REND. SEM. MAT. UNIVERS. POLITECN. TORINO Vol. 44', 2 (1986) ZivRan(*) ON A THEOREM OF MARTENS The celebrated theorem of Torelli (cf. [1], [6]) is usually stated as saying that an algebraic curve C is uniquely determined, up to birational equivalence by its Jacobian J(C) together with the canonical polarization of J(C). As Andreotti [1] has observed, this theorem is equivalent to the statement that the birational equivalence class of C is determined by that of the symmetric product S g ~ l (C), where g is the genus of C. This formulation leads naturally to the question as to whether an analogous result is true for other symmetric products. Now there are examples (see [2]) of birationally inequivalent curves of genus 2 with isomorphic Jacobians, hence isomorphic second symmetric products; and for d>g, S d (C) is birational to W d ~ 8 x /(C), so one had better restrict to d<g. In this case, as it turns out, a rather precise form of this result is true: this is the content of the following theorem due to Martens [4]: THEOREM. Let Ci and C 2 be nonsingular curves of genus g over a field k of characteristic 0, and suppose 4>: S d (C 1 )->'S d (C 2 ) is a birational correspondence defined over k, where d<g 2. Then v is induced by a birational map $ : C x -> C 2 defined over k. REMARK. If d =g~ 1 then Weil [6] shows that either $ or its "negative" is induced by a y>. Classificazioneper soggetto AMS (MOS, 1980): 14H40, 14E05 (*) Partially supported by NSF.
288 COROLLARY. Let Jf be a moduli space (defined up to birational equivalence) for deformations of the symmetric product S d (C) of a non-hyperelliptic curve of genus g, g<d, and let S d : J? g -+ jvbe the natural map from the moduli space of curves. Then S d is a birational isomorphism. Proof. A theorem of Kempf [ 3 ] says that a deformation of S d (C) is induced by a unique deformation of C, whence S d is dominant and unramified. By the Theorem, S d is one-to-one. Hence S d is birational. Martens' original proof relied on the Boolean calculus of special subvarieties of Jacobians, as in [5] or [6] (as well as Torelli's theorem). In this note we give a simpler proof of Martens theorem using, apart from Torelli's theorem, nothing more than Poincare's formula. Proof of theorem. To being with, we will reduce to the case k = C. Note that if y exists, then by definition <p(c) +... -I- ip(c) = v(c +... + c) for any R-valued point c and any ^-algebra R\ thus $ determines <p. Hence if y exists over the algebraic closure k, then it is automatically defined over k. Thus we may assume k = k algebraically closed, so that the Lefschetz_principle applies, whence we may assume k = C. Now according to Andreotti [1], the Albanese variety Alb (S d (Ci))~J(d)), 1=1,2, hence J(C 1 )^J(C 2 ), call them X. Thus we have a commutative diagram s d (c 2 y h Let W =ji(s d (C 1 ))=j 2 (S d (C 2 )) i and let 0,- be the canonical polarization which X carries as the Jacobian of Q. Poincare's formula computes the cohomology class of W on X: [W] = [BiV-'/ig-dy = [e 2 r d /(g-d)\. In particular, [ t ] g ~ d = [ 2 ] 8 ~ d. Let H ir i=l,2, be the positivedefinite, Hermitian form on T x 0 induced by [,-]. Then we may identify
289 Hj with the standard Hermitian form on <C*, and then H 2 may be represented by a positive-definite Hermitian matrix A i.e. H 2 (z,w) = H 1 (Az,w). By linear algebra, A may be unitarily diagonalized. Translating this fact in terms of [ J and [0 2 ] we conclude the following. There is a basis sp Xy... y\p g of H 1 ' 0^) such that 8 - [0J = 2 <# A <A and [0 2 ] = 2 a { <# A <A', «, > 0. 1=1 1=1 Then the fact that \% x \ 8 ' d = [ 2 ~ d yields 2 ^A^A.-.A^A^.^ 2 «i<...<^-rf 8 a h ' ' %-d Wi h Vh A... A w d A w d, hence a^... tf;, =; 1 for each i x <... < i g -d> which clearly implies a 1 =... =a g = l, i.e. [0J = [0 2 ] By Torelli's theorem, it follows that C t and C 2 are isomorphic. More precisely, Weil [6] shows that there exists a birational isomorphism <p: Q -> C 2 such that at least one of the following diagrams commutes: _ (2) *> X or Ci -> X (3) <P -1 x * X where the maps C,- - X are (translates of) the natural ones, and - l x multiplication by -1. denotes Now if (2) commutes then by comparing with (1) we see that y induces $. Suppose then that (3) commutes. Comparing again with (1), we conclude that W is invariant under -l x - * n term s of C = Ci, this conclusion has the following interpretation: identify X with the set of all divisor classes of degree d on C. Then W becomes identified with the subset of X consisting of effective divisor classes. Thus W being invariant under - l x means
290 that there is a line bundle L of degree 2d on C, such that h (L(-oi))J=0 for any divisor a of degree d. This implies that h (L)>d + 1. By Clifford's theorem (*) it follows that C is hyper elliptic. Let r denote the hyperelliptic involution. Then» X "I* commutes, from which we conclude that «±> is induced by r <. o F n It is perhaps worth stating separately the result which the last part of the argument proves: PROPOSITION. If C is a nonhyperelliptic curve and d<g-2 then the subvariety Wd ^/(C) does not coincide with a translate of -W^, its image under -l x - Note. Unaware of Martens' work, the author found the above argument in response to a question by Roy Smith, to whom he is grateful. (*) Clifford's theorem is sometimes stated only for special line bundles; however, it is valid for all line bundles of degree < 2g ~ 2, the nonspecial case being a direct consequence of Riemann-Roch.
291 REFERENCES [1] A. Andreotti, On a theorem of Torelli. Amer. J. Math. 80 (1958), 801-828. [2]* M.G.Humbert, Sur les fonctions abeliennes singulieres, J. Math. Pures et Appl. V (1899), 233-250; VI (1900), 279-386 (esp. section 178). [3] G. Kempf, Deformations of symmetric products. In: Riemann surfaces ans related topics (Proceedings), Annals of Math Studies, Princeton 1980. [4] H. Martens, An extended Torelli theorem. Amer. J. Math. 87 (.1965), 257-260. [5] T. Matsusaka, On a characterization of a Jacobian variety. Mem. Coll. Sci. Kyoto, Ser A, 23 (1959), 1-19. [6] A. Weil, Zum beweis des Torellischen Satzes. Nachr. Akad. Wiss. Gottingen (1957), 33-53. ZIV RAN - Mathematics Department - University of California Riverside CA 92521 Lavoro pervenuto in redazione il 28/X/1985 (*) I am grateful to the referee for this reference.