The theorem of Torelli for singular curves
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- by Thomas Jambois PDF
- Trans. Amer. Math. Soc. 239 (1978), 123-146 Request permission
Abstract:
Let C be a compact (singular) curve embedded in a surface. Then C carries a canonical sheaf $\Omega$ which is locally free of rank 1. Moreover, C has a generalized Jacobian J which fits in an exact sequence \begin{equation}\tag {$\ast $} 0 \to F \to J \to A \to 0\end{equation} of algebraic groups such that A is an abelian variety and $F = {({{\mathbf {C}}^\ast })^r} \times {{\mathbf {C}}^s}$. Let $\underline {C}$ be the set of nonsingular points of C and let $\theta$ = Zariski-closure of the image of $(\underline {C})^{(g - 1)}$ in J. Then: Theorem. If C is irreducible and sections of $\Omega$ map C onto X in ${P^{g - 1}}$ then the isomorphism class of J together with the translation class of the divisor $\theta$ on J determine the isomorphism class of X. As a corollary, if $\psi :C \to X$ is an isomorphism (in which case we call C nonhyperelliptic) the above data determine the isomorphism class of C. I do not know if this remains true when C is hyperelliptic. It should be noted that the linear equivalence class of $\theta$ is not enough to determine X. The principal idea of the proof is that of Andreotti, that is, to recover the curve as the dual of the branch locus of the Gauss map from $\theta$ to ${P^{g - 1}}$; however our arguments are usually analytic. The organization of this paper is as follows: In §1 we prove a stronger than usual version of Abel’s theorem for Riemann surfaces and in §2 we extend this theorem to apply to singular curves. In succeeding sections we construct the generalized Jacobian as a complex Lie group J and embed J in an analytic fibre bundle over A with projective spaces as fibre. This we use to endow J with the structure of an algebraic group. §7 contains a miscellany of facts about branch loci and dual varieties, and in §8 the main theorems are stated and proved. We should mention here that the variations on Abel’s theorem proved in this paper (1.2.4 and 3.0.1) were proved by Severi, at least in the special case corresponding to ordinary double points [12].References
- Aldo Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801–828. MR 102518, DOI 10.2307/2372835
- R. C. Gunning, Lectures on Riemann surfaces, Jacobi varieties, Mathematical Notes, No. 12, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0357407
- K. Kodaira, On compact complex analytic surfaces. I, Ann. of Math. (2) 71 (1960), 111–152. MR 132556, DOI 10.2307/1969881
- K. Kodaira, On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. of Math. (2) 60 (1954), 28–48. MR 68871, DOI 10.2307/1969701
- Henrik H. Martens, On the varieties of special divisors on a curve, J. Reine Angew. Math. 227 (1967), 111–120. MR 215847, DOI 10.1515/crll.1967.227.111
- A. L. Mayer, Special divisors and the Jacobian variety, Math. Ann. 153 (1964), 163–167. MR 164969, DOI 10.1007/BF01360314
- Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337, DOI 10.1007/BFb0077071
- Maxwell Rosenlicht, Generalized Jacobian varieties, Ann. of Math. (2) 59 (1954), 505–530. MR 61422, DOI 10.2307/1969715
- B. Saint-Donat, On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann. 206 (1973), 157–175. MR 337983, DOI 10.1007/BF01430982
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175, DOI 10.5802/aif.59
- Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Hermann, Paris, 1959 (French). MR 0103191
- Francesco Severi, Funzioni quasi abeliane, Pontificiae Academiae Scientiarum Scripta Varia, v. 4, Publisher unknown, Vatican City, 1947 (Italian). MR 0024985
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 239 (1978), 123-146
- MSC: Primary 14H15; Secondary 14K30
- DOI: https://doi.org/10.1090/S0002-9947-1978-0498584-3
- MathSciNet review: 0498584