Orthogonal bundles on curves and theta functions. Arnaud BEAUVILLE

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1 Orthogona bundes on curves and theta functions Arnaud BEAUVILLE Introduction Let C be a curve of genus g 2, G an amost simpe compex Lie group, and M G the modui space of semi-stabe G-bundes on C. For each component M G of M G, the Picard group is infinite cycic; its positive generator L G can be described expicitey as a determinant bunde. Then a natura question, which we wi address in this paper for the cassica groups, is to describe the space of generaized theta functions H 0 (M G, L G ) and the associated rationa map ϕ G : M G L G. The mode we have in mind is the case G = SL r. Let J g 1 be the component of the Picard variety of C parameterizing ine bundes of degree g 1 ; it is isomorphic to the Jacobian of C, and carries a canonica theta divisor Θ consisting of ine bundes L in J g 1 with H 0 (C, L) 0. For a genera E M SLr, the ocus Θ E = {L J g 1 H 0 (C, E L) 0} is in a natura way a divisor, which beongs to the inear system rθ on J g 1. We thus obtain a rationa map ϑ : M SLr rθ. The main resut of [BNR] is that there exists an isomorphism L SLr rθ which identifies the rationa maps ϕ SLr and ϑ. This gives a reasonaby concrete description of ϕ SLr, which aows to get some information on the behaviour of this map, at east for sma vaues of r or g (see [B3] for a survey of recent resuts). Let us consider now the case G = SO r with r 3. The modui space M SOr parametrizes oriented orthogona bundes (E, q) on C of rank r ; it has two components M + SO r and M SO r. Let θ : M SOr rθ be the map (E, q) Θ E. We wi see that θ maps M + SO r and M SO r into the subspaces rθ + and rθ corresponding to even and odd theta functions respectivey. Our main resut is: Theorem. There are canonica isomorphisms L ± SO r rθ ± which identify ϕ ± SO r : M ± SO r L ± SO r with the map θ ± : M ± SO r rθ ± induced by θ. This is easiy seen to be equivaent to the fact that the pu-back map θ : H 0 (J g 1, O(rΘ)) H 0 (M SOr, L SOr ) is an isomorphism. We wi prove that it is injective by restricting to a sma subvariety of M SOr ( 1). Then we wi use the Verinde formua ( 2 and 3) to show that the dimensions are the same. This is somewhat artificia since it forces us for instance to treat separatey the cases r even 6, r odd 5, r = 3 and r = 4. It woud be interesting to find a more direct proof, perhaps in the spirit of [BNR]. 1

2 In the ast section we consider the same question for the sympectic group. Here the theta map does not invove the Jacobian of C but the modui space N of semi-stabe rank 2 vector bundes on C with determinant K C. Let L be the determinant bunde on N. For (E, ϕ) genera in M Sp2r, the reduced subvariety E = {F N H 0 (E F) 0} is a divisor on N, which beongs to the inear system L r ; this defines a map M Sp2r L r which shoud coincide, up to a canonica isomorphism, with ϕ Sp2r. This is a particuar case of the strange duaity conjecture for the sympectic group, which we discuss in 4. Unfortunatey even this particuar case is not known, except in a few cases that we expain beow. 1. The modui space M SOr (1.1) Throughout the paper we fix a compex curve C of genus g 2. For G a semi-simpe compex Lie group, we denote by M G the modui space of semi-stabe G-bundes on C. It is a norma projective variety, of dimension (g 1) dim G. Its connected components are in one-to-one correspondence with the eements of the group π 1 (G). (1.2) Let us consider the case G = SO r (r 3). The space M SOr is the modui space of (semi-stabe) oriented orthogona bundes, that is tripes (E, q, ω) where E is a semi-stabe 1 vector bunde of rank r, q : S 2 E O C a non-degenerate quadratic form, and ω a section of det E with q(ω) = 1, where q is the quadratic form on det E deduced from q. The two components M + SO r and M SO r are distinguished by the parity of the second Stiefe- Whitney cass w 2 (E, q) H 2 (C, Z/2) = Z/2. This cass has the foowing property (see e.g. [Se], Thm. 2): for every theta-characteristic κ on C and orthogona bunde (E, q) M SOr, w 2 (E, q) h 0 (C, E κ) + rh 0 (C, κ) (mod. 2). (1.3) The invoution ι : L K C L 1 of J g 1 preserves Θ, hence ifts to an invoution of O J g 1(Θ). We denote by rθ + and rθ the two corresponding eigenspaces in rθ, and by θ : M SOr rθ the map (E, q) Θ E. Lemma 1.4. The rationa map θ : M SOr rθ maps M + SO r in rθ + and M SO r in rθ. 1 By [R], 4.2, an orthogona bunde (E, q) is semi-stabe if and ony if the vector bunde E is semi-stabe. 2

3 Proof : For any E M SLr we have ι Θ E = Θ E, so θ(m SOr ) is contained in the fixed ocus rθ + rθ of ι. Since M ± SO r is connected, it suffices to find one eement (E, q) of M + SO r (resp. M SO r ) such that Θ E is a divisor in rθ + (resp. rθ ). Let κ J g 1 be an even theta-characteristic of C ; a symmetric divisor D rθ is in rθ + (resp. rθ ) if and ony if mut κ (D) is even (resp. odd) see [M1], 2. Let J[2] be the 2-torsion subgroup of Pic(C) ; we take E = α 1... α r, where α 1,..., α r J[2] and α i = 0. We endow E with the diagona quadratic form q deduced from the isomorphisms αi 2 = O C. Then Θ E = Θ α Θ αr. By the Riemann singuarity theorem the mutipicity at κ of Θ α is h 0 (α κ). Thus by (1.3) mut κ (Θ E ) = i h 0 (α i κ) = h 0 (E κ) w 2 (E, q) mod. 2. (1.5) Let L SOr be the determinant bunde on M SOr, that is, the pu back of L SLr by the map (E, q) E, and et L + SO r and L SO r be its restrictions to M + SO r and M SO r. It foows from [BLS] that for r 4, L ± SO r generates Pic(M ± SO r ). Proposition 1.6. The map θ : H 0 (J g 1, O(rΘ)) H 0 (M SOr, L SOr ) induced by θ : M SOr rθ is an isomorphism. By Lemma (1.4) θ spits as a direct sum (θ + ) (θ ), where (θ ± ) : ( H 0 (J g 1, O(rΘ)) ±) H 0 (M ± SO r, L ± SO r ). The Proposition impies that (θ + ) and (θ ) are isomorphisms, and this is equivaent to the Theorem stated in the introduction. Proof of the Proposition : We wi show in 3 that the Verinde formua gives dim H 0 (M SOr, L SOr ) = dim H 0 (J g 1, O(rΘ)) = r g. It is therefore sufficient to prove that θ is injective, or equivaenty that θ(m SOr ) spans the projective space rθ. We consider again the orthogona bundes (E, q) = α 1... α r for α 1,..., α r in J[2], αi = 0. This bunde has a theta divisor Θ E = Θ α Θ αr. We caim that divisors of this form span rθ. To prove it, et us identify J g 1 with the Jacobian J of C (by choosing a divisor cass of degree g 1 ). For a J, the divisor Θ a is the ony eement of the inear system O J (Θ) ϕ(a), where ϕ : J Ĵ is the isomorphism associated to the principa poarization of J. Therefore our assertion foows from the foowing easy emma: 3

4 Lemma 1.7. Let A be an abeian variety, L an ampe ine bunde on A, Â[2] the 2-torsion subgroup of Pic(A). The mutipication map is surjective. Proof : Let 2 A α 1,...,αr Â[2] α αr =0 H 0 (A, 2 AL) = H 0 (A, L α 1 )... H 0 (A, L α r ) H 0 (A, L r ) be the mutipication by 2 in A. We have canonica isomorphisms α Â[2] H 0 (L α), H 0 (A, 2 AL r ) = β Â[2] H 0 (L r β) ; through these isomorphisms the product map m r : H 0 (A, 2 A L) r H 0 (A, 2 A Lr ) is the direct sum over β Â[2] of the maps m β r : α 1,...,αr Â[2] α αr =β H 0 (A, L α 1 )... H 0 (A, L α r ) H 0 (A, L r β). Since the ine bunde 2 A L is agebraicay equivaent to L4, the map m r is surjective [M2], hence so is m β r for every β. The case β = 0 gives the emma. 2. The Verinde formua (2.1) We keep the notation of (1.1); we denote by q the number of simpe factors of the Lie agebra of G (we are mainy interested in the case q = 1 ). To each representation ρ : G SL r is attached a ine bunde L ρ on M G, the pu back of the determinant bunde on M SLr by the morphism M G M SLr associated to ρ. The Verinde formua expresses the dimension of H 0 (M G, L k ρ), for each integer k, in the form dim H 0 (M G, L k ρ) = N kdρ (G), where d ρ N q is the Dynkin index of ρ. For q = 1 the number d ρ is defined and computed in [D], 2. In the genera case the universa cover of G is a product G 1... G q of amost simpe factors, and we put d ρ = (d ρ1,..., d ρq ), where ρ i is the pu back of ρ to G i. We wi need ony to know that the Dynkin index is 2 for the standard representation of SO r (r 5), 4 for that of SO 3, and (2, 2) for that of SO 4. N (G) is an integer depending on G, the genus g of C, and N q. We wi now expain how this number is computed. Our basic reference is [AMW]. 4

5 (2.2) The simpy connected case Let us first consider the case where G is simpy connected and amost simpe (that is, q = 1 ). Let T be a maxima torus of G, and R = R(G, T) the corresponding root system (we view the roots of G as characters of T ). We denote by T the (finite) subgroup of eements t T such that α(t) +h = 1 for each ong root α, and by T reg the subset of reguar eements t T (that is, such that α(t) 1 for each root α ). It is stabe under the action of the Wey group W. For t T, we put (t) = α R(α(t) 1). Then the Verinde formua is N (G) = t T reg /W ( T ) g 1. (t) (2.3) This number can be expicitey computed in the foowing way. Let t be the Lie agebra of T. The character group P(R) of T embeds naturay into t. We endow t with the W-invariant biinear form ( ) such that (α α) = 2 for each ong root α, and we use this product to identify t with t. Let θ be the highest root of R ; we denote by P the set of dominant weights λ P(R) such that (λ θ). Let ρ P(R) be the haf-sum of the positive roots. The number h := (ρ θ) + 1 is the dua Coxeter number of R. We have T = ( + h) s fν, where s is the rank of R, f the order of the center of G, and ν a number depending on R ; it is equa to 1 for R of type D s and to 2 for B s ([B2], 9.9). For λ P we put t λ = exp 2πi λ + ρ + h The map λ t λ is a bijection of P onto T reg (α λ + ρ) /W ([B2], 9.3.c)). For λ P, we have α(t λ ) = exp 2πi, and + h therefore (t λ ) = 4 sin 2 (α λ + ρ) π + h α R + (2.4) The non-simpy connected case We now give the formua for a genera amost simpe group, foowing [AMW]. Let Z be the center of G. An eement t of T beongs to Z if and ony if α(t) = 1 for a α R, or equivaenty w(t) = t for a w W. It foows that Z acts on the set T reg by mutipication; this action commutes with that of W and thus defines an action of Z on T reg /W. Through the bijection P T reg /W the action of Z on P is the one deduced from its action on the extended Dynkin diagram (see [O-W], 3 or [Bo], 2.3 and 4.3). Now et Γ be a subgroup of Z, and et G = G/Γ. We denote by P the subattice of weights λ P such that λ Γ = 1. The action of Γ on P preserves 5

6 P ; we denote by Γ λ the orbit of a weight λ in P. The Verinde formua for G is: N (G ) = Γ λ P Γ λ 2g( T ) g 1. (t λ ) Each term in the sum is invariant under Γ, so we may as we sum over P /Γ provided we mutipy each term by Γ λ : N (G ) = Γ λ P /Γ Γ λ 1 2g ( T ) g 1. (2.5) (t λ ) (2.6) The genera case The above formua actuay appies to any semi-simpe group G = G/Γ, where G is a product of simpy connected groups G 1,..., G q [AMW]. We choose a maxima torus T (i) in G i for each i and put T = T (1)... T (q). Let := ( 1,..., q ) be a q-upe of nonnegative integers. We put T = T (1) 1... T (q) q ; the subset T reg of reguar eements in T is the product of the subsets (T (i) i ) reg. For each i, et P (i) i be the set of dominant weights of T (i) associated to G i and i as in (2.3), and et P = P (1) 1... P (q) q. For λ = (λ 1,..., λ q ) P, we put t λ = (t λ1,..., t λq ) T ; this defines a bijection of P onto T reg /W. The eements of P are characters of T, and we denote by P the subset of characters which are trivia on Γ. The group Γ is contained in the center Z 1... Z q of G, which acts naturay on P and P. Then ( N (G ) = Γ Γ λ 1 2g T ) g 1 (2.7) (t λ ) λ P /Γ with T q (t λ ) = T i i (t λi ) i=1, i (t) = (α(t) 1) for t T (i). α R(G i,t (i) ) 3. The Verinde formua for SO r We now appy the previous formuas to the case G = SO r. We wi rest very much on the computations of [O-W]. We wi borrow their notation as we as that of [Bo]. (3.1) The case G = SO 2s, s 3 The root system R is of type D s. Let (ε 1,..., ε s ) be the standard basis of R s. The weight attice P(R) is spanned by the fundamenta weights ϖ j = ε ε j (1 j s 2), ϖ s 1 = 1 2 (ε ε s 1 ε s ), ϖ s = 1 2 (ε ε s 1 + ε s ). 6

7 For λ P(R), we write λ + ρ = i t i ϖ i = i u i ε i with u 1 = t t s (t s 1 + t s ),..., u s 2 = t s (t s 1 + t s ) u s 1 = 1 2 (t s 1 + t s ), u s = 1 2 ( t s 1 + t s ) (u i 1 2 Z, u i u i+1 Z). Put k = + 2s 2. The condition λ P becomes: u 1 >... > u s, u 1 + u 2 < k and u s 1 + u s > 0 ; the condition λ P imposes moreover t s 1 t s (mod. 2), that is, u i Z for each i. Thus we find a bijection between P U = {u 1,..., u s } of Z satisfying the above conditions. and the subsets The group Z is canonicay isomorphic to P(R)/Q(R) (note that R = R this case); its nonzero eements are the casses of ϖ 1, ϖ s 1 and ϖ s. The nonzero eement γ which vanishes in SO 2s is represented by the ony weight in this ist which comes from SO 2s, namey ϖ 1. It corresponds to the automorphism of the extended Dynkin diagram which exchanges α 0 with α 1 and α s 1 with α s (see [Bo], Tabe D ); it acts on P by γ(u 1,..., u s ) = (k u 1, u 2,..., u s 1, u s ). Thus the subsets U as above with u s 0, and moreover u 1 k 2 in if u s = 0, form a system of representatives of P /Γ. The corresponding orbit has one eement if u 1 = k 2 u s = 0, and 2 otherwise. For a subset U corresponding to the weight λ we have [O-W] (t λ ) = Π k (U) = 1 i<j s 4 sin 2 π k (u i u j ) 4 sin 2 π k (u i + u j ). Now we restrict ourseves to the case = 2, so that k = r = 2s. Put V = {s, s 1,..., 0}. The subsets U to consider are those of the form U j := V for 0 j s. We have Π r (U j ) = 4r s 1 Π r (U 0 ) = Π r (U s ) = r s 1 for 1 j s 1 by Coroary 1.7 (ii) in [O-W]; by Coroary 1.7 (iii) in [O-W]. We have T 2 = 4r s (2.3). Mutipying the terms U 0 and U s by 2 1 2g and summing, we find: N 2 (SO 2s ) = 2[(s 1).r g g [2.(4r) g 1 ] = r g. (3.2) The case G = SO 2s+1, s 2 Then R is of type B s. Denoting again by (ε 1,..., ε s ) the standard basis of R s, the weight attice P(R) is spanned by the fundamenta weights ϖ 1 = ε 1, ϖ 2 = ε 1 + ε 2,..., ϖ s 1 = ε ε s 1, ϖ s = 1 2 (ε ε s ). 7 and {j}

8 For λ P(R), we write λ + ρ = i t i ϖ i = i u i ε i with u 1 = t t s 1 + t s 2,..., u s 1 = t s 1 + t s 2, u s = t s 2, with u i 1 2 Z and u i u i+1 Z for each i. Put k = + 2s 1. The condition λ P becomes u 1 >... > u s > 0 and u 1 + u 2 < k. Since ϖ s is the ony fundamenta weight which does not come from SO 2s+1, the condition λ P is equivaent to t s odd, that is, u s Z Thus we find a bijection between P and the subsets U = {u 1,..., u s } of Z satisfying u 1 >... > u s > 0, u 1 + u 2 < k. The non-trivia eement γ of Γ acts on P by γ(u 1,..., u s ) = (k u 1, u 2,..., u s ). Thus the eements U as above with u 1 k 2 form a system of representatives of P /Γ. The corresponding orbit has one eement if u 1 = k 2 and 2 otherwise. For a subset U corresponding to the weight λ we have [O-W] (t λ ) = Φ r (U) = 1 i<j s 4 sin 2 π r (u i u j ) 4 sin 2 π r (u i + u j ) s 4 sin 2 π r u i. Now we restrict ourseves to the case = 2, so that k = r = 2s + 1. Put V = {s + 1 2, s 1 2,..., 1 2 }. The subsets U to consider are the subsets U j := V {j } for 0 j s. We have Φ r (U j ) = 4r s 1 Φ r (U s ) = r s 1 i=1 for 0 j s 1 by Coroary 1.9 (ii) in [O-W]; by Coroary 1.9 (ii) in [O-W]. We have again T 2 = 4r s (2.3). Mutipying the term U s by 2 1 2g and summing, we find: N 2 (SO 2s+1 ) = 2[s.r g g (4r) g 1 ] = r g. (3.3) The case G = SO 3 In that case G = SL 2 has a unique fundamenta weight ρ, and a unique positive root θ = 2ρ. The Dynkin index of the standard representation of SO 3 is 4, so we want to compute N 4 (SO 3 ). We have T 4 = 12 (2.3). The set P 4 contains the weights kρ with 0 k 4 ; the weights with k even come from SO 3, and Γ exchanges kρ and (4 k)ρ. Thus a system of representatives of P 4/Γ is {0, 2ρ}, with Γ 0 = 2 and Γ 2ρ = 1. Formua (2.5) gives: N 2 (SO 3 ) = 2.[2 1 2g 12 g g 1 ] = 3 g. 8

9 (3.4) The case G = SO 4 In that case G = SL 2 SL 2 and the nontrivia eement of Γ is ( I, I). The Dynkin index of the standard representation of SO 4 is (2, 2). We have T 2 = 8 for SL 2, hence T (2,2) = 8 2. The set P (2,2) contains the weights (kρ, ρ) with 0 k, 2, and P (2,2) is defined by the condition k (mod. 2). The eement ( I, I) exchanges (kρ, ρ) with ((2 k)ρ, (2 )ρ). Thus P (2,2)/Γ consists of the casses of (0, 0), (0, 2ρ) and (ρ, ρ), the atter being the ony one with a nontrivia stabiizer. Formua (2.7) gives N (2,2) (SO 4 ) = 2 [ g 4 2g g 2 ] = 4 g. Therefore for each r 3 we have obtained dim H 0 (M SOr, L SOr ) = r g. This achieves the proof of Proposition 1.6, and therefore of the Theorem stated in the introduction. 4. The modui space M Sp2r (4.1) Let r be an integer 1. The space M Sp2r is the modui space of (semistabe) sympectic bundes, that is pairs (E, ϕ) where E is a semi-stabe 2 vector bunde of rank 2r and trivia determinant and ϕ : Λ 2 E O C a non-degenerate aternate form. It is connected. To aeviate the notation we wi denote it by M r. The determinant bunde L r generates Pic(M r ) ([K-N], [L-S]). To describe the strange duaity in an intrinsic way we need a variant of this space, namey the modui space M r of semi-stabe vector bundes F of rank 2r and determinant K r C, endowed with a sympectic form ψ : Λ2 F K C. If κ is a thetacharacteristic on C, the map E E κ induces an isomorphism M r M r. We denote by L r the ine bunde corresponding to L r under any of these isomorphisms. Simiary, we wi consider for t even the modui space M SO t vector bundes E of rank t and determinant K t/2 C of semi-stabe, endowed with a quadratic form q : S 2 E K C. It has two components M ± SO t depending on the parity of h 0 (E) ; if κ is a theta-characteristic on C, the map E E κ induces isomorphisms M ± SO t M ± SO t (1.3). The space M + SO t reduced subvariety D = {(E, q) M + SO t H 0 (C, E) 0} ; carries a canonica Wei divisor, the 2D is a Cartier divisor, defined by a section of the generator L SO t of Pic(M + SO t ) ([L-S], 7). 2 By the same argument as in the orthogona case (footnote 1), a sympectic bunde (E, ϕ) is semi-stabe if and ony if E is semi-stabe as a vector bunde. 9

10 (4.2) The strange duaity for sympectic bundes Let r, s be integers 2, and t = 4rs. Consider the map π : M r M s M SO t which maps ((E, ϕ), (F, ψ) to (E F, ϕ ψ). Since M r is connected and contains the trivia bunde O 2r with the standard sympectic form, the image ands in M + SO t. For (E, ϕ) M r, the pu back of L SOt to {(E, ϕ)} M s is the ine bunde associated to 2r times the standard representation, that is L 2r s ; simiary its pu back to M r {(F, ψ)}, for (F, ψ) M s, is L 2s r. It foows that π L SOt = L 2s r L 2r s. If κ is a theta-characteristic on C with h 0 (κ) = 0, we have π(o 2r C, κ2s ) / D ( O 2r C and κ2s are endowed with the standard aternate forms). Thus := π D is a Wei divisor on M r M s, whose doube is a Cartier divisor defined by a section of (L s r L r s ) 2 ; but this modui space is ocay factoria ([S], Thm. 1.2), so that is actuay a Cartier divisor, defined by a section δ of L s r L r s, wedefined up to a scaar. Via the Künneth isomorphism we view δ as an eement of H 0 (M r, L s r) H 0 (M s, L r s ). The strange duaity conjecture for sympectic bundes is Conjecture 4.3. The section δ induces an isomorphism δ : H 0 (M r, L s r) H 0 (M s, L r s ). If the conjecture hods, the rationa map ϕ L s r : M r L s r is identified through δ to the map M r L r s given by E E, where E on {E} M s ; set-theoreticay: is the trace of E = {(F, ϕ) M s H 0 (C, E F) 0}. By [O-W], we have dim H 0 (M r, L s r) = dim H 0 (M s, L r s). Therefore the conjecture is equivaent to: (4.4) The inear system L s r is spanned by the divisors E, for E M r. We now speciaize to the case s = 1. The space M 1 is the modui space N of semi-stabe rank 2 vector bundes on C with determinant K C ; its Picard group is geenrated by the determinant bunde L. The conjecture becomes: 10

11 Conjecture 4.5. The isomorphism δ : H 0 (M r, L r ) H 0 (N, L r ) identifies the map ϕ Lr : M r L r with the rationa map E E of M r into L. By (4.4) this is equivaent to saying that the inear system L r on N spanned by the divisors E for E M r. is (4.6) Let G be a semi-stabe vector bunde of rank r and degree 0. To G is associated a divisor Θ G L r, supported on the set provided this set is N Θ G = {F N H 0 (C, G F) 0} [D-N]. Put E = G G, with the standard sympectic form. We have Θ G = Θ G by Serre duaity, hence E = 1 2 Θ E = 1 2 (Θ G + Θ G ) = Θ G ; thus conjecture 4.5 hods if the inear system L r on N is spanned by the divisors Θ G for G semi-stabe of degree 0. In particuar, it suffices to prove that L r is spanned by the divisors Θ L Θ Lr, for L 1,..., L r J. As a consequence of [BNR], the divisors Θ L for L in J span L, so Conjecture (4.5) hods if the mutipication map m r : S r H 0 (N, L) H 0 (N, L r ) is surjective. Proposition 4.7. Conjecture 4.5 hods in the foowing cases: (i) r = 2 and C has no vanishing thetanu; (ii) r 3g 6 and C is genera enough; (iii) g = 2, or g = 3 and C is non-hypereiptic. Proof : In each case the mutipication map m r : S r H 0 (N, L) H 0 (N, L r ) is surjective. This foows from [B1], Prop. 2.6 c) in case (i), and from the expicit description of M SL2 in case (iii). When C is generic, the surjectivity of m r for r even 2g 4 foows from that of m 2 together with [L]. We have H i (N, L j ) = 0 for i 1 and j 3 by [K-N], Thm By [M2] this impies that the mutipication map H 0 (N, L) H 0 (N, L k ) H 0 (N, L k+1 ) is surjective for k dim N 3 = 3g 6. Together with the previous resut this impies the surjectivity of m r Coroary 4.8. isomorphism L 2 for r 3g 6, and therefore by (4.6) the Proposition. Suppose C has no vanishing thetanu. There is a canonica 4Θ + which identify the maps ϕ L2 : M 2 L 2 with θ : M 2 4Θ + such that θ(e, ϕ) = Θ E. Proof : Let i : J g 1 N be the map L L ι L. The composition H 0 (M 2, L 2 ) δ H 0 (N, L 2 ) i H 0 (J g 1, O(4Θ)) + is an isomorphism by Prop. 4.7 (i) and Prop. 2.6 c) of [B1]; it maps ϕ L2 (E, ϕ) to i E. Using Serre duaity again we find i E = 1 2 (Θ E + Θ E ) = Θ E, hence the Coroary. 11

12 (4.9) Remarks. 1) The coroary does not hod if C has a vanishing thetanu: the image of θ is contained in that of i, which is a proper subspace of 4Θ +. 2) The anaogous statement for r 3 does not hod: the Verinde formua impies dim H 0 (N, L r ) > dim H 0 (J g 1, O(2rΘ)) + for g 3, or g = 2 and r 4. REFERENCES [AMW] A. ALEXEEV, E. MEINRENKEN, C. WOODWARD: Formuas of Verinde type for non-simpy connected groups. Preprint math.sg/ [B1] [B2] [B3] [BLS] [BNR] A. BEAUVILLE: Fibrés de rang 2 sur es courbes, fibré déterminant et fonctions thêta II. Bu. Soc. math. France 119 (1991), A. BEAUVILLE: Conforma bocks, Fusion rings and the Verinde formua. Proc. of the Hirzebruch 65 Conf. on Agebraic Geometry, Israe Math. Conf. Proc. 9 (1996), A. BEAUVILLE: Vector bundes on curves and theta functions. Preprint math.ag/ A. BEAUVILLE, Y. LASZLO, C. SORGER: The Picard group of the modui of G-bundes on a curve. Compositio math. 112 (1998), n o 2, A. BEAUVILLE, M.S. NARASIMHAN, S. RAMANAN: Spectra curves and the generaised theta divisor. J. Reine Angew. Math. 398 (1989), [Bo] N. BOURBAKI: Groupes et agèbres de Lie, Chap. VI. Hermann, Paris (1968). [D-N] [D] J.-M. DREZET, M.S. NARASIMHAN: Groupe de Picard des variétés de modues de fibrés semi-stabes sur es courbes agébriques. Invent. Math. 97 (1989), n o 1, E. DYNKIN: Semisimpe subagebras of semisimpe Lie agebras. Amer. Math. Soc. Transations (II) 6 (1957), [K-N] S. KUMAR, M.S. NARASIMHAN: Picard group of the modui spaces of G - bundes. Math. Ann. 308 (1997), n o 1, [L] Y. LASZLO: À propos de espace des modues de fibrés de rang 2 sur une courbe. Math. Ann. 299 (1994), n o 4, [L-S] Y. LASZLO, C. SORGER: The ine bundes on the modui of paraboic G - bundes over curves and their sections. Ann. Sci. Écoe Norm. Sup. (4) 30 (1997), n o 4, [M1] [M2] D. MUMFORD: On the equations defining abeian varieties I. Invent. Math. 1 (1966), D. MUMFORD: Varieties defined by quadratic equations. Questions on Agebraic Varieties (1970), ; Cremonese, Rome. 12

13 [O-W] [R] [Se] [S] W. OXBURY, S. WILSON: Reciprocity aws in the Verinde formuae for the cassica groups. Trans. Amer. Math. Soc. 348 (1996), n o 7, S. RAMANAN: Orthogona and spin bundes over hypereiptic curves. Proc. Indian Acad. Sci. Math. Sci. 90 (1981), n o 2, J.-P. SERRE: Revêtements à ramification impaire et thêta-caractéristiques. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), n o 9, C. SORGER: On modui of G -bundes of a curve for exceptiona G. Ann. Sci. Écoe Norm. Sup. (4) 32 (1999), no 1, Arnaud BEAUVILLE Institut Universitaire de France & Laboratoire J.-A. Dieudonné UMR 6621 du CNRS UNIVERSITÉ DE NICE Parc Varose F NICE Cedex 02 13