the effective divisor x + y is reduced. Let (4.5) θ 0 := θ U : U Quot(r, d p, d z ) be the restriction of the map θ in (4.2). Also, consider the restriction (4.6) δ 0 := δ δ 1 (U) : δ 1 (U) U. is the projective space parametrizing the hyperplanes in C r and P r 1 C is the projective space parametrizing the lines in C r (so P r 1 C parametrizes the hyperplanes in (C r ) ). From the homotopy exact sequence associated to δ 0 it follows that the induced homomorphism of fundamental groups Every fiber of δ 0 is identified with (P r 1 C (P r 1 )dp C, where P r 1 )dz C δ 0, : π 1 (δ 1 (U)) π 1 (U) is an isomorphism. The variety Quot(r, d p, d z ) is smooth, and δ 1 (U) is a nonempty Zariski open subset of it. Therefore, the homomorphism ι : π 1 (δ 1 (U)) π 1 (Quot(r, d p, d z )) induced by the inclusion ι : ϕ 1 (U) Quot(r, d p, d z ) is surjective. isomorphism, this implies that the homomorphism θ 0, : π 1 (U) π 1 (Quot(r, d p, d z )) Since δ 0, is an induced in θ 0 in (4.5) is surjective. Since θ 0 extends to θ, this immediately implies that the homomorphism θ : π 1 (Sym dp () Sym dz ()) π 1 (Quot(r, d p, d z ) induced in θ in (4.2) is surjective. Since θ is surjective, and the composition δ θ is injective (see (4.3)) we conclude that δ is also injective.

8 I. BISWAS, A. DHILLON, J. HURTUBISE, AND R. A. WENTWORTH 5. Cohomology of Quot(r, d p, d z ) 5.1. Generalization of a theorem of Bifet. Let S 1, S 2,, S k be a smooth connected projective varieties over C. Fix some line bundles L i on S i of relative degree d i over S i. In other words deg(l i s ) = d i for each point s S i. Set S = S 1 S k. Let be the natural projection. Define Let π Si : S S i L i := π S i L i. φ : Quot( i Li /S, d) S be the relative Quot scheme that parametrizes the torsion quotients of degree d. So for any s = (s 1,, s k ) S, the fiber φ 1 (s) parametrizes the torsion quotients of k i=1l i si of degree d. By deformation theory, φ is a smooth morphism of relative dimension kd, so Quot( i Li /S, d) is smooth of dimension dim(s) + kd. The torus G k m acts on Quot( i Li /S, d) via its action on k L i=1 i. For any positive integer p, let Quot(L i /S i, p) S i denote the relative Quot scheme parametrizing the torsion quotients of L i /S i of degree p. So the fiber of Quot(L i /S i, p) over any s S i parametrizes the torsion quotients of L i s of degree p. Proposition 5.1. There is a bijection between the partitions P = (p 1, p 2,, p k ) of d of length k and the connected components of the fixed point loci of the G k m action on Quot( i Li /S, d). The component corresponding to the partition k i=1 p i = d is the product of Quot schemes Quot(L /S, P) := Quot(L 1 /S 1, p 1 ) Quot(L 2 /S 2, p 2 )... Quot(L k /S k, p k ) with the obvious structure morphism to S. Proof. On applies the argument used to prove Lemme 1 in [Bif]. As all schemes and morphisms are assumed to be projective it is possible to choose a one parameter subgroup G m G k m so that Quot( i Li /S, d) Gm = Quot( i Li /S, d) Gk m. Further, the above one-parameter subgroup can be chosen to be given by an increasing sequence of weights λ 1 < λ 2 <... < λ k. There is an induced action of G m on the tangent space at a fixed point x. The action preserves the normal space to the fixed point locus and we wish to describe the subspace of positive weights. Take a partition P = (p 1,, p k ) of D. As before, let Quot(L /S, P) Quot( i Li /S, d) Gk m

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES 9 be the connected component corresponding to P. For a point x Quot(L /S, P), its image in S i will be denoted by x i. The line bundle L i xi on will be denoted by L x i. The point x i is given by the exact sequence 0 L x i O ( D i ) L x i O Di 0 where D i is an effective divisor on with deg D i = p i. The relative tangent bundle for the projection φ is k T x Quot( i Li /S, d)/s = Hom(L x i O ( D i ), O Dj ). i,j=1 On the other hand, the relative tangent space to the fixed point locus Quot(L /S, P) is k T x Quot(L /S, P)/S = Hom(L x i O ( D i ), O Di ). Consequently, the normal bundle N to Quot(L /S, P) Quot( i Li /S, d) is i=1 N x = i j Hom(L x i O ( D i, O Dj ). Also, the subspace of positive weights is N + x = i<j Hom(L x i O ( D i, O Dj ) because the torus acts on Hom(L x i O ( D i, O Dj ) with weight λ j λ i. It follows that k d(p) := dim N x + = (i 1)p i. Proposition 5.2. The Quot schemes for line bundles Quot(L i /S i, p i ) = Quot(O/S i, p i ) = Sym p i () S i. The Poincaré polynomial of Quot( i Li /S, d) is given by i=1 P (Quot( i Li /S, d), t) = P = P t 2d(P) P (Quot(L i /S i, p i ), t) t 2d(P) P (S i, t)p (Sym p i (), t), where the sum is over all partitions of d of length k. Proof. The isomorphism Quot(O/S i, p i ) Quot(L i /S i, p i ) is by tensoring exact sequences with L i. The second equality is via (2.1). We need to recall the theorems of [Bia] and [Ki] in our present context. The torus action determines two stratifications of the variety Quot( i Li /S, d). The strata are in bijection with connected components of the fixed point locus which are in turn in bijection with partitions of d of length k. Given such a partition P, its corresponding strata are Quot(L /S, P) + := {x lim t 0 t.x Quot(L /S, P)}

10 I. BISWAS, A. DHILLON, J. HURTUBISE, AND R. A. WENTWORTH and Quot(L /S, P) := {x lim t.x Quot(L /S, P)}. t Both of these stratifications are known to be perfect. There are affine fibrations Quot(L /S, P) + Quot(L /S, P) and Quot(L /S, P) Quot(L /S, P) of relative dimensions dim N x + and dim Nx respectively, where x Quot(L /S, P) is an arbitrary closed point. It follows that the codimension of Quot(L /S, P) is dim N x + which gives the above formula for the Poincare polynomial. 5.2. The cohomology of Quot(r, d p, d z ). In this subsection we describe the Poincaré polynomial of Quot(r, d p, d z ). Consider the morphism ϕ in (3.9). There is a natural action of the torus G r m on the target Quot(O r, d p ) = Q(r, d p ). This action clearly lifts to the domain Quot(r, d p, d z ) for ϕ. The previous subsection provides us with a decomposition and an induced formula for the Poincaré polynomial of Quot(r, d p, d z ). Let us recall it quickly in the present context. There is a bijection between connected components of fixed point locus and partitions of d p of length r. Given a partition P = (p 1, p 2,, p r ), the corresponding component of Quot(O r, d p ) Gm is Quot(O, p 1 ) Quot(O, p r ) = Sym p 1 () Sym pr () There are universal divisors Dp univ i corresponding to P, that is = Sym P. inside Sym p i (). The component of Quot(r, d p, d z ) Gr m φ 1 (Sym p 1 () Sym p 2 () Sym pr ()) is then identified with Quot( i O Sym p i() (Dp univ i )/ Sym p i (), d z ). As the morphism ϕ in (3.9) is smooth, and smooth morphisms preserve codimension, we obtain the following formula for the Poincaré polynomial: (5.1) P (Quot(r, d p, d z ), t) = t 2d(P) P (Quot( i O Sym p i() (Dp univ i )/ Sym p i (), d z ), t). P To complete the calculation we need to compute the Poincaré polynomials of Quot( i O Sym p i() (D univ p i )/ Sym P (), d z ). Once again Proposition 5.2 applies. The connected components of the fixed point loci are in bijection with partitions of d z of length r. Given a partition Q = (q 1,, q r ), the corresponding connected component is Quot(O Sym p 1 () ( D p1 )/ Sym p 1 (), q 1 ) Quot(O Sym pr () ( D pr )/ Sym pr (), q r ) which is canonically isomorphic to Sym P,Q := Sym p 1 () Sym pr () Sym q 1 () Sym qr (). We obtain the following formula: P (Quot( i OSym p i () (Dp univ i )/ Sym P (), d z )) = Q t 2d(Q) P (Sym P,Q (), t). Putting this all together we obtain the following:

A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES 11 Theorem 5.3. The Poincaré polynomial for Quot(r, d p, d z ) is P (Quot(r, d p, d z ), t) = P t 2[d(P)+d(Q)] P (Sym P (), t)p (Sym Q (), t), Q where P varies over all partitions of d p of length r and Q varies over all partitions of d z of length r. Poincaré polynomial of Sym n () is the coefficient of t n in (1 + tx) 2g (1 t)(1 tx 2 ), where g is the genus of [Ma, p. 322, (4.3)]. Using this and Theorem 5.3 we get an explicit expression for P (Quot(r, d p, d z ), t). Acknowledgements We thank the National University of Singapore for its hospitality. References [Ba] J.M. Baptista, On the L 2 -metric of vortex moduli spaces, Nucl. Phys. B 844 (2011), 308 333. [BDW] A. Bertram, G. Daskalopoulos and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, Jour. Amer. Math. Soc. 9 (1996) 529 571. [Bia] A. Bia lynicki-birula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480 497. [Bif] E. Bifet, Sur les points fixes schéma Quot O /,k sous l action du tore G r m,k, Com. Ren. Math. Acad. Sci. Paris 309 (1989), 609 612. [BR] I. Biswas and N. M. Romão, Moduli of vortices and Grassmann manifolds, Comm. Math. Phy. 320 (2013), 1 20. [Br] S. Bradlow, Vortices in holomorphic line bundles over closed Kähler manifolds, Commun. Math. Phys. 135 (1990), 1 17. [EINOS] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: the moduli matrix approach, J. Phys. A: Math. Gen. 39 (2006), 315 392. [Gr] A. Grothendieck, Techniques de construction et théorèmes d existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Exp. No. 221, 249 276. [Ki] F. Kirwan, Intersection homology and torus actions, Jour. Amer. Math. Soc. 1 (1988), 385 400. [Ma] I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319 343.

12 I. BISWAS, A. DHILLON, J. HURTUBISE, AND R. A. WENTWORTH School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India E-mail address: indranil@math.tifr.res.in Department of Mathematics, Middlesex College, University of Western Ontario, London, ON N6A 5B7, Canada E-mail address: adhill3@uwo.ca Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke St. W., Montreal, Que. H3A 0B9, Canada E-mail address: jacques.hurtubise@mcgill.ca Department of Mathematics, University of Maryland, College Park, MD 20742, USA E-mail address: raw@umd.edu