The theorem of Torelli for singular curves

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].