On a proof of Torelli's theorem

Abstract

A well known theorem of Torelli states that any compact Riemann surface is determined by its principally polarized jacobian variety. Torelli's original proof (see [T]) has been revisited and adapted to any characteristic of the base field by Matsusaka (see [M]). This Author slightly modified Torelli's argument avoiding the use of some Schubert's enumerative formulas, rigourosly proved only later over by Macdonald (see [MC]), and over any algebraically closed field by Ghione (unpublished). On the other hand Matsusaka's proof uses a suitable version of Castelnuovo­Humbert's theorem and a weak form of Riemann's singularity theorem. A second proof of Torelli's theorem was given in 1952 by Andreotti (see [A1]) and adapted to any by Weil and Andreotti himself (see [A2], [W]). There is however a third proof due to Comessatti (see [C]), which seems to have been ignored in the current literature. It is, in our opinion, a very elegant proof, and we believe it useful to give an account of it in this note. Comessatti's argrnnent (see § 3) basically relies upon two results (theorems (1.8) and (2.5)), both somewhat hidden in the, rather obscure, original exposition. We have worked on an algebraically closed field of any characteristic. This required some care which, in spite of the plainess of the involved ideas, has made the exposition longer than we originally believed. Anyhow tne proof can be made strictkingly simple in characteristic zero, avoiding the use of Castelnuovo­Humbert's theorem, present in comessatti's argument and in our theorem (1.8).This simplification (see 3) has been achieved by virtue of a result (theorem (9.1)) very easy to prove, but only in characteristic zero.